DYNAMICS OF PLANETS, STARS, AND BLACK
HOLE BINARIES: TIDES, SPIN RESONANCES,
AND GRAVITATIONAL WAVES
A Dissertation
Presented to the Faculty of the Graduate School
of Cornell University
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
by
Yubo Su
August 2022
© 2022 Yubo Su
ALL RIGHTS RESERVED
DYNAMICS OF PLANETS, STARS, AND BLACK HOLE BINARIES: TIDES,
SPIN RESONANCES, AND GRAVITATIONAL WAVES
Yubo Su, Ph.D.
Cornell University 2022
Simply by nature of their scale and complexity, astrophysical systems are almost
always studied by simplifications and effective, parameterized models. Yet even
after this considerable simplification, the resulting systems can still host unusual
and exciting phenomena. In this dissertation, I study and characterize analyti-
cally and numerically many of these behaviors in few-body systems consisting of
planets, stars, and black holes (BHs). My work falls into three broad categories: (i)
I carry out a comprehensive theoretical analysis of the spin dynamics of planets by
studying the “Colombo’s Top” system (a rotating planet whose spin axis precesses
around its orbital axis, which itself varies in time). In the course of this analysis, I
examine: the dynamics of planetary obliquities when surrounded by a dissipating
protoplanetary disk, and the capture probabilities of planetary spins into high-
obliquity, stable equilibria due to tidal dissipation in two and many-planet sys-
tems. I apply these results to assess the spin dynamics and evolution of planets in
various, observationally-relevant architectures, such as super Earths with nearby
or distant companions. My work is the first to describe comprehensively the com-
plex resonance capture process responsible for the excitation of large obliquities in
both integrable, two-planet systems and chaotic, three-planet systems. My results
show that super Earths around solar-type stars have a significant chance of cap-
ture into various stable high obliquity states; this will have a large impact on their
surface/atmosphere conditions. (ii) I use detailed hydrodynamical simulations to
understand the nonlinear internal gravity wave breaking process that is responsi-
ble for tidal dissipation in white dwarf and early-type stellar binaries. I then use
my results to understand the circularization of highly eccentric, massive stellar bi-
naries and to calculate the observational effects of tidal heating in compact white
dwarf binaries. (iii) I study the observational signatures of the tertiary-induced
merger channel for merging BH binaries, in which a distant companion drives
the BH binary into a high-eccentricity orbit via the von Zeipel-Lidov-Kozai mech-
anism, and the binary then merges due to gravitational wave emission. I study
both the spin dynamics of the merging BHs when the tertiary is circular as well
as the mass ratio distribution when the tertiary is eccentric, with observational
comparisons to the BH mergers seen by the LIGO/VIRGO Collaboration.
BIOGRAPHICAL SKETCH
Yubo Su was born in Chengdu, China, but grew up in the USA in Atlanta,
Georgia. At River Trail Middle School, he was one of three students in the state of
Georgia to receive a perfect score of 25 on the American Mathematics Contest 8,
and he continued to participate in local and national math competitions through
middle and high school. While completing his secondary education at Northview
High School, he ultimately decided to pursue a degree in physics after two years
of AP Physics with Mr. Chris Kemp. Around this time, he also conducted research
under the supervision of Professor Ken Brown at the Georgia Institute of Technol-
ogy.
In 2012, he began his undergraduate studies at the California Institute of
Technology (Caltech). Here, he completed a double major in physics and com-
puter science while conducting research under the supervision of: (i) Professors
William A. Goddard III (Caltech) and Jose L. Mendoza Cortes (Michigan State
University) in approximate numerical methods for intermolecular forces, (ii) Drs.
Paulett C. Liewer and Eric De Jong (Jet Propulsion Laboratory) in visualization of
coronal mass ejections, and (iii) Professor Sunil R. Golwala (Caltech) in modeling
systematic uncertainties in surveys using the kinetic Sunyaev-Zeldovich due to
foreground subtraction, resulting in a senior undergraduate thesis.
In 2017, after a one-year period of employment at Blend Labs as a software en-
gineer, he began to pursue a PhD in Astronomy under Professor Dong Lai, working
first on hydrodynamical simulations then on careful studies of few-body dynamics.
Upon the defense of this dissertation, he will begin work as a Lyman Spitzer, Jr.
Fellow at Princeton University in September 2022.
Much as with his research interests, Yubo also has dabbled in a large array of
hobbies. His spare time is currently largely occupied by running, cooking, practic-
ing piano, and playing Starcraft, but an updated list can (hopefully!) be found on
iii
his current website.
iv
To my parents, who have supported me on every step of this journey. To my late
grandfather, who was looking forward so much to seeing a PhD among his
grandkids. And to my late and great friend Cassidy Yang, who was one of my
partners in crime at Caltech and who left with so much to give the world still.
v
ACKNOWLEDGEMENTS
I must begin by thanking my adviser, Dong Lai. There is no way to put into
words the impact Dong has had on the trajectory of my academic career, let alone
in such a brief space. He has been tirelessly dedicated to my growth, providing
consistent and pointed guidance. Whether I am working through a month’s work
in a week or a week’s work in a month, he is always nudging me back onto the
right track. He has a ceaseless eye for improvement, continually reminding me
of all the things I still need to learn to become a successful researcher while still
helping me bring my strengths to the questions at hand. He has corrected the
unending typos in my equations that I can never seem to find; has consistently
found ways to clarify and streamline my writing, whether individual words or
entire sections; and has, against my every predisposition, found a way to teach me
the value of “telling a story” when communicating my work. A large part of my
success as a graduate student is thanks to Dong’s advising, and I feel both deeply
fortunate and grateful to have worked with him for the past five years.
Next, I thank the other members of the Cornell faculty and staff who have
taught me much and given me many opportunities. First, I thank the members of
my special committee for their very helpful input and feedback over the years: Jim
Cordes, Nikole Lewis, and Ira Wasserman. I thank Phil Nicholson for listening to
my research many, many times and for giving me opportunities both to grade for
his class and to have many enjoyable conversations. I thank Henrik Spoon for
the chance to help build out the IDEOS web portal, which was a unique and fun
experience. I thank our department’s staff for their infinite patience in the face
of my many questions, which have made my travel and other issues so much less
stressful: Monica Carpenter, Jessica Jones, Lynda Sovocool, Jill Tarbell, and Bez
Thomas.
I thank the collaborators I was fortunate enough to collaborate and correspond
vi
with. I am grateful to Daniel Lecoanet for his patience on my first project in
graduate school, in which I rapidly found myself out of my depth and only began
to understand the best approach to the problem with his extensive guidance. I am
grateful to Bin Liu and Laetitia Rodet for their insightful and careful comments
during our collaborations.
I thank my lovely parents, Xiaoli Fu and Ziqiang Su, for twenty-eight long
(and at times, difficult) years of raising and supporting me. Thank you for always
telling me what you think, even when I do not want to hear it, and supporting me
even when I do not listen. Through the hardest and most confusing parts of my
PhD journey, you have always lent me your open ears and your candid thoughts,
and I hope you know that I would not have been able to make the decisions I
did without your support and patience. I also thank my self-proclaimed “coolest
cousin” Will Su and his lovely wife Cindy Guan for much helpful advice and for
having me over many times over the past few years.
I thank my previous research mentors (listed in the Biographical Sketch). In
particular, I thank Sunil Golwala for taking a chance on me when I knew little
more than how to LATEX quickly and for patiently working with me as I struggled
to learn how to tackle a class of problems I had never known before. I thank Eric
Black and Ken Libbrecht for the support and many long and helpful discussions
over the years; I was so fortunate to have had the chance to be a TA for Ph3. I
thank Steven Strogatz for incredible applied mathematics classes that taught me
tools that I’ve been able to bring with me to astrophysical dynamics.
I thank Maria Paulene Abundo for the many long hours and late nights to-
gether and for all the support through our many triumphs and crises; we will get
GM one day, I am sure of it. I thank my fellow group members over the years: J.J.
Zanazzi, Kassandra Anderson, Bonan Pu, Michelle Vick, Jiaru Li, Chris O’Connor,
and J.T. Laune, all of whom have provided advice, support, and unforgettable com-
vii
pany at different points during my PhD. I thank the friends I’ve made at Cornell
who have made my days and my nights out memorable ones, even if it may have
felt like pulling teeth at times: Paul Corlies, Eiden and Maomao Leung, Jack Mad-
den, Dante Iozzo, Eamonn O’Shea, Gabriel Bonilla, Abhinav Jindal, Cristóbal
Armanza, Ishan Mishra, Bo Peng, Christopher Rooney, Akshay Suresh, Ngoc
Truong, Catie Ball, and Stella Ocker. I thank Michael Lam for helping me prepare
for the Philadelphia Marathon in 2019, one the most meaningful things outside
that research that I have done in the past few years. I thank Jiseon Min for intro-
ducing me to many new ideas, hobbies, and gizmos, and to Arthur Min for being
the best boy. I thank the many friends who have helped make research through a
pandemic bearable through their companionship, both virtual or in-person: Andy
Huang, Gordon KK Huang, Willy Xiao, Jesse Zhao, Cathy An, Jonathan Austin,
Maryyann Landlord Crichton and Will Crichton, Sudarshan Muralidhar, Diana
Wang and Justin Wang, Dhruva Bavadekar, Richard Kang, Jessica Lim and Chap-
lin Yi, Joseph Berleant, Dan Chui, Jacob Quinn Shenker, and Patrick Yu. I thank
my mentors and colleagues at Blend Labs who have shaped how I approach and
solve problems far beyond my former desk at 100 Montgomery: Eugene Marinelli,
Eduardo Lopez, Chris Nguyen-Blaser, Nitya Subramanian, Devon Yang, Michael
Millerick, Jeff Chen, Albert Lee, and Chao Xue. I thank the so many other friends
who I have not been able to thank by name here, of which there are still many;
they say that you see farther by standing on the shoulders of giants, and I’ve been
fortunate to have had so many such friends over the years.
Finally, I thank Cornell University for funding the first two years of my the-
sis work, and the NASA Future Investigators in Earth and Space Science and
Technology (FINESST) program for funding the following three.
viii
TABLE OF CONTENTS
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 Introduction 1
1.1 The Dynamics of Exoplanetary Spins . . . . . . . . . . . . . . . . . . . 2
1.2 Tidal Evolution in Stellar Binaries . . . . . . . . . . . . . . . . . . . . 5
1.3 Tertiary-Induced Formation of Black Hole Binaries . . . . . . . . . . 8
Part I: Dynamics of Exoplanetary Spins 11
2 Dynamics of Colombo’s Top: Generating Exoplanet Obliquities from
Planet-Disk Interactions 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Colombo’s Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.2 Planetary Obliquities from Planet-Disk Interaction . . . . . . 12
2.1.3 Goal of This Paper . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Cassini States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Separatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Adiabatic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Adiabatic Evolution Outcomes . . . . . . . . . . . . . . . . . . 24
2.3.2 Analytical Theory for Adiabatic Evolution . . . . . . . . . . . 25
2.4 Nonadiabatic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.1 Transition to Non-adiabaticity: Results for ϵ≲ ϵc . . . . . . . 33
2.4.2 Non-adiabatic Evolution: Result for ϵ≳ ϵc . . . . . . . . . . . . 40
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Dynamics of Colombo’s Top: Tidal Dissipation and Resonance Cap-
ture, With Applications to Oblique Super-Earths, Ultra-Short-Period
Planets and Inspiraling Hot Jupiters 48
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Spin Evolution Equations and Cassini States: Review . . . . . . . . 50
3.3 Spin Evolution with Alignment Torque . . . . . . . . . . . . . . . . . . 54
3.3.1 Modified Cassini States . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . 56
3.3.3 Spin Obliquity Evolution Driven by Alignment Torque . . . . 58
3.3.4 Analytical Calculation of Resonance Capture Probability . . 62
3.4 Spin Evolution with Weak Tidal Friction . . . . . . . . . . . . . . . . 69
ix
3.4.1 Tidal Cassini Equilibria (tCE) . . . . . . . . . . . . . . . . . . . 69
3.4.2 Spin and Obliquity Evolution as a Function of Initial Spin
Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4.3 Semi-analytical Calculation of Resonance Capture Probabil-
ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.4 Spin Obliquity Evolution for Isotropic Initial Spin Orienta-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.5.1 Obliquities of Super-Earths with Exterior Companions . . . 94
3.5.2 Formation of Ultra-short-period Planet Formation via Obliq-
uity Tides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.5.3 Orbital decay of WASP-12b Driven by Obliquity Tides . . . . 100
3.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 Dynamics of Colombo’s Top: Non-Trivial Oblique Spin Equilibria of
Super-Earths in Multi-planetary Systems 110
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.2 Spin Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3 Steady States Under Tidal Alignment Torque. . . . . . . . . . . . . . 115
4.3.1 Summary of Various Outcomes . . . . . . . . . . . . . . . . . . 127
4.4 Weak Tidal Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Part II: Tidal Evolution in Stellar Binaries 132
5 Physics of Tidal Dissipation in Early-Type Stars and White Dwarfs:
Hydrodynamical Simulations of Internal Gravity Wave Breaking in
Stellar Envelopes 132
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2 Problem Setup and Equations . . . . . . . . . . . . . . . . . . . . . . . 135
5.3 Internal Gravity Waves: Theory . . . . . . . . . . . . . . . . . . . . . . 136
5.3.1 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.2 Wave Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.3.3 Wave Breaking Height . . . . . . . . . . . . . . . . . . . . . . . 139
5.3.4 Critical Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.4 Numerical Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4.1 Parameter Choices . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.4.2 Damping Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5 Weakly Forced Numerical Simulation . . . . . . . . . . . . . . . . . . 144
5.6 Numerical Simulations of Wave Breaking . . . . . . . . . . . . . . . . 148
5.6.1 Numerical Simulation Results . . . . . . . . . . . . . . . . . . 148
5.6.2 Kelvin-Helmholtz Instability and Critical Layer Width . . . . 151
5.6.3 Flux Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.6.4 Critical Layer Propagation . . . . . . . . . . . . . . . . . . . . . 156
x
5.6.5 Non-absorption at Critical Layer . . . . . . . . . . . . . . . . . 157
5.6.6 Resolution Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.7 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.7.1 Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6 Dynamical Tides in Eccentric Binaries Containing Massive Main-
Sequence Stars: Analytical Expressions 166
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.2 Dynamical Tides in Massive Stars . . . . . . . . . . . . . . . . . . . . 167
6.2.1 Circular Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.2.2 Eccentric Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.3 Analytic Evaluation of Tidal Torque and Energy Transfer Rates . . 171
6.3.1 Approximating Hansen Coefficients . . . . . . . . . . . . . . . 172
6.3.2 Approximate Expressions for Torque and Energy Transfer . 176
6.4 PSR J0045-7319/B-Star Binary . . . . . . . . . . . . . . . . . . . . . . 186
6.5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.5.1 Key Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.5.2 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Part III: Tertiary-Induced Formation of Black Hole Bina-
ries 195
7 Spin-Orbit Misalignments in Tertiary-Induced Black-Hole Binary
Mergers: Theoretical Analysis 195
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.2 LK-Induced Mergers: Orbital Evolution . . . . . . . . . . . . . . . . . 198
7.2.1 Analytical Results Without GW Radiation . . . . . . . . . . . 203
7.2.2 Behavior with GW Radiation . . . . . . . . . . . . . . . . . . . 204
7.3 Spin Dynamics: Equations . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.3.1 Nondissipative Spin Dynamics . . . . . . . . . . . . . . . . . . 205
7.3.2 Spin Dynamics With GW Dissipation . . . . . . . . . . . . . . 209
7.3.3 Spin Dynamics Equation in Component Form . . . . . . . . . 210
7.4 Analysis: Approximate Adiabatic Invariant . . . . . . . . . . . . . . . 211
7.4.1 The Adiabatic Invariant . . . . . . . . . . . . . . . . . . . . . . 212
7.4.2 Deviation from Adiabaticity . . . . . . . . . . . . . . . . . . . . 213
7.4.3 Estimate of Deviation from Adiabaticity from Initial Conditions214
7.4.4 Origin of the θ ◦sl,f = 90 Attractor . . . . . . . . . . . . . . . . . 220
7.5 Analysis: Effect of Resonances . . . . . . . . . . . . . . . . . . . . . . . 224
7.5.1 Intuitive Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7.5.2 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.5.3 Effect of Resonances in LK-Induced Mergers . . . . . . . . . . 229
7.5.4 Effect of Resonances in LK-Enhanced Mergers . . . . . . . . . 230
7.6 Stellar Mass Black Hole Triples . . . . . . . . . . . . . . . . . . . . . . 235
xi
7.7 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 239
8 The Mass Ratio Distribution of Tertiary Induced Binary Black Hole
Mergers 244
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
8.2 Von Zeipel-Lidov-Kozai (ZLK) Oscillations: Analytical Results . . . 248
8.2.1 Quadrupole Order . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.2.2 Octupole Order: Test-particle Limit . . . . . . . . . . . . . . . 251
8.2.3 Octupole Order: General Masses . . . . . . . . . . . . . . . . . 254
8.3 Tertiary-Induced Black Hole Mergers . . . . . . . . . . . . . . . . . . 255
8.3.1 Merger Windows and Probability: Numerical Results . . . . 261
8.3.2 Merger Probability: Semi-analytic Criteria . . . . . . . . . . . 263
8.4 Merger Fraction as a Function of Mass Ratio . . . . . . . . . . . . . . 266
8.4.1 Merger Fraction for Fixed Tertiary Eccentricity . . . . . . . . 266
8.4.2 Merger Fraction for a Distribution of Tertiary Eccentricities 270
8.4.3 q≪ 1 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
8.4.4 Limitations of semi-analytic Calculation . . . . . . . . . . . . 274
8.5 Mass Ratio Distribution of Merging BH Binaries . . . . . . . . . . . . 277
8.5.1 Initial q-distribution of BH Binaries . . . . . . . . . . . . . . . 278
8.5.2 q-distribution of Merging BH Binaries . . . . . . . . . . . . . . 279
8.6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 280
A Appendicies to Dynamics of Colombo’s Top: Dissipating Disk 285
A.1 Cassini State Local Dynamics . . . . . . . . . . . . . . . . . . . . . . . 285
A.1.1 Canonical Equations of Motion and Solutions . . . . . . . . . 285
A.1.2 Stability and Frequency of Local Oscillations . . . . . . . . . . 286
A.2 Approximate Adiabatic Evolution . . . . . . . . . . . . . . . . . . . . . 287
B Appendicies to Colombo’s Top: Weak Tidal Friction 289
B.1 Convergence of Initial Conditions Inside the Separatrix to CS2 . . 289
B.2 Approximate TCE2 Probability for Small ηsync . . . . . . . . . . . . . 290
C Appendicies to Colombo’s Top: Multiplanetary Systems 293
C.1 Inclination Modes of 3-Planet Systems . . . . . . . . . . . . . . . . . . 293
C.2 Additional Comments and Results on Mixed Mode Equilibria . . . . 297
D Appendicies to Physics of Tidal Dissipation 301
D.1 Derivation of Fluid Equations . . . . . . . . . . . . . . . . . . . . . . . 301
D.2 Forcing Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
D.3 Equation Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 304
E Appendicies to Dynamical Tides in Eccentric Binaries 307
E.1 Stellar Structure calculations with MESA . . . . . . . . . . . . . . . . 307
xii
F Appendicies to Spin Dynamics of Tertiary-Induced Black Hole Merg-
ers 311
F.1 Floquet Theory Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 311
F.1.1 Without Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . 311
F.1.2 With Nutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
F.1.3 Quantitative Effect . . . . . . . . . . . . . . . . . . . . . . . . . 315
G Appendicies to Mass Ratio Distribution of Tertiary-Induced Black
Hole Mergers 319
G.1 Origin of Octupole-Inactive Gap . . . . . . . . . . . . . . . . . . . . . . 319
xiii
LIST OF TABLES
5.1 Spectral resolutions and Reynolds numbers of simulations of wave
breaking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
xiv
LIST OF FIGURES
2.1 Definition of angles in the Cassini state configuration and the adopted
sign convention for θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Cassini state obliqu( ities as)a function of η for I = 5◦. . . . . . . . . . 19
2.3 Level curves of H φ,cosθ for I = 5◦, where warmer colors denote
more positive values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Plot of fractional areas of each of the zones Aj(η)/4π for I = 5◦. . . . 22
2.5 Top: The final spin obliquity θf as a function of the initial spin-disk
misalignment angle θsd,i for systems evolving from initial ηi ≫ 1 to
ηf ≪ 1, for I = 5◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Same as the top panel of Fig. 2.5 but for I = 20◦ and with fewer
annotations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 An example of the II→ I evolutionary track for I = 5◦ and θ ◦sd,i = 17.2 . 34
2.8 Same as Fig. 2.7 but for the II → III track. . . . . . . . . . . . . . . . 35
2.9 Same as Fig. 2.7 but for the III → I track. . . . . . . . . . . . . . . . 36
2.10 Same as Fig. 2.7 but for the III → II → I track. . . . . . . . . . . . . 37
2.11 Same as Fig. 2.5 but for ϵ = 10−2.5 and restricting θ ◦sd,i < 90 (blue
dots). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.12 Same as Fig. 2.11 but for ϵ = 10−1.5 (i.e. larger non-adiabaticity ef-
fect). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.13 Same as Fig. 2.7 but for a nonadiabatic case, with ϵ= 0.3. . . . . . 41
2.14 Final obliquity θf as a function of ϵ for θ ◦sd,i = 0 and I = 5 . . . . . . 44
2.15 Final obliquity θf vs θsd,i for I = 5◦ and ϵ= 0.3 (firmly in the nonadi-
abatic regime). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.16 Schematic picture for understanding nonadiabatic obliquity evolu-
tion when θsd,i > 0 assuming ηi ≫ 1. . . . . . . . . . . . . . . . . . . . 45
3.1 Cassini State obliquities θ as a function of η ≡ −g/α (Eq. 8.5) for
I = 20◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Level curves of the Hamiltonian (Eq. 3.8) for I = 20◦, for which ηc ≈
0.57 (Eq. 3.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Modified CS obliquities (top) and azimuthal angles (bottom) for I =
20◦ and η= 0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 λ20 (Eq. 3.16) as a function of η for the four CSs, for three different
values of the shift in φcs (e.g. for ∆φ = 60◦cs , the phase angles are
φ = 120◦cs for CS2 and φcs = 60◦ for CSs 1, 3, and 4). . . . . . . . . . 59
3.5 Asymptotic outcomes of spin evolution driven by an alignment torque
for different initial spin orientations (θi and φi) for a system with
η= 0.2 and I = 20◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6 Plot illustrating the probabilistic origin of separatrix (resonance)
capture for a system with η= 0.2 and I = 20◦. . . . . . . . . . . . . . 66
3.7 Zone III to Zone II transition probability PIII→II upon separatrix
encounter as a function of η driven by an alignment torque. . . . . 67
xv
3.8 Schematic depiction of the effect of tidal friction on the planet’s spin
evolution in the θ-Ωs plane. . . . . . . . . . . . . . . . . . . . . . . . . 72
3.9 Same as Fig. 3.8 but for ηsync = 0.5. . . . . . . . . . . . . . . . . . . . 73
3.10 Same as Figs. 3.8 but for ηsync = 0.7. . . . . . . . . . . . . . . . . . . 74
3.11 Obliquities of the two tCE as a function of ηsync for I = 20◦ (top) and
I = 5◦ (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.12 Phase space evolution of the pink trajectory in Fig. 3.8, for which
ηsync = 0.06 and I = 20◦ (corresponding to ηc = 0.574). . . . . . . . . 76
3.13 Same as Fig. 3.12 but for φ ◦i = 286 , corresponding to the cyan tra-
jectory in Fig. 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.14 Same as Fig. 3.12 but for η = 0.7 and θ = 10◦sync i , corresponding to
the purple trajectory shown in Fig. 3.10. . . . . . . . . . . . . . . . . 78
3.15 Left: Asymptotic outcomes of spin evolution in the presence of weak
tidal friction for different initial spin orientations (θi and φi) for a
system with ηsync = 0.06 and I = 20◦. . . . . . . . . . . . . . . . . . . 83
3.16 Same as Fig. 3.15 but for ηsync = 0.2. . . . . . . . . . . . . . . . . . . 84
3.17 Same as Fig. 3.15 but for ηsync = 0.5. . . . . . . . . . . . . . . . . . . 85
3.18 Comparison of the fraction of systems converging to tCE2 obtained
via numerical simulation (red dots) and obtained via a semi-analytic
calculation (blue line) for ηsync = 0.06, I = 20◦, and Ωs,i = 10n (see
right panel of Fig. 3.15). . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.19 Same as Fig. 3.18 but for ηsync = 0.2, corresponding to the right
panel of Fig. 3.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.20 Obliquities and probabilities of attaining the two tidally-stable tCEs. 91
3.21 Same as Fig. 3.20 but for I = 5◦. . . . . . . . . . . . . . . . . . . . . . . 92
3.22 PtCE2 as a function of ηsync for I = 5◦ and I = 20◦ shown on a log-log
plot, to emphasize the scaling at small ηsync. . . . . . . . . . . . . . 93
3.23 Depiction of the values of ηsync for the Super Earth + Cold Jupiter
systems as a function of a and ap for I = 20◦ (top) and I = 5◦ (bot-
tom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.24 Constraints on the companion of WASP-12b and the values of ηsync,i
(Eq. 3.61) in the obliquity tidal decay scenario. . . . . . . . . . . . . 105
4.1 Two evolutionary trajectories of Ŝ showing capture into mode I res-
onances (CSs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.2 Three evolutionary trajectories for a system with the same param-
eters as in Fig. 4.1, except for g(II) =−α= 10g(I). . . . . . . . . . . . 118
4.3 Asymptotic outcomes of spin evolution driven by tidal alignment
torque for different initial spin orientations. . . . . . . . . . . . . . . 120
4.4 Same as the bottom right panel of Fig. 4.3 but zoomed-in to a nar-
row range of initial obliquities near θeq ≈ 57◦∣ . .∣ . . . . . . . . . . . . 1214.5 “Equilibrium” obliquity as a function of the r∣eson∣ant frequency (Eq. 4.9)for a system with I(I) = 10◦, I = 1◦(II) , α= 10 g(I) , and g(II) = 10g(I). 123
xvi
4.6 Final outc∣omes of spin evolution under tidal alignment torque for a3-planet s∣yste∣∣m with inclination mode parameters I = 10◦(I) , I(II) =1◦, α= 10 g(I) (same as Figs. 4.1–4.5) and varying g(II)/g(I). . . . . 124
4.7 Same as Fig. 4.3 but for I(II) = 3◦. . . . . . . . . . . . . . . . . . . . . 125
4.8 Same as Fig. 4.5 but for I = 3◦(II) . . . . . . . . . . . . . . . . . . . . . . 126
4.9 Same as Fig. 4.6 but for I = 3◦(II) . . . . . . . . . . . . . . . . . . . . . 126
4.10 Similar to Figs. 4.7 b∣∣ut i∣∣ncluding full tidal effects on the planet’sspin, with αsync = 10 g(I) (Eq. 4.19), I 3◦(II) , and initial spin Ωs,0 =
3n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.11 Similar to Fig. 4.9 but with weak tidal friction. . . . . . . . . . . . . 130
5.1 Amplitude of the excited IGW over time (in units of N−1) in the
weakly forced simulation, c(omp)uted using Eq. (5.28). . . . . . . . . 1465.2 F/Fan plotted at select times t (in units of N−1). . . . . . . . . . . . 147
5.3 Snapshots of ux and Υ≡ ln ρ/ρ̄ in the fiducial simulation illustrat-
ing distinct phases of the evolution of the flow. . . . . . . . . . . . . 150
5.4 The mean horizontal flow velocity U(z, t) (Eq. (5.16)) and the di-
mensionless momentum flux F(z, t)/Fan (Eqs. (5.11) and (5.29)) in
our fiducial simulation plotted at the same times as in Fig. 5.3. . . 152
5.5 Local Richardson number (Eq. (5.33)) of the flow at the critical layer
over time (in units of N−1) in our fiducial simulation. . . . . . . . . 154
5.6 Momentum flux decomposition calculated from the simulation. . . 156
5.7 Propagation of the critical layer over time. . . . . . . . . . . . . . . . 158
5.8 The incident wave amplitude A i(t) (solid black) and the reflected
wave amplitude Ar(t) (dashed red) just above the forcing zone. . . . 160
5.9 R2
−A
, F̂r, and redistribution flux F̂s as a function of time (in units of
N 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.10 Convergence of the median F̂r, R2A, F̂s, and Ri (Eqs. (5.43)–(5.45)
and (5.21) respectively) in simulations with varying resolution and
viscosity as given in Tab. 5.1. . . . . . . . . . . . . . . . . . . . . . . . 163
6.1 Plot of Hansen coefficients FN2 for e= 0.9. . . . . . . . . . . . . . . . 175
6.2 Plot of FN0 (black circles) for e= 0.9. . . . . . . . . . . . . . . . . . . 176
6.3 The tidal torque on a non-rotating (top) and rapidly rotating (bot-
tom) star due to a companion with orbital eccentricity e. . . . . . . 180
6.4 The tidal torque as a function of the stellar spin for a highly eccen-
tric e= 0.9 companion. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.5 The spin frequencies Ωps normalized by the pericentre frequency
Ωp (Eq. 6.17) as a function of eccentricity. . . . . . . . . . . . . . . . 182
6.6 The tidal energy transfer rate Ėin for a non-rotating (top) and a
rapidly rotating (bottom) star. . . . . . . . . . . . . . . . . . . . . . . 185
6.7 The tidal energy transfer rate Ėin as a function of spin (normalized
by Ωp) for a highly eccentric e= 0.9 companion. . . . . . . . . . . . . 186
xvii
6.8 The orbital decay ratio Ṗ/P as a function of Ωs for the canonical
parameters of the PSR J0045-7319 binary system, as evaluated by
Eq. (6.46), for four different values of rc (legend, in units of R⊙). . 191
7.1 An example of the “90◦ spin attractor” in LK-induced BH binary
mergers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.2 The merger time and the final spin-orbit misalignment angle θsl,f
as a function of the initial inclination I0 for LK-induced mergers. . 200
7.3 Bifurcation diagram for the BH spin orientation during LK oscilla-
tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.4 Definition of angles in the problem, shown in plane of the two an-
gular momenta Lout and L. . . . . . . . . . . . . . . . . . . . . . . . . 210
7.5 The same simulation as depicted in Fig. 7.1 but showing several
calculated quantities relevant to the theory of the spin evolution. . 215
7.6 The same simulation as Fig. 7.1 but zoomed in on the region around
A ≡ΩSL/ΩL ≃ 1 and showing a wide range of relevant quantities. . 216
7.7 Same as Fig. 7.6∣ excep∣ t for I
◦
0 = 90.2 (and all other parameters are
the same as in F∣∣ ig. 7.1∣∣ ), corresponding to a faster coalescence. . . . 2177.8 Comparison of I˙̄e/Ωe obtained from simulations and from the
max
analytical expression Eq. (7.52), where we take f = 2.72 in Eq. (7.49).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
7.9 Net change in θ̄e over the binary inspiral as a function of initial
inclination I0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.10 emax andΩe/ΩLK as a function of I(emin), the inclination of the inner
binary at eccentricity minimum, for varying values of emin (differ-
ent colors as labeled) for a LK-induced merger (with the parameters
the same as in Figs. 7.1 and 7.2). . . . . . . . . . . . . . . . . . . . . 231
7.11 The merger time (top), the magnitude of the initial adiabaticity pa-
rameter |A | ≡ΩSL/|ΩL| (middle), and the final spin-orbit misalign-
ment angle θsl,f (bottom) for LK-enhanced mergers. . . . . . . . . . 232
7.12 Same as Fig. 7.10, but for a LK-enhanced merger (with the param-
eters of Fig. 7.11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
7.13 Definition of angles in the case where L/Lout is nonzero. . . . . . . 235
7.14 Similar to Fig. 7.2 but for stellar-mass tertiary m3 = 30M⊙ and ã3 =
4500 AU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.15 Similar to Fig. 7.11 except for a stellar mass tertiary m3 = 30M⊙
and ãout = 3 AU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
8.1 Histogram of the mass ratios q ≡ m2/m1 of binary BH mergers in
the O3a data release, excluding the two NS-NS mergers but includ-
ing GW190814, whose 2.5M⊙ secondary may be a BH [Abbott et al.,
2021]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
8.2 The maximum eccentricity achieved for an inner binary in the test-
particle limit as a function of the initial inclination angle I0. . . . . 251
xviii
8.3 An example of the triple evolution for a system with significant oc-
tupole effects and finite η (see Eq. 8.5). . . . . . . . . . . . . . . . . . 253
8.4 Eccentricity excitation and merger windows for the fiducial BH triple
system (a = 100 AU, aout,eff = 3600 AU, m12 = 50M⊙, m3 = 30M⊙)
with q = 0.5 and eout = 0.6, corresponding to η ≈ 0.087 and ϵoct ≈
0.007. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
8.5 Same as Fig. 8.4 but for q= 0.3, corresponding to η≈ 0.07 and ϵoct ≈
0.011. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.6 Same as Fig. 8.4 but for q = 0.2, corresponding to η ≈ 0.054 and
ϵoct ≈ 0.014. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
8.7 Same as Fig. 8.4 but for eout = 0.9 while holding aout,eff the same,
corresponding to η= 0.118 and ϵoct = 0.019. . . . . . . . . . . . . . . 259
8.8 Same as Fig. 8.4 but for a more compact inner binary; the param-
eters are a0 = 10 AU, aout,eff = 700 AU, m12 = 50M⊙, m3 = 30M⊙,
eout = 0.9, and q= 0.4, corresponding to η= 0.118 and ϵoct = 0.029. 260
8.9 Merger fraction (Eq. 8.17) of BH binaries in triples as a function of
mass ratio q (left panel) for several values of outer binary eccentric-
ities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.10 Same as Fig. 8.9 but for a0 = 50 AU. . . . . . . . . . . . . . . . . . . . 269
8.11 Binary BH merger fraction, merger times, and eccentricities as a
function of mass ratio q for the fiducial triple systems. . . . . . . . . 272
8.12 Same as Fig. 8.11 but for aout,eff = 5500 AU. . . . . . . . . . . . . . . 273
8.13 Same as blue dashed line of the top panel of Fig. 8.11 but extended
to very small q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
8.14 Completeness of the semi-analytical merger fraction, defined as f anmerger/ fmerger,
as a function of the integration time used for the non-dissipative
simulations, in the fiducial parameter regime while eout is fixed at
a few values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
8.15 Mass ratio distributions of the initial BH binaries (solid lines) and
merging BH binaries (dotted lines) when using α= 2.35 for the MS
stellar initial mass function (see Sections 8.5.1 and 8.5.2). . . . . . 281
8.16 Same as Fig. 8.15 but for α= 2, i.e. a nearly uniform distribution of
the main sequence binary mass ratio. . . . . . . . . . . . . . . . . . . 281
A.1 λ2, given by Eq. (A.3), evaluated at each of the Cassini States. . . 287
C.1 Inclination mode frequencies and amplitudes for the 3SE (left) and
2SE + CJ (right) systems. . . . . . . . . . . . . . . . . . . . . . . . . . 295
C.2 Same as Fig. C.1 but for a2 = 0.2 AU. . . . . . . . . . . . . . . . . . . . 296
C.3 Same as Fig. C.1 but for a2 = 0.25 AU. . . . . . . . . . . . . . . . . . . 296
C.4 Same as Fig. 4.2 but for g ◦(II) = 10g(I) and I(II) = 3 . . . . . . . . . . . 298
C.5 Same as Figs. 4.5 and 4.8 but for I ◦(II) = 3 and g(II) = 8.5g(I), where
ranges of oscillation in θsl have been suppressed for clarity. . . . . . 299
C.6 Same as Fig. 4.4 but for I = 3◦(II) . . . . . . . . . . . . . . . . . . . . . 300
C.7 Same as Fig. 4.6 but for I ◦(II) = 9 . . . . . . . . . . . . . . . . . . . . . . 300
xix
E.1 Plots illustrating the procedure used to find the best-fitting stellar
model to the properties of PSR J0045-7319 with no convective over-
shooting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
E.2 Same as Fig. E.1 but with a convective overshoot parameter of 0.04
(see MESA documentation). . . . . . . . . . . . . . . . . . . . . . . . . 310
E.3 Same as Fig. E.1 but with a convective overshoot parameter of 0.04
(see MESA documentation) and with Z = 0.2Z⊙ = 0.004 . . . . . . . 310
F.1 Definition of angles for numerical study of the monodromy matrix
rotation axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
F.2 Comparison of the orientation R obtained from numerical simula-
tions of Eq. (7.22) in the η ̸= 0, LK-enhanced parameter regime with
the analytic resonance formula given by Eq. (F.15) as a function of
initial inclination, in the absence of GW dissipation. . . . . . . . . . 318
G.1 Octupole-active windows and amplitude of oscillation of K (Eq. 8.4). 320
G.2 The left panel is the same as Fig. 8.3 but includes the evolution of
the azimuthal angle of the eccentricity vector, Ωe. . . . . . . . . . . 322
G.3 Plot of ∆Ωe, the change in Ωe over a single ZLK cycle, for q = 0.2
and the fiducial parameters using different initial conditions. . . . 324
xx
CHAPTER 1
INTRODUCTION
The diversity and intractability of phenomena in astrophysics owe their origin
to the breadth of scales over which astrophysical systems evolve. Despite these dif-
ficulties, the dynamical evolution of such systems is important to understand, as
they shape the universe as we observe it today. These systems, consisting of bodies
such as planets, stars, and black holes, evolve over characteristic time scales rang-
ing from minutes to years due to effects such as tidal evolution and gravitational
wave emission. In principle, these effects are sufficiently well understood that as-
trophysical dynamics can be solved via direct numerical simulation. In practice,
however, numerical integration over times as long as the age of the observable
universe and for the wide range of observed system parameters is far beyond the
capability of modern computers. As a result, many effective models are developed
to bridge the gap between the “microphysics” of these systems, their behavior over
these short time scales, and their long-term evolution.
The focus of this dissertation concerns the dynamics of various effective mod-
els that are derived under the so-called “secular” approximation. In systems con-
sisting of multiple bodies, their mutual gravitational interaction causes the con-
stituent bodies to orbit one another over a wide range of orbital periods. The
secular approximation concerns the dynamics when averaged over the system’s
orbital period(s), turning individual bodies into massive rings. The evolution of
these rings occurs over substantially longer time scales than the orbital periods
of the original bodies, and so long term study of the system becomes possible. In
this dissertation, I apply both analytical and numerical techniques to study the
secular dynamics of three classes of systems: (i) the evolution of a planet’s spin
in a planetary system with multiple planets, (ii) the formation of merging black
1
hole binaries (detectable by LIGO/VIRGO) from systems consisting of three black
holes, and (iii) the evolution of stars in binaries due to their mutual tidal interac-
tion. Below, I motivate and describe my results for each of these three classes of
systems.
1.1 The Dynamics of Exoplanetary Spins
First, I introduce my work on the dynamics of an exoplanet’s spin. The spin state
of a planet is characterized by two attributes, its spin period and orientation. The
spin period of a planet determines the length of a day on the planet, while its obliq-
uity (the angle between a planet’s spin axis and orbital axes) gives rise to its sea-
sons. Both of these attributes are of general interest, as together they determine
the atmospheric and surface conditions of the planet. They are also of dynamical
interest as they reflect the evolutionary history of a planet. Within the solar sys-
tem, the planets’ obliquities take on a wide range of values, from nearly zero for
Mercury and 3.1◦ for Jupiter, to 23◦ for Earth and 26.7◦ for Saturn, to 30◦ for Nep-
tune and 98◦ for Uranus. The origin of these nonzero obliquities has motivated
many hypotheses for the formation of each of these planets: for instance, multiple
giant impacts are traditionally invoked to generate the large obliquities of the ice
giants [Safronov and Zvjagina, 1969, Benz et al., 1989, Korycansky et al., 1990,
Morbidelli et al., 2012]. In the case of the gas giants, obliquity excitation may be
achieved via certain spin-orbit resonances, where the spin and orbital precession
frequencies of the planet become commensurate as the system evolves [Ward and
Hamilton, 2004, Hamilton and Ward, 2004, Ward and Canup, 2006, Vokrouhlickỳ
and Nesvornỳ, 2015, Saillenfest et al., 2020, 2021]. These resonances were first
studied in the seminal work of Colombo [1966] and is sometimes referred to as
“Colombo’s Top”, while the equilibria of this system are often known as “Cassini
2
States” (CSs). Subsequent works have uncovered the rich dynamics of this system
[Peale, 1969, 1974, Ward, 1975, Henrard and Murigande, 1987].
Shifting focus beyond the solar system, there are about 5000 confirmed exo-
planets as of May 20221. Which among these planets may be habitable has re-
cently become a question of great interest. A key factor in assessing a planet’s
habitability is its obliquity, and so it is important to understand the dynamical
processes that affect the obliquity. Observationally, these obliquities are chal-
lenging to measure, and to date only loose constraints have been obtained for the
obliquities of faraway planetary-mass companions [Bryan et al., 2020, 2021]. Nev-
ertheless, there are prospects for better constraints on exoplanetary obliquities in
the coming years [Snellen et al., 2014, Bryan et al., 2018, Seager and Hui, 2002].
Furthermore, there is ongoing work towards inferring exoplanetary obliquities via
thermal phase curve modeling [Adams et al., 2019, Ohno and Zhang, 2019] and
via indirect dynamical evidence [Millholland and Laughlin, 2018, 2019, Millhol-
land and Spalding, 2020]. From a theory perspective, substantial obliquities may
arise even during the earlest phase of planet formation if protoplanetary disks
are turbulent and twisted [Tremaine, 1991, Jennings and Chiang, 2021]. They
can also arise due to giant impacts or planet collisions as a result of dynamical
instabilities of planetary orbits [Safronov and Zvjagina, 1969, Benz et al., 1989,
Korycansky et al., 1990, Dones and Tremaine, 1993, Morbidelli et al., 2012, Li
and Lai, 2020, Li et al., 2021]. However, many processes can still change these
obliquities before they reach their present-day values.
In this dissertation, I consider three mechanisms by which this primordial
obliquity can be related to present-day obliquities. In Chapter 2, I consider the
obliquity evolution of an oblate planet due to gravitational interactions both with
1https://exoplanetarchive.ipac.caltech.edu/
3
its host star and with an exterior, dissipating (mass-losing) protoplanetary disk.
Obliquity excitation occurs as the system passes through a resonance between
the planet’s secular spin and orbital precession frequencies as the disk dissipates.
This scenario was recently studied by Millholland and Batygin [2019], who focused
on the special case of small initial obliquities. However, the primordial obliquity is
not necessarily small, as argued above; the objective of Chapter 2 is to determine
the final obliquity of the planet as a function of an arbitrarily oriented initial spin.
When the disk dissipation is sufficiently gradual (adiabatic), I show that the prin-
ciple of conservation of phase space area combined with the theory of probabilistic
separatrix encounter [Henrard, 1982] is able to completely predict the final obliq-
uities and their associated probabilities for arbitrary initial conditions. On the
other hand, when the disk dissipates rapidly, I show that the obliquity is instead
excited impulsively. I provide analytical expressions for the obliquity excitation in
both of these regimes as well as for the boundary of the two regimes.
In Chapter 3, I consider a different mechanism for obliquity excitation, where
an inner planet with a single outer companion planet experiences tidal dissipa-
tion in addition to gravitational torques from its host star and companion. The
effect of tidal dissipation both acts to drive the planet’s obliquity towards zero
(spin-orbit alignment) and to drive the planet’s spin rate towards synchroniza-
tion (i.e. drive the spin frequency towards the planet’s orbital frequency). In the
absence of spin-orbit resonances, planets experiencing tidal dissipation are thus
expected to evolve towards low obliquities and become tidally locked. However,
in the presence of additional planets, the planet may remain in a high-obliquity,
asynchronously-rotating spin state if it is evolves into the so-called “Cassini State
2” (CS2) that is stable to tidal dissipation [Fabrycky et al., 2007, Levrard et al.,
2007, Peale, 2008]. For a general initial spin orientation, the planet evolves into
one of either CS2 or the low-obliquity CS1 via a probabilistic process. I show that
4
the probabilities of reaching each of the two CSs outcomes can be analytically
calculated via a generalization of Henrard’s seminal work of resonance capture
[Henrard, 1982]. I derive a simple closed-form expression for the case where the
initial spin orientation is isotropically distributed. I apply my results to show that
the common super Earth-cold Jupiter systems (Zhu and Wu, 2018, Bryan et al.,
2019; but see Rosenthal et al., 2021) may often host an oblique super Earth. I also
show that the proposed formation scenarios for WASP-12b and ultra-short-period
planets via obliquity-enhanced tidal dissipation [Millholland and Laughlin, 2018,
Millholland and Spalding, 2020] are disfavored due to the low probability of being
captured into CS2.
In Chapter 4, I generalize the discussion in Chapter 3 to include an additional
planet, considering systems of either three super Earths or two super Earths and
one cold Jupiter. In systems consisting of three or more planets, it is well-known
that the planetary obliquities can evolve chaotically in the absence of dissipation
[e.g. that of Mars Touma and Wisdom, 1993, Laskar and Robutel, 1993]. However,
the obliquity dynamics in the presence of tidal dissipation are not well-understood.
Via numerical integration, I show that the occurence rate of oblique planets is
generally enhanced by the presence of additional planets. I find that this is due
to a novel class of resonances that are not CSs, and that I term “mixed-mode
resonances.” I derive an analytic relation for the obliquities of these resonances.
1.2 Tidal Evolution in Stellar Binaries
Second, I introduce my work on the tidal evolution of stellar binaries, including
those consisting of two white dwarfs (WDs). About half of stars detected today
have detectable binary companions [e.g. Gao et al., 2014, Yuan et al., 2015]. In a
5
stellar binary, the changing tidal gravitational field between the two stars gener-
ates fluid motion within both component stars. As this fluid motion experiences
dissipation due to radiative and turbulent damping, energy is transferred between
the binary orbit and the individual stars. The first theoretical models for tidal
dissipation studied the equilibrium tide [Alexander, 1973, Hut, 1981]. Here, the
stars develop a hydrostatic bulge due to their mutual gravitational force, and the
motion of this bulge results in dissipation.
While simple and sufficient for weak tidal interactions, this model becomes
inaccurate for sufficiently close binaries [see Ogilvie, 2014 for a review]. In par-
ticular, internal gravity waves (IGWs), oscillations in the stellar fluid restored by
buoyancy, play an important role in several types of binary systems. In solar-type
stars with radiative cores and convective envelopes, IGWs are excited by tidal
forcing at the radiative-convective boundary and propagate inward; as the wave
amplitude grows due to geometric focusing, nonlinear effects can lead to efficient
damping of the wave [Goodman and Dickson, 1998, Barker and Ogilvie, 2010,
Essick and Weinberg, 2015]. In early-type main-sequence stars, with convec-
tive cores and radiative envelopes, IGWs are similarly excited at the convective-
radiative interface but travel toward the stellar surface; nonlinearity develops
as the wave amplitude grows, leading to efficient dissipation [Zahn, 1975, 1977].
WDs do not have convective regions, and IGWs are instead generated at changes
in the WD’s composition. For these outward-traveling IGWs, one important non-
linear effect arises when the fluid displacement becomes comparable to the wave-
length of the wave. When this threshold is reached, the waves break, like ocean
waves on the shore, and deposit their energy and angular momentum. This wave
breaking process is complex and turbulent, requiring numerical simulations to be
well characterized. However, despite its ubiquity and importance, wave breaking
has typically only been included in a parameterized fashion [e.g. Fuller and Lai,
6
2012a, Burkart et al., 2013, Vick et al., 2017].
In Chapter 5, I use well-resolved numerical simulations to study the detailed
wave breaking process. While I primarily consider the case of binary WDs [fol-
lowing the work of Fuller and Lai, 2011, 2012a, 2013], the qualitative behavior is
likely to be similar for early-type stars. Specifically, I develop quantitative pre-
scriptions for both the location and the spatial extent of the energy deposited by
the breaking IGWs. In my simulations, I generate IGWs from the bottom of a 2D
isothermal atmosphere. As these IGW propagate upwards, they grow in ampli-
tude and eventually break. I find that this wave breaking naturally generates a
sharp critical layer (CL), a thin region of fluid separating the lower stationary re-
gion (with no mean flow) and the upper “synchronized” region (with the mean flow
velocity equal to the horizontal IGW phase speed). Waves incident on the CL are
absorbed efficiently, depositing their energy and momentum flux within the CL. I
show that the width of this CL is mediated by the onset of the Kelvin-Helmholtz
instability (carried by the IGW). Furthermore, the CL propagates downwards to
accommodate the absorbed momentum flux, and the fluid synchronizes from top-
to-bottom, as first hypothesized by Goldreich and Nicholson [1989]. However, the
IGW absorption at the CL is not absolute, and I compute reflection and transmis-
sion coefficients for the IGW at the CL from my simulations.
In Chapter 6, I then present analytical expressions for the tidal evolution of a
massive star in a compact, eccentric binary. Due to the compactness of the binary,
the IGWs excited from the convective core of the star are expected to propagate
outwards through the radiative envelope and break near the stellar surface. The
tidal torque in circular binaries has recently been revised into a simpler and more
accurate form [Kushnir et al., 2017]. However, the inclusion of eccentricity signif-
icantly complicates the evaluation of the tidal torque, generally requiring a sum
7
over many Fourier harmonics [Vick et al., 2017]. I show that these two results
can be combined to yield an accurate closed-form approximation to the eccentric
tidal torque. I apply my results to the system PSR J0045-7319, which consists
of a pulsar and massive main-sequence star and exhibits detectable orbital decay
[Kaspi et al., 1994, Bell et al., 1995]. I conclude that the star must be rapidly
and differentially rotating in order for the tidal torque to be consistent with the
observed orbital decay rate.
1.3 Tertiary-Induced Formation of Black Hole Binaries
Third, I introduce my work on the formation of black hole (BH) binaries. Since
it first detected gravitational waves (GWs) in 2015, the LIGO/VIRGO Collabora-
tion (LVC) has since detected a total of 90 compact object coalescences to date, the
large majority of which are mergers of two BHs [The LIGO Scientific Collabora-
tion et al., 2021b]. As the catalog of events grows, it is increasingly fruitful to pose
the question of how these BH binaries are formed. The canonical scenario posits
that these BH binaries initially form as wider stellar binaries that undergo a com-
mon envelope phase and subsequently experience orbital decay to form compact
BH binaries that can then merge via GW emission [e.g. Belczynski et al., 2016].
However, such a process has numerous uncertainties, and many studies have ex-
plored alternatives that do not require an efficient common envelope phase. I
focus on the so-called “tertiary-induced” merger scenario, where a wide black hole
binary experiences von Zeipel-Lidov-Kozai [ZLK von Zeipel, 1910, Lidov, 1962,
Kozai, 1962] oscillations due to gravitational interactions with a distant tertiary
companion over very long, secular time scales. During these ZLK oscillations, the
eccentricity of the inner binary varies between small values to values very near
8
unity. If the maximum eccentricity is sufficiently large, the enhanced GW emis-
sion near pericenter can dramatically shrink the binary orbit, and the binary can
then merge in isolation via GW radiation alone.
In this dissertation, I consider the distributions both of the spin orientations
and of the mass ratio of the component BHs for binaries formed via the tertiary-
induced merger channel. In Chapter 7, I study the spin evolution of BH binaries
undergoing ZLK oscillations. Previously, Liu and Lai [2018] used numerical inte-
grations to study the final spin orientations of BHs in binaries that form via the
tertiary-induced channel. They found that, when the spins are initially aligned
with the orbital angular momentum axis and the octupole-order corrections to the
ZLK effect are negligible (this is accurate when either the tertiary’s orbit is cir-
cular or the inner binary has equal masses), the spins of the BHs tend to lie in
the orbital plane (i.e. normal to the angular momentum axis) upon merger. I show
via analytical arguments that this behavior is due to conservation of an approxi-
mate adiabatic invariant, which is well-conserved as long as the system coalesces
over many ZLK cycles. If this condition is satisfied, I solve for the final spin ori-
entation for an arbitrary initial spin orientation. I also quantify the deviation
from this prediction for systems that merge non-adiabatically. While the spin dis-
tribution from the LVC significantly prefers spin-orbit alignment, there is still
ongoing study concerning whether there may be a sub-population of BH mergers
with perpendicular spin and orbit axes [Roulet et al., 2021, The LIGO Scientific
Collaboration et al., 2021a]. Such a sub-population is consistent with formation
via the tertiary-induced channel.
In Chapter 8, I study the mass-ratio distribution in the tertiary-induced merger
channel. This distribution is sensitive to the octupole-order corrections neglected
in Chapter 7. In systems where the octupole-order terms are more important,
9
the inner binary can more readily attain extreme eccentricities. The strength of
these octupole-order terms scales inversely with the binary’s mass ratio (i.e. bina-
ries with more unequal masses experience stronger octupole-order effects). As a
result, it is evident that the tertiary-induced merger scenario is more efficient at
producing BH binaries with unequal component masses. This trend is opposite to
that seen by the LVC, which has observed more systems with equal masses than
systems with unequal masses. This may not entirely rule out tertiary-induced
mergers as a formation channel for BH binaries, however: the mass-ratio distri-
bution of wide BH binaries is unknown, so the enhanced formation rate may be
unable to overcome a natural paucity of small-mass-ratio BH binaries. I also show
that the formation rate enhancement for unequal-mass binaries is much weaker
when the inner binary is somewhat compact, which also can slightly ease the
tension with the observed mass-ratio distribution. Finally, note that the octupole-
order corrections are negligible when the tertiary is sufficiently distant, e.g. when
the binary is on a distant orbit around a supermassive BH.
10
CHAPTER 2
DYNAMICS OF COLOMBO’S TOP: GENERATING EXOPLANET
OBLIQUITIES FROM PLANET-DISK INTERACTIONS
Originally published in:
Yubo Su and Dong Lai. Dynamics of Colombo’s Top: Generating exoplanet obliq-
uities from planet-disk interactions. ApJ, 903(1):7, Oct 2020. doi: 10.3847/
1538-4357/abb6f3
2.1 Introduction
2.1.1 Colombo’s Top
A rotating planet is subjected to gravitational torque from its host star, making
its spin axis precess around its orbital (angular momentum) axis. Now suppose
the orbital axis precesses around another fixed axis—such orbital precession could
arise from gravitational interactions with other masses in the system (e.g. planets,
external disks, or binary stellar companion). What is the dynamics of the plan-
etary spin axis? How does the spin axis evolve as the spin precession rate, the
orbital precession rate, or their ratio, gradually changes in time?
Colombo [1966] was the first to point out the importance of the above simple
model in the study of the obliquity (the angle between the spin and orbital axes) of
planets and satellites. Subsequent works [Peale, 1969, 1974, Ward, 1975, Henrard
and Murigande, 1987] have revealed rich dynamics of this model. With appropri-
ate modification, this model can be used as a basis for understanding the evolution
of rotation axes of celestial bodies. Indeed, many contemporary problems in plane-
tary/exoplanetary dynamics can be cast into a form analogous to this simple model
or its variants [e.g. Ward and Hamilton, 2004, Fabrycky et al., 2007, Batygin and
11
Adams, 2013, Lai, 2014, Anderson and Lai, 2018, Zanazzi and Lai, 2018].
In this paper we present a systematic investigation on the secular evolution of
Colombo’s top, starting from general initial conditions. Our study includes several
new analytical results that go beyond previous works. While our results are gen-
eral, we frame our study in the context of generating exoplanet obliquities from
planet-disk interaction with a dissipating disk.
2.1.2 Planetary Obliquities from Planet-Disk Interaction
It is well recognized that the obliquity of a planet may provide important clues to
its dynamical history. In the the Solar System, a wide range of planetary obliq-
uities are observed, from nearly zero for Mercury and 3.1◦ for Jupiter, to 23◦ for
Earth and 26.7◦ for Saturn, to 98◦ for Uranus. Multiple giant impacts are tra-
ditionally invoked to generate the large obliquities of ice giants [Safronov and
Zvjagina, 1969, Benz et al., 1989, Korycansky et al., 1990, Morbidelli et al., 2012].
For gas giants, obliquity excitation may be achieved via spin-orbit resonances,
where the spin and orbital precession frequencies of the planet become commen-
surate as the system evolves [Ward and Hamilton, 2004, Hamilton and Ward,
2004, Vokrouhlickỳ and Nesvornỳ, 2015]. Such resonances may also play a role
in generating the obliquities of Uranus and Neptune [Rogoszinski and Hamilton,
2020]. For terrestrial planets, multiple spin-orbit resonances and their overlaps
can make the obliquity vary chaotically over a wide range [e.g. Laskar and Robu-
tel, 1993, Touma and Wisdom, 1993, Correia et al., 2003]
Obliquities of extrasolar planets are difficult to measure. So far only loose
constraints have been obtained for the obliquity of a faraway (≳ 50 au) planetary-
mass companion [Bryan et al., 2020]. But there are prospects for constraining
12
exoplanetary obliquities in the coming years, such as using high-resolution spec-
troscopy to obtain vsin i of the planet [Snellen et al., 2014, Bryan et al., 2018] and
using high-precision photometry to measure asphericity of the planet [Seager and
Hui, 2002]. Finite planetary obliquities have been indirectly inferred to explain
the peculiar thermal phase curves [see e.g. Adams et al., 2019, Ohno and Zhang,
2019] and tidal dissipation in hot Jupiters [Millholland and Laughlin, 2018] and
in super Earths [Millholland and Laughlin, 2019].
It is natural to imagine some of the mechanisms that generate planetary obliq-
uities in the Solar System may also operate in exoplanetary systems. Recently,
Millholland and Batygin [2019] studied the production of planet obliquities via a
spin-orbit resonance, where a dissipating protoplanetary disk causes resonance
capture and advection. In their work, a planet is accompanied by an inclined
exterior disk; as the disk gradually dissipates, the planetary obliquity increases,
reaching 90◦ for what the authors characterize as adiabatic resonance crossings.
The Millholland & Batygin study assumes a negligible initial planetary obliq-
uity. This assumption is intuitive, since the planet attains its spin angular mo-
mentum from the disk. But it may not always be satisfied. In particular, the for-
mation of rocky planets through planetesimal accretion can lead to a wide range of
obliquities, especially if the final spin is imparted by a few large bodies [Dones and
Tremaine, 1993, Lissauer et al., 1997, Miguel and Brunini, 2010]. Such “stochas-
tic” accretion likely happened for terrestrial planets in the Solar System. Giant
impacts may have also played a role in the formation of the close-in multiple-
planet systems diskovered by the Kepler satellite [e.g. Inamdar and Schlichting,
2015, Izidoro et al., 2017].
13
2.1.3 Goal of This Paper
In this paper, we consider a wide range of initial planetary obliquities in the
Millholland-Batygin dissipating disk scenario, and examine how the obliquity
evolves toward the “final” value as the exterior disk dissipates. We provide an
analytical framework for understanding the final planetary obliquity for arbi-
trary initial spin-disk misalignment angles. We also consider various dissipa-
tion timescales, and examine both “adiabatic” (slow disk dissipation) and “non-
adiabatic” evolution. We calibrate these analytical results with numerical calcula-
tions. On the technical side, our paper extends previous works [such as Henrard,
1982, Henrard and Murigande, 1987, Millholland and Batygin, 2019] in several
aspects. Two of our main results are: (i) a careful accounting of the phase space
area across sepratrix to analytically describe the rich dynamics of adiabatic evolu-
tion, and (ii) using the concept of “partial adiabatic resonance advection” to fully
capture the dynamics in the non-adiabatic limit.
It is important to note that while we focus on a specific scenario of gener-
ating/modifying planetary obliquities from planet-disk interactions, our analysis
and results have a wide range of applicability. For example, a dissipating disk is
dynamically equivalent to an outward-migrating external companion.
The paper is organized as follows. In Section 2.2, we review the relevant spin-
orbit dynamics and key concepts that are used in the remainder of the paper. In
Sections 2.3 and 2.4, we study the evolution of the system when the disk dissi-
pates on different timescales, from highly adiabatic to nonadiabatic. Analytical
results are presented to explain the numerical results in both limits. We discuss
the implications of our results in Section 2.5. Our primary physical results con-
sist of Fig. 2.5 in the adiabatic limit and Fig. 2.12 in the nonadiabatic limit. Some
14
detailed calculations are relegated to the appendices, including a leading-order es-
timate of the final planetary obliquities given small initial spin-disk misalignment
angles in Appendix A.2.
2.2 Theory
2.2.1 Equations of Motion
We consider a star of mass M⋆ hosting an oblate planet (mass Mp, radius Rp and
spin angular frequency Ωp) at semimajor axis ap, and a protoplanetary disk of
mass Md. For simplicity, we treat the disk as a ring of radius rd, but it is simple to
generalize to a disk with finite extent [see Millholland and Batygin, 2019]. Denote
S the spin angular momentum and L the orbital angular momentum of the planet,
and Ld the angular momentum of the disk. The corresponding unit vectors are
ŝ≡S/S, l̂≡L/L, and l̂d ≡Ld/Ld.
The spin axis ŝ of the planet tends to precess around its orbital (angular mo-
mentum) axis l̂, driven by the gravitational torque from the host star acting on
the planet’s rotational bulge. On the other hand, l̂ and the disk axis l̂d precess
around each other due to gravitational interactions. We assume S ≪ L ≪ Ld, so
l̂d is nearly constant and l̂ experiences negligible backreaction torque from ŝ. The
equations of motion for ŝ and l̂ in this limit are [Anderson and Lai, 2018]
dŝ = ( )( ) ( )( )ω
dt sl
ŝ · l̂ ŝ× l̂ ≡α ŝ · l̂ ŝ× l̂ , (2.1)
dl̂ = ( · )( )ωld l̂ l̂d l̂× l̂d ≡− ( × )g l̂ l̂d , (2.2)dt
15
where
3GJ 2
≡ 2
MpRpM⋆ = 3k
(
qp M⋆ R
)3
p
ωsl (3 Ω , (2.3)2apIp)Ωp 2kp mp a pp3
≡ 3Md apωld n. (2.4)4M⋆ rd
In Eq. (4.4), Ip = kpM 2pRp (with kp a constant) is the moment of inertia and
J2 = kqpΩ2p(R3p/GMp) (with kqp a constant) the rotation-induced (dimensionless)
quadrupole of the planet [for a body with uniform density, kp = 0.4,kqp = 0.5;
for giant planets, kp ≃ 0.25 and kqp ≃ 0.17 [e.g. Lainey, 2016]]. In other stud-
ies, 3kqp/2kp √is often notated as k2/2C [e.g. Millholland and Batygin, 2019]. In
Eq. (2.4), n≡ GM /a3⋆ p is the planet’s orbital mean motion, and we have assumed
rd ≫ ap and included only the leading-order (quadrupole) interaction between the
planet and disk. We define three relative inclination angles via
ŝ · l̂≡ cosθ, ŝ · l̂d ≡ cosθsd, l̂ · l̂d ≡ cos I. (2.5)
In our model, I is a constant. Following standard notation [e.g. Colombo, 1966,
Peale, 1969, Ward and Hamilton, 2004], we have defined α≡ωsl and g≡−ωld cos I.
We can combine Eqs. (7.20) and (2.2) into a single equation by transforming
into a frame rotating about l̂d with frequency g. In this frame, l̂d and l̂ are fixed,
and ŝ evolves as: ( )
dŝ = ( · )( × )+ ( )α ŝ l̂ ŝ l̂ g ŝ× l̂d . (2.6)dt rot
We define the dimensionless time τ as
τ≡αt, (2.7)
16
and the frequency ratio η
η≡ − g
α ( k )( )( )ρ M ( a )9/2= 2.08 p p d p
( kqp ) g/(cm3 ) 0.0(1M⊙ )5 AU
× rd −3 M
−3/2
⋆ Pp cos I, (2.8)
30 AU M⊙ 10 hrs
where ρp = 3Mp/(4πR3p) and Pp = 2π/Ωp is the planet’s rotation period. In Eq. (2.8),
we have introduced the fiducial values of variable parameters for the application
considered in this paper. E( q. )(3.5) then becomes
dŝ = ( · )( ) ( )ŝ l̂ ŝ× l̂ −η ŝ× l̂d . (2.9)dτ rot
Throughout this paper, we consider α constant, but allow g to vary in time.
In the dispersing disk scenario of Millholland and Batygin [2019], |g| decreases
in time due to the decreasing disk mass. We consider a simple exponential decay
model
Md(t)=Md(0)e−t/td , (2.10)
with td constant. This implies
dη =− dηη/td, or =−ϵη, (2.11)dt dτ
where
≡ 1ϵ
αtd ( k )( )( ) ( )( )ρ −1= 0.106 p p ap 3 Pp td . (2.12)
kqp g/cm3 5 AU 10 hrs Myr
Eqs. (2.9) and (2.11) together constitute our system of study.
In the next two subsections, we summarize the theoretical background rele-
vant to our analysis of the evolution of the system.
17
Figure 2.1: Definition of angles in the Cassini state configuration and the adopted
sign convention for θ. Traditionally, θ ∈ [−π,π].
2.2.2 Cassini States
Spin states satisfying (dŝ/dτ)rot = 0 are referred to as Cassini States (CSs) [Colombo,
1966, Peale, 1969]. They require that ŝ, l̂, and l̂d be coplanar. There are either two
or four CSs, depending on the value of η. They are specified by the obliquity θ
and the precessional phase of ŝ around l̂, denoted by φ. Following the standard
convention and nomenclature (see Figs. 2.1 and 3.1), CSs 1, 3, 4 have φ = 0 and
θ < 0, corresponding to ŝ and l̂d being on opposite sides of l̂, while CS2 has φ = π
and θ > 0, corresponding to ŝ and l̂d being on the same side of l̂. The CS obliquity
satisfies
sinθ cosθ−ηsin(θ− I)= 0. (2.13)
When η< ηc, where ( )−3/2
η ≡ sin2/3I+cos2/3c I , (2.14)
all four CSs exist, and when η > ηc, only CSs 2, 3 exist. The CS obliquities as a
function of η are shown in Fig. 3.1.
Of the four CSs, 1, 2, 3 are stable while 4 is unstable. Appendix A.1 gives the
libration frequencies and growth rates, respectively, near these CSs.
18
Figure 2.2: Cassini state obliquities as a function of η for I = 5◦. The thin verti-
cal dashed line indicates ηc (= 0.766 for I = 5◦), where CS1 and CS4 merge and
annihilate, and the thin horizontal dashed lines indicate θ = I and I −180◦, the
asymptotic values for CSs 2 and 3 for η≫ ηc.
2.2.3 Separatrix
The Hamiltonian (in the rotating frame) of the system is
( )=−1 ( ) ( )H φ,cos ŝ · l̂ 2θ +η ŝ · l̂
2 d
=−1 2 + ( − )cos θ η cosθ cos I sin I sinθ cosφ . (2.15)
2
Here, (φ and c)osθ are canonically conjugate variables. Trajectories in the phase
space φ,cosθ satisfy H = constant (see Fig. 2.3).
When η< ηc, CS4 exists and is a saddle point. The two trajectories originating
and ending at CS4 are the only two infinite-period orbits in the phase space. To-
gether, these two critical trajectories are referred to as the separatrix and divide
phase space into three zones. In Fig. 2.3, we show the separatrix, the three zones,
and their relations to the CSs. Trajectories in zone II librate about CS2 while
19
those in zones I and III circulate.
( ) ∮
Since φ,cosθ are canonically conjugate, the integral cosθ dφ(a∣∫long a tra∣j)ec-
tory is an adiabatic invariant (see Section 2.3). The unsigned areas ∣ cosθ dφ∣ of
the three zones (as defined in Fig. 2.3) can be compu√ted analytically. If we define
= tan
3θ
z0 ηcos I, = − 4χ −1, (2.16a)tan I
2
= sin θ4 cosθ4 cosθ4ρ χ
χ2 cos2
, T = 2χ , (2.16b)
θ4+1 χ2 cos2θ4−1
then the areas for η< ηc are given by [Ward and Hamilton, 2004]
AI =
A
2 IIπ (1− z0)− , (2.17a)2
AII = 8ρ+4arctanT−
1
8z0 arctan , (2.17b)
χ
= A2 (1+ z )− IIAIII π 0 . (2.17c)2
These are plotted as a function of η in Fig. 2.4. While the zones are not for-
mally defined for η > ηc since the separatrix disappears, a natural extension ex-
ists: evolve an initial phase space point p under adiabatic decrease of η until the
separatrix appears at η = ηc, then identify p with the zone it is in at ηc. Since
pha(se spa)ce area is conserved under adiabatic evolution, this extension implies
Aj η> ηc =Aj(ηc). The boundary between these extended zones is denoted by the
dashed black line in panel (a) of Fig. 2.3, where no separatrix exists.
2.3 Adiabatic Evolution
In this section, we study the evolution of the planetary obliquity θ when the pa-
rameter η [or the disk mass Md; see Eqs. (2.8) and (2.11)] decreases sufficiently
slowly that the evolution is adiabatic. Intuitively, this requires the disk evolu-
20
( )
Figure 2.3: Level curves of H φ,cosθ [Eq. (3.8)] for I = 5◦, where warmer colors
denote more positive values. The black solid line is the separatrix, which only
exists for η < ηc = 0.766. The three zones (I, II, III), divided by the separatrix,
are labeled. The Cassini states are denoted by filled circles and have the same
colors as in Fig. 3.1. The interior of the separatrix, shaded in grey, is formally
only defined for η < ηc, but we may identify the points in phase space that flow
into zone II when evolved forward in time (decreasing η adiabatically); this is the
shaded region in panel (a), bounded by the black dotted line.
21
Figure 2.4: Plot of fractional areas of each of the zones Aj(η)/4π as given by
Eqs. (2.17) for I = 5◦. The colored dotted lines correspond to small η approxima-
tions used in Appendix A.2. The colored dashed lines for η > ηc are the effective
values of AII,AIII for η > ηc, denoting the points that would flow into either area
under adiabatic decrease of η from η> ηc (see the text). The vertical black dashed
lines correspond to η = ηc [Eq.(2.14)] and the values of η for which AII is maxi-
mized (ηmax,II) and for which AIII is minimized (ηmin,III, Eq. (2.30)).
22
tion time td [Eq. (2.10)] be much larger than the spin precession period, 2π/α, i.e.
ϵ= 1/(αtd)≪ 1/(2π).
More rigorously, adiabaticity requires td be much larger than all timescales of
the dynamical system governed by the Hamiltonian [Eq. (3.8)]. This is of course
not possible in all cases, as the motion along the separatrix has an infinite period.
In practice, as η evolves, the system only crosses the separatrix once or twice,
while it spends many orbits inside one of the three zones and far from the sepa-
ratrix. Thus, a weak adiabaticity criterion is that, for all equilibria/fixed points,
the local circulation/libration periods are much shorter than the timescale for the
motion of the equilibria due to changing η. If this criterion is satisfied, then the
system will evolve adiabatically for most of its evolution save one or two separatrix
crossings.
As shown in Appendix A.1.2, libration about CS2 is slower than that about
CS1 or CS3. As such, it has the smallest characteristic frequency in the system.
The weak adiabaticity criterion is equivalent to requiring that, at all times other
than separatrix crossing, the obliquity of CS2 (θ2) evolve over a longer timescale
than the local libration period abou∣∣ t CS∣∣2, i.e.∣∣dθ2 ∣∣≪ ωlib , (2.18)dτ 2π
where √
= ( )ωlib ηsin I sinθ2 1+ηsin I csc3θ2 , (2.19)
is the libration frequency about CS2 for a given η (Appendix A.1.2). This formula
differs from that given in Millholland and Batygin [2019], where the csc3θ2 term
is neglected and the square root is missing1. Differentiating Eq. (2.13) gives
dθ2 =− ηsin(θ2− I)ϵ , (2.20)
dτ cos(2θ2)−ηcos(θ2− I)
1The missing csc3θ2 term can be traced to a θ≫ I approximation made in Eq. (3) of Hamilton
and Ward [2004]. Since θ2 ∼ I for η≫ 1 (Fig. 3.1), this approximation is not always valid.
23
where ϵ = −d(lnη)/dτ [Eq. (2.12)]. Eq. (2.18) is most constraining at η ∼ 1, i.e. it
will be satisfied for all η if it is satisfied near η∼ 1, where |dθ2/dτ| ∼ ϵ. Thus, weak
adiabiticity requires
(
≪ ≡ ω
)
lib 1 √
ϵ ϵc ≃ p sin I (1+8sin I), (2.21)2π η=1 2π 2
where in the last equality we have used sinθ2 ≃ 1/2 at η= 1 (e.g. when I = 5◦ and
η = 1, θ ≈ 31◦2 ). For I = 5◦, we obtain ϵc ≈ 0.0433. Since our criterion is only a
weak condition for adiabaticity, we use ϵ= 3×10−4 in our “adiabatic” calculations
below. We explore the consequences of nonadiabatic evolution in Section 2.4.
2.3.1 Adiabatic Evolution Outcomes
We consider the evolution of a system with arbitrary initial spin-disk misalign-
ment angle θsd,i and initial ηi ≫ 1. We are interested in the final spin obliquities
θf after η gradually decreases to ηf ≪ 1 (i.e. after the disk has dissipated to a neg-
ligible mass). Note that when ηi ≫ 1, l̂ precesses around l̂d much faster than the
spin-orbit precession (|ωld| ≫ |ωsl|), and the spin obliquity θ varies rapidly. It is
thus more appropriate to use θsd,i rather than θ to specify the initial spin orienta-
tion. We explore the entire range θsd,i ∈ [0,π] and choose ϵ= 3×10−4 (see above).
To obtain the distribution of the final obliquities θf, we evenly sample 101
values of θsd,i, and for each θsd,i value, we pick 101 evenly spaced orientations of
ŝ approximately from the ring of initial conditions having angular distance θsd,i
to l̂ 2d . To be concrete, we choose ηi = 10ηc where ηc is given by Eq. (2.14) and
2The actual procedure we adopt to choose the initial conditions is the natural extension of this
description to finite ηi. Note that the center of libration of ŝ is CS2, which, since ηi is finite, is
different from l̂d. Furthermore, the libration is not exactly circular. As a result, the libration
trajectories for initial conditions on the circular ring of points having angular distance θsd,i from
l̂d are not the same and will each enclose slightly different initial phase space areas Ai. Since
our analytical theory assumes exact conservation of the initially enclosed phase space area Ai
24
evolve Eqs. (2.9) and (2.11) until η reaches its final value 10−5. At such a small
η, ŝ is strongly coupled to l̂ and the final obliquity θf is frozen. The mapping
between θsd,i and θf is our primary result, and is shown for I = 5◦, 10◦, and 20◦ in
Figs. 2.5 and 2.6 respectively. The blue dots represent the results of the numerical
calculation. The colored tracks are calculated semi-analytically using the method
discussed in the following subsection.
2.3.2 Analytical Theory for Adiabatic Evolution
The evolutionary tracks that govern the θf-θsd,i mapping correspond to various
sequences of separatrix crossings. They can be understood using the principle of
adiabatic invariance, combined with (i) how the enclosed phase space area by the
trajectory evolves across each separatrix crossing, and (ii) the associated probabil-
ities with each separatrix crossing.
Governing Principle: Evolution of Enclosed Phase Space Area
First, we consider how the enclosed phase space area by a trajectory evolves over
t∮ime. In the absence of separatrix encounters, the enclosed phase space area
cosθ dφ is an adiabatic invariant.∮We adopt convention where
A ≡ (1−cosθ) dφ. (2.22)
Note that A can be negative when dφ/dt< 0, unlike the unsigned areasAi [Eqs. (2.17)]
which are positive by definition. This definition of A has two advantages: (i) it is
for each θsd,i (see Section 2.3.2), this diskrepanc(y introduces)an extra(deviation from the analyticalprediction. To guarantee all points for a particular θsd,i have the same Ai, w)e instead choose initial
conditions on the libration cycle going through θ2+θsd,i,φ2 [where θ2,φ2 are the coordinates of
CS2]. This ensures that all initial conditions for a given θsd,i enclose the same initial Ai. As
ηi →∞, this procedure generates initial conditions on the ring having angular distance θsd,i to l̂d,
recovering the procedure given in the text.
25
Figure 2.5: Top: The final spin obliquity θf as a function of the initial spin-disk
misalignment angle θsd,i for systems evolving from initial ηi ≫ 1 to ηf ≪ 1, for
I = 5◦. The blue dots are results of numerical calculations (Section 2.3.1), and the
colored tracks are semi-analytical results (Section 2.3.2). Bottom: The probabili-
ties of different outcomes. Where a particular θsd,i corresponds to multiple tracks,
the system evolves probabilistically. The track that a particular system evolves
along in a numerical simulation can be measured by examining its final obliquity.
The dots represent the inferred probabilities from measured final obliquities in
our simulations, while the colored tracks denote the semi-analytic probability of
the system evolving along each track. There are five regimes of θsd,i values for
which different tracks are accessible. In both plots, the vertical dashed black lines
denote semi-analytical calculations of the boundaries of these regimes (see Sec-
tion 2.3.2), while the black dotted lines represent analytical approximations valid
in the small-θsd,i limit (see Appendix A.2).
26
Figure 2.6: Same as the top panel of Fig. 2.5 but for I = 20◦ and with fewer anno-
tations.
27
continuous across transitions from circulating to librating that cross the North
pole (cosθ = 1), and (ii) the areas of the three zones are equal in absolute value to
the expressions given in Eqs. (2.17). The path over which the integral is taken is
either a libration or circulation cycle. When ηi ≫ 1, trajectories librate about l̂d
with constant θsd, meaning they enclose initial phase space area
(
Ai = 2π 1−
)
cosθsd,i . (2.23)
Complications arise when considering finite ηi, as trajectories near CS2 or CS3
librate about these equilibria, rather than l̂d, and Eq. (2.23) is no longer exact. In
practice, Eq. (2.23) holds very well when defining θsd,i as the angular distance to
CS2; an exception is discussed in Section 2.3.2.
Beginning at the last separatrix crossing, the final enclosed phase space area
Af will be conserved for all time. As η→ 0, trajectories circulate about l̂ at constant
obliquity θf, related to Af by
2π (1−cosθf)= Af. (2.24)
The enclosed phase space area is not conserved when the trajectory encounters
the separatrix. However, the change is easily understood [Henrard, 1982]. In
essence, when the trajectory crosses the separatrix, it continues to evolve adjacent
to the separatrix. So if a separatrix crossing results in a zone I trajectory (see
Fig. 2.3), the new area can be approximated by integrating Eq. (2.22) along the
upper leg of the separatrix. Pictorially, this can be seen in the bottom panels of
Fig. 2.7.
28
Governing Principle: Probabilistic Separatrix Crossing
When a trajectory experiences separatrix crossing, it transitions into nearby zones
probabilistically. This process is studied in the adiabatic limit by Henrard [1982]
and Henrard and Murigande [1987]. Their results may be summarized as follows:
if zone i is shrinking while adjacent zones j,k are expanding such that the sum of
their areas is constant, the probabilities of transition from zone i to zones j and k
are given by
→ =−∂Aj/∂ηPr(i j) , (2.25a)
∂Ai/∂η
Pr(i→ k)=−∂Ak/∂η . (2.25b)
∂Ai/∂η
Note that Pr(i→ j)+Pr(i→ k) = 1. Eqs. (2.25) can be used in conjunction with
Eqs. (2.17) to understand for what initial conditions each track can be observed
and with what probabilities.
As an example, consider a system in zone II in panel (d) of Fig. 2.3. As η de-
creases, zone II will shrink while zones I and III will expand until the trajectory
crosses the separatrix. Suppose the trajectory exits zone II at some η⋆, then the
probability of the II → I transition is Pr(II→ I)=−ȦI/ȦII, while the II → III tran-
sition occurs with probability Pr(II→ III)=−ȦIII/ȦII.
Evolutionary Trajectories
Returning to the evolution of ŝ, we can classify trajectories by the sequence of
separatrix encounters. Initially, in the η> ηc regime, only zones II and III exist; as
η→ 0, only zones I and III exist (see Fig. 2.3). There are five distinct evolutionary
tracks:
29
1. II→ I (see Fig. 2.7 for an example). The spin axis ŝ initially circulates in zone
II (snapshot a), and then starts librating about CS2 as η decreases (snap-
shot b), enclosing some initial phase space area Ai. This libration continues
until the separatrix expands (due to decreasing η) to “touch” the trajectory
(snapshot c), at which AII(η⋆) = Ai. As ŝ moves to a circulating trajectory
in zone I immediately bordering the separatrix, it will encompass −AI(η⋆)
phase spa(ce )area. The final obliquity θf is then given by Eq. (2.24), with
Af =−AI η⋆ . An analytical approximation to θf is derived in Appendix A.2
and is ( 2 )2πθ 2
cos ≃ sd,i
θsd,i
( θf)II→I cot I+ . (2.26)16 4
The transition probability is ( )
Pr(II→ =− ∂AI/∂ηI) . (2.27)
∂AII/∂η η=η⋆
This track can only occur wh(en)the initial condition begins in zone II, requir-
ing Ai <AII(ηc), where AII ηc is given by Eq. (2.17b) evaluated at η = ηc.
Since ∂AI/∂η< 0 everywhere, while ∂AII/∂η> 0 at all possible η⋆ for an ini-
tial condition starting in zone II, this track always has nonzero probability.
2. II → III (see Fig. 2.8). This track is similar to the II → I track; the only
difference is that, upon separatrix encounter, the trajectory follows the cir-
culating trajectory in zone III bordering the separatrix, upon which it will
encompass area AI(η⋆)+AII(η⋆) = Af. The final obliquity is still given by
Eq. (2.24), and the analytical appro(ximati)on derived in Appendix A.2 is
πθ2
2
θ2
cos ≃ sd,i cot I− sd,i( θf)II→III . (2.28)16 4
The transition probability is ( )
∂A
Pr(II→ III)=− III/∂η . (2.29)
∂AII/∂η η=η⋆
30
Again, this track can only occur when Ai <AII(ηc), but a further constraint
arises when we consider the transition probability. Upon examination of
Fig. 2.4, it is clear that ∂AIII/∂η> 0 for a large range of η, which would give
a negative transition probability—implying a forbidden transition. Define
ηmin,I I I ≡ argminAIII(η), (2.30)
which is labeled in Fig. 2.4. Thus, the II → III track is permitted only if
η⋆ <,ηmin,III.
3. III→ I (see Fig. 2.9). The trajectory encounters the separatrix whenAI(η⋆)+
AII(η⋆)= Ai, upon which it transitions to a zone I trajectory enclosing Af =
−AI. The final obliquity is again given by Eq. (2.24).
This track can only occur if Ai >AII(ηc), but is also constrained by requir-
ing Ai be sufficiently small so that it will encounter the separatrix (if Ai
is too large, it will never encounter the separatrix, and we simply have a
III → III transition). This condition is Ai <max(AI+AII) = 4π−min(AIII).
Since ∂AI/∂η < 0 and ∂AIII/∂η > 0 for all accessible η⋆, this track is always
permitted.
4. III → II → I (see Fig. 2.10). That AII(η) is not a monotonic function of
η (see Fig. 2.4) is key to the existence of this track. Consider a trajec-
tory originating in zone III that first encounters the separatrix at η1, when
AI(η1)+AII(η1)= Ai, such that it transitions into zone II enclosing interme-
diate phase space area Am =AII(η1). S(uch a tra)nsition has probability
→ =− ∂AII/∂ηPr(III II) , (2.31)
∂AIII/∂η η=η1
which is nonnegative (i.e. the tr[ans(itio)n is perm] itted) if η1 ∈ [ηmax,II,ηc].
Equivalently, this requires Ai ∈ AII ηc ,AII,max . Then, as η continues to
decrease, a second η2 value exists for which Am =AII(η2), upon which the
31
trajectory is ejected to zone I and Af =−AI(η2). Note that η2 < ηmax,II neces-
sarily, as zone II must be shrinking in order for the trajectory to be ejected.
The final obliquity is given by Eq. (2.24). Graphical inspection of Fig. 2.4
shows that ∂AII/∂η and ∂AIII/∂η have the same signs for η < ηmax,II, and
therefore the complementary II → I transition is guaranteed. Overall, the
III →[ II →( I)track is p]ermitted so long as the first transition is permitted, or
Ai ∈ AII ηc ,AII,max .
5. III→ III. This track is the trivial case where no separatrix encounter occurs,
and A is constant throughout the evolution (Af = Ai) except for a jump by 4π
when crossing the South pole (cosθ = −1) due to the coordinate singularity.
This requires Ai >max(AI+AII). In the limit of ηi →∞ and ηf → 0 we have
θf = θsd,i. For finite ηi, the initial enclosed phase space area for III → III tra-
jectories is not given exactly by using θ = θsd,i in Eq. (2.23). This is because
the initial orbits for such trajectories are better described as librating about
CS3 with angle of libration ∆θ−θsd,i rather than about CS2 with angle of
libration θsd,i. Here, ∆θ is the angular distance between CS2 and CS3 and
is not equal to 180◦ except when ηi →∞. This finite-ηi effect is responsible
for the small cusp at the very right (θsd,i → 180◦) of Figs. 2.5 and 2.6.
In summary, starting from an initial condition with phase space area Ai at
η= ηi ≫ 1, the five evolutionary tracks are:
∈ [ ( )]1. Ai [0,A∈ (
II ηmin,I)II : Both] the II → III and the II → I tracks are possible.
2. Ai [AII ηmin,III ,A]II(ηc) : Only the II → I track.
3. Ai ∈ [AII(ηc),AII,max : Both∈ + ]
the III → I and III → II → I are possible.
4. Ai AII,max,max(AI AII) : Only the III → I track.
5. Ai >max(AI+AII): Only the III → III track.
32
In all cases, the corresponding ranges for θsd,i can be computed via Eq. (2.23). The
boundaries between these ranges are overplotted in Fig. 2.5, where they can be
seen to agree well with the numerical results.
2.4 Nonadiabatic Effects
In Section 2.3, we have examined the spin axis evolution in the limit where ϵ≪ ϵc
[see Eq. (7.37)] and the evolution is mostly adiabatic (except at separatrix cross-
ings). We now consider nonadiabatic effects.
2.4.1 Transition to Non-adiabaticity: Results for ϵ≲ ϵc
To illustrate the transition to nonadiabaticity, we carried out a suite of numerical
calculations for several values of ϵ. The results for two of these values are shown
in Figs. 2.11 and 2.12.
As ϵ increases (see Fig. 2.11), nonadiabaticity manifests as a larger scatter of
final obliquities near the tracks predicted from adiabatic evolution. This scatter
first sets in for trajectories starting in zone III, as these trajectories encounter the
separatrix at larger η compared to those originating in zone II. This means the
obliquity of CS2 θ2 is smaller for these trajectories, and the adiabaticity criterion
is stricter [see Eq. (7.37)]. Physically, approaching the adiabaticity criterion cor-
responds to the separatrix crossing process becoming sensitive to the phase of the
libration/circulation cycle at the crossing: if the trajectory crosses the separatrix
when the obliquity is at its maximum, the final obliquity will also be relatively
larger.
33
Figure 2.7: An example of the II → I evolutionary track for I = 5◦ and θ = 17.2◦sd,i .
Upper panel: The thin green line shows cosθ as a function of η, obtained by nu-
merical integration (with ϵ= 3×10−4). Overlaid are the location of Cassini State 2
(dashed red) and the upper and lower bounds on the separatrix (dotted black). The
trajectory tracks CS2 to a final obliquity of 88.57◦. The black vertical dashed lines
denote instants in the simulation portrayed in bottom panels. Middle panel: The
enclosed separatrix area obtained by integrating the simulated tr(ajectory) (green
dots) and adiabatic theory (red line). Lower plot: Snapshots in cosθ,φ phase
space of one circulation/libration cycle of the trajectory, shown in dark green with
an arrow indicating direction. The snapshots correspond to the start of the sim-
ulation (a), the appearance of the separatrix (b), two panels depicting the separa-
trix crossing process (c-d), and a final snapshot at η= 10−3.5 (e). The separatrices
at the beginning and end of the portrayed cycle in each snapshot are shown in
solid/dashed black lines respectively. Also labeled is CS2 at the start of each cycle
(filled red circle). Finally, the enclosed phase space area is shaded in grey (A > 0)
and red (A < 0).
34
Figure 2.8: Same as Fig. 2.7 but for the II → III track. θ = 17.2◦sd,i and ϵ= 3.01×
10−4.
35
Figure 2.9: Same as Fig. 2.7 but for the III → I track. θsd,i = 89.1◦, and ϵ= 3×10−4.
36
Figure 2.10: Same as Fig. 2.7 but for the III → II → I track. θ = 60◦sd,i , and
ϵ= 3.14×10−4. Two separatrix crossings are shown, in panels (c-d) and (e-f).
37
Figure 2.11: Same as Fig. 2.5 but for ϵ = 10−2.5 and restricting θ ◦sd,i < 90 (blue
dots). The colored solid lines are analytical adiabatic results (same as shown in
Fig. 2.5). A larger spread from the adiabatic tracks is observed in the numerical
results due to the non-adiabaticity effect.
As ϵ increases further (see Fig. 2.12) but still marginally satisfies the weak adi-
abaticity criterion [Eq. (7.37)], the scatter in θf continues to widen. The horizontal
banded structure of the final obliquities is a consequence of even stronger phase
sensitivity during separatrix crossing: trajectories cross the separatrix at similar
phases evolve to similar final obliquities that only depend weakly on on θsd,i. Fi-
nally, in Fig. 2.12, the bottom edge of the data and the III → I track deviate very
noticeably. This non-adiabatic effect is the result of the separatrix evolving signif-
icantly within the separatrix-crossing orbit, as the tracks computed in Section 2.3
assume that η is constant throughout the separatrix-crossing orbit.
38
Figure 2.12: Same as Fig. 2.11 but for ϵ = 10−1.5 (i.e. larger non-adiabaticity ef-
fect). Some small resemblance to the adiabatic tracks remains, and the deviations
appear to have a banded structure.
39
A sample trajectory following in the style of Fig. 2.7 but for ϵ = 0.3 (violating
even weak adiabaticity) is provided in Fig. 2.13. It is clear that the trajectory
does not track the level curves of the Hamiltonian during each individual snap-
shot. This results from CS2 migrating more quickly than the trajectory can librate
about CS2, violating the weak adiabaticity criterion.
2.4.2 Non-adiabatic Evolution: Result for ϵ≳ ϵc
In general, numerical calculations are needed to determine the non-adiabatic
obliquity evolution (ϵ≳ ϵc). However, some analytical results can still be obtained
when the obliquity change is small.
We start from Eq. (2.9), which governs the evolution of the spin axis in the
rotating frame. We choose coordinate axes such that l̂= ẑ and l̂d = ẑcos I+ x̂sin I,
giving ( )
dŝ = [( ) ]ηcos I−cosθ ẑ+ηsin I x̂ × ŝ. (2.32)
dτ rot
Defining S = ŝx+ iŝy, we find
dS = ( )i ηcos I−cosθ S− iηsin I cosθ. (2.33)
dτ
To proceed, we assume the obliquity is roughly constant, cosθ ≈ cosθi. Eq. (2.33)
can then be solved explicitly, starting from the in∫itial value S(τi):τ
S(τ)e−iΦ(τ)−S(τi)≃−
′
isin I cosθ η(τ′)e−iΦ(τ )i dτ′, (2.34)
τi
where ∫ τ
≡ ( ′ )Φ(τ) η(τ )cos I−cosθ ′i dτ . (2.35)
τi
We now invoke the stationary phase approximation, so thatΦ(τ)≃Φ(τ0)+(1/2)Φ̈(τ0)(τ−
τ )20 , where τ0 is determined by Φ̇ = 0, occurring when η0 = cosθi/cos I. We then
40
Figure 2.13: Same as Fig. 2.7 but for a nonadiabatic case, with ϵ= 0.3. In the top
panel, it is evident that the libration cycle about CS2 is unable to keep up with the
swift migration of CS2 as η changes, decreasing the obliquity excitation compared
to the adiabatic simulation. In the middle panel, the trajectory only undergoes six
libration/circulation cycles before η < 10−5, and the enclosed phase space area is
clearly not conserved. In the bottom panel, we can see that individual trajectories
do not lie along level curves of the Hamiltonian, as the Hamiltonian phase space
changes quickly compared to the period of circulation cycles.
41
find, for τ≫ τ0, √
S(τ)e−iΦ(τ)−S(τi)≃−
2π
iη(τ0)sin I cosθ e−iΦ(τ0)i . (2.36)iΦ̈(τ0)
Using η̇=−ϵη [Eq. (2.11)] and Φ̈(τ0)= η̇(τ0)cos I =−ϵcosθi, w√e have
S(τ)e−iΦ(τ)−S(τ )≃−i3/2 tan I(cosθ )3/2e−iΦ(τ0) 2πi i . (2.37)
ϵ
The final obliquity θf is then∣∣ given by √ ∣∣
sinθ ≃ ∣∣∣∣sinθ + e−iϕ0 tan I(cosθ )3/2 2π ∣f i i ∣∣∣ , (2.38)ϵ
where ϕ0 =Φ(τ0)+π/4 is a constant phase. If the initial obliquity is much smaller
than the final obliquity (sinθi ≪ sin√θf), we obtain
2π
sinθ ≃ tan I(cosθ )3/2f i . (2.39)
ϵ
This expression is valid only if cosθ ≈ cosθi throughout the evolution. This corre-
sponds to the limit where θf is not much larger than θi, which requires ϵ not to be
too small. Numerically, this is consistent with the system being in the nonadia-
batic regime ϵ≳ ϵc (see Fig. 2.14).
The above calculation applies for a specific initial θi, but, as discussed at the
beginning of Section 2.3.1, the initial spin orientation is more appropriately de-
scribed by θsd,i since ηi ≫ 1. The correct way to predict the final obliquity for a
given θsd,i using Eq. (2.38) is somewhat subtle but yields good agreement with
numerical results.
First consider the case with θsd,i = 0. This corresponds to a well-defined initial
obliquity θi = I (more precisely, the initial condition is CS2). The final obliquity in
this case, denoted θ0f, is given by ∣∣∣∣ √ ∣≃ ∣ ∣∣ + −iϕ 2πcos I ∣sinθ0f sin I 1 e 0 ∣∣∣ ,√ ϵ
≈ 2πcos Isin I , (2.40)
ϵ
42
p
where the second equality assumes 2π/ϵ≫ 1. Fig. 2.14 shows the final obliq-
uity as function of ϵ for θsd,i = 0 and I = 5◦. We see that the agreement between
the numerical results and Eq. (2.40) is excellent. For ϵ≪ ϵc, we find θf ≃ 90◦, in
agreement with the result of adiabatic evolution (see Fig. 2.5).
When θsd,i ̸= 0, we find that the final obliquity θf spans a range of values for a
given θsd,i, as can be seen in Fig. 2.15. The range can be described by
∣∣ ∣θ0f−θsd,i∣≲ θf ≲ θ0f+θsd,i. (2.41)
Eq. (2.41) can be understood as follows (see Fig. 2.16). In the beginning (η= ηi ≫
1), the initial spin vector precesses around l̂d on a cone with opening half-angle
θsd,i (more precisely, the cone is centered on CS2, which coincides with l̂d as ηi →
∞). Note that for η≫ 1, the adiabaticity condition is easily satisfied: using θ2 ≃
I +η−1 sin I cos I (see Section A.1), Eq. (2.19) gives ωlib ≃ η while Eq. (2.20) gives
|dθ2/dτ| ≃
( )
ϵ/η sin I cos I ≪ ωlib. As η decreases, the system will transition from
being adiabatic to being nonadiabatic, since ϵ ≳ ϵc. The evolution of the system
can thus be decomposed into two phases: (i) when the evolution is adiabatic, the
spin vector will precess around the slowly-moving CS2; (ii) when the evolution
becomes nonadiabatic, the spin vector stops tracking the quickly-evolving CS2.
During the adiabatic evolution of phase (i), the angle between CS2 and the spin
vector is approximately unchanged due to conservation of phase space area3. Once
the evolution enters phase (ii), the precession axis quickly (on timescale ≪ 1/ωlib)
changes to l̂ (as η decreases to ηf ≪ 1). Precession about l̂ does not change the
obliquity, so the range of obliquities at the end of phase (i) is frozen in as the
range of final obliquities. We refer to this two-phase evolution as partial adiabatic
3This approximation assumes sufficiently small θsd,i such that libration about CS2 remains
approximately circular throughout phase (i) (initially, when η→∞, all librations are circular about
l̂d). This assumption breaks down when θsd,i is sufficiently large that librating orbits become non-
circular as η decreases before the end of phase (i) (Fig. 2.3 illustrates that the libration cycles
farther from CS2, corresponding to a larger θsd,i, are less circular for a given η). This causes the
deviation of the numerical results in Fig. 2.15 from Eq. (2.41) for θsd,i ≳ 45◦.
43
Figure 2.14: Final obliquity θf as a function of ϵ for θ ◦sd,i = 0 and I = 5 . The
shaded area, bordered by the black line, corresponds to the adiabatic regime esti-
mated by Eq. (7.37). The blue dots are numerical results, and the red dashed line
corresponds to Eq. (2.40), which is in good agreement with numerical results for
ϵ > ϵc ≈ 0.1 (the nonadiabatic regime). Note that θf ≃ 90◦ in the adiabatic regime
(ϵ≪ ϵc).
resonance advection.
Fig. 2.15 shows the numerical result of θf vs θsd,i for I = 5◦ and ϵ = 0.3. We
see that Eq. (2.41) provides good lower and upper bounds of the final obliquity for
θsd,i ≲ 45◦ (see footnote 3).
2.5 Summary
In this paper, we have studied the excitation of planetary obliquities due to grav-
itational interaction with an exterior, dissipating (mass-losing) protoplanetary
disk. Obliquity excitation occurs as the system passes through a secular reso-
44
Figure 2.15: Final obliquity θf vs θ ◦sd,i for I = 5 and ϵ= 0.3 (firmly in the nonadi-
abatic regime). The blue dots represent numerical results, and the two red lines
show the analytical lower and upper bounds given by Eq. (2.41).
Figure 2.16: Schematic picture for understanding nonadiabatic obliquity evolution
when θsd,i > 0 assuming ηi ≫ 1. The figure shows a projection onto the plane
containing both l̂ and l̂d. When θsd,i = 0, the initial spin vector points along l̂d and
evolves into the final spin vector (grey), which has obliquity θ0f [Eq. (2.40)]. When
θsd,i ̸= 0, the set of initial conditions for the spin vector forms a cone (solid red
area) centered on l̂d with opening half-angle θsd,i. Under nonadiabatic evolution,
the set of final spin vectors forms a new cone, still with opening half-angle θsd,i,
centered on θ0f (light red area).
45
nance between spin precession and orbital (nodal) precession. This scenario was
recently studied by Millholland and Batygin [2019], who focused on the special
case of small initial obliquities. In contrast, we consider arbitrary initial misalign-
ment angles in this paper, motivated by the fact that planet formation through
core accretion can lead to a wide range of initial spin orientations. We present our
result as a mapping from θsd,i to θf, where θsd,i is the initial misalignment angle
between the planet’s spin axis and the disk’s orbital angular momentum axis, and
θf is the final planetary obliquity. We have derived analytical results that capture
the behavior of this mapping in both the adiabatic and nonadiabatic limits:
1. In the adiabatic limit (i.e. the disk dissipates at a sufficiently slow rate),
we reproduce the known result θ ≃ 90◦f for θsd,i ≃ 0. We demonstrate (via
numerical calculation and analytical argument) the dual-valued behavior of
θf for nonzero θsd,i (see Fig. 2.5). We show for the first time that both the final
θf values and the probabilities of achieving each value can be understood
analytically via careful accounting of adiabatic invariance and separatrix
crossing dynamics.
2. As the disk dissipates more rapidly, the adiabatic condition [Eq. (7.37)] breaks
down, we find that a broad range of final obliquities can be reached for a
given θsd,i (see Fig. 2.15). We understand this result via the novel concept of
partial adiabatic resonance advection and provide an analytical expression
of the bounds on θf in Eq. (2.41).
As noted in Section 2.1, while in this paper we have examined a specific sce-
nario of generating/modifying planetary obliquities from planet-disk interactions,
the dynamical problem have studied is more general [Colombo, 1966, Peale, 1969,
1974, Ward, 1975, Henrard and Murigande, 1987]. Our work goes beyond these
previous works and provides the most general solution to the evolution of “Colombo’s
46
top” as the system evolves from the “weak spin-orbit coupling” regime (η≫ 1) to
the “strong spin-orbit coupling” regime (η≪ 1). The new analytical results pre-
sented in this paper can be adapted to other applications.
Concerning the production of planetary obliquity with a dissipating disk, when
there are multiple planets in a system, the nodal precession rate g for the planet
of interest never decays below the Laplace-Lagrange rate driven by planet-planet
secular interactions [Millholland and Batygin, 2019]. Therefore, η has a minimum
value at late times. This does not affect the methodology of our analysis, but
can affect the detailed results. For example, the adiabaticity criterion must be
modified slightly as dlnη/dt is no longer constant but asymptotes to zero as η
decreases; the planet may never undergo separatrix crossing if their η⋆ (which
depends on θsd,i) in the absence of the companions is too small; the planetary
obliquity will oscillate even when the disk has fully evaporated (as l̂ is no longer
constant). The spin dynamics can be even more complex if the two planets are in
mean motion resonance [e.g. Millholland and Laughlin, 2019].
47
CHAPTER 3
DYNAMICS OF COLOMBO’S TOP: TIDAL DISSIPATION AND
RESONANCE CAPTURE, WITH APPLICATIONS TO OBLIQUE
SUPER-EARTHS, ULTRA-SHORT-PERIOD PLANETS AND
INSPIRALING HOT JUPITERS
Originally published in:
Yubo Su and Dong Lai. Dynamics of Colombo’s Top: Tidal dissipation and reso-
nance capture, with applications to oblique super-Earths, ultra-short-period plan-
ets and inspiraling hot Jupiters. MNRAS, 509(3):3301–3320, November 2021.
ISSN 0035-8711. doi: 10.1093/mnras/stab3172
3.1 Introduction
It is well recognized that the obliquity of a planet, the angle between the spin and
orbital axes, likely reflects its dynamical history. In our Solar System, planetary
obliquities (hereafter just “obliquities”) range from 3.1◦ for Jupiter to 26.7◦ for Sat-
urn to 98◦ for Uranus. The obliquities of exoplanets are challenging to measure,
and so far only loose constraints have been obtained for the obliquity of a faraway
(≳ 50 AU) planetary-mass companion [Bryan et al., 2020]. Nevertheless, there are
prospects for better constraints on exoplanetary obliquities in the coming years,
such as using high-resolution spectroscopy to measure vsin i for planetary rota-
tion [Snellen et al., 2014, Bryan et al., 2018] and using high-precision photome-
try to measure the asphericity of a planet [Seager and Hui, 2002]. Substantial
obliquities are of increasing theoretical interest for their proposed role in explain-
ing peculiar thermal phase curves [see e.g. Adams et al., 2019, Ohno and Zhang,
2019], in enhancing tidal dissipation in hot Jupiters [Millholland and Laughlin,
2018] and super Earths [Millholland and Laughlin, 2019], and in the formation of
ultra-short-period planets [USPs; Millholland and Spalding, 2020].
48
While nonzero obliquities are sometimes attributed to one or many giant im-
pacts/collisions [e.g. Safronov and Zvjagina, 1969, Benz et al., 1989, Korycansky
et al., 1990, Dones and Tremaine, 1993, Morbidelli et al., 2012, Li and Lai, 2020,
Li et al., 2021], some studies suggest that large planetary obliquities may be pro-
duced by spin-orbit resonances. In this scenario, a rotating planet is subjected to
a gravitational torque from its host star, making its spin axis precess around its
orbital (angular momentum) axis. At the same time, the orbital axis precesses
around another fixed axis under the gravitational influence of other masses in the
system, e.g. additional planets or a protoplanetary disk. When the two precession
frequencies become comparable, a resonance can occur that excites the obliquity
to large values. This model is known as “Colombo’s Top” after the seminal work of
Colombo [1966], and subsequent works have investigated the rich dynamics of this
system [Peale, 1969, 1974, Ward, 1975, Henrard and Murigande, 1987]. Such res-
onances have been invoked to explain the obliquities of both the Solar System gas
giants [Ward and Hamilton, 2004, Hamilton and Ward, 2004, Ward and Canup,
2006, Vokrouhlickỳ and Nesvornỳ, 2015, Saillenfest et al., 2020, 2021] and the ice
giants [Rogoszinski and Hamilton, 2020].
In a previous paper [Su and Lai, 2020, hereafter Paper I], we presented a
systematic and general investigation of the dynamics of Colombo’s Top when the
two precession frequencies of the system evolve through a commensurability. We
obtained a semi-analytic mapping between the (arbitrary) planetary spin orienta-
tion and the final obliquity after a resonance encounter. We applied our results to
investigate the generation of exoplanetary obliquities via a dissipating protoplan-
etary disk. However, our model did not consider the effect of additional torques
in the system. In particular, tidal dissipation in the planet can cause the planet’s
spin frequency to approach its orbital frequency and drive the planet’s spin axis
towards its orbital axis, complicating the evolution of Colombo’s Top [Fabrycky
49
et al., 2007, Levrard et al., 2007, Peale, 2008]. In this paper, we extend these pre-
vious works to present a comprehensive study on how tidal dissipation influences
the equilibria (called “Cassini States”) of the system and drive its long-term evo-
lution. Our new results (summarized in Section 3.6) include a stability analysis
of tide-modified Cassini States and a novel, analytic description/calculation of the
resonance encounter process. We apply our general theoretical results to assess
how obliquity tides may affect different types of exoplanetary systems.
Our paper is organized as follows. In Section 3.2, we briefly review the ba-
sic setup and non-dissipative dynamics of Colombo’s Top. In Section 3.3, we in-
vestigate the effect of adding a simple alignment torque to Colombo’s Top. The
resulting dynamics captures the essential behavior that emerges due to tidal dis-
sipation. In Section 3.4, we solve for the dynamics of the system including the full
effect of tidal dissipation. In Section 3.5, we apply our results to three exoplane-
tary systems/scenarios of interest: (i) a super Earth with an exterior companion,
(ii) the formation of USPs via obliquity tides, and (iii) the rapid orbital decay of
the hot Jupiter WASP-12b. We summarize and discuss in Section 3.6.
3.2 Spin Evolution Equations and Cassini States: Review
In this section, we briefly review the spin dynamics of a planet in the presence of
a distant perturber and introduce our notations; see Paper I for more details. We
consider a star of mass M⋆ hosting an inner oblate planet of mass m and radius
R on a circular orbit with semi-major axis a and an outer perturber of mass mp
on a circular orbit with semi-major axis ap. The two orbits are mutually inclined
by the angle I. Denote S the spin angular momentum and L the orbital angular
momentum of the planet, and Lp the angular momentum of the perturber. The
50
corresponding unit vectors are ŝ≡ S/S, l̂≡L/L, and l̂p ≡Lp/Lp. The spin axis ŝ of
the planet tends to precess around its orbital (angular momentum) axis l̂, driven
by the gravitational torque from the host star acting on the planet’s rotational
bulge. On the other hand, l̂ and l̂p precess around each other due to gravitational
interactions. Assuming S≪ L, the equations of motion for ŝ and l̂ are
dŝ = ( )( ) ( )( )ωsl ŝ · l̂ ŝ× l̂ ≡α ŝ · l̂ ŝ× l̂ , (3.1)dt
dl̂ = ( · )(ωlp l̂ l̂p l̂× )l̂p ≡− ( )g l̂× l̂p , (3.2)dt
where
2 ( )3
ωsl ≡α=
3GJ2mR M⋆ = 3kq M⋆ R3 Ωs, (3.3)2a IΩs 2k m a
≡− g = 3m
( )3
p a
ωlp n. (3.4)cos I 4M⋆ ap
In Eq. (4.4), Ωs is the spin frequency of the inner planet, I = kmR2 (with k the
normalized moment of inertia, often notated as CN) is its moment of inertia and
J2 = kqΩ2s (R3/Gm) (with kq a constant, related to the hydrostatic Love number
k2 by kq = k2/3) is its rotation-induced (dimensionless) quadrupole moment [for
a fluid body with uniform density, k = 0.4,kq = 0.5; for the Earth, k ≃ 0.331 and
kq ≃ 0.31; for Jupiter, k ≃ 0.27 and kq ≃ 0.18 [e.g. Groten, 2004, Lainey, 2016]].
In other studies, 3kq/2k is o√ften notated as k2/2CN [e.g. Millholland and Baty-
gin, 2019]. In Eq. (3.4), n ≡ GM⋆/a3 is the inner planet’s orbital mean motion,
and we have assumed ap ≫ a and included only the leading-order (quadrupole)
interaction between the inner planet and perturber (Section 3.5.2 discusses mod-
ifications to Eq. 3.4 when ap ≳ a). Eq. (3.2) neglects the back-reaction torque on l̂
from ŝ; this is justified since L≫ S (see Anderson and Lai, 2018 for the case when
L ∼ S). In Eq. (4.4) (and throughout Sections 3.2–3.4), we assume Lp ≫ L so that
l̂p is a constant (Section 3.5.3 discusses the case of L≃ Lp). Following the standard
notations, we have defined α=ωsl and g≡−ω1p cos I [e.g. Colombo, 1966].
51
As in Paper I, we combine Eqs. (4.2)–(3.2) into a single equation by transform-
ing into a frame rotating about l̂p with frequency g. In this frame, l̂p and l̂ are
both fixed, and ŝ evolves(as )
dŝ = ( )( ) ( )α ŝ · l̂ ŝ× l̂ + g ŝ× l̂p . (3.5)dt rot
We choose the coordinate system such that ẑ = l̂ and l̂p lies in the x̂-ẑ plane. We
describe ŝ in spherical coordinates using the polar angle θ, the planet’s obliquity,
and φ, the precessional phase of ŝ about l̂, defined so that when φ = 0◦, l̂p and ŝ
are on opposite sides of l̂.
The equilibria of Eq. (3.5) are referred to as Cassini States [CSs; Colombo,
1966, Peale, 1969]. We follow the notation of Paper I and introduce the parameter
≡− g = 1 k mpm
( )
a 3 ( a )3 n
η 2 cos I. (3.6)α 2 kq M⋆ ap R Ωs
For a given value of η, there can be either two or four CSs, all of which require ŝ lie
in the plane of l̂ and l̂p. In the standard nomenclature, CSs 1, 3, and 4 have θ < 0,
implying that ŝ and l̂p are on opposite sides of l̂, while CS2 has θ > 0, implying
that ŝ and l̂ are on the same side of l̂. We depart from the standard convention
and simply label the CSs using the polar angles θ and φ (with θ ∈ [0,π]): Figure 3.1
shows the CS obliquities as a function of η. CS1 and CS4 do not exist when η> ηc,
where ( )
η ≡ sin2/3I+cos2/3 −3/2c I . (3.7)
The Hamiltonian corresponding to Eq. (3.5) is
H =−α (ŝ · )l̂ 2− ( )g ŝ · l̂
2 p
=−α ( )cos2θ− g cosθ cos I−sin I sinθ cosφ . (3.8)
2
52
Figure 3.1: Cassini State obliquities θ as a function of η ≡ −g/α (Eq. 8.5) for I =
20◦. The vertical dashed line denotes ηc, where the number of Cassini States
changes from four to two (Eq. 3.7). The y-axis labels on the right of the plot show
the asymptotic obliquities for CS2 and CS3, I and 180◦− I respectively. Note that
θ does not follow the standard convention (e.g. Colombo, 1966, Paper I) and is
simply the angle between ŝ and l̂, while φ = 0 corresponds to ŝ and l̂p being on
opposite sides of l̂. While CSs 1–3 are “dynamically” stable, only CS1 and CS2
are stable and attracting in the presence of the spin-orbit alignment torque (see
Section 3.3.2).
53
Here, cosθ and φ form a canonically conjugate pair of variables. Figure 3.2 shows
the level curves of this Hamiltonian for I = 20◦, for which ηc ≈ 0.574 (Eq. 3.7).
When η < ηc, CS4 exists and is a saddle point. The infinite-period orbits origi-
nating and ending at CS4 form the separatrix and divide phase space into three
zones. The angle φ librates for trajectories in zone II and circulates for trajectories
in zones I and III. On the other hand, when η > ηc, the separatrix is absent and
all trajectories circulate. When the separatrix exists, we divide it into two curves:
C+, the boundary between zones I and II, and C−, the boundary between zones II
and III.
3.3 Spin Evolution with Alignment Torque
In this section, we consider a simplified dissipative torque that isolates the impor-
tant new phenomenon presented in this paper. We assume that the spin magni-
tude of the planet is constant, so α and g are both fixed, while the spin orientation
ŝ experiences an alignment to(rqu)e towards l̂ on the alignment timescale tal:
dŝ = 1 ( )ŝ× l̂× ŝ . (3.9)
dt tide tal
The full equations of motion for ŝ in the coordinates θ and φ can be written as
dθ =−gsin I sinφ− 1 sinθ, (3.10)
dt tal
dφ =− ( )αcosθ− g cos I+sin I cotθ cosφ . (3.11)
dt
3.3.1 Modified Cassini States
If the alignment torque is weak (|g| tal ≫ 1), then the fixed points of Eqs. (3.10)–
(3.11) are slightly modified CSs. To leading order, all of the CS obliquities θcs are
54
Figure 3.2: Level curves of the Hamiltonian (Eq. 3.8) for I = 20◦, for which ηc ≈
0.57 (Eq. 3.7). For η < ηc, there are four Cassini States (labeled), while for η >
ηc there are only two. In the former case, the existence of a separatrix (solid
black lines) separates phase space into three numbered zones (I/II/III, labeled).
We denote the upper and lower legs of the separatrix by C± respectively, as shown
in the upper two panels.
55
unchanged while the azimuthal angle φcs for each CS now satisfies
sinθ
sin = csφcs sin I |g| . (3.12)tal
We can see that if tal is longer than the critical alignment timescale tal,c, given for
a particular θcs by
≡ sinθt csal,c | | , (3.13)g sin I
then Eq. (3.12) will always have solutions for φcs, and the alignment torque does
not change the number of fixed points of the system. If tal is decreased below
tal,c ∼ |gsin I|−1, CS2 and CS4 cease to be fixed points when η ≲ 1 [as noted in
Levrard et al., 2007, Fabrycky et al., 2007], as θcs ∼ 90◦ for these (see Fig. 3.1). On
the other hand, the other CSs have small sinθcs and are only slightly modified.
Figure 3.3 shows the obliquity and azimuthal angle for each of the CSs when
η = 0.2, obtained via numerical root finding of Eqs. (3.10–3.11), where it can be
seen that CS2 and CS4 collide and annihilate when tal reaches tal,c. The phase
shifts φcs for CS2 and CS4 for tal > tal,c can be predicted to good accuracy using
Eq. (3.12) and θcs ≈ π/2−ηcos I ≈ 79◦ [Su and Lai, 2020]; these are shown as the
dashed lines in the bottom panel of Fig. 3.3. For the remainder of this section, we
will consider the case where tal ≫ tal,c and the CSs only differ slightly from their
unmodified locations.
3.3.2 Linear Stability Analysis
We next seek to characterize the stability of small perturbations about each of the
CSs in the presence of the weak alignment torque. We can linearize Eqs. (3.10–
56
Figure 3.3: Modified CS obliquities (top) and azimuthal angles (bottom) for I = 20◦
and η= 0.2, where the CS1 and CS3 obliquities have been offset (as labeled) to im-
prove clarity of the plot. In both panels, the solid lines give the result when apply-
ing a numerical root finding algorithm to the full equations of motion, Eqs. (3.10–
3.11), while the dotted lines in the bottom panel give the CS2 and CS4 azimuthal
angles according to Eq. (3.12). At |gtal sin I| = 1, CS2 and CS4 collide and annihi-
late (see Eq. 3.13).
57
3.11) about a shifted CS,yelding   d ∆θ= − cosθt −gsin I cosφal  ∆θ , (3.14)
dt sin + g sin I cosφ∆φ α θ 2 0 ∆φsin θ cs
where the “cs” subscript indicates evaluating at a CS, ∆θ = θ−θcs, and ∆φ=φ−φcs.
The eigenvalues λ of Eq. (3.14) sa(tisfy the eq)uation
= + cosθ0 csλ λ−λ20, (3.15)tal
where
λ20 ≡
( )( )
αsinθcs+ gsin I csc2θcs cosφcs −gsin I cosφcs . (3.16)
When tal is large, we can simplify Eq. (3.15) t√o
≈−cosθcsλ ± λ2
t 0
. (3.17)
al
The stability of a CS depends on the real part of λ in Eq. (3.17). Equation (3.17)
λ20 is a generalization of Eq. (A4) in Paper I and generally has the same behavior:
it is negative for CSs 1–3 and positive for CS4, as shown in Fig. 3.4. Thus, CS4 is
always “dynamically” unstable (i.e. unstable even in the limit of tal →∞), as there
will always be at least one positive solution for λ. On the other hand, CSs 1–3 are
dynamically stable, and their overall stabilities in the presence of the alignment
torque are determined by the sign of cosθcs. Using Fig. 3.1, we conclude that CS1
and CS2 are stable and attracting while trajectories near CS3 are driven away
by the alignment torque. These calculations quantify the results long used in the
literature [e.g. Ward, 1975, Fabrycky et al., 2007].
3.3.3 Spin Obliquity Evolution Driven by Alignment Torque
With the above results, we are equipped to ask questions about the dynamics of
Eqs. (3.10–3.11): what is the long-term evolution of ŝ for a general initial ŝi?
58
Figure 3.4: λ20 (Eq. 3.16) as a function of η for the four CSs, for three different
values of the shift in φcs (e.g. for ∆φ ◦cs = 60 , the phase angles are φcs = 120◦ for
CS2 and φcs = 60◦ for CSs 1, 3, and 4). The values of ∆φcs are labeled (∆φcs = 0
corresponds to the unmodified CSs).
59
For η > ηc, the only stable (and attracting) spin state is CS2, and all initial
conditions will evolve asymptotically towards it.
For η < ηc, both CS1 and CS2 are stable (assuming the alignment torque is
sufficiently weak that CS2 remains a fixed point; see Section 3.3.1), and spin evo-
lution may involve separatrix crossing. To explore the fate of various initial ŝ
orientations, we numerically integra(te Eqs. ()3.10–3.11) for many random initial
conditions uniformly distributed in cosθi,φi and determine the nearest CS for
each integration after 10tal. In Fig. 3.5, we show the results of this procedure for
η = 0.2, and I = 20◦ (we use tal = 103/ |g|, but the results are unchanged as long
as tal ≫ |g|−1). It is clear that initial conditions in zone I evolve into CS1, those
in zone II evolve into CS2, while those in zone III have a probabilistic outcome.
These can be understood as follows:
For initial conditions in zone I, the spin orientation circulates, and θ̇ is nega-
tive everywhere during the cycle. Thus θ decreases until the trajectory has con-
verged to CS1. This is intuitively reasonable, as CS1 is stable and attracting (see
Section 3.3.2).
For initial conditions in zone II, our stability analysis in Section 3.3.2 shows
that when ŝ is sufficiently near CS2, it will converge to CS2 since CS2 is stable
and attracting. In fact, this result can be extended to all initial conditions inside
the separatrix, as shown in Appendix B.1.
For initial conditions in zone III, since there are no stable CSs in zone III,
the system must evolve through the separatrix to reach either CS1 or CS2. The
outcome of the separatrix encounter is probabilistic and determines the final CS.
Intuitively, this can be understood as probabilistic resonance capture, as first stud-
ied in the seminal work of Henrard [1982]: for η≲ ηc ∼ 1, we have that α≳−g, but
60
Figure 3.5: Asymptotic outcomes of spin evolution driven by an alignment torque
for different initial spin orientations (θi and φi) for a system with η = 0.2 and
I = 20◦. Each dot represents an initial spin orientation, and the coloring of the
dot indicates which stable Cassini State (legend) the system evolves into: initial
conditions in Zone I evolve into CS1, those in Zone II evolve into CS2, and those
in Zone III have a probabilistic outcome.
61
αcosθ can become commensurate with −g if cosθ becomes small. This is achieved
as θ evolves from an initially retrograde obliquity through 90◦ towards 0◦ under
the influence of the alignment torque.
While similar in behavior to previous studies of probabilistic resonance capture
[Henrard, 1982, Su and Lai, 2020], the underlying mechanism is different: In
these previous studies, the phase space structure itself evolves and causes the
system to transition among different phase space zones; here in the problem at
hand, a non-Hamiltonian, dissipative perturbation causes the system to transition
among fixed phase space zones. In the following subsection, we present an analytic
calculation to determine the probability distribution of outcomes upon separatrix
encounter. Readers not interested in the technical details can simply examine the
resulting Fig. 3.7.
3.3.4 Analytical Calculation of Resonance Capture Proba-
bility
Before discussing our quantitative calculation, we first present a graphical under-
standing of the separatrix encounter process. Figure 3.6 shows how the perturba-
tive alignment torque generates the two outcomes upon separatrix encounter, i.e.
the zone III to zone II and the zone III to zone I transition. The critical trajecto-
ries in Fig. 3.6 are calculated numerically by integrating from a point infinitesi-
mally close to CS4 forward and backward in time. In the absence of the alignment
torque, these trajectories would evolve along the separatrix, but in the presence
of the alignment torque, they are perturbed slightly and cease to overlap. It can
be seen in Fig. 3.6 that this splitting opens a path from zone III into both zones I
and II: the coloring scheme indicates that the trajectories within the orange and
62
green regions of phase space stay within their respective colored regions.
To understand this process more concretely, and to compute the associated
probabilities of the two possible outcomes, we consider the evolution of the value
of the unperturbed Hamiltonian (Eq. 3.8) as the spin evolves due to the alignment
torque. A point in zone III evolves such that H is increasing until H ≈Hsep, where
Hsep is the value of H along the separatrix, given by
≡ ( )Hsep H cosθ4,φ4
≈ g
2 ( )
gsin I+ cos2 I+O η2 , (3.18)
2α
where
θ4 ≃π/2−ηcos I (3.19)
(see Section A.1 of Paper I) and φ4 = 0 are the coordinates of CS4. As the system
evolves closer to the separatrix, the change in H over each circulation cycle can
be approximated by ∆H−, the change in H along C− (see Fig. 3.2). In general, we
define the quantities ∆H± ∮
≡ dH∆H± dt. (3.20)dt
C±
Using
dH = ∂H d(cosθ) + ∂H dφ
dt (∂(cosθ) ) dt ∂φ dt
= d(cosθ) dφ (3.21)
dt tide dt
and Eq. (3.10), we find
∫2π
∆H± =∓ 1 sin2θ dφ, (3.22)t
( ) al 0
where θ = θ φ is evolved along C±. Thus, if we evaluate H every time that a
trajectory originating in zone III crosses φ= 0, we see that will initially be <Hsep
and increase for each circulation cycle until the system encounters the separatrix.
63
At the beginning of the separatrix-crossing orbit, the initial value of H, denoted
by Hi, must be greater than Hsep−∆H− to encounter the separatrix on the current
orbit. We thus require
∈ [ − ]Hi Hsep ∆H−,Hsep . (3.23)
The values of cosθ corresponding to the lower and upper bounds in this range are
shown as the black and purple dots on the left of Fig. 3.6 respectively.
During the separatrix-crossing orbit, the trajectory first evolves approximately
along C− and then along C+, after which the final value of H, denoted by Hf, is
approximately equal to
Hf =Hi+∆H++∆H−. (3.24)
There are two outcomes depending on the value of Hf:
• If Hf < Hsep, then, since H < Hsep corresponds to the exterior of the sepa-
ratrix, this implies that the trajectory has ended outside of the separatrix.
This outcome thus corresponds to a zone III to zone I transition. In Fig. 3.6,
the evolution within the orange shaded regions exhibits such an outcome.
• If Hf > Hsep, then the trajectory has instead ended inside of the separatrix
and has executed a zone III to zone II transition. This corresponds to evolu-
tion within the green shaded regions in Fig. 3.6.
These two possibilities can be re-expressed in terms of Hi: if Hi is in the interval
[Hsep−∆H−, Hsep−∆H−−∆H+], then the system executes a III → I transition,
and if it is in the interval [Hsep−∆H−−∆H+, Hsep], then the system executes a
III → II transition. We see that there is a critical value of Hi,
(Hi)crit =Hsep−∆H−−∆H+, (3.25)
64
that separates the two possible outcomes of the separatrix encounter within the
interval given by Eq. (3.23). The value of cosθ for which H is equal to Hsep−∆H−−
∆H+ is shown as the blue dot on the left of Fig. 3.6. Finally, if the alignment
torque is weak, then |∆H±| ∝ t−1al is small compared to any variation in the value
of H (e.g. when changing the initial φi or θi by a small amount), and Hi can be
effectiv[ely considered as] randomly chosen from a uniform distribution over the
range Hsep−∆H−,Hsep . As a consequence we obtain the probability of the III →
II transition:
= ∆H−+∆HP +III→II . (3.26)
∆H−
To evaluate Eq. (3.26) analytically, we use the approximate expression for the
separatrix η≪ 1 (see Eq. B5 of Paper I)1√: ( )
(cosθ)C± ≈ ηcos I± 2ηsin I 1−cosφ . (3.27)
Using Eq. (3.22), we find
≈ 2π ( − )∆H− 1 2ηsin I +O (η3/2), (3.28)tal
32η3/2
p
+ ≈ cos I sin I
( )
∆H+ ∆H− +O η5/2 , (3.29)tal
and thus
p
≈ 16η(3/2 cos I sin IPIII→II
π 1− ) . (3.30)2ηsin I
To compare Eq. (3.30) with numerical results, we perform numerical integra-
tions of Eqs. (3.10–3.11) while restricting the initial conditions to those in zone III.
In Fig. 3.7, we display Eq. (3.30) alongside the computed PIII→II using 1000 initial
conditions in zone III for each of 60 values of η. Excellent agreement is observed.
1A more exact expression valid for all η ≤ ηc can be obtained by using the exact analytical
solution to Colombo’s Top, see Ward and Hamilton [2004]. We forgo this approach due to the
significant complexity of the expression involved for a small extension in the regime of validity:
our expression is sufficiently accurate when η≲ 0.3, while ηc ≲ 1.
65
Figure 3.6: Plot illustrating the probabilistic origin of separatrix (resonance) cap-
ture for a system with η = 0.2 and I = 20◦. The orange regions converge to CS1,
and the green to CS2. The purple dots denote CS4, a saddle point. The boundaries
separating the CS1 and CS2-approaching regions consist of four critical trajecto-
ries (labeled in the legend) that are evolved starting from infinitesimal displace-
ments from CS4 along its stable and unstable eigenvectors going forwards and
backwards in time: e.g. the trajectory labeled CS4+φ=0 starts at φ = ϵ (for some
small, positive ϵ) and evolves forwards in time (with tal = 103 |g|−1), while the
trajectory labeled CS4− ◦φ=360 starts at φ = 360 − ϵ and evolves backwards in time.
The blue and black dots denote the intersections of CS4−φ=0 and CS4
−
φ=360 critical
trajectories with the vertical line φ = 0. These critical trajectories can be used to
understand the probabilistic outcomes that trajectories originating in zone III ex-
perience upon separatrix encounter, illustrated by the tightly spaced orange and
green bands in zone III; see Section 3.3.4 for additional details.
66
Figure 3.7: Zone III to Zone II transition probability PIII→II upon separatrix en-
counter a(s a fu)nction of η driven by an alignment torque. For each η, 1000 initial
random θi,φi values in zone III are evolved until just after separatrix encounter,
where the outcome of the encounter is recorded. The red line shows the analytical
result, Eq. (3.30).
67
The rigorous connection between the above calculation, focusing on the evolu-
tion of H along the two legs of the separatrix C±, and the graphical picture illus-
trated in Fig. 3.6 is provided by Melnikov’s Method [Guckenheimer and Holmes,
1983]. Melnikov’s Method is a general calculation that gives the degree of split-
ting of a “homoclinic orbit” (here, the separatrix) of a Hamiltonian system induced
by a small, possibly time-dependent, perturbation. At a qualitative level, we can
state the connection succinctly (see Fig. 3.6):
• The trajectory labeled CS4−φ=360 is evolved backwards in time from CS4
(where the Hamiltonian has the value Hsep) along C−, and thus the black
dot labels the start of a separatrix-crossing orbit with the initial value of the
Hamiltonian Hi =Hsep−H−. According to Eq. (3.23), this is exactly the min-
imum Hi such that a trajectory experiences a separatrix-crossing orbit. This
is consistent with Fig. 3.6, where it is clear that any trajectories below the
black dot at φ = 0 will not experience a separatrix encounter on its current
circulation cycle.
• The trajectory labeled CS4−φ=0 is the one evolving backwards in time from
CS4 along first C+ then C−, and thus the blue dot labels the start of a
separatrix-crossing orbit with Hi =Hsep−∆H−−∆H+. According to Eq. (3.25),
this is exactly the critical value of Hi that separates trajectories executing
a III→II transition and a III→I transition. This is also consistent with
Fig. 3.6, where the region above CS4−φ=0 is colored green while the region
below is colored orange.
68
3.4 Spin Evolution with Weak Tidal Friction
3.4.1 Tidal Cassini Equilibria (tCE)
Having understood the effect of the alignment torque on the spin evolution (Sec-
tion 3.3), we now implement the full effect of tidal dissipation, including both tidal
alignment and spin synchronization. We use the weak friction theory of equilib-
rium tides [e.g. Alexander, 1973, Hut, 1981]. In this model, tides cause both the
spin orientation ŝ and frequency Ωs to evolve on the characteristic tidal timescale
ts following [see Lai, 2012(]: ) [ ]
dŝ
( ) =
1 2n − ( ) ( )ŝ · l̂ ŝ× l̂× ŝ , (3.31)
dt tide ts [Ωs
1 dΩs = 1 2n ( · ) ( ) ]ŝ l̂ −1− ŝ · l̂ 2 , (3.32)
Ωs dt tide ts Ωs
where ts is given by ( )( )
1 ≡ 1 3k M
3
2 ⋆ R n, (3.33)
ts 4k Q m a
with k2 and Q the tidal Love number2 and tidal quality factor, respectively. We
neglect orbital evolution (thus, ts is a constant) in this section since the time scale
is longer than ts by a factor of ∼ L/S≫ 1 (we discuss the effect of orbital evolution
in Section 3.5.3). We will continue to consider the case where tidal dissipation is
slow, i.e. |g| ts ≫ 1. The full equations of motion including weak tidal friction can
2Note that for rocky planets, the tidal k2 and the hydrostatic k2 (which is equal to the 3kq)
need not be equal, e.g. for the Earth, ktidal2 ≈ 0.29 [Lainey, 2016] while the hydrostatic krotational2 =
0.94 [Fricke, 1977]. This is due to the Earth’s appreciable rigidity. For higher-mass, more “fluid”
planets, ktidal ≃ krotational2 2 .
69
be written in component form as ( )
dθ = 1 2ngsin I sinφ− sinθ −cosθ , (3.34)
dt
dφ =− − (
ts Ωs )
α[cosθ g cos I+sin I c]otθ cosφ , (3.35)dt
1 dΩs = 1 2n − ( + 2 )cosθ 1 cos θ . (3.36)
Ωs dt ts Ωs
Equation (3.36) shows that, at a given obliquity, tides tend to drive Ωs towards
the pseudo-synchronous equilibrium value, given by
Ωs = 2cosθ ( )+ 2 Ω̇s = 0 . (3.37)n 1 cos θ
On the other hand, Eq. (3.34) shows that the spin-orbit alignment timescale tal is
related to ts by ( )
t−1 = t−1 2nal s −cosθ . (3.38)Ωs
Thus, θ̇tide < 0 for 2n/Ωs > cosθ and θ̇tide > 0 for 2n/Ωs < cosθ.
To understand the long-term evolution of the system, we first consider its be-
havior near a CS. Specifically, we wish to understand whether initial conditions
near a CS stay near the CS as the evolution of Ωs causes the CSs (and separatrix)
to evolve. We first note that the evolution of Ωs alone does not drive ŝ towards or
away from CSs: As long as it evolves sufficiently slowly (adiabatically; see Paper
I), conservation of phase space area ensures that trajectories will remain at fixed
distances to stable equilibria of the system. Thus, Eq. (4.16) or (3.34) alone deter-
mine whether the system evolves towards or away from a nearby CS asΩs evolves.
Then, from Eq. (3.38), we see that CS2 is still always stable (and attracting), while
CS1 is becomes unstable for Ωs > 2ncosθ1 ≈ 2n, where θ1 ≈ ηsin I (Paper I) is the
obliquity of CS1.
With this consideration, we can identify the long-term equilibria of the system
when tidal torques drive the evolution of both the obliquity and Ωs (and thus η):
70
these equilibria must satisfy Ω̇s = 0 and be a CS that is stable in the presence of
the tidal torque (i.e. satisfying dŝ/dt = 0); we call such long-term equilibria tidal
Cassini Equilibria (tCE). Figure 3.8 depicts the evolution of the system following
Eqs. (4.16–4.17) in (Ωs,θ) space starting from several representative initial con-
ditions, along with the locations of CS1 and CS2. The circled points in Fig. 3.8
denote the two tCEs (tCE1 and tCE2, depending on whether it lies on CS1 or
CS2).
The obliquities of the tCE and the evolutionary track in the θ-Ωs plane depend
on the parameter
( ) Ωs
ηsync ≡ η Ωs=n = η( n )
k m m a 3 ( a )= p 32 cos I. (3.39)2kq M⋆ ap R
In Fig. 3.8, ηsync = 0.06; Figs. 3.9–3.10 illustrate the cases with ηsync = 0.5 and 0.7
respectively.
The tCE obliquities as a function of ηsync are shown in Fig. 3.11 for I = 20◦ and
I = 5◦. In fact, an analytical expression for the tCE2 obliquity and rotation rate
for ηsync ≪ 1 can be obtained using Eqs. (3.37)–(3.39) and cosθ2 ≃ ηcos I (valid for
η≪ 1; see Appendix of Paper I): √
≃ ηsync cos IcosθtCE2 √ , (3.40)2Ωs,tCE2 ≃ 2ηsync cos I. (3.41)n
This approximation for θtCE2 is shown as the blue dashed line in Fig. 3.11, in-
dicating good agreement with the numerical result obtained via root finding of
Eqs. (3.34)–(3.36) while assuming |g| ts ≫ 1.
There are two important conditions that can influence the existence and sta-
bility of the tCE. First, if ηsync > ηc (where ηc is given by Eq. 3.7), then tCE1 will
71
Figure 3.8: Schematic depiction of the effect of tidal friction on the planet’s spin
evolution in the θ-Ωs plane for a system with I = 20◦ (corresponding to ηc = 0.574;
see Eq. 3.7) and ηsync = 0.06 (see Eq. 3.39). The black and blue lines denote where
the tidal Ω̇s and θ̇ change signs (see Eqs. 3.34 and 3.37). The orange and green
lines give the CS1 and CS2 obliquities respectively (these are the two CSs that
can be stable in the presence of tidal dissipation). Note that when θ̇tide > 0, CS1
becomes unstable, denoted by the dashed orange line. The points that lie on CSs
and satisfy Ω̇s = 0 are called tidal Cassini Equilibria (tCE), which are circled and
labeled. The various colored crosses and their associated colored lines represent
a few characteristic examples of the spin evolution under weak tidal friction (for
illustrative purposes, we have used |g| ts = 102 and evolved each example for 5ts).
The phase space evolution of the two thicker evolutionary trajectories (cyan and
pink; those beginning at θi = 120◦) are shown in Figs. 3.12–3.13.
72
Figure 3.9: Same as Fig. 3.8 but for ηsync = 0.5. The crosses and lines correspond
to evolutionary trajectories using the same initial conditions as those shown in
Fig. 3.8.
73
Figure 3.10: Same as Figs. 3.8 but for ηsync = 0.7. Note that ηsync = 0.7> ηc = 0.574
and tCE1 does not exist. The phase space evolution of the thick purple trajectory
(starting at θ ◦i = 10 ) is shown in Fig. 3.14.
74
Figure 3.11: Obliquities of the two tCE as a function of ηsync (defined in Eq. 3.39)
for I = 20◦ (top) and I = 5◦ (bottom). The blue dashed lines denote the analytical
approximation given by Eq. (3.40) and is only valid for ηsync ≪ 1. The vertical
dashed lines denote where ηsync = ηc(I), above which tCE1 ceases to exist.
75
Figure 3.12: Phase space evolution of the pink trajectory in Fig. 3.8, for which
ηsync = 0.06 and I = 20◦ (corresponding to ηc = 0.574). The initial conditions are
Ωs,i = 2.5n, θ = 120◦i , and φi = 0◦, and we have used |g| ts = 102 and have evolved
the system for 5ts. In the left-most panel, the trajectory’s evolution in the θ-Ωs
plane along with the curves indicating CS1, CS2, and Ω̇s = 0 are re-displayed from
Fig. 3.8. The vertical dashed lines denote the values of Ωs/n for which a few phase
space snapshots of the system are displayed in the right four panels. In each of
these right four panels, the trajectory’s evolution for a single circulation/libration
cycle is displayed in the cosθ-φ plane for the labeled value of Ωs (and the corre-
sponding value of η). The system encounters the separatrix, undergoes a III → I
transition, and converges to tCE1.
76
Figure 3.13: Same as Fig. 3.12 but for φi = 286◦, corresponding to the cyan tra-
jectory in Fig. 3.8. The system encounters the separatrix, undergoes a III → II
transition, and converges to tCE2. The small φ offset of tCE2 from 180◦ arises
from the alignment torque (see Fig. 3.3). Note that the initial condition of this
trajectory and that displayed in Fig. 3.12 have the same initial θi and Ωs,i but dif-
ferent precessional phases φi.
77
Figure 3.14: Same as Fig. 3.12 but for ηsync = 0.7 and θi = 10◦, corresponding to the
purple trajectory shown in Fig. 3.10. Here, the system evolves along CS1 until the
separatrix disappears, upon which it experiences large obliquity variations that
damp due to tidal dissipation. The highly asymmetric shape in the third panel
arises due to the strong tidal dissipation used in this simulation (|g| ts = 102). The
system finally converges to tCE2, the only tCE that exists for this value of ηsync.
78
not exist (Fig. 3.10 gives an example)3. Second, tCE2 may not be stable if the
phase shift due to the alignment torque (see Section 3.3.1) is too large. Applying
the results of Section 3.3.2 (see Eqs. 3.12–3.13), we find that tCE2 is stable as long
as ts ≥ ts,c where ( )
≡ sinθt tCE2 2ns,c | | −cosθ . (3.42)g sin I tCE2Ωs,tCE2
When ηsync ≪ 1, we can use Eqs. (3.40–3.41) to√further simplify ts,c to
4
t ≃ tanθtCE2 1 2s,c |g| ≈sin I | | . (3.43)g sin I ηsync cos I
3.4.2 Spin and Obliquity Evolution as a Function of Initial
Spin Orientation
We can now study the final fate of the planet’s spin as a function of the initial
condition. We begin by examining the example trajectories shown in Figs. 3.8, for
which we have integrated the equations of motion (combining Eqs. 3.5 and 4.16 to
give dŝ/dt, and Eq. 4.17) and set I = 20◦, t = 100 |g|−1s . We discuss each of the six
trajectories in turn:
• The trajectory with the initial condition Ωs,i = 2.5n and θi = 10◦ (purple) has
an initially prograde spin (i.e. in zone I, see Fig. 3.2) and directly evolves
to tCE1, with the final Ωs/n ≃ 1 and θ = θCS1 ≃ ηsync sin I (for ηsync ≪ 1; see
Appendix A of paper I).
• The trajectory with Ω ◦s,i = 2.5n and θi = 90 (red) has an initial condition
inside the resonance / separatrix (zone II) and evolves to tCE2. Note that
3Strictly speaking, ηsync can be slightly smaller than ηc, as the planet’s spin is slightly subsyn-
chronous at tCE1 (see Eq. 3.37).
4Note that Eq. (3.43) for the critical ts agrees with Eq. (16) of Levrard et al. [2007].
79
the obliquity is trapped in a high value due to the stability of CS2 under the
alignment torque, as shown in Section 3.4.1.
• We have chosen two trajectories, both with the initial condition Ωs,i = 2.5n
and θi = 120◦, but with different initial precessional phases φi. Consider
first the pink trajectory, for which φi = 0◦. It originates in zone III, evolves
towards the separatrix as tidal friction damps the obliquity, and crosses
the resonance (separatrix) without being captured, upon which the obliquity
continues to damp until the system converges to tCE1. The detailed phase
space evolution of this trajectory is shown in Fig. 3.12, where the outcome of
the separatrix encounter is very visible.
• The light blue trajectory also has Ωs,i = 2.5n and θi = 120◦ (like the pink tra-
jectory) but with the initial precessional phase φi ≈ 286◦. It also originates in
zone III, encounters the separatrix but is captured into the resonance (zone
II), upon which tidal friction drives the system towards tCE2. The detailed
phase space evolution of this trajectory is shown in Fig. 3.13, where the res-
onance capture is displayed. Also visible in the final panel of Fig. 3.13 is the
slight phase offset of CS2, i.e. φcs < 180◦, in agreement with the result of
Section 3.3.1 (see Fig. 3.3).
• For completeness, we also examine some trajectories for initially subsyn-
chronous spin rates. The trajectory with Ωs,i = 0.1n and θ = 35◦i (blue)
has its obliquity rapidly damped to zero by tidal friction as it spins up to
spin-orbit synchronization, eventually converging to tCE1. A subtlety of ini-
tially subsynchronous spins can be seen here: since the initial ηi = 0.6 > ηc
(= 0.574), the separatrix and CS1 do not exist initially. As such, naively,
one expects initial convergence to CS2 and subsequent obliquity evolution
along CS2 as the spin increases. However, due to the strong tidal dissipa-
tion adapted in the calculation and the proximity of ηi to ηc, CS1 appears
80
within a single circulation cycle, and the obliquity quickly damps to, and
continues to evolve along CS1.
• The trajectory with Ωs,i = 0.1n and θi = 100◦ (teal) also has its obliquity
damped toward tCE1 as it approaches spin-orbit synchronization. We note
that if we adopt t = 103 |g|−1s , the same initial condition will converge to
tCE2, agreeing with the intuitive analysis given in the previous paragraph.
In Figs. 3.9–3.10 we show, for each of the six initial conditions, the evolutionary
trajectories for the ηsync = 0.5 and ηsync = 0.7 cases. The qualitative behaviors of
these six examples change in several important ways, so we will discuss a few
points of interest:
• For both ηsync = 0.5 and ηsync = 0.7, we see that the initial conditions with
θi = 120◦ (φi = 0, pink; and φi = 286◦, blue) converge to tCE2. In fact, for
these values of ηsync, all initial conditions with θ ◦i = 120 will converge to
tCE2 regardless of φi.
• The two subsynchronous initial conditions evolve to tCE2 for both ηsync = 0.5
and ηsync = 0.7, as in both cases ηi ≫ ηc and CS2 is the only low-obliquity
spin equilibrium. The system then continues to evolve along CS2 toward
tCE2.
• Of particular interest is the trajectory starting from the initial condition
Ωs,i = 2.5n and θi = 10◦ (purple) in the case of ηsync = 0.7. Figure 3.14 shows
the detailed phase space evolution of this trajectory, where it can be seen
that the system initially evolves along the stable CS1, but is ejected when
Ωs becomes sufficiently small that CS1 ceases to exist, upon which large
obliquity variations eventually lead to convergence to tCE2, the only tCE
that exists.
81
From the above examples, we see that the spin evolution driven by tides can be
complex and varies greatly depending on the various system parameters and ini-
tial conditions. In the case where the initial spin is subsynchronous, the detailed
outcome depends sensitively on the initial value of η and the tidal dissipation rate.
In the following, we restrict our discussion to the more astrophysically common
regime of Ωs,i ≫ n, and we adopt the fiducial value Ωs,i = 10n.
Having developed an intuition for a few different possible evolutionary tra-
jectories, we can attempt to draw general conclusions about the final fate of the
planet’s spin as a function of its initial conditions. We do this by again integrating
Eqs. (3.5, 4.16–4.17) for many initial θi and φi and examining the final outcomes.
In contrast to the examples shown in Figs. 3.8–3.14, we use a more gradual tidal
dissipation rate of |gt 3s| = 10 . In Fig. 3.15, we show the final outcome for many
randomly chosen θ and φ for η = 0.06 and I = 20◦i i sync . We see that the behaviors
seen in the example trajectories of Fig. 3.8 are general: tCE1 is generally reached
for spins initially in zone I (like the purple trajectory in Fig. 3.8), tCE2 is gener-
ally reached for spins initially in zone II (like the red trajectory in Fig. 3.8), and
a probabilistic outcome is observed for spins initially in zone III (like the light
blue and pink trajectories in Fig. 3.8). Figures 3.16 and 3.17 show similar results
for ηsync = 0.2 and ηsync = 0.5. As ηsync is increased, more initial conditions reach
tCE2. This is both because there are more systems initially in zone II and be-
cause systems initially in zone III have a higher probability of executing a III → II
transition upon separatrix encounter. Note also that in Fig. 3.17, even initial con-
ditions in zone I are able to reach tCE2; we comment on the origin of this behavior
in the next section.
82
Figure 3.15: Left: Asymptotic outcomes of spin evolution in the presence of weak
tidal friction for different initial spin orientations (θi and φi) for a system with
ηsync = 0.06 and I = 20◦. Each dot represents an initial spin orientation, and the
coloring of the dot indicates which tCE (legend) the system evolves into. Similarly
to Fig. 3.5 initial conditions in Zone I evolve into CS1, those in Zone II evolve into
CS2, and those in Zone III have a probabilistic outcome. Right: Histogram of the
final tCE that a given initial obliquity θi evolves into, averaged over φi.
83
Figure 3.16: Same as Fig. 3.15 but for ηsync = 0.2.
84
Figure 3.17: Same as Fig. 3.15 but for ηsync = 0.5. Note that even points above the
separatrix can evolve towards tCE2 here.
85
3.4.3 Semi-analytical Calculation of Resonance Capture Prob-
ability
Even when including the evolution of Ωs, and therefore the parameter η (see
Eq. 8.5), the probabilities of the III → I and III → II transitions upon separa-
trix encounter can still be obtained semi-analytically. The calculation resembles
that presented in Section 3.3.4 but involves several new ingredients. We describe
the calculation below.
In Section 3.3.4, we found that the evolution of H, the value of the unperturbed
Hamiltonian, allowed us to calculate the probabilities of the various outcomes of
separatrix encounter. Specifically, the outcome upon separatrix encounter is de-
termined by the value of H at the start of the separatrix-crossing orbit relative
to Hsep, the value of H along the separatrix. However, when the spin Ωs is also
evolving, Hsep also changes during the separatrix-crossing orbit, and the calcula-
tion in Section 3.3.4 must be generalized to account for this. Instead of focusing
on the evolution of H along a trajectory, we instead follow the evolution of
K ≡H−Hsep. (3.44)
Note that K > 0 inside the separatrix, and K < 0 outside. With this modification,
the outcome of the separatrix-crossing orbit can be determined in the same way
as in Section 3.3.4. First, we must compute the change in K along the legs of the
separatrix. We define ∆K± by gen∮eral[izing Eq. (3.2]0) in the natural way:
∆K± = dH −
dHsep dt. (3.45)
C± dt dt
Here, however, note that the contours C± depends on the value of Ωs at separatrix
encounter (or the corresponding value η = ηcross). Since there is no closed form
solution for Ωs(t), the probabilities of the various outcomes cannot be expressed
86
as a simple function of the initial conditions.
Continuing the argument presented in Section 3.3.4, we consider the outcome
of the separatrix-crossing orbit as a function of K i, the value of K at the start
(φ= 0) of the separatrix-crossing orbit. We find that if −∆K+−∆K− < Ii < 0, then
the system undergoes a III → II transition and eventually evolves towards tCE2,
and if −∆K− < K i <−∆K−−∆K+, then the system undergoes a III → I transition
and ultimately evolves towards tCE1. Thus, we find that the probability of a III
→ II transition is given by
= ∆K++∆KP −III→II . (3.46)
∆K−
Again, since ∆K± are evaluated at resonance encounter, and Ωs is evolving, there
is no way to express ∆K± in a closed form of the initial conditions. In fact, since
many resonance encounters occur when η = ηcross is ≳ 0.2, even an approximate
calculation of ∆K± using Eq. (3.27) (which is valid only for η≪ 1) is inaccurate,
and we instead calculate ∆K± along the numerically-computed C± for arbitrary
η. Note that Eqs. (3.45, 3.46) are equivalent to the separatrix capture result of
Henrard [1982] when θ̇tide = 0 (see also Henrard and Murigande, 1987 and Paper
I). In other words, we argue that this classic calculation can be unified with the
calculation given in Section 3.3.4 to give an accurate prediction of separatrix en-
counter outcome probabilities in the presence of both dissipative perturbation and
parametric evolution of the Hamiltonian.
We note that Levrard et al. [2007] also presented an analytical expression for
the resonance capture probability (their Eq. 14) following the method of Goldreich
and Peale [1966]. However, their expression is incomplete, as it does not account
for the contribution of the tidal alignment torque to the change of the integral of
motion over a single orbit.
87
To validate the accuracy of Eq. (3.46), we can compare with direct numerical
integration of Eqs. (3.5, 4.16–4.17) for many initial conditions while evaluating
PIII→II (and thus also obtaining PIII→I = 1− PIII→II) for each simulation at the
moment it encounters the separatrix, if it does so. If the theory is correct, the total
numbers of systems converging to each of tCE1 and tCE2 should be equal to those
predicted by the calculated probabilities. In Fig. 3.18, we show the agreement
of this semi-analytic procedure with the numerical result displayed in the right
panel of Fig. 3.15. Figure 3.19 depicts the same for the parameters of Figs. 3.16,
also showing satisfactory agreement. Thus, we conclude that the outcomes of
separatrix encounter are accurately predicted by Eq. (3.46).
With the above calculation, we can understand why even some initial condi-
tions in zone I may converge to tCE2 in certain situations (see Fig. 3.17). As
long as the initial spin is sufficiently large (≥ 2n), Eq. (3.34) shows that when
cosθ > 2n/Ωs, the obliquity can increase. In particular, when the critical obliq-
uity cosθ = 2n/Ωs is inside the separatrix, tidal alignment acts to drive initial
conditions in both zones I and III towards the critical obliquity and into the sepa-
ratrix, and also towards larger H. As such, when this effect is sufficiently strong,
Eq. (3.45) shows that both ∆K± > 0, and both III → II and I → II transitions are
guaranteed upon separatrix encounter (Eq. 3.46).
3.4.4 Spin Obliquity Evolution for Isotropic Initial Spin Ori-
entations
In Sections 3.4.2–3.4.3, we considered the outcome of the spin evolution driven
by tidal torque as a function of the initial spin orientation, specified by θi and φi.
Here, we calculate the probability of evolution into tCE2 when averaging over a
88
Figure 3.18: Comparison of the fraction of systems converging to tCE2 obtained
via numerical simulation (red dots) and obtained via a semi-analytic calculation
(blue line) for ηsync = 0.06, I = 20◦, andΩs,i = 10n (see right panel of Fig. 3.15). The
semi-analytic calculation is performed by numerically integrating Eqs. (3.5, 4.16–
4.17) on a grid of initial conditions uniform in cosθi and φi until the system reaches
the separatrix, then calculating the probability of reaching tCE2 for each integra-
tion using Eq. (3.46). The green dashed line in the top panel shows the result of
using the analytical expression (Eq. B.7) for ∆K±, and the bottom panel shows
the distribution of values of ηcross, the value of η when a trajectory starting at θi
encounters the separatrix.
89
Figure 3.19: Same as Fig. 3.18 but for ηsync = 0.2, corresponding to the right panel
of Fig. 3.16. Note that the analytical equation (Eq. B.7; green dashed line) exhibits
significantly poorer agreement than in Fig. 3.18 when ηcross ≳ 0.2.
90
Figure 3.20: Top: Same as top panel of Fig. 3.11. Bottom: Total probability of
the system ending up in tCE2 (PtCE2; red dots) as a function of ηsync (Eq. 3.39),
averaged over an isotropic initial spin orientation and taking Ωs,i = 10n. The red
dashed line shows the analytical prediction Eq. (3.47). The three shaded regions
denote the contributions of initial conditions in zones I/II/III (labeled) to the total
tCE2 probability. For example, among systems that converge to tCE2 for ηsync =
0.06, more originate in zone III than zone II, and none originate in zone I.
distribution of initial spin orientations, which we denote by PtCE2. For simplicity,
we assume ŝ to be isotropically distributed (see Section 3.6 for discussions con-
cerning impact of more physically realistic distributions of ŝ). The bottom panel
of Fig. 3.20 shows PtCE2 for I = 20◦ as a function of ηsync. We see that, e.g., tCE2
with a large obliquity (∼ 70◦) can be reached with substantial probability (≳ 50%).
When ηsync ≪ 1 and Ωs,i ≳ n, an approximate analytical formula for PtCE2 can
91
Figure 3.21: Same as Fig. 3.20 but for I = 5◦.
be obtained (see Appendix√B.2): [ ]
≃ 4 η
√
sync sin I 3
PtCE2 n/Ωs,i+ ( √ ) . (3.47)
π 2 1+ n/Ωs,i
Eq. (3.47) is shown in Figs. 3.20–3.21 as the red dashed lines; it agrees well with
the numerical results (red dots) for ηsync ≲ 0.4. To illu[strate the] predicted values
of PtCE2 for small ηsync, we display PtCE2 for ηsync ∈ 10−4,0.4 for both I = 20◦
and I = 5◦ in Fig. 3.22. Note that for η ≤ 10−2sync , numerical results for PtCE2 are
difficult to obtain, as the integration of Eqs. (3.34)–(3.36) slows down dramatically
due to the rapid precession of ŝ about l̂.
92
Figure 3.22: PtCE2 as a function of η ◦ ◦sync for I = 5 and I = 20 shown on a log-
log plot, to emphasize the scaling at small ηsync. The crosses are the results of
numerical integrations as shown in Figs. 3.20–3.21, the solid lines are Eq. (3.47)
for Ωs,i = 10n and the dashed lines are for Ωs,i = 3n.
93
3.5 Applications
3.5.1 Obliquities of Super-Earths with Exterior Companions
Consider a system consisting of an inner Super-Earth (SE) with semi-major axis
a≲ 0.5 AU and an exterior companion. For concreteness, we assume the compan-
ion (with mass mp) to be a cold Jupiter (CJ) with ap ≳ 1 AU. Such systems are
quite abundant [Zhu and Wu, 2018, Bryan et al., 2019]. A phase of giant impacts
may occur in the formation of such SEs [Inamdar and Schlichting, 2015, Izidoro
et al., 2017], leading to a wide range of initial obliquities for the SEs. We are
interested in the “final” obliquities of the SEs driven by tidal dissipation.
For typical SE parameters, the spin evolution timescale due to tidal dissipation
is given by ( )( )( )
1 1 1 2k /Q M 3/2
( )
m −1≃ 2 ⋆
ts 3×( 107 y)r 4k 1
R 3 ( a )0−3 M⊙ 4M⊕× −9/2 . (3.48)
2R⊕ 0.4 AU
This occurs well within the age of SE-CJ systems. On the other hand, the orbital
evolution of the SE occurs on th(e ti)mes(cale [e.g. La)i, 2012]
ȧ 3k 5− = 2 M⋆ R Ωn 1− s cosθ
a Q m a( n )( )( )
≃ 1 −Ωs 2k2/Q M
3/2
⋆
7×( 1014 ) 1 cosθyr n 10−3 M⊙
m −1
( )
R 5 ( a )× −13/2 . (3.49)
4M⊕ 2R⊕ 0.4 AU
Thus, a does not evolve within the age of the SE-CJ system (for a≳ 0.06 AU), and
we shall treat a as a constant in this subsection (but see Sections 3.5.2–3.5.3).
94
With typical SE-CJ parameters(, Eq).((3.39))(can b)e(evalu)ated:
k m −2 ( )= 0.303cos I p m M⋆ a 6ηsync ( ) k( q M) J 4M⊕ M⊙ 0.4 AUap −3 −3× R . (3.50)
5 AU 2R⊕
We see from Figs. 3.20 and 3.21 that this value of ηsync can lead to a high-obliquity
tCE2 with significant probability, assuming the SE has a wide range of initial
obliquities. In addition, Eq. (3.43() sh)o(ws th)at (tCE2 i)s stable if ts ≳ ts,c, where
1 sin I cos2 I k m 3/2 m 1/2= p
ts,c 3(×105)yr (kq MJ 4M⊕−3/2 ( )
× M⋆ a
)6 ( a )p −9/2 R −3/2 . (3.51)
M⊙ 0.4 AU 5 AU 2R(⊕ )
In Fig. 3.23, we show the value of ηsync in the regions of a,ap parameter space
that satisfy the stability condition for tCE2. We see that a generous portion of
parameter space is able to generate and sustain SEs in stable tCE2 with signifi-
cant obliquities. In summary, we predict that a large fraction of SEs with exterior
CJ companions can have long-lived, significant obliquities (≳ 60◦) due to being
trapped in tCE2.
3.5.2 Formation of Ultra-short-period Planet Formation via
Obliquity Tides
Ultra-short period planets (USPs), Earth-sized planets with sub-day periods, con-
stitute a statistically distinct subsample of Kepler planets [e.g. Winn et al., 2018,
Dai et al., 2018]. It is generally thought that USPs evolved from close-in SEs
through orbital decay, driven by tidal dissipation in their host stars [Lee and Chi-
ang, 2017] or in the planets [Petrovich et al., 2019, Pu and Lai, 2019]. In particu-
lar, Pu and Lai [2019] showed that a “low-eccentricity migration” mechanism can
95
Figure 3.23: Depiction of the values of ηsync for the Super Earth + Cold Jupiter
systems as a function of a and ap for I = 20◦ (top) and I = 5◦ (bottom). The SE is
taken to have m= 4M⊕ and R = 2R⊕ while the CJ is taken to have mp = MJ, and
we have taken k≈ kq for the SE. We only show the regions satisfying ts ≥ ts,c (the
stability condition for tCE2; Eqs. 3.48–3.51). The line satisfying ηsync = ηc (Eq. 3.7)
is shown as the black dashed line. Systems with ηsync ≳ 0.1 have appreciable
probabilities of being captured in permanent tCE2 with significant obliquities (see
Figs. 3.20–3.22).
96
successfully produce USPs with the observed properties. In this scenario, USPs
evolve from a subset of SE systems: a low-mass planet with an initial period of
a few days maintains a small but finite eccentricity due to secular forcings from
exterior companion planets (SEs or sub-Neptunes) and evolve to become a USP
due to orbital decay driven by tidal dissipation.
Millholland and Spalding [2020] proposed an alternative formation mecha-
nism of USPs based on obliquity tides (instead of eccentricity tides as in Pu and
Lai, 2019). This mechanism consists of three stages:
• A proto-USP (with two external companions) is assumed to be rapidly cap-
tured into CS2 with appreciable obliquity and a pseudo-synchronous spin
rate.
• The inner planet undergoes runaway tidal migration as a result of the de-
creasing semi-major axis and increasing obliquity while following CS2.
• The inward migration stalls when the tidal torque becomes sufficiently strong
to destroy CS2.
Here, we evaluate the viability of the obliquity-driven migration scenario for
USPs using the general results presented earlier in this paper.
First, the spin evoluti(on t)i(mescale)(for ty)pica(l pr)oto-USP parameters is
1 = 1 1 2k /Q M
3/2 ρ −1 ( )2 ⋆ a −9/2 . (3.52)
ts 1200 yr 4k 10−3 M⊙ ρ⊕ 0.035 AU
where ρ is the density of the proto-USP, ρ⊕ is the density of the Earth, and we
have adopted the (approximately) largest possible value for a (the semi-major axis
of the proto-SE) to ensure that orbital decay can happen within the age of the
system (see Eq. 3.56). This is much shorter than the age of the system, and so the
proto-USP can quickly evolve into one of the stable tCE (either tCE1 or tCE2).
97
Next, to determine which tCE the planet evolves into, we need to evaluate
ηsync (see Eq. 3.39). For simplicity, we consider the case where the proto-USP is
surrounded by a single external planetary companion with ap ≳ a (typical of Ke-
pler multi-planet systems) but with Lp ≫ L (this condition can easily be relaxed;
see Section 3.5.3). To account for such a close-by companion, Eq. (3.4) must be
modified to [see e.g. Lai and Pu, 2017]:
3m ( )a 3p
ωlp = nf (α), (3.53)4M⋆ ap
where α= a/ap and
b(1)≡ 3/2(α) ≈ + 15 2+ 175f (α) 1 α α4 . . . (3.54)
3α 8 64
with b(1)3/2 the Laplace coefficient. With this modification, ηsync (Eq. 3.39) is given
by ( )( )
k ρ ( a )= 3ηsync 0.0(11 f (α)m )(kq ρ)⊕3 ( 0.0)
cos I
35 AU
× p 1.3a M
−2
⋆ , (3.55)
10M⊕ ap M⊙
where we have normalized ap/a to 1.3 (corresponding to a period ratio Pp/P = 1.5),
for which f (α)≈ 5.5. As k/kq ∼ 1 (see footnote 1) for the close-in proto-USP, we have
ηsync ≲ 0.06, much less than ηc ∼ 1 under most conditions5. As such, if the initial
planetary obliquity is prograde, the planet is guaranteed to evolve into tCE1, and
not tCE2 (see Figs. 3.15, 3.18–3.19). If we assume instead a randomly oriented
initial planetary spin, Figs. 3.22–3.20 suggest that the probability of capture into
tCE2 is small (≲ 20%). A more sophisticated calculation including the effect of a
third planet does not greatly modify these results.
5One can make ηsync larger by choosing a larger initial value for a, e.g. a = 0.05 AU. However,
the planet would not be able to experience orbital decay for such a large value, see Eq. (3.56). Also
note that Kepler systems of SEs have adjacent period ratios in the range of 1.3–4 [Fabrycky et al.,
2014], corresponding to semi-major axis ratios of 1.2–2.5.
98
The second stage of the proposed mechanism, runaway inward migration after
attaining tCE2, requires that the initial orbital decay timescale be sufficiently
fast. Evaluating Eq. (3.49) for the(relevant ph)y(sical pa)r(amet)ers, we find
− ȧ = 1 Ω 2k /Q M
3/2
s 2 ⋆
×( 8) (1− ) (cosθa 8 10 yr n 10−3 M⊙m −1 R 5× a )−13/2 . (3.56)
M⊕ R⊕ 0.035 AU
For ηsync ≪ ηc, Eqs. (3.41) imply that Ωs cosθ/n ≪ 1 in tCE2, so indeed the orbit
of the proto-USP is able to decay within the lifetime of the system. On the other
hand, in tCE1, ωs ≈ n and cosθ ≃ 1−η2sync sin2 I/2, so ȧ/a is suppressed by a factor
of ∼ η2sync sin2 I. This shows that a proto-USP in tCE1 is unable to initiate runaway
orbital decay within the age of the system. Note that this constraint also implies
ηsync (Eq. 3.55) cannot be increased by considering proto-USPs with larger values
of a, as the initial orbital decay will become too slow.
Finally, we compute the orbital separation at which tCE2 becomes unstable
when the tidal alignment torque is too strong. Evaluating Eq. (3.43), we find that
tCE2 breaks (ts ≲ ts,c) w[hen the ]semi(-major axis is smaller than abreak, where
≃ k
1/18 )
q 2k 1/92 ( 2 )a −1/9break (3 sin I cos Ik f 3(α) Q
M2
)1/6
× ⋆ ( )Ra 1/2
mpm ( p)
2k /Q 1/9
( )
≃ 2 ( 2 )−1/9 M 1/30.0(28 AU sin I cos I
⋆
10−3 M⊙
× m
)−1/6 ( )
p ρ
−1/6 ( a )p 1/2 , (3.57)
10M⊕ ρ⊕ 0.05 AU
where we have used k ∼ kq ∼ 0.4 and α= 0.028/0.05. Once the system exits tCE2,
it rapidly evolves to tCE1, in which orbital decay is severely suppressed (Eq. 3.56).
This final orbital separation does not qualify as a USP (P ≲ day). To reduce abreak
to 0.0195 AU (corresponding to a 1 day orbital period for M⋆ = 1M⊙) would require
the value of a /m1/3p p to be ∼ 2 times smaller than that adopted in Eq. (3.57) (e.g. for
99
mp to be larger by a factor of 8 for the same ap). Note that observed USPs almost
always have ap/a≳ 3 [Steffen and Farr, 2013, Winn et al., 2018].
In summary, our results suggest that only proto-USPs with large primordial
obliquities have a nonzero probability of evolving into tCE2 initially6. More impor-
tantly, proto-USPs that successfully initiate runaway tidal migration after reach-
ing tCE2 will likely cease their inward migration before becoming a USP.
3.5.3 Orbital decay of WASP-12b Driven by Obliquity Tides
WASP-12b is a hot Jupiter (HJ) with mass m = 1.41MJ and radius R = 1.89RJ
orbiting a host star (with mass M⋆ = 1.36M⊙ and radius R⋆ = 1.63R⊙) on a P =
1.09 day (a = 0.023 AU) orbit [Hebb et al., 2009, Maciejewski et al., 2013]. Long-
term observations have revealed that its orbit is undergoing decay with P/Ṗ =
−3.2 Myr [Maciejewski et al., 2016, Patra et al., 2017, Patra et al., 2020, Turner
et al., 2021]. Such a rapid orbital decay puts useful constraints on the physics of
tidal dissipation in the host star [e.g. Weinberg et al., 2017, Barker, 2020].
Millholland and Laughlin [2019] considered the possibility that the measured
orbital decay of WASP-12b is caused by tidal dissipation in the HJ trapped in
a high-obliquity CS due to an undetected planetary companion. We now evalu-
ate the plausibility of this scenario. We begin with the planetary spin evolution
timescale, which is given by (se(e Eq).((4.16))): ( )
1 = 1 1 2k2/Q M
3/2
⋆
ts 60(00 yr 4k) (10−6 )1.36M⊙
× m
−1 R 3 ( a )−9/2
. (3.58)
1.41MJ 1.89RJ 0.023 AU
6The probability is small even for isotropic primordial obliquities. This low probability may not
be an issue, as the occurence rate of USPs is only ∼ 0.5% around solar type stars [Sanchis-Ojeda
et al., 2014, Winn et al., 2018].
100
Thus, the spin of WASP-12b has plenty of time to find a tCE. We also wish to
calculate ηsync, but there are two uncertainties: (i) the properties of the hypothet-
ical planet companion (mass mp) to WASP-12b are unknown, and it is likely that
Lp is smaller than L; and (ii) we should evaluate ηsync using the “primordial” /
initial value of a for WASP-12b at the start of its orbital migration, not neces-
sarily its present day value. Concerning (i), we express the precession of l̂ about
J= J ȷ̂ ≡L+Lp, the total angular momentum axis, as
dl̂ = J ( )ωlp l̂× ȷ̂ cos I, (3.59)dt Lp
where ωlp is given by Eq. (3.53). Thus, we see that the precession frequency g in
Sections 3.2–3.4 is changed to (cf. Eq. 3.4) ( )
J 3m a 3
g=− cos I =− p Jωlp nf (α) cos I. (3.60)Lp 4M⋆ ap Lp
Concerning (ii), we use the fiducial values for the initial semi-major axis ai =
0.038 AU and initial semi-major axis ratio ap/ai = 1.29, to be justified a posteriori.
Assuming J/Lp ≃ L/Lp (i.e. (L≫ )Lp(), w)e have2 7/2 ( )
ηsync,i ≃
k m ai ai 3
( ) ( f (αi)co)s I,2kq M⋆ ap R
= m
2 M −2
0.0 ⋆(15 f (αi) 1).4(1MJ )1.36(M⊙a 3 a −7/2 )−3× i p R cos I, (3.61)
0.038 AU 1.29ai 1.89RJ
where we have used k/kq ≃ 1. For the adopted fiducial of ai and ap, αi = ai/ap and
f (αi)≃ 5.
We next work towards justifying these choices of fiducial parameters. There
are four physical and observational constraints on the “WASP-12b + companion”
system (see Fig. 3.24):
(i) The HJ must have had a sufficiently small initial semi-major axis such that
its orbital decay timescale is less than the age of the system. The orbital decay
101
rate is given by ( ) ( )( )
ȧ 1 2k /Q M 3/2
( )−1
− = 2 ⋆ m
a i Gy(r 10−6) ( 1.36M⊙ ) 1.(41MJ5 )
× R ai −13/2 1−Ωs cosθ . (3.62)
1.89RJ 0.038 AU n
Thus, the initial semi-major axis for the HJ cannot exceed 0.038 AU even when
(1−Ωs cosθ/n)≈ 1.
(ii) The exterior planet must be sufficiently massive to keep the HJ in the
high-obliquity tCE2 today, i.e. the tCE2 must be stable under the influence of the
exterior planet. With the amended precession frequency |g| given by Eq. (3.60),
the stability of tCE2 requires (see Eq. 3.43) √
1 J ηsync cos I
≲ωlp cos I sin I
J
J , (3.63)ts Lp 2
where cos IJ ≡ l̂ ·
( )
ȷ̂. U[sing sin IJ = Lp/J sin I ≪ 1 for L≫ Lp,]this yields4/19
ap (kf (α))3/2 2 ( )3/2 Q mpm a 9/2≲ 1/2 sin I cos Ia [k 3q 2k2 M⋆ R
k3 f 3
]
( 2/19α) ( ) ( )4/19 m 8/19≃ 3.5( ) ( s)in I cos
3/2 I
kq 1.41MJ
4/19 ( )
× mp M
−12/19
⋆ 2k /Q −4/192
(80M⊕ ) M⊙( )10−6a 18/19 R −18/19× , (3.64)
0.023 AU 1.89RJ
where in the second equality(, we hav)e used the currently observed values for a,
m, R, and M⋆, and have set k3 f 3/k
2/19
q ≃ 1.
(iii) The RV signal of the exterior planet must be smaller than the residuals
of the published RVs, ∼ 16 m/s [Hebb et al., 2009, Husnoo et al., 2011, Knutson
et al., 2014, Bonomo et al., 2017(]. This)r(equires( a )p −1/2 m )M −1/2 ( )p ⋆ sinpip ≲ 1, (3.65)
0.076 AU 80M⊕ 1.36M⊙ 1/ 2
102
where ip is the line-of-sight inclination angle of mp. Here, we have taken ap to be
the maximum value (3.3×0.023 AU) permitted by Eq. (3.64). For these extreme
values of ap and mp, we still have Lp/L ≃ 0.4, and so L ≫ Lp is satisfied for the
permitted parameter space.
(iv) Finally, we require that the initial orbital configuration of the two planets
be dynamically stable. We use the Hill stability criterion [e.g. Gladman, 1993,
Petit et al., 2020],
p (a +a )(m+m )1/3
a −a > 2 3 p i pp i . (3.66)2 3M⋆
Assuming mp ≪m, this yields
ap > 1.29. (3.67)
ai
The combination of the two constraints in Eqs. (3.62, 3.67) justify the fiducial
parameters used in Eq. (3.64).
We next address the implications of the rather small “initial” ηsync value found
in Eq. (3.61). When evaluating ηsync,i, it is possible that R is larger today than its
“primordial” value (at semi-major axis ai > a) due to inflation induced by increased
stellar irradiation. However, if a smaller value of R is used in Eq. (3.61), the value
of ai must also be decreased such that R5/a13/2i is constant in order to maintain
the same (ȧ/a)i (see Eq. 3.62), which further decreases ηsync,i.
For ηsync,i ∼ 0.075 (corresponding to the fiducial parameters used in Eq. 3.61),
we can infer that prograde primordial obliquities will evolve towards tCE1 (see
Figs. 3.15, 3.20–3.21). On the other hand, if the primordial obliquity of the HJ
is assumed to be isotropically distributed, then Fig. 3.20 suggests that the proba-
bility of entry into tCE2 is ≲ 2(5% ev)en if the perturbing planet is misaligned by
I ∼ I ∼ 20◦J . In reality, sin IJ = Lp/J sin I ≪ sin I for Lp ≪ L, so the probability is
likely much smaller (see Eq. 3.47 with I replaced by IJ).
103
Figure 3.24 illustrates the joint constraints on the possible companion to WASP-
12b and the resulting range of ηsync,i values. These small ηsync,i values suggest
that capture of WASP-12b into the high-obliquity tCE2 is unlikely from either an
isotropic or prograde-favoring initial obliquity distribution, and the observed or-
bital decay of WASP-12b is unlikely to be driven by obliquity tides in the planet.
For obliquity tides to be operating today, we would have to imagine a scenario
where dynamical effects when the WASP-12b system was young may have prefer-
entially generated tCE2-producing systems, i.e. systems with θi ≃ 90◦. While the
scenario considered by Millholland and Batygin [2019] and Su and Lai [2020] with
an exterior, dissipating protoplanetary disk does not directly apply here due to the
slow disk dispersal time scale, a similar effect (decreasing η) can be accomplished
by simultaneous disk-driven migration of an inner HJ and exterior companion.
The exploration of such a scenario in the context of HJ formation is beyond the
scope of this paper.
3.6 Summary and Discussion
We have presented a comprehensive study on the evolution of a planet’s spin
(both magnitude and direction) due to the combined effects of tidal dissipation
and gravitational interaction with an exterior companion/perturber. This paper
extends our previous study [Su and Lai, 2020] of Colombo’s Top (“spin + compan-
ion” system) to include dissipative tidal effects, for which we have adopted the
weak friction theory of the equilibrium tide. Our paper contains several new gen-
eral theoretical results that can be adapted to various situations, as well as three
applications to exoplanetary systems of current interest.
We summarize our general theoretical results and provide a guide to the key
104
Figure 3.24: Constraints on the companion of WASP-12b and the values of ηsync,i
(Eq. 3.61) in the obliquity tidal decay scenario. The right dashed line is from
Eq. (3.64), required for the current system to be locked in a stable tCE2; the left
vertical line is from Eq. (3.67), required for the dynamical stability of the “primor-
dial” system; the red line is from Eq. (3.65). The probability of capture into tCE2
(starting from a isotropic distribution of spin orientation) is proportional to η1/2sync,i
(see Eq. 3.47 with I replaced by IJ, and note that IJ ≪ 1 for the WASP-12b sys-
tem). The plot adopts the largest possible value of ai (= 0.038 AU> 0.023 AU= a);
using a smaller ai would significantly reduce ηsync,i, making capture into tCE2
even less likely.
105
equations and figures as follows:
1. In the presence of a spin-orbit alignment torque (such as that arising from
tidal dissipation), our linear analysis (Section 3.3.2) shows explicitly that
only two of the equilibrium spin orientations (called “Cassini States”, CSs)
are stable and attracting (see Fig. 3.1): the “simple” CS1 (which typically has
a low obliquity) and the “resonant” CS2 (which can have a large obliquity).
The latter arises from the spin-orbit resonance, which occurs when the spin
precession frequency of the planet is comparable to the orbital precession
frequency driven by the companion. However, when the alignment torque
is too strong (or the alignment timescale tal too short), the CSs themselves
can be significantly modified. In particular, when tal is shorter than a criti-
cal value (of order the planet’s orbital precession period; see Eq. 3.13), CS2
becomes destabilized and ceases to exist.
2. We compute the long-term evolution of the planetary spin obliquity driven by
the alignment torque for an arbitrary initial spin orientation (Section 3.3.3).
When neglecting the evolution of the planet’s spin magnitude, which im-
plies that the spin and orbital precession frequencies α, g (see Eqs. 4.4–3.4)
and the ratio η = −g/α are held constant, the asymptotic outcomes of the
obliquity evolution (CS1 or CS2) can be analytically determined from the
initial spin orientation (see Fig. 3.5), and we have obtained a new analytical
expression for the probability of resonance capture into CS2 (Eq. 3.30 and
Fig. 3.7).
3. In general, tidal torques act on both the obliquity and magnitude of the
planetary spin, thus the ratio η = −g/α (which determines the phase-space
structure of the system) evolves in time. Still, there are at most two equi-
librium configurations (spin magnitude and obliquity) that are stable under
106
the effect of tidal dissipation. We call these tidal Cassini Equilibria (tCE;
see Fig. 3.8). The locations of these equilibria are determined by the system
architecture and are parameterized by ηsync (Eq. 3.39), the ratio η evaluated
for Ωs = n (fully synchronized spin rate).
4. We show that if tCE1 exists (which requires ηsync < ηc, where ηc is given
by Eqs. 3.7; Section 3.4.1), which tCE a given initial planetary spin configu-
ration asymptotically evolves towards depends on which of the phase space
zones (see Fig. 3.2) the initial spin orientation belongs to (see Figs. 3.15–
3.17): (i) If the spin originates in zone I, then it generally evolves towards
tCE1 (unless ηsync very near ηc, e.g. see Fig. 3.17); (ii) if the spin originates in
zone II, then it evolves towards tCE2 (which has a nontrivial obliquity); and
(iii) if the spin originates in zone III, the outcome is generally probabilistic.
5. For initial conditions in zone III, the probability of approaching either tCE
can be determined by careful study of the dynamics upon separatrix en-
counter (Sections 3.3.4 and 3.4.3); Figs. 3.18 and 3.19 give two example re-
sults. Assuming that the initial spin orientation is isotropically distributed,
we have computed the overall probability of the system evolving into tCE2
as a function of ηsync: Figs. 3.20 and 3.21 give the results for two different
planet mutual inclinations, and Eq. (3.47) gives an approximate analytical
expression valid for ηsync ≪ 1.
Applying our general theoretical results to three types of exoplanetary sys-
tems, our key findings are (see Section 3.5):
1. We show that over a wide range of parameter space, a super Earth (SE)
with an exterior cold Jupiter companion (or other types of companions with a
similar mp/a3p) has a substantial probability of being trapped in a permanent
107
tCE2 with a significant obliquity, assuming that SEs have a wide range of
primordial obliquities (e.g. due to giant impacts or collisions).
2. We show that, in general, the formation of ultra-short-period planets (USPs)
via runaway orbital decay driven by obliquity tides is difficult due to the low
probability of capture into the high-obliquity tCE2. More importantly, proto-
USPs that happen to be captured into tCE2 and initiate runaway tidal mi-
gration will likely break away from tCE2 and cease their inward migration
before becoming a USP.
3. The hot Jupiter WASP-12b is unlikely to be undergoing enhanced orbital
decay due to obliquity tides, as the capture into tCE2 has a low probability
or requires rather special initial conditions.
Finally, we mention some possible caveats of our study. We have adopted dis-
sipative tidal torques according to the (parameterized) weak friction theory of the
equilibrium tide. Other mechanisms of tidal dissipation may be dominant, de-
pending on the internal property of the planet and the nature of tidal forcing [e.g.
Papaloizou and Ivanov, 2010, Ogilvie, 2014, Storch and Lai, 2013]. We expect that,
with proper parameterization and rescaling, our theoretical results presented in
Sections 3.3–3.4 are largely unaffected by the details of the tidal model. In any
case, a different tidal model is amenable to the same analysis as presented in this
paper: The tCEs can still be found by an analysis similar to that shown in Fig. 3.8,
and the probabilistic outcome of a separatrix encounter can still be solved using
the techniques developed in Sections 3.3.4 and 3.4.3.
Some of results presented in Section 3.4, such as Figs. 3.20–3.22, pertain to
the probabilistic outcomes of an initially isotropic distribution of spin orienta-
tions, assuming that giant impacts or planet collisions effectively randomize a
planet’s primordial spin. More physically accurate distributions can be used in
108
the case of planetary mergers [Li and Lai, 2020] or many smaller impacts [Dones
and Tremaine, 1993]. Figures 3.20–3.21 can be updated accordingly by convolving
any initial obliquity distribution with the tCE2 capture probability distributions,
such as those shown in the right panels of Figs. 3.15–3.17 or the upper panels
of Figs. 3.18–3.19. The qualitative results are unlikely to change, though the de-
tailed probabilities for tCE2 capture can increase (decrease) if the initial obliquity
distribution favors (disfavors) θi ≈ 90◦ compared to the isotropic distribution.
109
CHAPTER 4
DYNAMICS OF COLOMBO’S TOP: NON-TRIVIAL OBLIQUE SPIN
EQUILIBRIA OF SUPER-EARTHS IN MULTI-PLANETARY SYSTEMS
Originally published in:
Yubo Su and Dong Lai. Dynamics of Colombo’s Top: Non-trivial oblique spin equi-
libria of super-Earths in multi-planetary systems. MNRAS, 2022a
4.1 Introduction
The obliquity of a planet, the angle between the spin and orbital axes, plays an
important role in the atmospheric dynamics, climate, and potential habitability
of the planet. For instance, the atmospheric circulation of a planet changes dra-
matically as the obliquity increases beyond 54◦, as the averaged insolation at the
poles becomes greater than that at the equator [Ferreira et al., 2014, Lobo and
Bordoni, 2020]. Variations in insolation over long timescales can also affect the
habitability of an exoplanet [Spiegel et al., 2010, Armstrong et al., 2014]. In the
Solar System, planetary obliquities range from nearly zero for Mercury and 3.1◦
for Jupiter, to 23◦ for Earth and 26.7◦ for Saturn, to 98◦ for Uranus. The obliquities
of exoplanets are challenging to measure, and so far only loose constraints have
been obtained for the obliquities of faraway planetary-mass companions [Bryan
et al., 2020, 2021]. Nevertheless, there are prospects for better constraints on exo-
planetary obliquities in the coming years [Snellen et al., 2014, Bryan et al., 2018,
Seager and Hui, 2002]. Substantial obliquities are of great theoretical interest for
their proposed role in explaining peculiar thermal phase curves of transiting plan-
ets [Adams et al., 2019, Ohno and Zhang, 2019] and various other open dynamical
puzzles [Millholland and Laughlin, 2018, 2019, Millholland and Spalding, 2020,
Su and Lai, 2021].
110
The obliquity of a planet reflects its dynamical history. Some obliquities could
be generated in the earlest phase of planet formation, when/if proto-planetary
disks are turbulent and twisted [Tremaine, 1991, Jennings and Chiang, 2021].
Large obliquities are commonly attributed to giant impacts or planet collisions
as a result of dynamical instabilities of planetary orbits [Safronov and Zvjagina,
1969, Benz et al., 1989, Korycansky et al., 1990, Dones and Tremaine, 1993, Mor-
bidelli et al., 2012, Li and Lai, 2020, Li et al., 2021]. Repeated planet-planet
scatterings could also lead to modest obliquities [Hong et al., 2021, Li, 2021]. Sub-
stantial obliquity excitation can be achieved over long (secular) timescales via
spin-orbit resonances, when the spin precession and orbital (nodal) precession fre-
quencies of the planet become comparable [Ward and Hamilton, 2004, Hamilton
and Ward, 2004, Ward and Canup, 2006, Vokrouhlickỳ and Nesvornỳ, 2015, Mill-
holland and Batygin, 2019, Saillenfest et al., 2020, Su and Lai, 2020, Saillenfest
et al., 2021]. Such resonances are likely responsible for the obliquities of the So-
lar System gas giants and may have also played a role in generating obliquities
of ice giants [Rogoszinski and Hamilton, 2020]. For terrestrial planets, multiple
spin-orbit resonances and their overlaps can make the obliquity vary chaotically
over a wide range [Laskar and Robutel, 1993, Touma and Wisdom, 1993, Correia
et al., 2003].
A large fraction (30−90%) of Sun-like stars host close-in super Earths (SEs),
with radii 1−4R⊕ and orbital distances ≲ 0.5 AU, mostly in multi-planetary (≥ 3)
systems [e.g. Lissauer et al., 2011, Howard et al., 2012, Zhu et al., 2018, Sandford
et al., 2019, Yang et al., 2020, He et al., 2021]. In these systems, the SE orbits are
mildly misaligned with mutual inclinations ∼ 2◦ [Lissauer et al., 2011, Tremaine
and Dong, 2012, Fabrycky et al., 2014], which increase as the number of planets
in the system decreases [Zhu et al., 2018, He et al., 2020]. In addition, ∼ 30–40%
of the SE systems are accompanied by cold Jupiters [CJs; masses ≳ 0.5MJ and
111
semi-major axes ≳ 1 AU Zhu and Wu, 2018, Bryan et al., 2019] with significantly
inclined (≳ 10◦) orbits relative to the SEs [Masuda et al., 2020].
SEs are formed in gaseous protoplanetary disks, and likely have experienced
an earlier phase of giant impacts and collisions following the dispersal of disks
[e.g. Liu et al., 2015b, Inamdar and Schlichting, 2016, Izidoro et al., 2017]. As a
result, the SEs’ initial obliquities are expected to be broadly distributed [Li and
Lai, 2020, Li et al., 2021]. However, due to the proximity of these planets to their
host stars, tidal dissipation can change the planets’ spin rates and orientations
substantially within the age of the planetary system. Indeed, the tidal spin-orbit
alignment timescale is given by ( )( )
Q/2k M −3/2
t 2 ⋆al ≃ (3(0 Myr))( 103)−3 (M⊙ )
× m R a 9/2 , (4.1)
4m⊕ 2R⊕ 0.4 AU
where m, R, k2, Q, and a are the planet’s mass, radius, tidal Love number, tidal
quality factor, and semi-major axis respectively, and M⋆ is the stellar mass. In a
previous paper [Su and Lai, 2021], we have studied the combined effects of spin-
orbit resonance and tidal dissipation in a two-planet system (i.e. a SE with a com-
panion), and showed that the planet’s spin can only evolve into to two possible
long-term equilibria (“Tidal Cassini Equilibria”), one of which can have a signif-
icant obliquity. In this paper, we extend our analysis to three-planet systems
consisting of either three SEs or two SEs and a CJ. In addition to the equilib-
ria analogous to those of the two-planet case, we discover a novel class of oblique
spin equilibria unique to multi-planet systems. Such equilibria can substantially
increase the occurrence rate of oblique SEs in these architectures.
This paper is organized as follows. In Section 4.2, we summarize the evolution
of a SE in a multi-planetary system. In Section 4.3, we introduce a tidal alignment
112
torque that damps the SE’s obliquity and investigate the resulting steady-state be-
havior. In Section 4.4, we evolve both the SE spin rate and orientation according
to weak friction theory of the equilibrium tide. We show that the qualitative dy-
namics are similar to the simpler model studied in Section 4.3. We summarize
and discuss in Section 4.5.
4.2 Spin Equations of Motion
The unit spin vector Ŝ of an oblate planet orbiting a host star precesses around
the planet’s unit angular momentum L̂ following the equation
dŜ = ( · )( × )α Ŝ L̂ Ŝ L̂ , (4.2)
dt
where the characteristic spin-orbit precession fre(que)ncy α is given by
= 3GJ2mR
2M⋆ = 3k
3
q M⋆ R
α Ω
2a3C( Ωs)( )2k m a s
1 k 3/2= q M⋆
15( (4.3)0 yr )k M⊙
m −1
( )
× R
3 ( a )−9/2 Ωs . (4.4)
2M⊕ 1.2R⊕ 0.1 AU n
In Eq. (4.4),Ωs is the rotation rate of the planet, C = kmR2 is its moment of inertia
(with k the normalized moment of inertia), J = k Ω2(R32 q s /Gm) (with k√q a constant)
is its rotation-induced (dimensionless) quadrupole moment, and n≡ GM⋆/a3 is
its mean motion. For a SE, we adopt k∼ kq ∼ 0.3 [e.g. Groten, 2004, Lainey, 2016].
The orbital axis L̂ also evolves in time, precessing and nutating about the
total angular momentum axis of the exoplanetary system, which we denote by
Ĵ. When there are just two planets, this precession is uniform (with rate g and
constant inclination angle between L̂ and Ĵ), and the spin dynamics of the planet
is described by the well-studied “Colombo’s Top” system [Colombo, 1966, Peale,
113
1969, 1974, Ward, 1975, Henrard and Murigande, 1987]. The spin equilibra of
this system are termed “Cassini States” (CSs), and the number of CSs and their
obliquities depend on the ratio η ≡ |g| /α. In the presence of a tidal spin-orbit
alignment torque, up to two equilibria are stable and attracting, as shown in [Su
and Lai, 2021; see also Fabrycky et al., 2007, Levrard et al., 2007, Peale, 2008]:
for η≫ 1, only CS2 is stable, with Ŝ nearly aligned with Ĵ; for η≲ 1, Ŝ can evolve
towards two possible states, the “trivial” CS1 with a small spin-orbit misalignment
angle θsl, or the “resonant” CS2 with significant θsl (which approaches 90◦ for
η≪ 1).
When the SE is surrounded by multiple companions, the precession of L̂ oc-
curs on multiple characteristic frequencies [see Murray and Dermott, 1999]. In
this case, the spin dynamics given by Eq. (4.2) is complex and can lead to chaotic
behavior [e.g. the chaotic obliquity evolution of Mars Laskar and Robutel, 1993,
Touma and Wisdom, 1993]. But what is the final equilibrium state of Ŝ in the
presence of tidal alignment? In this paper, we focus on the case where the SE has
two planetary companions. If the mutual inclinations among the three planets are
small, then the explicit solution for L̂(t) can be written as [Murray and Dermott,
1999]
I ≡ I exp(iΩ[ ) ] [ ]= I(I) exp ig(I)t+ iφ(I√) + I(II) exp ig t+ iφ(II)(II) , (4.5)
L̂=ℜ (I )X̂+ℑ (I )Ŷ+ 1−|I |2 Ẑ. (4.6)
Here, I is the inclination of L̂ relative to Ĵ, I is the complex inclination, the
quantities I(I,II), g(I,II) and φ(I,II) are the amplitudes, frequencies, and phase offsets
of the two inclination modes, indexed by (I) and (II), and the Cartesian coordinate
system XY Z is defined such that Ĵ = Ẑ. Without loss of generality, we denote
mode I as the dominant mode, with I(I) ≥ I(II). For simplicity, we fix φ(I) =φ(II) = 0.
114
4.3 Steady States Under Tidal Alignment Torque.
Since SEs are close to their host stars, tidal torques tend to drive Ŝ towards align-
ment with L̂ and Ωs towards synchronization with the mean motion (see Eq. 4.1).
As the evolution of Ωs also changes α (Eq. 4.4) and the underlying phase-space
structure, we first consider the dynamics when ignoring the spin magnitude evo-
lution. In this case, the planet’s spin orientation experiences an alignment torque,
which we describe by ( )
dŜ = 1 ( )Ŝ× L̂× Ŝ , (4.7)
dt al tal
where tal is given by Eq. (4.1). Note that tal is significantly longer than all preces-
sion timescales in the system.
With two precessional modes for L̂(t), we expect that the tidally stable spin
equilibria (steady states) correspond to the stable, attracting CSs for each mode,
when they exist. In other words, if we denote the CS2 corresponding to mode I by
CS2(I), then we expect that the tidally stable equilibria are among the four CSs:
CS1(I) (if it exists), CS2(I), CS1(II) (if it exists), and CS2(II). The corresponding
CS obliquities θsl ≡ cos−1(Ŝ · L̂) are given by
αcosθsl =−
( )
g(I,II) cos I(I,II)+sin I(I,II) cotθsl cosφsl , (4.8)
where the azimuthal angle φsl of Ŝ around L̂ is φsl = 0 (corresponding to Ŝ and Ĵ
being coplanar but on opposite sides of L̂) for CS1 and φsl = π for CS2 [note that
because of the tidal alignment torque, the actual φsl value is slightly shifted from
0 or π; see Su and Lai, 2021].
To confront this expectation, we integrate Eqs. (4.2), (4.6), and (4.7) numeri-
cally starting from various spin orientations. Figu∣∣ re ∣∣4.1 shows two∣∣ evo∣∣lutionarytrajectories for a system with I(I) = 10◦, I(II) = 1◦, g(I) = 0.1α, and g(II) = 0.01α.
115
Such a system can be realized, for instance, by three SEs with masses M⊕, 3M⊕,
and 3M⊕ and semi-major axes 0.1 AU, 0.15 AU, and 0.4 AU (see the left panels of
Fig. C.1 in the Appendix1). We see that the initially retrograde spin is eventually
captured into a steady state centered around CS1 or CS2 of the dominant mode
(i.e. mode I), with φsl librating around 0 or π, respectively. The small oscillation of
the final θsl is the result of perturbations from mode II.
In addition to CS1(I) and CS2(I), we find that the spin can also settle down
into other equilibria (steady states) with different librating angles. In general, we
define the resonant phase angle
φres ≡φsl− grest. (4.9)
The examples shown in Fig. 4.1 correspond to gres = 0. In Fig. 4.2, we show three
evolutionary trajectories (with three different initial spin orientations) of a system
with the same parameters as in Fig. 4.1 but with g(II) = −α. Such a system can
be realized, for instance, by two warm SEs orbited by a cold Jupiter (see the right
panels of Fig. C.3 in the Appendix where a3 is small). Among these three exam-
ples, the first is captured into a resonance with gres = 0 [i.e. CS2(I)], the second
is captured into a resonance with gres =∆g ≡ g(II)− g(I), and the third is captured
into a resonance with gres =∆g/2.
To explore the regimes under which various resonances are important, we nu-
merically determine [by integrating Eqs. (4.2), (4.6), and (4.7)] the final spin equi-
libria (steady states) for systems with different mode parameters (I(I), I(II), g(I),
and g(II)), starting from all possible initial spin orientations. Fig. 4.3 shows some
examples of such a calculation for I(I) = 10◦, I ◦(II) = 1 , g(I) = −0.1α (the same as
1Note that the mode amplitudes I(I) and I(II) in Fig. C.1 are closer in magnitude than the case
we consider here. We exaggerate the inclination hierarchy for a more intuitive physical picture,
and explore the case where the modes are of comparable amplitudes later in the paper and in
Appendix C.2.
116
Figure 4.1: Two evolutionary trajectories of Ŝ showing capture into mode I reso-
nances (CSs). In both cases, the mode parameters describing the evolution of L̂
(see Eq. 4.6) are I(I) = 10◦, I ◦(II) = 1 , g(I) =−0.1α, and g(II) = 0.1g(I), while the ini-
tial spin orientations differ. In the top group of plots showing capture into CS1(I)
[i.e. Cassini State 1 of mode I], the left four panels show the evolution of the spin
obliquity θsl and the resonant phase angle φres; in this case, φres equals φsl, the
azimuthal angle of Ŝ around L̂, defined so that φsl = 0 corresponds to Ŝ and Ĵ be-
ing coplanar with L̂ but on opposite sides of L̂. The horizontal black dashed line
shows the theoretically predicted obliquity of CS1(I), given by Eq. (4.8). The right
panel shows the final steady-state spin axis projected onto the orbital plane (per-
pendicular to L̂). In these coordinates, L̂ (black dot) is stationary, while Ĵ (green
line) librates with a fixed orientation, and Ŝ is shown in blue. The bottom group
of panels shows the same but for capture into the CS2(I) resonance.
117
Figure 4.2: Three evolutionary trajectories for a system with the same parame-
ters as in Fig. 4.1, except for g(II) = −α = 10g(I). The three examples correspond
to capture into CS2(I) (with φres = φsl), a resonance with φres = φsl−∆gt [correp-
sonding to CS2(II)], and a “mixed mode” resonance with φres = φsl−∆gt/2, where
∆g= g(II)− g(I). In the bottom two groups of plots, φsl is not the resonant angle, so
the spin axis encircles L̂ in the top-right plots. For these two cases, we also display
in the bottom-right panels the projection of the steady-state spin axis onto the L̂
plane but with φres as the azimuthal angle. In these two panels, Ĵ encircles L̂ (as
shown by the green ring), but the spin can be seen not to encircle L̂, indicating that
φres is indeed librating. Finally, for the mixed-mode example (bottom group), the
vertical red lines in the bottom-middle panel are separated by 2π/ |gres| = 4π/ |∆g|,
denoting the period of oscillation in θsl.
118
in Figs. 4.1–4.2), but with the four different values of g(II) = {0.1,2.5,3.5,10}× g(I).
We identify three qualitatively different regimes:
∣
• When ∣ ∣ ∣ ∣g(II)∣ ≪ ∣g(I)∣ (top-left panel of Fig. 4.3), mode II serves as a slow,
small-amplitude perturbation to the dominant mode I, and the spin evo-
lution is very similar to that studied in Su and Lai [2021]: prograde ini-
tial conditions (ICs) outside of the mode-I resonance evolve towards CS2(I),
ICs inside the resonance evolve to CS2(I), and retrograde ICs outside of the
mode-I∣reso∣nan∣ce r∣each one of the two probabilistically.
• When ∣g(II)∣ ∼ ∣g(I)∣ (see the top-right and bottom-left panels of Fig. 4.3),
the resonances corresponding to the two modes overlap, chaotic obliquity
evolution occurs [see Touma and Wisdom, 1993, Laskar and Robutel, 1993],
and we expect that CS2(I) becomes less stable2. Indeed we see that in this
regime, fewer ICs evolve into the high-obliquity CS2(I) equilibrium of the
domina∣nt m∣ode∣I, a∣nd most ICs lead to the low-obliquity CS1(I).
• When ∣g(II)∣≫ ∣g(I)∣ (see the bottom-right panel of Fig. 4.3), the separatrix
for mode II does not exist, we see that all ICs inside the separatrix of mode
I again converge successfully to CS2(I), and CS2(II) becomes the preferred
low-obliquity equilibrium. Additionally, a narrow band of ICs with cosθsl,0 ∼
0.6 and some other scattered ICs with cosθsl,0 ≲ 0.6 evolve to the mixed-
mode equilibrium with gres = ∆g/2, which has θ ◦eq ≈ 60 . A second mixed-
mode resonance with gres =∆g/3 is also observed (with θeq ≈ 67◦) for a sparse
set of ICs.
Towards a better understanding of how systems are captured into these mixed-
2An more precise resonance overlap condition can be obtained by comparing the separatrix
widths, as the mode-II resonance is much narrower even when g(I) = g(II). Such a condition would
require g(I) sin I(I) ∼ g(II) sin I(II). Thus, the onset of chaos due to resonance overlap occurs some-
where in the range I(II)/I(I) ≲ g(II)/g(I) ≲ 1.
119
Figure 4.3: Asymptotic outcomes o∣f spin evolution driven by tidal alignmenttorque for different initial spin orient∣atio∣ns (described by θsl,0 and φsl,0) for a sys-tem with I = 10◦(I) , I = 1◦(II) , α = 10 g(I)∣, and g(II) = {0.1,2.5,3.5,10}× g(I) in the
top-left, top-right, bottom-left, and bottom-right panels respectively (as labeled).
Each dot represents an initial spin orientation, and the coloring of the dot indi-
cates which Cassini State (CS) and which mode (legend) the system evolves into.
The obliquity and φsl values of the mode-I CSs are labeled as the circled stars,
with the same colors as in the legend. The obliquities of the other equilibria are
labeled as the boxed stars at the left edges of the right-bottom panel, with the
same colors as in the legend; a small, arbitrary offset in φsl is added to reflect the
fact that these equilibria do not have fixed φsl values. The separatrices for the
mode I and mode II resonances are given in the black and blue lines respectively.
120
Figure 4.4: Same as the bottom right panel of Fig. 4.3 but zoomed-in to a narrow
range of initial obliquities near θeq ≈ 57◦ (horizontal dashed line) for the gres =
∆g/2 mixed mode equilibrium as predicted by Eq. (4.15). The horizontal dash-
dotted lines indicate the amplitude of oscillation of the mode (see Fig. 4.2). It is
clear that there are two basins of attraction for this mixed-mode resonance near
φsl,0 = 0 and φsl,0 = 180◦.
mode equilibria, we numerically calculate the “basin of attraction” by repeating
the procedure for producing Fig. 4.3 but instead use a fine grid of ICs with θsl,0
near the average obliquity of the equilibrium. This results in a “zoomed-in” ver-
sion of the bottom-right panel of in Fig. 4.3 and doubles as a numerical stability
analysis of the equilibrium. Figure 4.4 shows the result of this procedure applied
to the gres =∆g/2 resonance, where we have zoomed in to θ ◦sl,0 near the θeq ≈ 60
associated with the resonance. We see(that th)e resonance is reached consistently
from some well-defined regions in the θsl,φsl space.
Figure 4.5 summarizes the equilibrium obliquities θeq of various resonances
achieved for the system depicted in the bottom-right panel of Fig. 4.3. For a tra-
jectory that reaches a (p∣∣arti∣∣cular )equilibrium, we compute θeq = 〈θsl〉 by averagingover the last 100/max g(I) , |∆g〈| of〉the integration. We obtain the corresponding
“resonance” frequency by gres ≃ φ̇sl .
121
In fact, the relation between θeq and gres can be described analytically. We
consider the equation of motion in the rotating frame where L̂= ẑ is constant and
Ĵ lies in the xz plane((i.e.)Ĵ=−sin Ix̂+cos Iẑ):
dŜ = ( · )( × )+ × ( + )α Ŝ L̂ Ŝ L̂ Ŝ Ω̇Ĵ İŷ . (4.10)
( dt rot )
Let Ŝ= sinθsl cosφslx̂+sinφslŷ +cosθslẑ. The evolution of φsl then follows
dφsl = − ( )αcosθsl− Ω̇ cos I+sin I cotθsl cosφdt sl
− İ cotθsl sinφsl. (4.11)
Note that the single-mode CSs satisfy Eq. (4.11) where Ω̇ = g, İ = 0, and φsl is
either equal to 0◦ or 180◦ (Eq. 4.8). For the general, two-mode problem, if φres =
φsl− grest is a resonant an〈gle, the〉n it〈must s〉atisfy
dφres = dφsl − gres = 0, (4.12)dt dt
where the angle brackets〈 denot〉e an〈 avera〉ge over a libration period. Since φsl
circulates when gres ≠ 0, cosφsl = sinφsl = 0. Furthermore, if I(II)/I(I) ≡ ϵ≪ 1,
we can expand Eq. (4.5) to obtain
Ω̇= g(I)(+
[ ]
∆gϵcos(∆gt)+O ϵ2[ ]) , (4.13)
I = I(I) 1+ϵcos(∆gt)+O ϵ2 , (4.14)
where ∆g≡ g(II)− g(I). To leading order, we have Ω̇≃ g(I) and I ≃ I(I), so Eq. (4.11)
reduces to
αcosθeq ≃−g(I) cos I(I)− gres. (4.15)
This is Eq. (4.15) in the main text, and is shown in Fig. 4.5. Good agreement be-
tween the analytic expression and numerical results is observed. Note that setting
gres =∆g in Eq. (4.15) does not yield the mode II CSs, as the mode I inclination is
still being used, and the φsl terms are averaged out in the mixed mode calculation
while being nonzero in the CS obliquity calculation.
122
Figure 4.5: “Equilibrium” obliquity as a function ∣of t∣he resonant frequency
(Eq. 4.9) for a system with I ◦ ◦(I) = 10 , I(II) = 1 , α = 10 ∣g(I)∣, and g(II) = 10g(I) (cor-
responding to the bottom-right panel of Fig. 4.3). The green line is the analytical
result given by Eq. (4.15). The error bars denote the amplitude of oscillation of
θsl when the spin has reached a steady state (e.g. Figs. 4.1–4.2). The blue crosses
denote the CSs for the two inclination modes.
The four systems shown in Fig. 4.3 demonstrate that the characteristic spin
evolution depends strongly on the ratio g(II)/g(I). To understand the transition
between these different regimes, we vary g(II) over a wide range of values (while
keeping the other parameters the same as in Fig. 4.3). For each g(II), we nu-
merically determine the steady-state (equilibrium) obliquities and compute the
probability of reaching each equilibrium by evolving 3000 initial spin orienta-
tions drawn randomly from an isotropic distribution. Figure 4.6 shows the result.
Two trends can be see∣∣n: th∣∣ e probability of lon∣∣g-live∣∣d ca∣∣ptu∣∣re in∣∣to th∣∣e C∣∣S2(I∣∣) res-onance decreases as g(II) is increased from g(II) ≪ g(I) to g(II) ∼ g(I) , and
m∣∣ ixe∣∣d-m∣∣ode ∣resonances become significant, though non-dominant, outcomes forg(II) ≫ g(I)∣.
Having discussed how the spin evolution changes when g(II) is varied, we now
explore the effect of different values of I(II). In Fig. 4.7, we display the final out-
123
Figure 4.6: Final outcomes of spin evolution under tidal alignment torque f∣or a∣
3-planet system with inclination mode parameters I = 10◦ ◦(I) , I(II) = 1 , α= 10 ∣g(I)∣
(same as Figs. 4.1–4.5) and varying g(II)/g(I). The top panel shows the probability
of ∣each∣ of the possible steady-state outcomes for 3000 initial spin orientationssam∣ pled∣ from an isotropic distribution. The vertical dashed line shows the valueof g(II) above which CS1(II) no longer exists. The bottom panel shows the final
equilibrium obliquities (open black circles) for each g(II)/g(I). For the mixed-mode
resonances (gres ≠ 0,∆g), the equilibrium obliquities are given by Eq. (4.15) and
are shown as the solid green and purple lines for the labeled values of gres. The
other lines are the equilibrium θeq for “pure” CSs (as labeled), whose equilibria
satisfy Eq. (4.8). Not all observed mixed-mode resonances are plotted (e.g. for
g(II) = 7.5g(I), there is an outcome with gres/∆g= 3/4).
124
Figure 4.7: Same as Fig. 4.3 but for I ◦(II) = 3 .
comes as a function of the initial spin orientation for the same g(II) values as in
Fig. 4.3, but for I ◦(II) = 3 . Comparing the bottom-left panels of Figs. 4.3 and 4.7
(with g(II) = 3.5g(I)), we find that the favored low-obliquity CS changes from CS1(I)
to CS1(II) when using I = 3◦(II) . In both cases, CS2(I) is destabilized such that most
initial conditions converge to the low-obliquity CS, either CS1(I) or CS1(II). In the
bottom-right panel, we find that many initial conditions converge to other mixed
modes than the gres =∆g/2 mode. The values of gres observed for the system are
shown in Fig. 4.8, where we find that many low-order rational multiples of gres/∆g
are obtained. While the amplitude of oscillation in the final θsl is substantial (and
larger than in Fig. 4.5), we find that the predictions of Eq. (4.15) are consistent up
to the range of oscillation of θsl. In Fig. 4.9, we summarize the outcomes of spin
evolution as a function of g(II)/g(I) for I ◦(II) = 3 . We identify the same two qualita-
tive trends as seen in Fig. 4.6: the∣∣ ins∣∣tabi∣∣lity ∣∣of CS2(I) when g(II) ∼ g(I) and theappearance of mixed modes when g(II) ≫ g(I) .
125
Figure 4.8: Same as Fig. 4.5 but for I = 3◦(II) .
Figure 4.9: Same as Fig. 4.6 but for I(II) = 3◦. Note that the agreement of the black
open circles with the theoretical obliquities in the bottom panel is slightly worse
than in Fig. 4.6 but still well within the ranges of oscillation of the obliquities (see
Fig. 4.8 for the characteristic ranges).
126
4.3.1 Summary of Various Outcomes
In summary, the spin evolution in a 3-planet system driven by a tidal alignment
torque depends largely on the frequency of the smaller-amplitude mode, ∣g(II),∣com-
pared to that of the larger-amplitude mode, g(I). In the regime where ∣g(I)∣ ≲ α,
we find that:
∣ ∣ ∣ ∣
• When ∣g(II)∣≪ ∣g(I)∣, the low-amplitude and slow (II) mode does not signifi-
cantly affect the spin evolution, and the results of [Su and Lai, 2021] are re-
covered. The two possible outcomes are the tidally stable CS1(I) (generally
low obliquity) and CS2(I) (generally high obliquity). Prograde initial spins
converge to CS1(I), spins inside the mode I resonance converge to CS2(I),
and retrograde initial spins attain one of these two tidally stable CSs proba-
bilistically. For the fiducial parameters used for Fig. 4.6, approximately 20%
of systems are trapped in the high-obliquity CS2(I).
• When g(II) ∼ g(I), CS2(I) is increasingly difficult to attain due to the interact-
ing mode I and mode II resonances, and the probability of attaining CS2(I)
can be strongly suppressed (see Fig. 4.6, where the probability of a high-
obliqui∣ty ou∣tco∣me ∣goes to zero for g(II)/g(I) = 3.5).
• When ∣g(II)∣≳ ∣g(I)∣, there are three classes of outcomes. The highest-obliquity
outcome is still CS2(I), and is attained for initial conditions inside the mode
I resonance (separatrix; see Fig. 4.5). The lowest-obliquity outcome is gener-
ally CS2(II)3 and is the most favored outcome (see Fig. 4.6). The third possi-
ble outcome are mixed modes with gres/∆g a low∣ -ord∣er r∣ation∣ al number (see
Eq. 4.9). These mixed modes only appear for ∣g(II)∣≫ ∣g(I)∣, and generally
have obliquities between those of CS2(I) and CS2(II) (see Fig. 4.3). For the
3This may not be the case when I(II) ≲ I(I) while g(II) ≫ g(I); see Fig. C.7 in the Appendix
127
fiducial parameters used for Fig. 4.6, the mixed-mode resonances increase
the probability of obtaining a substantial (≳ 45◦) obliquity from ∼ 20% to
∼ 30%.
4.4 Weak Tidal Friction
We now briefly discuss the spin evolution of the system incorporating the full tidal
effects. In the weak friction theory of the equilibrium tide, the spin orientation and
frequency jointly evolve (follo)wing [Ale[xander, 197]3, Hut, 1981, Lai, 2012]
dŜ = 1 2n − ( ) ( )( ) [ Ŝ · L̂ Ŝ× L̂× Ŝ] , (4.16)dt tide ts Ωs
1 dΩs = 1 2n ( · )− − ( )Ŝ L̂ 1 Ŝ · L̂ 2 , (4.17)
Ωs dt tide ts Ωs
where ( )( )
1 ≡ 1 3k2 M
3
⋆ R n, (4.18)
ts 4k Q m a
(see Eq. 4.1, but with 4k= 1).
Since α∝ Ωs evolves in time, we describe the spin-orbit coupling by the pa-
rameter
αsync ≡ [α]Ωs=n . ∣ (∣4.19)
∣∣To f∣∣acilitate comparison with the previous results, we use αsync = 10
∣g(I)∣ and
g(I) ts = 300. Note that for the physical parameters used in Eqs. (4.4) and (4.18),
g 4(I)ts ∼ 10 ; we choose a faster tidal timescale to accelerate our numerical inte-
grations. The initial spin is fixed Ωs,0 = 3n. We then integrate Eqs. (4.2), (4.6),
and (4.16–4.17) starting from various initial spin orientations and determine the
final outcomes. Figure 4.10 shows the results for a few select values of g(II) and
for I(II) = 3◦. Similar behaviors to Figs. 4.3 and 4.7 are observed. The probabilities
128
Figure 4.10: Similar∣∣ to ∣∣Figs. 4.7 but including full tidal effects on the planet’sspin, with αsync = 10 g(I) (Eq. 4.19), I ◦(II)3 , and initial spin Ωs,0 = 3n. In the three
panels, g(II) = {0.1,2,10}×g(I) respectively. Note that the plotted separatrices (blue
and black lines) use the initial value of α.
and obliquities of the various equilibria are shown in Fig. 4.11. Note that each
equilibrium obliquity has a corresponding equilibrium rotation rate, given by
Ωeq = 2cosθeq+ . (4.20)n 1 cos2θeq
The probabilities shown in Fig. 4.11 exhibit qualitative trends that are quite sim-
ilar to
when ∣∣
thos∣∣e se∣∣en ∣∣for the evolution driven by the tidal alignment torque alone:g(II) ≪ g(I) , the results of [Su and Lai, 2021] are recovered; when g(II) ∼
∣∣g(I), ∣∣the∣∣pro∣∣bability of attaining CS2(I) is significantly suppressed; and wheng(II) ≫ g(I) , mixed modes appear.
4.5 Summary and Discussion
In this work, we have shown that the planetary spins in compact systems of mul-
tiple super Earths (SEs), possibly with an outer cold Jupiter companion, can be
129
Figure 4.11: Similar to Fig. 4.9 but with weak tidal friction.
trapped into a number of spin-orbit resonances when evolving under tidal dissi-
pation, either via a tidal alignment torque (Section 4.3) or via weak tidal fric-
tion (Section 4.4). In addition to the well-understood tidally-stable Cassini States
associated with each of the orbital precession modes, we have also discovered a
new class of “mixed mode” spin-orbit resonances that yield substantial obliquities.
These additional resonances constitute a significant fraction of the final states
of tidal evolution if the planet’s initial spin orientation is broadly distributed, a
likely outcome for planets that have experienced an early phase of collisions or
giant impacts. For instance, for the fiducial system parameters shown in Fig. 4.6,
these mixed-mode equilibria increase the probability that a planet retains a sub-
stantial (≳ 45◦) obliquity from 20% to 30%. A large equilibrium obliquity has a
significant influence on the planet’s insolation and climate. For planetary systems
surrounding cooler stars (M dwarfs), the SEs (or Earth-mass planets) studied in
this work lie in the habitable zone [e.g. Dressing et al., 2017, Gillon et al., 2017],
130
and the nontrivial obliquity can impact the habitability of such planets.
In a broader sense, the mixed-mode equilibria discovered in our study repre-
sent a novel astrophysical example of subharmonic responses in parametrically
driven nonlinear oscillators. In equilibrium, the planetary obliquity oscillates
with a period that is an integer multiple of the driving period 2π/ |∆g| (see Ap-
pendix C.2 for further discussion). Such subharmonic responses are often seen in
nonlinear oscillators [e.g. in the classic van der Pol and Duffing equations Leven-
son, 1949, Flaherty and Hoppensteadt, 1978, Hayashi, 2014].
∣ ∣
Throughout this paper, we have adopted fiducial parameters where ∣∣g(I)∣∣≲ α,
which is generally expected for the SE systems being studied. If instead ∣g(I)∣≳α,
then there is no resonance fo∣∣r th∣∣e dominant (larger-amplitude) mode I. There arethen a few possible cases: if g(II) ≲α, mode II is both slower than mode I and has
a smaller amplitu∣∣de, s∣∣o it will not affect the mode I dynamics significantly. Onthe other hand, if g(II) ≳α, then mode II also has no resonance, and both CS2(I)
and CS2(II) have low obliquities, implying that the system will always settle into
a low-obliquity state.
131
CHAPTER 5
PHYSICS OF TIDAL DISSIPATION IN EARLY-TYPE STARS AND WHITE
DWARFS: HYDRODYNAMICAL SIMULATIONS OF INTERNAL GRAVITY
WAVE BREAKING IN STELLAR ENVELOPES
Originally published in:
Yubo Su, Daniel Lecoanet, and Dong Lai. Physics of tidal dissipation in early-
type stars and white dwarfs: Hydrodynamical simulations of internal gravity wave
breaking in stellar envelopes. MNRAS, 495(1):1239–1251, May 2020. ISSN 0035-
8711. doi: 10.1093/mnras/staa1306
5.1 Introduction
The physical processes responsible for tidal evolution in close binaries often in-
volve the excitation and dissipation of internal waves, going beyond the “weak
friction” of equilibrium tides [see Ogilvie, 2014, for a review]. In particular, in-
ternal gravity waves (IGWs), arising from buoyancy of stratified stellar fluid, play
an important role in several types of binary systems. In solar-type stars with
radiative cores and convective envelopes, IGWs are excited by tidal forcing at
the radiative-convective boundary and propagate inward; as the wave amplitude
grows due to geometric focusing, nonlinear effects can lead to efficient damping
of the wave [Goodman and Dickson, 1998, Barker and Ogilvie, 2010, Essick and
Weinberg, 2015]. In early-type main-sequence stars, with convective cores and ra-
diative envelopes, IGWs are similarly excited at the convective-radiative interface
but travel toward the stellar surface; nonlinearity develops as the wave ampli-
tude grows, leading to efficient dissipation [Zahn, 1975, 1977]. As the outgoing
wave deposits its angular momentum to the stellar surface layer, a critical layer
may form and the star is expected to synchronize from outside-in [Goldreich and
Nicholson, 1989].
132
Tidal dissipation can also play an important role in compact double white
dwarf (WD) binary systems (with orbital periods in the range of minutes to hours).
Such binaries may produce a variety of exotic astrophysical systems and phenom-
ena, ranging from isolated sdB/sdO stars, R CrB stars, AM CVn binaries, high-
mass neutron stars and magnetars (created by the accretion-induced collapse of
merging WDs), and various optical transients (underluminous supernovae, Ca-
rich fast transients, and type Ia supernovae) [e.g. Livio and Mazzali, 2018, Toloza
et al., 2019]. The outcomes of WD mergers depend on the WD masses and com-
position, but tidal dissipation can strongly affect the pre-merger conditions of the
WDs and therefore the merger outcomes. Tidal dissipation may also influence the
evolution of eccentric WD-massive black hole binaries prior to the eventual tidal
disruption of the WD [Vick et al., 2017].
Recent studies have identified nonlinear dissipation of IGWs as the key tidal
process in compact WD binaries [Fuller and Lai, 2012a, 2013, 2012b, Burkart
et al., 2013]: IGWs are tidally excited mainly at the composition transitions of the
WD envelope; as these waves propagate outwards towards the WD surface, they
grow in amplitude until they break, and transfer both energy and angular momen-
tum from the binary orbit to the outer envelope of the WD. However, these pre-
vious works parameterized the wave breaking process in an ad hoc manner. The
details of dissipation, namely the location and spatial extent of the wave break-
ing, affect the observable outcomes: dissipation near the surface of the WD can be
efficiently radiated away and simply brightens the WD, while dissipation deep in
the WD envelope causes an energy buildup that results in energetic flares [Fuller
and Lai, 2012b]. An important goal of this paper is to elucidate the details of the
nonlinear IGW breaking process; the result of this “microphysics” study will help
determine the thermal evolution and the observational manifestations of tidally
heated binary WDs.
133
In this paper, we perform numerical simulations of IGW breaking in a plane-
parallel stratified atmosphere (a simple model for a stellar envelope). We use
the pseudo-spectral code Dedalus [Burns et al., 2016, Burns et al., 2020] and a
2D Cartesian geometry, and consider IGWs propagating into an isothermal fluid
initially at rest. We find that, after an initial transient phase, a critical layer nat-
urally develops, separating a lower zone that has no horizontal mean flow and an
upper zone with mean flow at the horizontal phase velocity of the IGW. The major
part of our paper is dedicated to characterizing the behavior of the critical layer
when interacting with a continuous train of IGW excited from the bottom of the
atmosphere. IGWs are generally anti-diffusive, in that they steepen shear flows
[Lindzen and Holton, 1968, Couston et al., 2018] and act to narrow the critical
layer. We find this steepening is counter-balanced by the Kelvin-Helmholtz insta-
bility and turbulence within the narrow critical layer. By careful accounting of the
momentum flux budget about the critical layer, we are able to model the reflection
and absorption of the incident IGW, and the slow downward propagation of the
critical layer.
While the motivation of our study is to understand tidal dissipation in WD
and early-type stellar binaries, the IGW breaking process studied in this paper is
also quite relevant to the circulation dynamics of planetary atmospheres [see e.g.
Lindzen, 1981, Holton, 1983, Baldwin et al., 2001].
This paper is organized as follows. In Section 5.2 we present the system of
equations used in our simulations. In Section 5.3, we review the existing un-
derstanding of wave breaking and present analytical results characterizing IGW
behavior near a critical layer. In Section 5.4 we describe our numerical setup and
in Section 5.5 we validate our method in the weak-forcing limit against linear the-
ory. In Section 5.6, we present the results of simulations of IGW breaking and our
134
characterization of the critical layer. We summarize and conclude in Section 5.7.
5.2 Problem Setup and Equations
We consider a incompressible, isothermally stratified fluid representing a stellar
envelope or atmosphere. We study dynamics in 2D, so that fluid variables depend
only on the Cartesian coordinates x and z. While it is well known that waves break
differently in 2D versus 3D [Klostermeyer, 1991, Winters and D’Asaro, 1994], the
dynamical effect of the breaking process is likely to be similar in 2D [Barker and
Ogilvie, 2010]. We approximate the gravitational field as uniform, pointing in the
(−ẑ) direction. The plane-parallel approximation is justified since wave breaking
generally occurs near the stellar surface. The background density stratification is
given by
ρ = ρ e−z/H0 , (5.1)
with ρ0 some reference density. We denote background quantities with overbars
and perturbation quantities with primes.
The Euler equations for an incompressible fluid in a uniform gravitational field
are
∇·u= 0, (5.2a)
Dρ = 0, (5.2b)
Dt
Du + ∇P + gẑ= 0, (5.2c)
Dt ρ
where D/Dt = ∂/∂t + (u ·∇) is the Lagrangian or material derivative, and u,ρ,P
denote the velocity field, density and pressure respectively. The constant gravita-
tional acceleration is (−gẑ). Note that these equations conserve the same wave en-
ergy as the commonly used anelastic equations [Ogura and Phillips, 1962, Brown
135
et al., 2012] and thus give the same wave amplitude growth. Appendix D.1 pro-
vides a derivation of these equations and justification for using them.
For this isothermal background, hydrostatic equilibrium implies P(z)= ρ(z)gH.
We assume there is initially no background flow, so u = u′. Physically, this as-
sumption corresponds to a non-rotating star.
For convenience, we introduce the dimensionless density variable Υ and the
reduced pressure ϖ [e.g. Lecoanet et al., 2014] via
ρ
Υ≡ ln , (5.3)
ρ̄
P
ϖ≡ . (5.4)
ρ
These variables automatically enforce ρ > 0 and eliminate the stiff term ∇P/ρ in
the Euler equation. In terms of Υ and ϖ, the second two equations in (5.2) become
DΥ + ∂ lnρuz = 0, (5.5a)Dt ∂z
Du +∇ + ϖϖ ϖ∇Υ− ẑ+ gẑ= 0. (5.5b)
Dt H
Hydrostatic equilibrium corresponds to Υ= 0,ϖ= gH.
5.3 Internal Gravity Waves: Theory
5.3.1 Linear Analysis
In the small perturbation limit, we may linearize Eq. (5.5). The resulting equa-
tions admit the canonical IGW solution [Drazin, 1977, Dosser and Sutherland,
2011b]
u′ (x, z, t)= Aez/2Hz cos(kxx+kzz−ωt) , (5.6)
136
where A is a constant amplitude, and the frequency ω and the wave number
(kx,kz) satisfy the dispersion relation
2 = N
2k2x
ω
k2x+k2z+ −2
. (5.7)
(2H)
Our equations are valid in the limit of large sound speed (cs →∞), in which the
Brunt-Väisälä frequency, N, is given (by )
2 ≡ 2 dρ − 1N g 2 =
g
, (5.8)
dP cs H
and is constant. Other dynamical quantities are simply related to u′z.
In the short-wavelength/WKB limit (|kzH| ≫ 1), the solution exhibits the fol-
lowing characteristics:
1. The amplitude of the wave grows with z as ez/2H . Thus, the linear approxi-
mation always breaks down for sufficiently large z.
2. The phase and group velocities are given by:
cp = ω(kx[x̂+kzẑ) k2x+] 2+ , (5.9)kz (2H)−2
k[2= z+ (2H)−2 x̂− (k]xkzẑ)cg N . (5.10)
k2x+k2 −2 3/2z+ (2H)
The additional (2H)−2 term in the denominator accounts for the growing
amplitude of the IGW in the z directio[n (as the] z wavenumber is effec-
tively kz − i/(2H)). We note cp · cg = O( (kzH) )−2 ≈ 0. In the Boussinesq
approximation where terms of order O H−2 are ignored, the phase and
group velocities are exactly orthogonal [Drazin, 1977, Dosser and Suther-
land, 2011a]. We use the convention where upward propagating IGW have
cg,z > 0, kz < 0,kx > 0.
137
3. The averaged horizontal momentum flux F (in the +ẑ direction) carried by
the IGW is defined by
〈 〉 ∫Lx1
F(z, t)≡ ρu′ ′ ′ ′xuz x ≡ ρuxuz dx. (5.11)Lx
0
The notation 〈. . .〉x denotes averaging over the x direction. For the linear
solution (Eq. 5.6), this evaluates to
≈−A
2 k
F zρ
2 0
, (5.12)
kx
Thus, indeed F > 0 for an upward propagating IGW (cg,z > 0).
5.3.2 Wave Generation
To model continuous excitation of IGWs deep in the stellar envelope propagat-
ing towards the surface, we use a volumetric forcing term to excite IGW near the
bottom of the simulation domain. Our forcing excites both IGWs propagating up-
wards, imitating a wave tidally excited deeper in the star, and downwards, which
are not physically relevant in binaries. In our simulations, these downward prop-
agating waves are dissipated by a damping zone described in Section 5.4.2.
As not to interfere with the incompressibility constraint, we force the system
on the density equation. We implement forcing with strength C localized around
height z0 with small width σ by replacing Eq. (5.5a) with
DΥ + ∂ lnρ
(z−z )2
= 0uz Ce− 2σ2 cos(kxx−ωt) . (5.13)Dt ∂z
Using a narrow Gaussian profile excites a broad z power spectrum, but only the kz
satisfying the dispersion relation (Eq. 5.7) for the given kx and ω will propagate.
In the linearized system, the effect of this forcing can be solved exactly (see
Appendix D.2). In the limit |kzH| ≫ 1,σ≪ H, the solution can be approximated
138
as two plane waves propagating away( from th)e forcing zone
≈ C gk
2 k2σ2 √
uz(x, z, t)
x
2 exp −
z 2πσ2
2kz ω 2
 z−z ( )0e 2H sin kxx+ − 2kz(z z0)− k σωt+ z
 ( 2H
for z> z0,
× (5.14)z−z )0 2e 2H sin kxx−kz(z− z kzσ0)−ωt+ 2H for z< z0.
The z > z0 region contains an upward propagating IGW wavetrain. The x compo-
nent of the velocity can be obtained by the incompressibility constraint (Eq. 5.2a).
5.3.3 Wave Breaking Height
As the upward propagating IGW grows in amplitude (|u|∝ ez/2H), it is expected to
break due to nonlinear effects. We can estimate the height of wave breaking using
the condition |u| ∼ω/ |k|. This can be rewritten using the Lagrangian displacement
ξ=u/ (−iω):
|ξzkz| ∼ 1. (5.15)
Drazin [1977], Klostermeyer [1991], Winters and D’Asaro [1994] describe the
onset of wave breaking in some detail. At intermediate amplitudes, wave break-
ing occurs via triadic resonances, transferring energy from the “parent” IGW to
“daughter” waves on smaller length scales that efficiently damp. The horizontal
momentum flux decreases from F to 0 over this breaking region. The lost flux is
deposited into a horizontal mean flow
U(z, t)≡ 〈ux〉x . (5.16)
As the mean flow grows, a critical layer may form, as discussed below.
139
5.3.4 Critical Layers
A horizontal shear flow U(z, t)x̂ enters the fluid equations via the Lagrangian
derivative, which can be decomposed as
D = ∂ + ∂U + ( )u′ ·∇ , (5.17)
Dt ∂t ∂x
where u′ is the velocity field without the shear flow. Thus, U has the effect of
Doppler shifting the time derivative into the frame comoving with the mean flow.
If U is roughly constant, then the behavior of a linear plane-wave perturbation
satisfies the modified dispersion relation
( ) 2 2
ω− 2 N kUkx = x
k2 2x+kz+
. (5.18)
(2H)−2
This is just Eq. (5.7) with ω→ω−Ukx. It is apparent that if U =U c, where
U c ≡ ω , (5.19)kx
then the dispersion relation is singular and the linear solution breaks down. Phys-
ically, this corresponds to the Doppler-shifted frequency of the IGW being zero.
Anywhere U =U c is called a critical layer.
The behavior of an IGW incident upon a critical layer was first studied in the
inviscid, linear regime in Booker and Bretherton [1967], which found nearly com-
plete absorption of the IGW. The amplitude reflection and transmission coeffi-
cients are given by ( √ ) ( √ )
1 1
R = exp −2π Ri− , T = exp −π Ri− , (5.20)
4 4
where Ri is the local Richardson number evalu∣ated at the critical layer height zc:∣
2 ∣
Ri≡ ( N ) ∣2 ∣∣ . (5.21)
∂U /∂z ∣
zc
140
In the Ri≫ 1 limit,R,T ≪ 1 and the incident wave is almost completely absorbed.
This result also applies to viscous fluids [Hazel, 1967]. However, weakly nonlinear
theory [Brown and Stewartson, 1982] and numerical simulations [Winters and
D’Asaro, 1994] suggest that nonlinear effects may significantly enhance reflection
and transmission.
Consider now the long-term evolution of the critical layer due to continuous
horizontal momentum transfer by IGWs. Any incident horizontal momentum flux
absorbed by the fluid, denoted Fa(t), must manifest as additional horizontal mo-
mentum of the shear flow. Additionally, as the mean flow U cannot grow efficiently
above U c (due to the breakdown of the linear solution), we assume U saturates at
U c, which holds to good accuracy (see Fig. 5.4). In this case, the critical layer must
propagate downward in response to the incident momentum flux. The horizontal
momentum of the shear flow∫satisfies
∂
ρ(z)U(z, t) dz−Fa(t)= 0. (5.22)
∂t
Assuming U(z> zc)≈U c and U(z< zc)≈ 0, this condition becomes
− dz(z )U cρ c c = F (t). (5.23)dt a
If Fa is constant in time, the height of the critical layer zc(t) has analytical solu-
tion: [ ( ) ]
=− − zc(t= 0) tFzc(t) H ln exp + a , (5.24)H U cHρ0
where zc(t= 0) is the initial critical layer height.
5.4 Numerical Simulation Setup
We use the pseudo-spectral code Dedalus [Burns et al., 2016, Burns et al., 2020]
to simulate the excitation and propagation of IGWs (Section 5.5) as well as their
141
nonlinear breaking and the formation of a critical layer (Section 5.6).
5.4.1 Parameter Choices
We solve Eqs. (5.2a), (5.5b), and (5.13) in a Cartesian box with size Lx,Lz. We
choose periodic boundary conditions in both the x and z direction. To mimic the
absence of physical boundaries at the top/bottom of the simulation domain, we
damp perturbations to zero near the top/bottom using damping zones (see Sec-
tion 5.4.2). We expand all variables as Fourier series with Nx and Nz modes, and
use the 3/2 dealiasing rule to avoid aliasing errors in the nonlinear terms [Boyd,
2001]. We choose Lz = 12.5H (z runs from −H to 11.5H), and the lower and upper
damping zones are active for z < 0.3H and z > 9.5H respectively. The forcing (see
Eq. (5.13)) is centered at z0 = 2H with width σ= 0.078H, sufficiently far from the
lower damping zone and permitting sufficient room for the upward propagating
wave to grow as ∝ ez/2H . Finally, we want similar grid spacing in the x and z
directions (i.e. Lx/Nx ∼ Lz/Nz), guided by the intuition that turbulence generated
by wave breaking is approximately isotropic, so we use Lx = 4H and Nz/Nx = 4.
The time integration uses a split implicit-explicit third-order scheme where
certain terms are treated implicitly and the remaining terms are treated explic-
itly. A third-order, four-stage DIRK-ERK scheme [Ascher et al., 1997] is used
with adaptive timesteps computed from the minimum of 0.1/N and the advective
Courant-Friedrichs-Lewy (CFL) time. The CFL time is given by∆t= 0.7min(∆x/ux,∆z/uz),
where the minimum is taken over every grid point in the domain, and ∆x≡ Lx/Nx
and ∆z≡ Lz/Nz are the grid spacings in the x and z directions respectively.
We non-dimensionalize the problem such that H = N = ρ0 = 1. The physics of
the simulation is then fixed by the four remaining parameters kx, ω, C, and the
142
viscosity ν. We describe our choices for these parameters below:
1. kx: Tidally excited waves in stars generally have ℓ = 2, corresponding to a
horizontal wavenumber k⊥ ∼ 1/R, where R is the radius of the star. We use
the smallest wavenumber in our simulation, kx = 2π/Lx.
2. ω: We choose ω by evaluating the dispersion relation ω(kx,kz) for a desired
kz (see Eq. (5.7)). We pick |kzH| = 2π to ensure the waves are very well re-
solved in all of our simulations. Note however that tidally forced IGWs typi-
cally have ω≪N, or equivalently kr/k⊥ ∼ krR≫ 1. This requires |kzH|≳ 1,
which is only marginally satisfied in our simulations.
3. C: In our weak forcing simulations (Section 5.5), we first choose the forcing
strength C (see Eq. (5.13)) to be sufficiently weak such that |ξzkz|≪ 1 is sat-
isfied everywhere in the simulation domain. This constrains C by Eq. (5.14).
In our wave breaking simulations (Section 5.6), we choose larger C.
4. ν: Nonlinear effects transfer wave energy from the injection wavenumber k
to larger wavenumbers. Our spectral method does not have any numerical
viscosity, so diffusivity must be introduced into the equations to regularize
the systems at large wavenumbers. We add viscosity and diffusivity to the
system in a way that conserves horizontal momentum (see Appendix D.3 for
details). We define the dimensionless Rey(nold)s number
≡ ω = ω H
2
Re 2 . (5.25)νkz ν 2π
We use Re≫ 1 in our simulations1.
Finally, we use initial conditions u(x, z,0)=Υ(x, z,0)= 0 and ϖ(x, z,0)= 1, cor-
responding to hydrostatic equilibrium and no initial fluid motion.
1This condition is always satisfied in stars. For example, in WDs, the dominant linear dissi-
pation mechanism of g-modes is radiative damping, with damping rate ranging from 10−11–10−4
of the mode frequency [Fuller and Lai, 2011]. This corresponds to a small effective viscosity or
Re≫ 1.
143
5.4.2 Damping Layers
We aim to damp disturbances that reach the vertical boundaries of the simulation
domain without inducing nonphysical reflection. To do so, we replace material
derivatives in Eq. (5.5) with:
D → D[+Γ(z), ] (5.26)Dt Dt
Γ(z)= 1 + z− z z − z2 tanh T + tanh B , (5.27)
2τ ∆z ∆z
where zB = 0.3H and zT = 9.5H are the boundaries of the lower and upper damp-
ing zones respectively. This damps perturbations below zB and above zT with
damping time τ and negligibly affects the dynamics between zB and zT . Most
importantly, horizontal momentum remains conserved between zB and zT , and
outgoing boundary conditions are imposed at zB, zT . We choose the transition
width ∆z = 0.25H and damping time τ = 1/(15N). This prescription is similar
to Lecoanet et al. [2016] and has the advantage of being smooth, important for
spectral methods. Further details of our implementation of the fluid equations in
Dedalus are described in Appendix D.3.
5.5 Weakly Forced Numerical Simulation
To test our numerical code and implementation, we carry out a simulation in
the weakly forced regime with C = 1.64×10−7. According to the linear solution
(Eq. (5.14)), this generates IGW with |ξzkz| ≈ 5×10−5 just above the forcing zone.
The IGW grows to |ξzkz| ≈ 7.4× 10−3 at the upper damping zone and satisfies
|ξzkz|≪ 1 in the entire simulation domain. We include a nonzero ν corresponding
to Re= 107.
144
We expect the waves to follow the analytical solution given by Eq. (5.14) and
the corresponding ux(x, z, t); we denote this analytical solution uan(x, z, t). The
amplitude of the observed IGW in the simulation field u relative to uan over some
region z ∈ [zb, zt] can be estimated f∫romzt L∫x
ρ (u ·uan) dxdz
z
A t = b∫0i( ) . (5.28)zt L∫x
ρ |u |2an dxdz
zb 0
The subscript i denotes the incident wave. If u = uan, then A i(t) = 1. The factor
of ρ̄ inside the integrands in Eq. (5.28) corrects for the ∝ ez/2H growth of uan;
without it, A i(t) would be dominated by the contribution near zt.
For the weakly forced simulation, we expect A i(t)= 1 when integrated between
the forcing and damping zones, i.e. zb ≳ z0 and zt ≲ zT (z0, zT are defined in
Eq. (5.13) and Eq. (5.27) respectively). For consistency with the nonlinear case
later, we choose zb = z0+3σ and zt = zb+H. Note that using a larger integration
domain by choosing zt = zT −∆z just below the upper damping zone instead does
not change the measured A i. The resulting measurement of A i(t) is shown in
Fig. 5.1, and indeed A i ≈ 1 after the initial transient.
The analytical theory (Section 5.3.1) also predicts that the horizontal momen-
tum flux F(z, t) is independent of z between the forcing zone where the wave is
generated and the damping zone where it is dissipated. The expected horizontal
momentum flux carried by the excited IGW in the linear theory can be computed
by simply evaluating Eq. (5.11) for u
〈an
and is a constant:
Fan ≡
〉
ρuan,xuan,z x . (5.29)
Denoting the momentum flux measured in the simulation by F(z, t) (use Eq. (5.11)
with velocities taken from the simulation), we expect F(z, t)= Fan between z0 and
zT . Fig. 5.2 shows agreement with this prediction.
145
Figure 5.1: Amplitude of the excited IGW over time (in units of N−1) in the weakly
forced simulation, computed using Eq. (5.28). A i(t) = 1 corresponds to perfect
agreement with the analytical estimate. After an initial transient phase, we ob-
serve A i(t) asymptotes to ≈ 1, implying continuous excitation of identical IGW
with the expected amplitude. The small deviation of A i(t) from unity may be due
to truncation error in our implicit timestepping scheme, as a relatively large fixed
step size ∆t= 0.1/N was used for this simulation.
146
Figure 5.2: F/Fan plotted at select times t (in units of N−1). As the initial transient
dies out, F/Fan ≈ 1 to a good approximation above the forcing zone z> z0 = 2H and
below the damping zone z ≲ zT = 9.5H. The horizontal momentum flux excited in
the forcing zone is transported without loss to the top of the domain, where it is
dissipated by the damping zone (see Section 5.4.2) without reflection.
147
Resolution Re
1024×4096 2048
768×3072 1024
512×2048 512
256×1024 341
256×1024 205
256×1024 146
Table 5.1: Spectral resolutions and Reynolds numbers of simulations of wave
breaking.
5.6 Numerical Simulations of Wave Breaking
To perform simulations of wave breaking phenomena, we use the same setup as
described in Section 5.4 and Section 5.5 except for different values of C and ν.
In particular, we choose C such that |ξzkz| = 0.1 in the forcing zone (z = z0). The
linear solution predicts |ξzkz| ∼ 4.25 at the upper damping zone zT . We choose
the viscosity ν for each resolution to be as small as possible while still resolving
the shortest spatial scales of the wave breaking. A table of our simulations can be
found in Table 5.1.
5.6.1 Numerical Simulation Results
A full video of our simulation with Nx = 768, Nz = 3072, Re = 1024 is available
online2. We take this to be our fiducial simulation for the remainder of this paper,
though other simulations show qualitatively similar behavior.
In Fig. 5.3, we present snapshots of ux and Υ at various phases of the simula-
tion. Note that Υ≲ 0.1, so the density stratification does not deviate significantly
from equilibrium. The flow evolves through several distinct stages:
2https://bit.ly/3vuVKNN
148
1. At early times (top left panel), the flow resembles a linear IGW lower in the
simulation domain but breaks down into smaller-scale features at higher z.
Some characteristic swirling motion can be seen in the advected scalar Υ,
indicating Kelvin-Helmholtz instabilities.
2. At a slightly later time (top right panel), the mean flow in ux becomes much
more prominent and the critical layer zc has become much more definite.
Small-scale fluctuations are still present in ux but at smaller amplitudes
due to being in a denser region of the fluid.
3. In the bottom left panel, the critical layer transition becomes very sharp, and
small swirls of limited vertical extent in Υ at the location of the critical layer
suggest that the Kelvin-Helmholtz instability is responsible for regulating
the width of this transition. More discussion can be found in Section 5.6.2.
4. At the end of the simulation (bottom right panel), the critical layer has ad-
vanced downwards, but otherwise the flow shows very few significant quali-
tative differences from the previous snapshot. This suggests that the latter
phase of the simulation has reached a steady state. Notably, the horizontal
banded structure of ux in the upper, synchronized fluid does not continue to
evolve (also visible in the top panel of Fig. 5.4), suggesting that momentum
redistribution and mixing within the synchronized fluid are negligible.
In Fig. 5.4, we plot the mean horizontal flow velocity U (Eq. (5.16)) and the
dimensionless momentum flux F/Fan (Eqs. (5.11) and (5.29)) as a function of z at
the times depicted in Fig. 5.3. At each time, U is close to zero below the critical
layer, but then sharply increases to U c at the critical layer (i.e. the flow is “spun-
up”). Above the critical layer, U varies slightly due to momentum transport within
the spun-up layer. This agrees with the expectation discussed in Section 5.3.4.
149
( )
Figure 5.3: Snapshots of ux and Υ≡ ln ρ/ρ̄ in the fiducial simulation illustrating
distinct phases of the evolution of the flow. See the online PDF for a color version.
Damping layers at the top and bottom of the simulation domain are shaded in
light grey (see Section 5.4.2), while the forcing zone in the lower middle portion
of the simulation domain (see Section 5.3.2) is shaded in light green (boundaries
are at z0±3σ). The four panels illustrate (i) the initial transient wave breaking
phase, (ii) formation of a distinct critical layer, (iii) steepening of the critical layer,
and (iv) downward advance of the critical layer.
150
Similarly, F ≲ Fan below the critical layer, and then decreases to about zero
above the critical layer. However, two notable deviations from the discussion in
Section 5.3.4 can be observed: (i) the incident flux on the critical layer fluctuates
somewhat temporally, and (ii) there is a small negative flux just above the critical
layer at later times. These are addressed in subsequent sections.
5.6.2 Kelvin-Helmholtz Instability and Critical Layer Width
The formation of the critical layer is associated with a strong shear flow. What is
the width of this layer? Inspection of Fig. 5.3 suggests the presence of the Kelvin-
Helmholtz Instability (KHI) in the critical layer. In a stratified medium, KHI
occurs when the Richardson number (Eq. (5.21)) satisfies Ri≲ 1/4 [e.g. Shu, 1991].
It is natural to suspect that the shear flow cannot steepen further than the onset
of KHI. To test this, we compute the local Ri for the shear flow around the critical
layer.
It is difficult to accurately measure the Richardson number, as it depends on
the derivative of the velocity. We measure Ri as follows: we first assign an Rix(x, t)
for every x in the critical layer, then take the median of Ri for the entire layer.
Rix is computed using the vertical distance over which the local ux increases from
0.3Ūc to Ūc (see Eq. (5.19)). The value 0.3 is necessary to exclude the small mean
flow generated in the weakly nonlinear regime far below the critical layer. This
151
Figure 5.4: The mean horizontal flow velocity U(z, t) (Eq. (5.16)) and the dimen-
sionless momentum flux F(z, t)/Fan (Eqs. (5.11) and (5.29)) in our fiducial simula-
tion plotted at the same times as in Fig. 5.3. The two distinct zones of mean flow
are separated by a critical layer. The critical layer propagates toward lower z due
to momentum transport (∂F/∂z).
152
procedure can be written:
{ }
zCL,min(x, t)≡min {z | ux(x, z, t)> 0.3U} c , (5.30)
zCL,max(x, t)≡ (max( z | ux(x, z, t)<U c) ,) (5.31)
≡ N
2 z 2
Ri (x, t) CL,max
− zCL,min
x , (5.32)
(0.7U c)2
Ri(t)≡medRi(x, t) . (5.33)
x
We use the background buoyancy frequency to compute Ri, as fluctuations do not
change N2 significantly (∼ 1%). To understand the variation in Ri over x, we also
compute minRix(x, t) (the maximum is very noisy). Both are shown in Fig. 5.5.x
Absorption of incident IGWs quickly decreases the Richardson number to between
0.25 and 0.5, characteristic of the onset of the KHI.
This result suggests that the critical layer width is regulated by the competi-
tion between steepening induced by IGW breaking and broadening due to shear
instability. This width does not vary significantly with resolution in our resolved
simulations (see Fig. 5.10). As such, Ri ∼ 0.5 can be used to calculate the critical
layer width in stars, where N2 and U c (corresponding to the tidal frequency) are
known.
5.6.3 Flux Budget
The downward propagation of the critical layer location zc(t) is driven by the ab-
sorption of horizontal momentum flux at zc, following Eq. (5.23). The flux budget
at the critical layer can be decomposed as
Fi(t)= Fa(t)+Fr(t)+Fs(t), (5.34)
153
Figure 5.5: Local Richardson number (Eq. (5.33)) of the flow at the critical layer
over time (in units of N−1) in our fiducial simulation. The solid red and dotted
black lines denote respectively the minimum and median of Rix(x, t). These num-
bers measure the mean and spread in width of the critical layer over x. Note that
Ri∼ 14 corresponds to the KHI, so this plot suggests the shear at the critical layer
does not steepen past the onset of the KHI.
154
where Fi is the incident flux, Fa is the absorbed flux, Fr is the reflected flux, and
Fs is some “redistribution” flux above the critical layer, responsible for momen-
tum redistribution within the synchronized upper layer. Careful accounting of Fs
turns out to be important to obtain the correct Fa and resulting critical layer prop-
agation. A more specific physical interpretation of Fs is unclear; it is somewhat
tempting but unfounded to identify Fs with the transmitted flux. In these simu-
lations, we find Fs < 0, corresponding to net momentum transport into the critical
layer from the synchronized layer above it.
After measuring zc (see Section 5.6.4) and F(z) (Eq. (5.11)) at each time step,
we determine each of Fi, Fa, Fr, Fs as follows:
Fi(t)= F 2anA i (t), ∫ (5.35)zc−∆z
1
Fr(t)= Fi(t)− F(z, t) dz, (5.36)H
zc∫ zc−∆z−H+∆z
Fs(t)= 1 min(F(z, t),0) dz, (5.37)
∆z
zc
Fa(t)= Fi−Fr−Fs. (5.38)
Fig. 5.6 depicts the four components of this flux decomposition. Below the critical
layer, we average over an interval of length H, also the vertical wavelength. The
offset ∆z is necessary to make the measurement of the incident flux unaffected
by the turbulence within the critical layer itself. The width of the critical layer is
limited by Ri ≲ 1 (see Section 5.6.2), which bounds its vertical extent ∼ 1|k . Wez|
empirically found an offset of ∆z = 3|k | was necessary to be sufficiently far fromz
strong fluctuations near the critical layer.
Above the critical layer, we observe that the Fs feature has varying width (com-
pare e.g. the t = 1171.4/N and t = 3437.8/N lines in the bottom panel of Fig. 5.4)
but contributes significantly to the total flux budget. We average only where F < 0
155
Figure 5.6: Momentum flux decomposition calculated from the simulation. Plot-
ted are the four components of the horizontal momentum flux budget over time
(see Eq. (5.34)), in units of the analytical estimate for the incident wave flux Fan
(Eq. (5.12)): Fi, the flux incident on the critical layer; Fa, the flux absorbed by the
critical layer; Fr, the flux reflected at the critical layer; and Fs, the flux inside the
synchronized layer.
so that Fs is robust to such width variations. We find that this is an accurate way
of measuring Fs and determining Fa.
5.6.4 Critical Layer Propagation
With a careful determination of Fa, we can make predictions for the propagation of
zc(t) and compare to the measured propagation in the simulation. In principle, zc
is the location where the incident flux significantly attenuates. In the simulation,
shear turbulence causes F to have significant spatial and temporal fluctuations
that translate to large temporal fluctuations in zc(t). To minimize these spurious
156
fluctuations, we measure the location of the critical layer using a spatial average
of where flux deposition occurs:
zc,min(t)≡min {z : F(z, t)> 0.3Fan} , (5.39)z
zc,max(t)≡max {z : F(z, t)< 0.3Fan} , (5.40)z
≡ zc,min(t)+ zz (t) c,max(t)c . (5.41)2
Measuring zc in other ways does not significantly change the results of the analy-
sis.
In Fig. 5.7 we plot the numerically measured zc against numerical integration
of Eq. (5.23) using the measured Fa(t). Since the critical layer is still forming
at early times, we solve Eq. (5.23) by integrating backwards from the end of the
simulation (t= t f ), using zc(t f ) as the initial condition. From Fig. 5.7, we see that
the agreement between the measured zc(t) and its estimate via Fa(t) is excellent.
By time-averaging the numerically measured Fa, we find 〈Fa〉t ≈ 0.71Fan. Note
that Fa < Fan, so momentum flux absorption at the critical layer is incomplete.
This is due to reflection of waves off the critical layer, which carry momentum
downward.
5.6.5 Non-absorption at Critical Layer
To further understand the behavior at the critical layer, we compare two reflective
behaviors observed in the simulation: (i) the presence of a reflected wave with the
same frequency as the incident wave (i.e. with wave vector kr = kxx̂− kzẑ), and
(ii) the reflected flux Fr. The reflected wave amplitude and flux need not agree
exactly if some reflected flux is in higher-order modes, which is indeed the case in
157
Figure 5.7: Propagation of the critical layer over time. Shown are: (solid black)
zc(t) from simulation data, and (dashed red) model for zc(t) using direct integra-
tion of Eq. (5.23) for Fa(t) measured from simulation data (described in Eq. (5.38)).
The model uses the end of the simulation as its initial condition and integrates
backwards, as the critical layer is still forming at earlier times. The agreement of
the model with the simulation shows Eq. (5.23) is a good description of the evolu-
tion of zc.
158
our simulations. Both are of physical interest, however: the reflected wave ampli-
tude is essential for setting up standing modes in a realistic star, while the flux
is important for accurately tracking angular momentum transfer during synchro-
nization.
To measure the reflected wave amplitude Ar(t), we use an approach similar to
the calculation of A i(t) (Eq. (5.28)):
∫zt L∫x ( ∣ )
ρ u · uan,k ∣r x=x+δx dxdz
z
Ar(t = 0) max b ∫ ∫ , (5.42)δx zt Lx
ρ |u |2an dxdz
zb 0
where zb = z0+3σ and zt = zb+H as before. The primary difference from Eq. (5.28)
is the introduction of free parameter δx, the horizontal phase offset of the reflected
wave. Since δx is unknown a priori, we choose δx ∈ [0,2π] that maximizes Ar(t).
In our simulation, the phase ∣∣offset φ∣∣r(t)≡ kxδx(t) is consistent with reflection offa moving boundary at zc, i.e. ∂φr/∂t ≃ 2 |∂(kzzc)/∂t|.
Fig. 5.8 illustrates the behaviors of A i and Ar. Both vary significantly in time
but their mean values appear to converge towards the end of the simulation.
Since A i(t), Ar(t) vary somewhat over time, we perform time averaging over
interval of four wave periods, denoted by angle brackets. We can then define the
amplitude reflectivity
〈A
(t)≡ r〉 (t)RA 〈 〉 . (5.43)A i (t)
We compare the square of the reflectivity to the ratios of Fr and −Fs to Fi, as
F ∝ A2 (Eq. (5.12)). We define
〈F 〉 (t)
F̂ ≡ rr 〈 〉 , (5.44)Fi (t)
≡−〈Fs〉 (t)F̂s 〈 . (5.45)Fi〉 (t)
159
Figure 5.8: The incident wave amplitude A i(t) (solid black) and the reflected wave
amplitude Ar(t) (dashed red) just above the forcing zone.
Fig. 5.9 shows R2A, F̂r, and F̂s as functions of time. The three quantities appear
to be roughly stationary for t ≳ 2500/N. Modest fluctuations (∼ 20%) in A i do not
affect our reflectivity results thanks to the time averaging used in Eqs. (5.43)–
(5.45).We see that in general F̂ ≳R2r A, conforming with the expectation that the
reflected flux consists of the simple reflected mode and higher order modes as well.
5.6.6 Resolution Study
Although throughout this paper we focused on our fiducial simulation with Re =
1024 and resolution Nx = 768, Nz = 3072, we also ran a suite of simulations vary-
ing the resolution and corresponding Reynolds number (Tab. 5.1). We find that
160
Figure 5.9: R2A, F̂r, and redistribution flux F̂s [Eqs. (5.43–5.45)] as a function of
time (in units of N−1). These quantities seem to become comparatively stable past
about t = 2500/N, indicating that an asymptotic value may have been reached.
That F̂r ≳ R2A implies a substantial fraction of reflected flux is in higher-order
modes than the reflected IGW.
161
our global, quantitative measurements in the simulations (R2A, Fr, Fs, and Ri)
are very similar for our highest Reynolds numbers (1024 and 2048).
For each simulation in Tab. 5.1, we compute the median values of R2A, F̂r, F̂s,
and Ri [Eqs. (5.43–5.45) and (5.21) respectively] over the last 1/4 of the simulation
time, when these quantities have reached their asymptotic values. These results
are shown in Fig. 5.10.
As the simulation resolution increases and the viscosity decreases, we find that
the Richardson number decreases, while the reflection and redistribution fluxes
increase. The Richardson number is roughly constant for Re > 200 with a value
of Ri ∼ 0.4. The behavior of the fluxes is more complicated. While the fraction of
reflected and redistributed flux is similar for our simulations with Re= 1024 and
2048, higher resolution simulations would be required to determine these flux
fractions in the limit Re→∞.
Nevertheless, the difference in behavior of Ri and the flux reflectivity as Re is
varied is in tension with Eq. (5.20). This tension is natural: Eq. (5.20) is derived
from a linear theory, while fluid motion within the critical layer is turbulent, so
reflection at the critical layer cannot be captured by the linear theory.
5.7 Summary and Discussion
5.7.1 Key Results
In this paper, we have performed numerical simulations of nonlinear breaking of
IGWs in a stratified isothermal atmosphere. Such a setup represents the plane-
parallel idealization of the outer stellar envelope. Our simulations use the spectral
162
Figure 5.10: Convergence of the median F̂ 2r, RA, F̂s, and Ri (Eqs. (5.43)–(5.45)
and (5.21) respectively) in simulations with varying resolution and viscosity as
given in Tab. 5.1. Vertical bars show the temporal variation of each measurement
between the 16% and 84% range. Small horizontal displacements are made for
data points at identical Re for readability. Note that simulations with larger Re
correspond to smaller viscosity and are more physically realistic. At the smallest
Re value, Ri≈ 50 is too large to fit on the plot.
163
code Dedalus [Burns et al., 2016, Burns et al., 2020], and are carried out in 2D. We
observe spontaneous formation of a critical layer that separates a “synchronized”
upper layer of fluid and a lower layer with no mean horizontal flow. This critical
layer then propagates downwards as incident IGWs break and deposit horizontal
momentum to the fluid (see Fig. 5.3 for snapshots from our fiducial simulation).
Our primary conclusions regarding the evolution of the critical layer are as fol-
lows:
1. The width of the turbulent critical layer is determined by requiring the local
Richardson number (Eq. (5.21)) Ri∼ 0.5 (see Fig. 5.5).
2. The location of the critical layer zc(t) can be predicted by careful measure-
ment of the absorbed horizontal momentum flux at the critical layer (see
Eq. (5.23) and Fig. 5.7).
3. The absorption of IGW momentum flux at the critical layer is incomplete.
The critical layer only absorbs ∼ 70% of the incident flux in our highest res-
olution simulations (see Fig. 5.10). The reflected flux is carried away from
the critical layer as both lowest-order reflected waves and waves with larger
z wavenumbers.
5.7.2 Discussion
In this paper, we have studied the nonlinear behavior of IGWs with |kx/kz| ∼ 1/(2π)
in a plane-parallel geometry. Tidally excited IGWs in binary stars have horizontal
wavenumber k⊥ ∼ 1/R (where R is the stellar radius) much smaller than the radial
wavenumber kr. While our simulations do not satisfy |kx/kz| ≪ 1, the qualitative
behavior is likely to be similar, as the turbulence driving the critical layer dynam-
ics occurs at scales significantly smaller than either 1/kx or 1/ |kz|. Simulating
164
IGWs with kx ≪|kz| is more challenging numerically and we defer its exploration
to future work.
It is interesting to compare our work with that of Barker and Ogilvie [2010],
who studied inward-propagating IGWs in solar-type stars and their nonlinear
breaking due to geometric focusing. In their numerical simulations in a 2D po-
lar geometry, they found no evidence for reflected waves, contrary to our result.
Note that their simulations were run with substantially higher viscosity, or lower
resolution, than explored here, and their effective Reynolds number (equal to 1/λ
in their notation) is of order 10. We also find at low Reynolds numbers that there
is negligible wave reflection.
Regardless of the limitations inherent in our simulations (e.g. plane-parallel
geometry), our results shed light on the physical mechanism of tidal heating in
close binaries. In particular, our simulations indicate that energy dissipation oc-
curs in a narrow critical layer. The star heats up from outside-in as the criti-
cal layer propagates inwards. This tidal heating profile differs flom that used by
Fuller and Lai [2012b]. We plan to study this issue in a future work.
165
CHAPTER 6
DYNAMICAL TIDES IN ECCENTRIC BINARIES CONTAINING
MASSIVE MAIN-SEQUENCE STARS: ANALYTICAL EXPRESSIONS
Originally published in:
Yubo Su and Dong Lai. Dynamical tides in eccentric binaries containing massive
main-sequence stars: Analytical expressions. MNRAS, 510(4):4943–4951, 2022b
6.1 Introduction
The physics of tidal dissipation in massive, main-sequence (MS) stars (i.e. having a
convective core and radiative envelope) under the gravitational influence of a com-
panion was first studied by Zahn [1975] [see also Savonije and Papaloizou, 1983,
Goldreich and Nicholson, 1989]. The dominant dissipation mechanism is through
the dynamical tide, in which the time-dependent tidal potential of the companion
excites internal gravity waves (IGWs) at the convective-radiative boundary (RCB).
As the wave propagates towards the surface, its amplitude grows, and the wave
dissipates efficiently [Zahn, 1975, Goldreich and Nicholson, 1989, Su et al., 2020].
The contribution due to viscous dissipation in the convective core is expected to be
subdominant.
The original expression describing the torque due to dynamical tides by Zahn
[1975] is very sensitive to the global properties of the star. Kushnir et al. [2017]
present an updated derivation of the tidal torque that depends only on the local
stellar properties near the RCB, eliminating many uncertainties from the Zahn’s
original expression. In Zaldarriaga et al. [2018], the authors use the new expres-
sion to study tidal synchronization in binaries consisting of a Wolf-Rayet star and
a black hole, the likely progenitors to merging black-hole binaries observed by
LIGO/VIRGO.
166
These previous works all apply to nearly circular binaries. However, massive
stars can often be found in high-eccentricity (high-e) systems, such as binaries
consisting of one MS star and one neutron star (NS). The NS is formed with a
large kick velocity [e.g. Lai et al., 2001, Janka et al., 2021], giving rise to a high-e
binary. Several such high-e MS-NS systems have been discovered [e.g. Kaspi et al.,
1994, Johnston et al., 1994, Champion et al., 2008]. These are the progenitors of
double NS systems [e.g. Tauris et al., 2017]. An important issue is to understand
whether such high-e systems can circularize prior to mass transfer or a common
envelope phase [Vigna-Gómez et al., 2020, Vick et al., 2021].
The purpose of this paper is to derive easy-to-use, analytical expressions for
the effects of dynamical tides for high-e binaries with massive MS stellar com-
panions. In Section 6.2, we summarize the equations of dynamical tides involving
IGWs in circular binaries and existing techniques for studying high-e systems. In
Section 6.3, we evaluate the effect of dynamical tides in high-e systems containing
massive MS stellar companions, including the torque and orbital decay rate. In
Section 6.4, we apply our results to the pulsar-MS binary PSR J0045-7319, for
which a non-zero orbital decay rate has been measured. Finally, we summarize
our results and discuss the uncertainties in Section 6.5.
6.2 Dynamical Tides in Massive Stars
6.2.1 Circular Binaries
We first review the case where the binary is circular. Let M be the mass of the
MS star, M2 the mass of the companion, a the semimajor axis of the binary, and
Ω the angular frequency (mean motion) of the binary. The tidal torque exerted on
167
the star by the companion due to tidal excitation of IGWs at the RCB is [Kushnir
et al., 2017]
∣∣ ωT ∣∣8/3circ(ω)=T0 sgn(ω) ∣ ∣ , (6.1)
Ω
where  8/3
≡ GM
2r5 √  ( )Ω ρ ρ 2T β 2 c c c0 2 6 1− , (6.2)
[ (a 2 ) G]M /r[
3 ρ̄c ρ̄c
c c
r dN −1/3 2 2
]
≡ c 3 Γ (1/3)β 22 α . (6.3)gc dln r r=rc 40π122/3
Here, ω≡ 2Ω−2Ωs is the tidal forcing frequency, Ωs is the spin of the MS star, N is
the Brunt-Väisälä frequency, r is the radial coordinate within the star, and rc, Mc,
gc, ρc, and ρ̄c are the radius of the RCB, the mass contained within the convective
core, the gravitational acceleration at the RCB, the stellar density at the RCB, and
the average density of the convective core respectively. Γ is the gamma function, α
is a numerical constant of order unity given by Eq. (A32) of Kushnir et al. [2017],
and β2 ≈ 1 for a large range of stellar models (Fig. 2 of Kushnir et al., 2017). In
Eq. (6.1), we have expressed the various factors such that T0 contains all the spin-
independent terms.
The above result (Eq. 6.1) assumes that tidally excited IGWs dissipate com-
pletely as they propagate outwards towards the stellar surface. Such dissipation
can happen either through radiative damping or nonlinear effects. Recent hydro-
dynamical simulations of the IGW breaking process in the stellar envelope [Su
et al., 2020] show that the nonlinear damping of the outward-propagating IGWs
is due to the development of a narrow critical layer. This critical layer divides
the star into a asynchronously-rotating interior and a synchronously-rotating ex-
terior [see Goldreich and Nicholson, 1989], and it efficiently absorbs the angular
momentum of the incident IGWs (∼ 70% of incident flux, Su et al., 2020). If the
168
IGWs instead reflect before dissipating completely, then standing waves are set up
in the stellar interior. These internal oscillations then dissipate due to radiative
damping and nonlinear mode couplings (for the former, see e.g. Lai, 1996, 1997,
Kumar and Quataert, 1997, 1998, and for the latter, see e.g. O’Leary and Burkart,
2014). Except when a tidal forcing frequency is resonant with a stellar oscilla-
tion mode, the traveling-wave assumption represents an upper bound on the tidal
torque (see e.g. Yu et al., 2020b, 2021 concerning tidal dissipation in white dwarf
binaries).
Note additionally that we use Eq. (6.1) from Kushnir et al. [2017] instead of
the classic expression from Zahn [1975], which( is given by:2 5 )8/3
T(Zahn)
3 GM2R ω
circ (ω)= E2 a6 2 p , (6.4)GM/R3
where M and R are the mass and radius of the MS star and E2 is a numerical
parameter obtained by integrating over the entire star. The fitting formula E2 =
1.592×10−9 (M/M⊙)2.84 as given by Hurley et al. [2002] is commonly used, which
varies by many orders of magnitude for different stars. Moreover, T(Zahn)circ depends
on M and R, properties of the entire star, whereas the tidal torque is entirely
generated at the RCB. For these reasons, the expression by Kushnir et al. [2017]
is preferred.
6.2.2 Eccentric Binaries
The gravitational potential of an eccentric companion at the quadrupole order can
be decomposed as a sum over circular orbits [e.g. Storch and Lai, 2013, Vick et al.,
169
2017]:
2
U (r, t)=
∑
U2m (r, t) , (6.5)
m=−2
2
U2m (r, t)=−GM2W2mr3 Y2m(θ,φ)e−imf(t),D(t)
=−GM2W
2
2mr ( ) ∑∞Y θ,φ F e−iNΩt3 2m Nm . (6.6)a N=−∞
Here, the coordinate system is centered on the MS star, (r,θ,φ) are the radial,
p
polar, and azimuthal coordinates of r respectively, W2±2 = 3π/10, W2±1 = 0, W20 =
−pπ/5, D(t) is the instantaneous distance to the companion, f is the true anomaly,
and Ylm denote the spherical harmonics. FNm denote the Hansen coefficients for
l = 2 [also denoted X N2m in Murray and Dermott, 1999], which are the Fourier
coefficients of the perturbing function, i.e.
a3
e−
∞
imf(t) = ∑3 FNme−iNΩt. (6.7)D(t) N=−∞
The FNm can be written explicitly as an integral over the eccentric anomaly [Mur-
ray and Dermott, 1999, Storc∫h and Lai, 2013]:π
= 1 cos[N (E− esinE)−mf (E)]FNm dE. (6.8)
π (1− ecosE)2
0
By considering the effect of each summand in Eq. (6.5), the total torque on
the star, energy transfer in the inertial frame, and energy transfer in the star’s
corotating frame (which is also the tidal heating rate) can be obtained [Storch and
Lai, 2013, Vick et al., 2017]:
= ∑∞T F2N{2(Tcirc)(NΩ−2Ωs) , (6.9)N=−∞
1 ∑∞ W 2
Ė 20in = NΩF2N0T}circ (NΩ)2 N=−∞ W22
+NΩF2N2Tcirc (NΩ−2Ωs) , (6.10)
Ėrot = Ėin−ΩsT. (6.11)
170
Here, dots indicate time derivatives.
Equations (6.9–6.10) can be used to express the binary orbital decay and cir-
cularization rates using
ȧ =− 2aĖin , (6.12)
a GMM2
ėe
− 2 =−
aĖin + T , (6.13)
[ 1 e G]MM2 Lorb
where Lorb =MM2 Ga(1− e2)/(M+M ) 1/22 is the orbital angular momentum. The
stellar spin synchronization rate can also be computed assuming that the star
rotates rigidly:
= TΩ̇s 2 , (6.14)kMR
where kMR2 is the moment of inertia of the MS star.
6.3 Analytic Evaluation of Tidal Torque and Energy Trans-
fer Rates
We can combine the expressions given in Sections 6.2.1 and 6.2.2 to compute the
torque and energy transfer rate due to dynamical tides in an eccentric binary.
The tidal torque is obtained by evaluating Eq. (6.9) with the circular torque set to
Eq. (6.1), giving:
N∑=∞ ( )∣ ∣= 2 2Ωs ∣∣ 2Ω 8/3T F s ∣∣N2T0 sgn N− ∣N− ∣ . (6.15)
N=−∞ Ω Ω
The energy transfer rate in the inertial frame is obtained by evaluating Eq. (6.10)
in the same way, giving: [ ∣ ∣
= T0 ∑∞ 2 − ∣∣∣ − 2Ω ∣∣∣8/3Ėin ( NΩFN2sgn(
s
2 N=−∞)2 ]
N 2Ωs/Ω) N
Ω
+ W20 ΩF2 |N|11/3N0 . (6.16)W22
171
These two expressions can be used to obtain the orbital decay, circularization, and
spin synchronization rates using Eqs. (6.12–6.14).
While exact, the two sums in Eqs. (6.15–6.16) are difficult to evaluate for larger
eccentricities, where one often must sum hundreds or thousands of terms, each of
which has a different FNm. In the following, we obtain closed-form approximations
to Eqs. (6.15–6.16) when the eccentricity is large.
6.3.1 Approximating Hansen Coefficients
To simplify Eqs. (6.15–6.16), we seek tractable approximations for both FN2 and
FN0. Note that while the Hansen coefficients can be evaluated using the integral
expression Eq. (6.8), this requires calculating a separate integral for each N. In-
stead, it is more convenient to use the discrete Fourier Transform of the left hand
side of Eq. (6.7) to calculate arbitrarily many N at once [as pointed out by Correia
et al., 2014]. Since F(−N)m = FN(−m), we will only study the Hansen coefficient
behavior for m≥ 0.
m= 2 Hansen Coefficients
Figure 6.1 shows FN2 vs. N when e = 0.9. First, we note that FN2 is much larger
when N ≥ 0 than for N < 0, so we focus on the behavior for N ≥ 0. Here, FN2 has
only one substantial peak. There are only two characteristic frequency scales: Ω
and Ωp, the pericentre frequency, defined byp
≡ 1+ eΩp Ω − 3/2 . (6.17)(1 e)
For convenience, we also define Np as the floor of Ωp/Ω, i.e.
Np ≡ ⌊Ωp/Ω⌋, (6.18)
172
We find that the peak of the FN2 occurs at N ∼ Np, the only characteristic scale
in N over which FN2 can vary. When N ≫ Np, the Fourier coefficients must fall
off exponentially by the Paley-Wiener theorem, as the left hand side of Eq. (6.7) is
smooth [e.g. Stein and Shakarchi, 2009]. When instead N ≪Np, there are no char-
acteristic frequencies between Ω and Ωp, so we expect the Hansen coefficients to
be scale-free between N = 1 and Np, i.e. a power law in N. The expected behaviors
in both of these regimes are in agreement with Fig. 6.1.
Motivated by these considerations, we approximate the Hansen coefficients byC N pe−N/η2 N ≥ 0,
FN2 ≈
2
(6.19)
0 N < 0,
for some fitting coefficients C2, p, and η2. By performing fits to FN2 for varying
eccentricities, we find that p ≈ 2 for modest-to-large eccentricities, and we take
p = 2 to be fixed for the remainder of this work. For e = 0.9, Fig. 6.1 shows that
p= 2 accurately captures the power-law behavior of the FN2 for N ≲ Nmax, where
Nmax is the value of N for which FN2 is maximized. While FN2 deviates apprecia-
bly from this power law for N ≪ Nmax, these terms are small in magnitude and
are sub-dominant in any sum over the FN2. Thus, our approximation for FN2 is
sufficiently accurate when evaluating expressions that involve sums over the FN2,
such as in Eqs. (6.15–6.16)1.
To constrain the remaining two free parameters η2 and C2, we use the well
1There is good reason to expect p= 2 for N ≲ Np as long as the eccentricity is sufficiently large.
In this regime, we first note that the left-hand side of Eq. (6.7) resembles the second derivative
of a Gaussian with width ∼Ω−1p , as it is both sharply peaked about t = 0 and has zero derivative
three times every period (at t = ϵ, t = P/2, and t = P − ϵ for some small ϵ ∼ Ω−1p ). Secondly, we
note that a Gaussian resembles a Dirac delta function over timescales longer than its width (here,
∼Ω−1p ). Since the Dirac delta function has a flat Fourier spectrum (∝N0), and time differentiation
multiplies by N in frequency space, the second derivative of a Gaussian has a Fourier spectrum
∝ N2 for N ≲ Np. As FN2 is the Nth Fourier coefficient for the left-hand side of Eq. (6.7), we do
indeed expect FN2 ∝N2 for N ≲ Np.
173
known Hansen coefficient moments [e.g. Eqs. 22, 23 of Storch and Lai, 2013]
∑∞ f
F2N2 = ( 5 )9/2 , (6.20)
N=−∞ 1− e2
3e4
∑ f5 ≡ 1+3e
2+ , (6.21)
8
∞ 2 f
F2N2N = ( 2 ) , (6.22)
N=−∞ 1− e2 6
≡ + 15e
2 45e4 5e6
f2 1 + + . (6.23)2 8 16
Applying Eq. (6.19) to Eqs. (6.20, 6.22), we obtain the coefficients η2 and C2
= ( 4 f2η2 [ )
1
5 f 1− e2 3/2
∼
]2(1− e)3/2
, (6.24)
5
1/2
C = ( 4 f52 )9/2 . (6.25)3 1− e2 η52
Figure 6.1 illustrates the agreement of Eq. (6.19) using these two values of η2
and C2 with the numerical FN2. The good agreement for N ≳ Np/10 (which is
where FN2 is large) is especially impressive as there are no fitting parameters in
Eq. (6.19), as C2, η2, and p are all analytically constrained. Finally, note that the
maximum of the FN2 occurs at N = ⌊2η2⌋ ∼ (1− e)−3/2 ∼Np.
m= 0 Hansen Coefficients
We now turn to the m = 0 Hansen coefficients, FN0, which are shown in Fig. 6.2.
We know that FN0 = F(−N)0, so we consider only N ≥ 0. From the figure, we see
that the FN0 decay exponentially. There is only one characteristic scale available
for this decay, namely Np. Therefore, we naturally assume the FN0 coefficients
can be approximated by a function of form:
F =C e−|N|/ηN 00 0 . (6.26)
174
Figure 6.1: Plot of Hansen coefficients FN2 for e = 0.9. The red circles denote
negative N, while the black circles and crosses denote positive and negative FN2.
The blue line is the formula given by Eq. (6.19) with p = 2 and with η2 and C2
given by Eqs. (6.24–6.25).
The two free parameters C0 and η0 are constrained by the identities
∑∞ f
F2 5N0 = ( ) , (6.27)
∑N=−∞ 1− e2
9/2
∞ 9e2
F2 2N0N = ( )
=−∞ 2 1− e2 15/2
f3, (6.28)
N
= 1 + 15e
2
+ 15e
4
+ 5e
6
f3 . (6.29)2 8 16 128
Applying Eq. (6.26) to Eqs. (6.27–6[.28), we obta]in
( 9e2= f) 1/23η0 3 , (6.30)[ 1− e2 f5 ]
C0 = ( 1/2f5)9/2 . (6.31)1− e2 η0
Figure 6.2 illustrates the agreement of Eq. (6.26) using these two values of η0 and
C0with the numerically computed FN0.
175
Figure 6.2: Plot of FN0 (black circles) for e= 0.9. Since FN0 = F(−N)0, we only show
positive N. The blue line is given by Eq. (6.26) with η0 and C0 given by Eqs. (6.30–
6.31).
6.3.2 Approximate Expressions for Torque and Energy Trans-
fer
Having found good approximations for the Hansen coefficients, we now apply
them to simplify the expressions for the torque and the energy transfer rate in
Eqs. (6.15–6.16).
Tidal Torque
To simplify Eq. (6.15), we replace FN2 with Eq. (6.19) and the sum with an inte-
gral, obtaining ∫ ∞
T ≈T0 C22N4e−2N/η2 sgn(N−2Ω /Ω) |N−2Ω /Ω|8/3s s dN. (6.32)
0
176
This expression is already easier to evaluate than Eq. (6.15), but we can use
further approximations to obtain a closed form. We first analyze Eq. (6.32) in the
small-spin limit, where it can be integrated analytically2, giving
( ) f (e)(η /2)8/3 Γ(23/3)
T | 5 2Ωs|≪Ωp ≃T0 (1− e2 9/2 . (6.33)) 4!
Note that this has the scaling T ∼ T0 (1− e)−17/2 ∼ TpΩ/Ωp, where Tp is the torque
exerted by a circular orbit with separation equal to the pericentre separation ap ≡
a(1− e), i.e.  8/3
2
= GM2r
5 √ ( )c Ωp  ρc − ρ 2T β 1 c ∼T (1− e)−10p 2 0 . (6.34)
a6p GM 3 ρ̄c ρ̄cc/rc
The top panel of Fig. 6.3 compares Eq. 6.33 to the integral of Eq. (6.32) and
to the direct sum of Eq. (6.15) as a function of the eccentricity. It can be see that
both the integral and the analytic closed form perform well for moderate-to-large
eccentricities, but both over-predict the torque at small e ≲ 0.2 (at e = 0.2, the
torque is over-predicted by ∼ 20%). This discrepancy is expected: there are only
a few non-negligible summands in Eq. (6.15) when e is small, so replacing the
sum over N with an integral is expected to introduce significant inaccuracy that
appears in both the integral and closed-form expressions.
∣ ∣
Eq. (6.33) is valid so long as |Ωs/Ω|≪Nmax (or equivalently, ∣Ωs ≪Ωp∣), where
Nmax = 10η2/3 is where the integrand is in Eq. (6.32) maximized. If instead
|Ωs/Ω| ≫ Nmax, the torque can be evaluated directly using Eq. (6.15) and the
known Hansen coefficient moments, giving:
( )
T |Ωs|≫Ωp ≃− f (e)T0 sgn(Ωs) |2Ω 8/3 5s/Ω| (1− e2)9/2 . (6.35)
The bottom panel of Fig. 6.3 compares this formula to the integral of Eq. (6.32) and
to the direct sum of Eq. (6.15) as a function of the ec∑centricity, where Ωs/Ω= 400.2The key to the success of our approach is that su∫ms of form ∞n=−∞F2N2N p can be approximated
for non-integer p in terms of the Γ function, since ∞ xp e−x0 dx=Γ(p−1).
177
Here, Nmax ≪ 400 for all eccentricities shown. We see that direct summation, the
integral expression, and Eq. (6.35) agree very well for all eccentricities.
Having obtained closed-form expressions of Eq. (6.32) for small and large spins,
we can further derive a single expression joining these two limits. To do this, we
first assume that the spin is small but non-negligible. In this regime, we make the
approximation ( )
N 2γ Ω
N−2 T sΩs/Ω≃ NN max− , (6.36)max Ω
for some free parameter γT . With this, we can integrate Eq. (6.32) in closed form.
We can fix γT by requiring our expression reproduce the large spin limit (Eq. 6.35)
when taking |Ωs| → ∞. This procedure gives an expression for the torque that
agrees with both limiting forms (Eq.(6.33–6.35))a∣∣nd i(s given by )∣f (e)(η /2)8/3 Ω ∣ 4 Ω ∣8/3T ≃T 5 2 s0 − 2 9/2 sgn 1−γT ∣ 1− sγT ∣∣ , (6.37)(1 e ) η2Ω γT η2Ω
where ( )
= 4!
3/8
γT 4 ≈ 0.691. (6.38)
Γ(23/3)
Figure 6.4 compares this expression to the integral of Eq. (6.32) and to the direct
sum of Eq. (6.15) at fixed e = 0.9 and varying Ωs. We see that Eq. (6.37) agrees
well with the integral and sum for both large and small spins, and is also reason-
ably accurate for intermediate spins. However, Eq. (6.32) is more accurate than
Eq. (6.37) when T changes signs and |T| is small. This is also expected: T changes
signs when the spin approaches pseudosynchronization (Section 6.3.2) because
large contributions to the sum in Eq. (6.15) have opposite signs and mostly cancel
out. Thus, small inaccuracies in the summand result in significant discrepancies
in the total torque. The integral approximation, Eq. (6.32), is expected to be in
good agreement with the direct sum, Eq. (6.15), as the accuracy of the Hansen co-
efficient approximation in Section 6.3.1 is good for large eccentricities, thus guar-
anteeing term-by-term accuracy. On the other hand, the closed-form expression,
178
Eq. (6.37), is more approximate whenΩs/Ω∼Nmax ∼ η2. In fact, Eq. (6.37) predicts
dT/dΩs ≈ 0 near pseudosynchronization. This is not accurate and is an artifact of
our factorization ansatz in Eq. (6.36).
In summary, the tidal torque must be evaluated with explicit summation (Eq. 6.15)
when e ≲ 0.2 (see discussion after Eq. 6.33), can be approximated by the integral
expression (Eq. 6.32) when e is large for all values of Ωs, and otherwise can be
approximated by the closed-form expression given by Eq. (6.37). Recall that when
e is small, the explicit summation of Eq. (6.15) is quite simple, as good accuracy
can be obtained with just the first few terms in the summation (e.g. for e = 0.2,
including just the three terms N ∈ [2,4] results in 98% accuracy).
Pseudosynchronization
In general, the exact torque as given by Eq. (6.15) vanishes for a single Ωs, which
we call the pseudo-synchronized spin frequency. An approximation for the pseudo-
synchronized spin can be directly read off from Eq. (6.37):
Ωps = η2 = 4 f2((e) )3/2 . (6.39)Ω γT 5γT f5(e) 1− e2
This has the expected scaling Ωps ∼ Ωp (the pericentre orbital frequency). Fig-
ure 6.5 compares this prediction for the pseudo-synchronized spin to the exact one
obtained by applying a root finding algorithm to Eq. (6.15). We see that Eq. (6.39)
is a reasonable approximation for the pseudo-synchronized spin frequency when
e ≳ 0.1.
In passing, we note that, in the standard weak friction theory of equilibrium
tides, the pseudo-synchronized spin is given by [Alexander, 1973, Hut, 1981]
(Eq)
Ωps = (f2(e) ) . (6.40)
Ω f5(e) 1− e2 3/2
179
Figure 6.3: The tidal torque on a non-rotating (top) and rapidly rotating (bottom)
star due to a companion with orbital eccentricity e. Black circles represent direct
summation of Eq. (6.15), green crosses are evaluated using the integral approx-
imation Eq. (6.32), and the blue line is Eq. (6.37). In the small and large spin
limits, Eq. (6.37) reduces to Eq. (6.33) and Eq. (6.35) respectively.
180
Figure 6.4: The tidal torque as a function of the stellar spin for a highly eccentric
e = 0.9 companion. The black circles represent direct summation of Eq. (6.15),
green crosses the integral approximation (Eq. 6.32), and the solid line the analytic
closed-form expression Eq. (6.37). The spin is normalized by the pericentre orbital
frequency Ωp ≈ 43Ω (Eq. 6.17).
181
Figure 6.5: The spin frequencies Ωps normalized by the pericentre frequency Ωp
(Eq. 6.17) as a function of eccentricity. The blue line is given by Eq. (6.39). The
black line shows the exact solution, obtained by using a root finding algorithm
to solve for the zero of Eq. (6.9). The red dashed line shows the spin frequency
predicted by the weak friction theory of equilibrium tides (Eq. 6.40).
182
Though describing a different tidal phenomenon, this only differs from Eq. (6.39)
by a factor of 4/(5γT) ≈ 1.15. We show it for comparison as the red dotted line in
Fig. 6.5.
Energy Transfer
We now turn our attention to Eq. (6.10) and replace FN2 and FN0 with their re-
spective approximations (Eqs. 6.19 and 6.26) to obtain the energy transfer rate
∫∞[
Ė = T0Ω C2N5e−2N/η2in 2 sgn(N−2Ωs/Ω) |N−2Ωs/Ω|8/32 ( 0 )
+ W
2 ]
2 20 C2e−2N/η00 N
11/3 dN. (6.41)
W22
We evaluate the m= 2 and m= 0 contributions to this expression separately.
We first examine the m = 2 contribution using the same procedure in Sec-
tion 6.3.2 for the torque. If the spin is moderate, i.e. |Ωs/Ω| ≲ Nmax where now
Nmax = 23η2/6, we make the approximatio(n )
− ≃ N − 2γEΩN 2Ωs/Ω N smax , (6.42)Nmax Ω
where γE is a free parameter. This lets us integrate the m = 2 contribution to
Eq. (6.41) analytically. We constrain γE by requiring agreement with the large-
spin limit: for |Ωs/Ω|≫Nmax, we have
( ) T
Ė(m=2) | |≫ ≃− 0Ω 2 f (e)in Ωs Ωp sgn(Ωs)|2 / |8/3
2
Ω Ω
2 s (1− 2 6 . (6.43)e )
This fixes γE and we obtain the complete m= 2 contribution to Eq. (6.41):
= T Ω f (e)(η /2)11/3Ė(m 2) 0 5 2in ≃ 2((1− e2)9/2 )∣
× − Ωs ∣sgn 1 γE ∣∣ ( )∣4 Ω ∣8/31− sγE ∣∣ , (6.44)η2Ω γE η2Ω
183
where ( )
5! 3/8
γE = 4 ≈ 0.5886. (6.45)
Γ (26/3)
Note that, when Ωs ≈ 0, Eq. (6.44) gives the expected scaling of Ė(m=2)in ∼ TpΩ
where Tp is the torque exerted by a circular companion at the pericentre separa-
tion, given by Eq. (6.34).
The m= 0 contribution to Eq. (6.41) can be straightforwardly integrated using
the parameterization Eq. (6.26). The sum of the two contributions then gives the
total energy transfer rate: [
= T0Ω f5(e)(η2/2)
11/3
Ėin 2 ( (1− e2)9/2)∣ ( )∣
× − Ω ∣sgn 1 sγE ∣( ) ∣
4 8/3
( 1−
Ωs
γE ∣∣
η Ω γ ∣2 E η2Ω
f (e)Γ(14/3) 3 8/3 2
)
e f (e) 11/6
]
+ 5 3− 2 10 . (6.46)(1 e ) 2 f5(e)
The two panels of Fig. 6.6 compare this expression with the integral form
Eq. (6.41) and the direct sum Eq. (6.16) for small and large spins. The agree-
ments are excellent except that, when the spin and eccentricity are small, both
the integral and closed form expressions over-predict the energy transfer rate.
Figure 6.7 compares these three expressions as a function of spin for e = 0.9. The
performance of Eq. (6.46) degrades when the system is near pseudosynchroniza-
tion (Ωs ≃ Ωp), but generally captures the correct behavior of the exact result,
while Eq. (6.41) is accurate for all spins. As was the case with the tidal torque,
we see that the evaluation of the energy transfer rate when e ≲ 0.2 requires direct
summation (Eq. 6.16), and the evaluation when e is substantial but the spin is
near pseudosynchronization can be performed using the integral approximation
(Eq. 6.41), and otherwise the evaluation can be performed using the closed-form
expression (Eq. 6.46).
184
Figure 6.6: The tidal energy transfer rate Ėin for a non-rotating (top) and a rapidly
rotating (bottom) star. The black circles represent direct summation of Eq. (6.16),
green crosses the integral form Eq. (6.41), and the blue line the closed-form ex-
pression Eq. (6.46).
185
Figure 6.7: The tidal energy transfer rate Ėin as a function of spin [normalized
by Ωp; Eq. (6.17)] for a highly eccentric e = 0.9 companion. The black circles rep-
resent direct summation of Eq. (6.10), green crosses the integral approximation
Eq. (6.41), and the blue line the analytic closed form Eq. (6.46).
6.4 PSR J0045-7319/B-Star Binary
As an application of our analytical results above, we consider the PSR J0045-
7319/MS binary system [Kaspi et al., 1994] and attempt to explain its orbital decay
via dynamical tides. This system is one of the few pulsar binaries discovered so
far that have massive MS companions [the other known binaries are, PSR B1259-
63, PSR J1740-3052, PSR J1638-4725, J2032+4127; Johnston et al., 1992, Stairs
et al., 2001, Lorimer et al., 2006, Lyne et al., 2015]. These systems evolve from
MS-MS binaries when one of the stars explodes in a supernova to form a neutron
star (NS), and will eventually become double NS systems when the MS companion
also explodes to form a NS. Thus, characterizing the evolution of such systems
is important for understanding the formation of double NS binaries [e.g. Tauris
et al., 2017]. The PSR J0045-7319 system contains a radio pulsar (M2 ≃ 1.4M⊙)
186
and a massive B star (M ≃ 8.8M⊙) companion in an eccentric (e= 0.808) orbit with
period P = 51.17 days, corresponding to a semi-major axis a = 126R⊙. Timing
observation shows that the orbit is decaying at the rate Ṗ =−3.03×10−7, or P/Ṗ ≃
−0.5 Myr [Kaspi et al., 1996]. The measured B-star luminosity L= 1.2×104L⊙ and
surface temperature Tsurf = (24000± 1000) K imply a radius of R = 6.4R⊙ [Bell
et al., 1995]. The B star has a projected surface rotation velocity vsin i = 113±
10 km/s [Bell et al., 1995], consistent with rapid rotation (the breakup velocity is
p
GM/R = 512 km/s). While there are uncertainties in some of these parameters
(e.g., the pulsar and companion masses could be larger by about 10%; see Thorsett
and Chakrabarty, 1999), we adopt the above values in our calculation below for
better comparison with previous works.
Lai [1996] and Lai [1997] explained the observed orbital decay of the PSR
J0045-7319 binary in terms of tidal excitations of discrete (rotation-modified) g-
modes followed by radiative damping. He showed that retrograde rotation of the
B-star [consistent with the observed nodal precession of the binary orbit; see Lai
et al., 1995, Kaspi et al., 1996] can significantly enhance the strength of mode ex-
citation, and potentially account for the observed orbital decay, although he did
not compute the mode damping rate quantitatively. Kumar and Quataert [1997,
1998] examined the damping of g-modes and showed that radiative damping in
rigidly rotating B-star is inadequate to explain the observed orbital decay rate,
suggesting that in addition to retrograde rotation, significant differential rota-
tion or nonlinear parametric mode decay may be required. Given the uncertain-
ties in their estimates, it was not clear to what extent the observed orbital decay
rate can be quantitatively explained. Moreover, both Lai et al. [1995] and Ku-
mar and Quataert [1997, 1998] considered discrete g-modes, whereas absorption
of tidally excited gravity waves near the stellar envelope, via either efficient radia-
tive damping or nonlinear breaking [see Su et al., 2020], implies outgoing waves
187
(as adopted in the works of Zahn, 1975, Goldreich and Nicholson, 1989, Kushnir
et al., 2017) rather than discrete g-modes.
We now consider the evolution of stellar spin and binary orbit for the PSR
J0045-7319 system. For the adopted system paramet√ers above, the pericentre dis-p
tance is ap ≃ 3.78R, and pericentre frequency isΩp = (1+ e)GMtot/a3p ≃ 0.20 GM/R3.
First we note that the tidal torque T and the energy transfer rate Ėin (dominated
by the m= 2 term) are related by
≃ fĖin T 2Ωp , (6.47)5(1+ e)2 f5
with f2 = 8.38, f5 = 3.12. Thus the spin evolution rate (for spin-orbit synchroniza-
tion and alignment) is given∣by∣∣∣ ∣∣∣∣ − 2 ∣∣∣∣ ∣Ṡ ≃ L 5(1 e ) f5 Ėin ∣∣∣ , (6.48)S S 2 f2 Eorb
where S = kMR2Ω̄s is the spin angular momentum of the star (with Ω̄s the mean
rotation rate, and k ≃ 0.1), L is the orbital angular momentum, and Eorb is the
orbital energy. Thus ∣∣∣∣ Ṡ ∣∣∣∣ ∣ ∣≃ Ω6.3 p ∣∣∣ ȧ ∣∣∣ . (6.49)S Ω̄s a
The observed nodal precession of the PSR J0045-7319 binary implies that the spin
of the B-star is far from spin-orbit alignment and synchronization driven by tides
p
[Lai et al., 1995]. Thus we require Ω̄s ≳ 6.3Ωp ∼ GM/R3, suggesting that the
internal rotation rate of the star is much larger that the surface rate.
Using Eq. (6.46) with e= 0.808 (and keeping only the m= 2 term), we find that
the orbital decay ra∣te is given by∣∣∣ ȧ ∣∣∣∣ ( )4/3≃ M M1.6(×10
5 2 totΩ
a (M M) c
× r
) 2
c 9 ρc − ρ1 c ∣∣∣∣ ( )∣4 γ Ω 8/31− E sa ρ̄ ρ̄ γ η ∣∣∣ . (6.50)c c E 2Ω
188
With η2 = 10.55, ∣a≃∣19.7R, and(adopt)ing(ρc/ρ̄c ≃)1/3, we find the orbital decay rate∣∣∣ Ṗ ∣∣∣ 1 3M⊙ 4/3≃ rc 9 ∣∣∣1− 0.89ΩsP 0.5Myr M 0.54R ∣ Ω ∣∣∣
∣8/3
. (6.51)
c p
NoteΩs in the above expression refers to the rotation rate at the radiative-convective
boundary (RCB) rc, and we expect Ωs ≳ Ω̄s ≳ 6Ωp.
To explain the observed orbital decay timescale |P/Ṗ| ≃ 0.5 Myr, the most crit-
ical parameter is the core radius rc. Kumar & Quataert (1998) adopted Mc ≃ 3M⊙
and rc ≃ 1.38R⊙ (or rc ≃ 0.22R) based on comparison with a Yale stellar evolution
p
model. This value of rc would require Ωs/Ωp ≃ −24, or Ωs ≃ −4.8 GM/R3, to ex-
plain the observed decay rate. If we use a slightly larger core radius, rc = 1.5R⊙ ≃
p
0.234R, the required Ωs would be Ωs ≃ −18Ωp ≃ −3.6 GM/R3. In either case,
extreme differential rotation (with |Ωs| at the RCB at least a factor of few larger
than the surface rotation) is required. In Fig. 6.8 we show Ṗ/P as a function of
Ωs, obtained using the exact expression Eq. (6.46) for four different values of rc.
p
In general, given that the surface rotation rate of the star is less than GM/R3,
our calculation suggests that the B-star must have large, retrograde differential
rotation in order to explain the observed orbital decay.
We note that our own exploration of stellar models using the Modules for Ex-
periments in Stellar Astrophysics [MESA; Paxton et al., 2011, 2013, 2015, 2018,
2019] generally finds a small core radius (rc ≲ R⊙) for the B-star in the PSR J0045-
7319 system. Since the system is in the Small Magellanic Cloud, we use a typical
metallicity Z = 0.1Z⊙, where Z⊙ is the solar metallicity. For a range of initial stel-
lar masses, we begin with a non-rotating zero-age main sequence (ZAMS) stellar
model, and evolve it until its core hydrogen is depleted. Among these evolutionary
tracks, we select the model and stellar age that best matches the observed L and
Tsurf and record the core radius at that age. We then repeat this procedure with
different values for the convective overshoot parameter, initial rotation rate (up to
189
nearly maximally rotating), and metallicity (up to Z = 0.2Z⊙). For all parameter
combinations, the predicted stellar masses lie in the range 8.8M⊙ ≤ M ≤ 10M⊙,
in agreement with the estimates in the literature. However, the core radius for
all stellar models is rc ≲ R⊙ ≈ 0.16R. This reflects the fact that our best-fitting
stellar models tend to be somewhat evolved from the ZAMS. For further details,
see AppendiX E.1.
We caution that applying standard isolated stellar evolution to the B-star in
the PSR J0045-7319 system can be fraught with uncertainties. The very rapid
rotation may induce instabilities and allow the star to transport material from the
hydrogen-rich envelope into the central burning region and vice versa, potentially
making the convective core larger [e.g. Maeder, 1987, Heger and Langer, 2000].
This effect may be enhanced by the large misaligned, differential rotation. Related
to the rotational effect is the mass transfer effect: The observed rapid orbital decay
rate and general orbital evolution modeling sets an upper limit of about 1.4 Myr
on the age of the binary system since the last supernova explosion [Lai, 1996],
much shorter than the canonical MS lifetime. Thus the B-star must have accreted
significant material from the pulsar progenitor in the recent past. Such accretion
can affect the structure and evolution of the B-star in a significant way. Given
these uncertainties, we think it is likely that the B-star has a larger core radius
than the canonical value, rendering a less extreme differential rotation3.
3We note that a recent study of intermediate and high-mass eclipsing binaries suggests that
convective core masses are underpredicted by stellar structure codes [Tkachenko et al., 2020a].
190
Figure 6.8: The orbital decay ratio Ṗ/P as a function of Ωs for the canonical pa-
rameters of the PSR J0045-7319 binary system, as evaluated by Eq. (6.46), for four
different values of rc (legend, in units of R⊙). The measured Ṗ/P = − (0.5 Myr)−1
is shown by the horizontal dashed line. The vertical shaded region is the region
where |Ωs| is less than the breakup rotation rate of the star as a whole, given by
(GM/R3)1/2. Each rc is only shown for |Ωs| ≤Ωs,c ≡ (GM /r3)1/2c c , the core breakup
rotation rate, and the colored crosses denote where Ωs =Ωs,c for each value of rc.
Note that for rc ≲ 1.3R⊙, even a maximally rotating core cannot generate enough
tidal torque to match the observed Ṗ/P.
191
6.5 Summary and Discussion
6.5.1 Key Results
The main goal of this paper is to derive easy-to-use, analytical expressions for
the tidal torque T and tidal energy transfer rate Ėin due to internal gravity wave
(IGW) dissipation in a massive, main-sequence (MS) star under the gravitational
influence of an eccentric companion. Tidal evolution in such systems plays an
important role in the formation scenarios of merging neutron-star (NS) binaries.
For general eccentricities, these expressions are given by the sums, Eqs. (6.15)
and (6.16), respectively. However, when the eccentricity is large, these sums re-
quire the evaluation of many terms to be accurate. We have derived approximate
expressions in two regimes:
• For e ≳ 0.2, we show that the tidal torque and energy transfer rate can be
accurately approximated by the integral expressions, Eqs. (6.32) and (6.41).
• If furthermore the spin of the stellar core is not very close to its pseudo-
synchronized value (i.e. where the torque vanishes; see Section 6.3.2), we
show that these two integral expressions can be well approximated by the
closed-form expressions, Eqs. (6.37) and (6.46).
These analytical expressions for T and Ėin (particularly the closed-form expres-
sions) can be easily applied to study the spin and orbital evolution of eccentric
binaries consisting of massive MS stars, such as those found in the evolution sce-
narios leading to the formation of merging neutron star binaries.
We then apply our analytical expressions to the PSR J0045-7319 binary sys-
tem, which has a massive B-star companion and a measured orbital decay rate.
192
This system provides a rare example to test/calibrate our theoretical understand-
ing of IGW-mediated dynamical tides in massive stars. We show that for the “stan-
dard” radius of the convective core based on isolated non-rotating stellar modeling,
the B-star must have a significant retrograde and differential rotation (with the
rotation rate at the convective-radiative boundary at least a few times larger than
the surface rotation rate of the star). Alternatively, we suggest that the convec-
tive core radius may be larger than the standard value as a result of rapid stellar
rotation and/or mass transfer to the B-star in the recent past during the post-MS
evolution of the pulsar progenitor. Overall, our attempt to explain the rapid orbital
decay of the PSR J0045-7319 binary highlights the critical importance of internal
stellar structure, particularly the size of the convective core, in determining the
tidal evolution of eccentric binaries containing massive MS stars.
6.5.2 Caveats
Concerning our analytical expressions for the tidal torque and energy transfer
rate, we note a few potential caveats:
(i) For simplicity, we have assumed that the stellar spin and orbit axes are
aligned or anti-aligned. For general stellar obliquities (as in the case of the PSR
J0045-7319 binary system), it is possible to decompose the tidal force into different
Fourier components (see Appendix A of Lai, 1997), and this would be unlikely to
yield qualitatively different result in terms of the orbital decay rate [cf. Lai, 2012].
(ii) More importantly, in our work we have assumed that all Fourier harmonics
(with different forcing frequencies NΩ in the inertial frame, where Ω is the mean
orbital frequency) of the tidal potential excite IGWs at the radiative-convective
boundary that damp completely as they propagate towards the stellar surface.
193
This is known to be the case when the normalized tidal forcing frequency ω (equal
p
to NΩ− 2Ωs for m = 2 and NΩ for m = 0) satisfies |ω| ≪ GM/R3 due to effi-
cient radiative damping (Zahn, 1975, Kushnir et al., 2017; see also the review
paper Ogilvie, 2014). Note that the dominant Fourier harmonics of the tidal po-
tential have NΩ ∼ Ωp (the orbital frequency at the pericentre). Thus, as long
p
as Ωp, |Ωs| ≪ GM/R3, the assumption of outgoing IGWs inherent in our the-
p
ory is valid. When |ω| is comparable to GM/R3, it is traditionally thought that
IGWs set up standing modes in the stellar interior. This is because IGWs are
evanescent sufficiently near stellar surface, where the pressure scale height be-
comes smaller than the radial wavelength of the wave [Goldreich and Nicholson,
1989]. Moreover, Su et al. [2020] found that sufficiently large-amplitude IGWs
can spontaneously cause a critical layer to form due to nonlinear wave breaking,
causing incident IGWs to be efficiently absorbed. After such a critical layer forms,
it can propagate to deep within the stellar interior and efficiently absorb IGWs
even before they reach breaking amplitudes. Wave breaking and other nonlinear
effects (such as nonlinear mode couplings) may be enhanced in an eccentric bi-
nary such as PSR J0045-7319, where IGWs with a wide range of frequencies are
excited. Burkart et al. [2012] found that the frequency cutoff for g-modes in the
eccentric heartbeat star system KOI-54 (with eccentricity e = 0.83, orbital period
P = 41.8 days, and stellar mass M ≈ 2.3M⊙) is in agreement with the frequency at
which IGWs become efficiently damped by thermal diffusion. However, in higher-
mass MS stars such as that in PSR J0045-7319, nonlinear wave breaking may be
an important source of IGW damping due to the larger tidal torque. More work is
needed to address whether our assumption of traveling IGWs in massive stars is
accurate.
194
CHAPTER 7
SPIN-ORBIT MISALIGNMENTS IN TERTIARY-INDUCED BLACK-HOLE
BINARY MERGERS: THEORETICAL ANALYSIS
Originally published in:
Yubo Su, Dong Lai, and Bin Liu. Spin-orbit misalignments in tertiary-induced bi-
nary black-hole mergers: Theoretical analysis. Phys. Rev. D, 103(6):063040, 2021a
7.1 Introduction
As LIGO/VIRGO continues to detect mergers of black hole (BH) binaries [e.g. Ab-
bott et al., 2016, 2019], it is increasingly important to systematically study various
formation channels of BH binaries and their observable signatures. The canonical
channel consists of isolated binary evolution, in which mass transfer and friction
in the common envelope phase cause the binary orbit to decay sufficiently that
it subsequently merges via emission of gravitational waves (GW) within a Hub-
ble time [e.g. Lipunov et al., 1997, 2017, Podsiadlowski et al., 2003, Belczynski
et al., 2010, 2016, Dominik et al., 2012, 2013, 2015]. BH binaries formed via iso-
lated binary evolution are generally expected to have small misalignment between
the BH spin axis and the orbital angular momentum axis [Postnov and Kuranov,
2019, Belczynski et al., 2020]. On the other hand, various flavors of dynamical
formation channels of BH binaries have also been studied. These involve either
strong gravitational scatterings in dense clusters [e.g. Zwart and McMillan, 1999,
O’leary et al., 2006, Miller and Lauburg, 2009, Banerjee et al., 2010, Downing
et al., 2010, Ziosi et al., 2014, Rodriguez et al., 2015, Samsing and Ramirez-Ruiz,
2017, Samsing and D’Orazio, 2018, Rodriguez et al., 2018, Gondán et al., 2018] or
more gentle “tertiary-induced mergers” [e.g. Liu and Lai, 2021, Liu and Lai, 2020,
Liu et al., 2019b,a, Liu and Lai, 2019, Blaes et al., 2002, Miller and Hamilton,
2002, Wen, 2003, Antonini and Perets, 2012, Antonini et al., 2017, Silsbee and
195
Tremaine, 2017a, Liu and Lai, 2017, 2018, Randall and Xianyu, 2018b, Hoang
et al., 2018]. The dynamical formation channels generally produce BH binaries
with misaligned spins with respect to the orbital axes.
GW observations of binary inspirals can put constraints on BH masses and
spins. Typically, the spin constraints come in the form of two dimensionless mass-
weighted combinations of the component BH spins: (i) the aligned spin parameter
≡ m1χ1 cosθs1l+m2χ2 cosθs2lχeff + , (7.1)m1 m2
where m1,2 are the masses of the BHs, θsil is the angle between the i-th spin
and the binary orbital angular momentum axis, and χi ≡ cS 2i/(Gmi ) is the dimen-
sionless Kerr spin parameter; and (ii) the perpendicular spin parameter [Schmidt
et al., 2015] {
≡ q (4q+
}
3)
χp max χ1 sinθs1l, χ sinθ , (7.2)4+3q 2 s2l
where q ≡ m2/m1 and m1 ≥ m2. The systems detected in the first and second ob-
serving runs (O1 and O2) of LIGO/VIRGO have χeff ∼ 0 (but see [Zackay et al.,
2019, Venumadhav et al., 2020] for exceptions). In the third observing run (O3) of
LIGO/VIRGO, two events exhibit substantial spin-orbit misalignment. In GW190412
[Abbott et al., 2020a], the two BH component masses are 29+5.0−5.3M⊙ and 8.4
+1.7
−1.0M⊙.
The primary (more massive) BH is inferred to have χ = 0.43+0.161 −0.26, and the effec-
tive spin parameter of the binary is constrained to be χ +0.09eff = 0.25−0.11, indicat-
ing a non-negligible spin-orbit misalignment angle. In GW190521 [Abbott et al.,
2020b], the two component BHs have masses of 85+21−14M⊙ and 66
+17
−18M⊙ and spins
of χ1 = 0.69+0.27 and χ = 0.72+0.24−0.62 2 −0.64. The binary’s aligned spin is χ +0.27eff = 0.08−0.36
while the perpendicular spin is χp = 0.68+0.25−0.37, again suggesting significant spin-
orbit misalignments.
Liu and Lai [2017, 2018, hereafter LL17, LL18], and Liu et al. [2019a] carried
196
out a systematic study of binary BH mergers in the presence of a tertiary com-
panion. LL17 pointed out the important effect of spin-orbit coupling (de-Sitter
precession) in determining the final spin-orbit misalignment angles of BH bina-
ries in triple systems. They considered binaries with sufficiently compact orbits
(so that mergers are possible even without a tertiary) and showed that the combi-
nation of LK oscillations (induced by a modestly inclined tertiary) and spin-orbit
coupling gives rise to a broad range of final spin-orbit misalignments in the merg-
ing binary BHs. We call these mergers LK-enhanced mergers. On the other hand,
LL18 considered the more interesting case of LK-induced mergers, in which an
initially wide BH binary (too wide to merge in isolation) is pushed to extreme ec-
centricities (close to unity) by a highly inclined tertiary and merges within a few
Gyrs. LL18 examined a wide range of orbital and spin evolution behaviors and
found that LK-induced mergers can sometimes exhibit a “90◦ attractor” in the
spin evolution: when the BH spin is initially aligned with the inner binary angu-
lar momentum axis (θsl,0 = 0), it evolves towards a perpendicular state (θ ◦sl,f ≃ 90 )
near merger (when the inner binary becomes gravitationally decoupled from the
tertiary, and θsl freezes). Qualitatively, they found that the attractor exists when
the LK-induced orbital decay is sufficiently “gentle” and the octupole LK effects
are unimportant. Figure 7.1 gives an example of a system evolving towards this
attractor, where θsl converges to ≈ 90◦ at late times in the bottom right panel. Fig-
ure 7.2 shows how θsl,f varies when the initial inclination of the tertiary orbit I0
(relative to the inner orbit) is varied. Note that for rapid mergers (when I0 is close
to 90◦), the attractor does not exist; as I0 deviates more from 90◦, the merger time
increases and θsl,f becomes close to 90◦. This “90◦ attractor” gives rise to a peak
around χeff = 0 in the final χeff distribution in tertiary-induced mergers [LL18; Liu
et al., 2019a]. This peak was also found in the population studies of Antonini et al.
[2018].
197
The physical origin of this “90◦ attractor” and under what conditions it can be
achieved are not well understood. LL18 proposed an explanation based on analogy
with an adiabatic invariant in systems where the inner binary remains circular
throughout the inspiral (LL17; see also [Yu et al., 2020a]). However, the validity of
this analogy is hard to justify, as significant eccentricity excitation is a necessary
ingredient in LK-induced mergers. In addition, the LK-enhanced mergers consid-
ered in LL17 show no 90◦ attractor even though the orbital evolution is slow and
regular.
In this paper, we present an analytic theory to explain the 90◦ attractor and
to characterize its regime of validity. More generally, we develop a theoretical
framework to help understand the BH spin evolution in LK-induced mergers. In
Sections 7.2 and 7.3, we set up the relevant equations of motion for the orbital and
spin evolution of the system. To simplify the theoretical analysis, we initially con-
sider the cases where the tertiary mass is much larger than the binary mass. In
Sections 7.4 and 7.5, we develop an analytic understanding of the spin evolution.
In Section 7.6, we generalize our results to stellar-mass tertiary companions. We
discuss and conclude in Section 7.7.
7.2 LK-Induced Mergers: Orbital Evolution
In this section we summarize the key features and relevant equations for LK-
induced mergers to be used for our analysis in later sections. Consider a black
hole (BH) binary with masses m1 and m2 having total mass m12, reduced mass µ=
m1m2/m12, semimajor axis a and eccentricity e. This inner binary orbits around a
tertiary with mass m3, semimajor axis aout and eccentricity eout in a hierarchical
configuration (aout ≫ a). Unless explicitly stated, we assume m3 ≫ m1,m2 (e.g.
198
Figure 7.1: An example of the “90◦ spin attractor” in LK-induced BH binary merg-
ers. The four panes show the time evolution of the binary semi-major axis a, ec-
centricity e, inclination I [the red line denotes the averaged Ī given by Eq. (7.29)],
and spin-orbit misalignment angle θsl. The unit of time tLK,0 is the LK timescale
[Eq. (7.9)] evaluated for the initial conditions. The inner binary has m1 = 30M⊙,
m2 = 20M⊙, and initial a0 = 100 AU, e0 = 0.001, I0 = 90.35◦ (with respect to
the outer binary), and θsl,0 = 0. The tertiary has aout = 2.2 pc, eout = 0, and
m3 = 3× 107M⊙ (The result depends only on m /ã33 out, provided that Lout ≫ L).
It can be seen that θsl evolves to ∼ 90◦ as a decays to smaller values, and we stop
the simulation when a = 0.5 AU as the LK oscillation “freezes” and θsl has con-
verged to a constant value.
199
Figure 7.2: The merger time and the final spin-orbit misalignment angle θsl,f as
a function of the initial inclination I0 for LK-induced mergers. The other param-
eters are the same as those in Fig. 7.1. For I0 somewhat far away from 90◦, the
resulting θ ◦sl,f are all quite near 90 . In the lower panel, the horizontal black solid
line shows the predicted θsl,f if θ̄e is conserved, i.e. Eq. (7.57), and the black dashed
line shows Eq. (7.61), which provides an estimate for the deviation from the 90◦
attractor.
200
the tertiary can be a supermassive black hole, or SMBH), although our analysis
can be easily generalized to comparable masses (see Section 7.6). We denote the
orbital angular momentum of the inner binary by L ≡ LL̂ and the angular mo-
mentum of the outer binary by Lout ≡ LoutL̂out. Since Lout ≫ L, we take Lout to be
fixed.
The equations of motion governing the orbital elements a, e, , I, ω (where
, I, ω are the longitude of the ascending node, inclination, and argument of
periapsis respectively) of(the)inner binary are
da = da , ( ) (7.3)dt dt GW
de = 15 dee j(e)(sin2ωsin2 I+ ) , (7.4)dt 8tLK dt GW
d = 3 cos I 5e
2 cos2ω−4e2−1
, (7.5)
dt 4tLK j(e)
dI =− 15 e
2 sin2ωsin2I
, (7.6)
dt 16tLK j(e)
dω 2= 3 2 j (e)+5sin
2ω(e2−sin2 I) +Ω , (7.7)
dt 4tLK j(e)
GR
where we have defined √
j(e)= (1− e2),( ) (7.8)
m a 3
t−1 3LK ≡ n , √ (7.9)√ m12 ãout
with n≡ Gm12/a3 the mean motion of the inner binary, and ã 2out = aout 1− eout.
The general relativity (GR) induced apsidal precession of the inner binary is given
by
3Gnm
ΩGR(e)= 122 2 . (7.10)c a j (e)
The dissipative term( s d)ue to gravitational radiation are
da
( ) =−
a
, ( ) (7.11)dt GW tGW(e)
de =−304 G
3µm212 1 1+ 121 e2 e, (7.12)
dt GW 15 c5a4 j5(e) 304
201
where ( )
−1 ≡ 64 G
3µm2
t (e) 12
1
GW 5 4 7 1+
73
e2+ 37 e4 . (7.13)
5 c a j (e) 24 96
Equations (7.4–7.7) include only the effects of quadrupole perturbations from
m3 on the binary. The octupole effects depend on the parameter
m1−m2 aeout
ϵoct = m12 a (1− e2
, (7.14)
out out)
[see Liu et al., 2015a]. Throughout this paper, we consider systems where ϵoct ≪ 1
so that the octupole effects are neglected.
In Fig. 7.1, we provide an example of an LK-induced merger for our fiducial pa-
rameters (described in the figure caption). The inner binary initially decays slowly
as its eccentricity undergoes LK oscillations, with a nearly constant maximum
eccentricity close to unity. Then, as a decreases and the minimum eccentricity
within each LK cycle increases, the orbital decay of the inner binary accelerates.
As apsidal precession further suppresses eccentricity oscillations, the eccentric-
ity rapidly decays. We terminate the simulation at a = 0.5 AU as the spin-orbit
misalignment angle θsl has reached its final value, even though the binary still
has non-negligible eccentricity. We refer to the example depicted in Fig. 7.1 as the
fiducial example, and much of our analysis in later sections will be based on this
example unless otherwise noted. Also note that the parameters of Fig. 7.1 give the
same tLK as Fig. 4 of LL18.
We next discuss the key analytical properties of the orbital evolution.
202
7.2.1 Analytical Results Without GW Radiation
First, neglecting the GW radiation terms, the system admits two conservation
laws, the “Kozai constant” and energy conservation,
j([e)cos I = const, ] (7.15)3
2e2+ j2(e)cos2 I−5e2 ϵsin2 I sin2 + GRω = const, (7.16)
8 j(e)
(see [Anderson et al., 2016] and LL18 for more general expressions when Lout is
comparable to L), where
2 3
ϵGR ≡ (ΩGRtLK)e=0 =
3Gm12ãout
c2m3a4
. (7.17)
The conservation laws can be combined to obtain the maximum eccentricity emax
as a function of the initial I0 (and initial e0 ≪ 1). The largest value of emax occurs
at I0 = 90◦ and is given by
j(emax)I0=90◦ = (8/9)ϵGR. (7.18)
Eccentricity excitation then requires ϵGR < 9/8. Our fiducial examples in Figs. 7.1
and 7.2 satisfy ϵGR ≪ 1 at a = a0, leading to emax ∼ 1 within a narrow inclination
window around I0 = 90◦.
Eqs. (7.15) and (7.16) imply that e is a function of sin2ω alone [see Kinoshita
and Nakai, 1999, Storch and Lai, 2015, for the exact form], so an eccentricity
maximum occurs every half period of ω. We define the LK period of eccentricity
oscillation PLK and its corresponding angular frequency ΩLK via
P∫LK
= dω ≡ 2ππ dt, ΩLK . (7.19)dt PLK
0
In LK cycles, the inner binary oscillates between the eccentricity minimum
emin and maximum emax. The oscillation is “uneven”: when emin ≪ emax, the bi-
203
nary spends a fraction ∼ j(emax) of the LK cycle, or time ∆t ∼ tLK j (emax), near
e≃ emax [see Eq. (7.7)].
7.2.2 Behavior with GW Radiation
Including the effect of GW radiation, orbital decay predominantly occurs at e ≃
emax with the timescale of tGW (emax) [see Eq. (7.13)]. On the other hand, Eq. (7.7)
implies that, when ϵGR ≪ 1, the binary spends only a small fraction (∼ j(emax))
of the time near e ≃ emax. Thus, we expect two qualitatively different merger
behaviors:
• “Rapid mergers”: When tGW (emax)≲ tLK j(emax), the binary is “pushed” into
high eccentricity and exhibits a “one shot merger” without any e-oscillations.
• “Smooth mergers”: When tGW (emax) ≳ tLK j(emax), the binary goes through
a phase of eccentricity oscillations while the orbit gradually decays. In this
case, the LK-averaged orbital decay rate is ∼ j(e −1max)tGW(emax). As a de-
creases, emax decreases slightly while the minimum eccentricity increases,
approaching emax (see Fig. 7.1). This eccentricity oscillation “freeze” (emin ∼
emax) is due to GR-induced apsidal precession (ϵGR increases as a decreases),
and occurs when ϵGR(a)≫ j(emax). After the eccentricity is frozen, the binary
circularizes and decays on the timescale tGW (emax).
7.3 Spin Dynamics: Equations
We are interested in the spin orientations of the inner BHs at merger as a function
of initial conditions. Since they evolve independently to leading post-Newtonian
204
order, we focus on the dynamics of Ŝ1 = Ŝ, the unit spin vector of m1. Since the spin
magnitude does not enter into the dynamics, we write S≡ Ŝ for brevity (i.e. S is a
unit vector). Neglecting spin-spin interactions, S undergoes de Sitter precession
about L as
dS =ΩSLL̂×S, (7.20)dt
with ( )
= 3Gn m2+µ/3ΩSL 2 2 . (7.21)2c a j (e)
In the presence of a tertiary companion, the orbital axis L̂ of the inner binary
precesses around L̂out with rate d/dt and nutates with varying I [see Eqs. (7.5)
and (7.6)]. To analyze the dynamics of the spin vector, we go to the co-rotating
frame with L̂ about L̂out, in whic(h Eq). (7.20) becomes
dS =Ωe×S, (7.22)dt rot
where we have defined an effective rotation vector
Ωe ≡ΩLL̂out+ΩSLL̂, (7.23)
with [see Eq. (7.5)]
ΩL ≡−
d
. (7.24)
dt
In this rotating frame, the plane spanned by L̂out and L̂ is constant in time, only
the inclination angle I can vary.
7.3.1 Nondissipative Spin Dynamics
We first consider the limit where dissipation via GW radiation is completely ne-
glected (tGW(e)→∞). Then Ωe is exactly periodic with period PLK [see Eq. (7.19)]
205
We can rewrite Eq. ((7.22)) in F[ourier components
dS ∑∞ ]= Ωe+ ΩeN cos(NΩdt LKt) ×S. (7.25)rot N=1
Note that Ωe is the zeroth Fourier component, where the bar denotes an average
over a LK cycle. We have adopted the convention where t = 0 is the time of max-
imum eccentricity of the LK cycle, so that Eq. (7.25) does not have sin(NΩLKt)
terms.
This system superficially resembles that considered in Storch and Lai [2015]
(SL15), who studied the dynamics of the spin axis of a star when driven by a
giant planet undergoing LK oscillations [see also Storch et al., 2014, 2017]. In
their system, the spin-orbit coupling arises from Newtonian interaction between
the planet (Mp) and the rotation-induced stellar quadrupole (I3− I1), and the spin
precession frequency is
(Newtonian) =−3GMp (I3− I1) cosθslΩSL 3 3 , (7.26)2a j (e) I3Ωs
where I3Ωs is the spin angular momentum of the star. SL15 showed that under
some conditions that depend on a dimensionless adiabaticity parameter (roughly
the ratio between the magnitudes of Ω(Newtonian)SL and ΩL when factoring out the
eccentricity and obliquity dependence), the stellar spin axis can vary chaotically.
One strong indicator of chaos in their study is the presence of irregular, fine struc-
ture in a bifurcation diagram [Fig. 1 of Storch and Lai [2015]] that shows the
values of the spin-orbit misalignment angle θsl when varying system parameters
in the “transadiabatic” regime, where the adiabaticity parameter crosses unity.
To generate an analogous bifurcation diagram for our problem, we consider
a sample system with m = 60M⊙, m = 3×10712 3 M⊙, a = 0.1 AU, e = 10−30 , I0 =
70◦, aout = 300 AU, eout = 0, and initial θsl = 0 (note that these parameters are
different from those in Fig. 7.1). We then evolve Eq. (7.20) together with the orbital
206
evolution equations [Eqs. (7.3–7.7) without the GW terms] while sampling both θsl
and θe at eccentricity maxima, where θe is the angle between Ωe and S, i.e.
cosθe = Ωe ·S, (7.27)
Ω
where ≡ ∣∣
e
Ωe ∣ ∣∣Ωe∣. We repeat this procedure with different mass ratios m1/m12 of
the inner binary, which only changes ΩSL without changing the orbital evolution
(note that the LK oscillation depends only on m12 and not on individual masses
of the inner binary). Analogous to SL15, we consider systems with a range of the
adibaticity parameterA [to be defined later in Eq. (7.31)] that crosses order unity.
Note that the fiducial system of Fig. 7.1 does not serve this purpose because the
initial ΩSL is too small. Our result is depicted in Fig. 7.3.
While our bifurcation diagram has interesting structure, the features are all
regular. This is in contrast to the star-planet system studied by SL15 (see their
Fig. 1). A key difference is that in our system, ΩSL does not depend on θsl, while
for the planet-star system, Ω(Newtonian)SL does, and this latter feature introduces
nonlinearity to the dynamics.
A more formal understanding of the dynamical behavior of our spin-orbit sys-
tem comes from Floquet theory[Floquet, 1883, Chicone, 2006], as Eq. (7.22) is a
linear system with periodic coefficients (the system studied in SL15 is nonlinear).
By Floquet’s theorem, when a linear system with periodic coefficients is integrated
over a period, the evolution can be described by the linear transformation
S (t+PLK)= M̃S(t), (7.28)
where M̃ is called the monodromy matrix and is independent of S.
For our system, while M̃ can be easily defined, it cannot be evaluated in closed
form. Instead, we can reason directly about the general properties of M̃: it must
207
Figure 7.3: Bifurcation diagram for the BH spin orientation during LK oscilla-
tions. The physical parameters are m = 60M⊙, m = 3× 10712 3 M⊙, a = 0.1 AU,
e = 10−3, I = 70◦0 0 , aout = 300 AU, eout = 0, and initial condition θsl,0 = 0. For each
mass ratio m1/m12, the orbit-spin system is solved over 500 LK cycles, and both
θsl (the angle between S and L̂) and θe [defined by Eq. (7.27)] are sampled at ev-
ery eccentricity maximum and are plotted. The top axis shows the adiabaticity
parameter A as defined by Eq. (7.31). Note that for a given m12, changing the
mass ratio m1/m12 only changes the spin evolution and not the orbital evolution.
208
be a proper orthogonal matrix, or a rotation matrix, as it represents the effect of
many infinitesimal rotations, each about the instantaneous Ω 1e . Therefore, over
each period PLK, the dynamics of S are equivalent to a rotation about a fixed axis,
prohibiting chaotic behavior.
Another traditional indicator of chaos is a positive Lyapunov exponent, ob-
tained when the separation between nearby trajectories diverges exponentially in
time. In Floquet theory, the Lyapunov exponent is the logarithm of the largest
eigenvalue of the monodromy matrix. Since M̃ is a rotation matrix in our prob-
lem, the Lyapunov exponent must be 0, indicating no chaos. We have verified this
numerically.
7.3.2 Spin Dynamics With GW Dissipation
When tGW is finite, the coefficientsΩeN, includingΩe [see Eq. (7.25)], are no longer
constant, but change over time. For “smooth” mergers (satisfying tGW (emax) ≫
tLK j(emax); see Section 7.2), the binary goes through a sequence of LK cycles, and
the coefficients vary on the LK-averaged orbital decay time tGW (emax) / j (emax).
After the LK oscillation freezes, we have Ωe ≃Ωe (and ΩeN ≃ 0 for N ≥ 1), which
evolves on timesale tGW(e) as the orbit decays and circularizes.
Once a is sufficiently small that ΩSL ≫ ΩL, it can be seen from Eqs. (7.22–
7.23) that θe = θsl is constant, i.e. the spin-orbit misalignment angle is frozen
(see bottom right panel of Fig. 7.1). This is the “final” spin-orbit misalignment,
although the binary may still be far from the final merger. Note that at such
1More formally, M̃ = Φ̃(PLK) where Φ̃(t) is the principal fundamental matrix solution: the
columns of Φ̃ are solutions to Eq. (7.22) and Φ̃(0) is the identity. By linearity, the columns of
Φ̃(t) remain orthonormal, while its determinant does not change, so M̃ is a proper orthogonal
matrix, or a rotation matrix.
209
Figure 7.4: Definition of angles in the problem, shown in plane of the two angular
momenta Lout and L. Here, Ωe is the LK-averaged Ωe, and Ωe1 is the first har-
monic component (see Eqs. (7.23) and (7.25)). Note that for I0 > 90◦, we choose
Īe ∈ (90◦,180◦) so that Ωe > 0 (since ΩL < 0). The bottom right shows our choice of
coordinate axes with ẑ∝Ωe.
separations, ϵGR ≫ 1 as well since ΩSL ∼ ΩGR, and so LK eccentricity excitation
is suppressed. For the fiducial examples depicted in Figs. 7.1–7.2, we stop the
simulation at a= 0.5 AU, as θsl has converged to its final value.
7.3.3 Spin Dynamics Equation in Component Form
For later analysis, it is useful to write Eq. (7.25) in component form. To do so,
we define the inclination angle Īe as the angle between Ωe and Lout as shown in
Fig. 7.4. To express Īe algebraically, we define the LK-averaged quantities
ΩSL sin I ≡ΩSL sin Ī, ΩSL cos I ≡ΩSL cos Ī. (7.29)
It then follows from Eq. (7.23) that
A sin Ī
tan Īe = + , (7.30)1 A cos Ī
210
where A is the adiabaticity parameter, given by
A ≡ ΩSL . (7.31)
ΩL
Note that in Eq. (7.30), Īe is defined in the domain [0◦,180◦], i.e. Īe ∈ (0,90) when
tan Īe > 0 and Īe ∈ (90,180) when tan Īe < 0.
We now choose a non-inertial coordinate system where ẑ∝Ωe and x̂ lies in the
plane of Lout and L (see Fig. 7.4). In this reference frame, the spin orientation is
specified by the polar angle θe as defined above in Eq. (7.27), and the spin evolution
equation becom(es ) [ ]
dS ∞= ∑Ωeẑ+ ΩeN cos(NΩLKt) ×S− I˙̄eŷ×S. (7.32)dt xyz N=1
One further simplification lets us cast this vector equation of motion into a scalar
form. Break S into components S= Sxx̂+Syŷ+cosθeẑ and define complex variable
S⊥ ≡ Sx+ iSy. (7.33)
Then, we can rewrite Eq. (7.32) as
dS⊥ ∞ [= ∑iΩ ˙̄
dt e
S⊥− Ie cosθe+ ] cos(∆IeN)S⊥N=1
− i cosθe sin(∆IeN) ΩeN cos(NΩLKt) , (7.34)
where I˙̄e = dĪe/dt and ΩeN is the magnitude of the vector ΩeN [see Eq. (7.25)] and
∆IeN = Ie√N − Īe, with IeN the angle between ΩeN and Lout (see Fig. 7.4). Since
cosθe =± 1−|S⊥|2, Eq. (7.34) is generally nonlinear in S⊥, but becomes approx-
imately linear when |θe|≪ 1.
7.4 Analysis: Approximate Adiabatic Invariant
In general, Eqs. (7.25) and (7.34) are difficult to study analytically. In this section,
we neglect the harmonic terms and focus on how the varyingΩe affects the evolu-
211
tion of the BH spin axis. The effect of the harmonic terms is studied in Section 7.5.
7.4.1 The Adiabatic Invariant
When neglecting the N ≥ 1 harm(onic)terms, Eq. (7.25) reduces to
dS =Ωe×S. (7.35)dt
rot
It is not obvious to what extent the analysis of Eq. (7.35) is applicable to Eq. (7.25).
From our numerical calculations, we find that the LK-average of S often evolves
following Eq. (7.35), motivating our notation S. Over timescales shorter than the
LK period PLK, Eq. (7.35) loses accuracy as the evolution of S itself is dominated
by the N ≥ 1 harmonics we have neglected. An intuitive interpretation of this
result is that the N ≥ 1 harmonics vanish when integrating Eq. (7.25) over a LK
cycle.
Eq. (7.35) has one desirable property: θ̄e, defined by
Ω
cos eθ̄e ≡ ·S, (7.36)
Ωe
is an adiabatic invariant. This follows from the fact that Sz, the projection of
S on the Ωe axis, and ϕ, the precessional angle of S around Ωe, form a pair of
action-angle variables. The adiabaticity condition requires that the precession
axis ẑ =Ωe/Ωe evolve slowly compared to the precession frequency at all times,
i.e. ∣∣∣∣ ∣dĪe ∣∣∣≪Ωe. (7.37)dt
For our fiducial example depicted in Fig. 7.1, the values of I˙̄e and Ωe are shown
in the top panel of Fig. 7.5, and the evolution of θ̄e in the bottom panel. The net
change in θ̄e in this simulation is 0.01◦, small as expected since |I˙̄e| ≪ Ωe at all
times.
212
7.4.2 Deviation from Adiabaticity
The extent to which θ̄e is conserved depends on how well Eq. (7.37) is satisfied.
In this subsection, we derive a bound on the total non-conservation of θ̄e, then
in the next subsection we show how this bound can be estimated from the initial
conditions.
When neglecting harmonic terms, the scalar evolution equation Eq. (7.34) be-
comes
dS⊥ = iΩ S − I˙̄ cos θ̄ . (7.38)
dt e ⊥ e e
This can be solved in closed form. Defini∫ngt
Φ(t)≡ Ωe dt, (7.39)
we obtain the solution at time t:
− ∣∣∣ tt ∫e iΦS⊥ =− e−iΦ(τ)I˙̄e cos θ̄e dτ. (7.40)
0
0
Recalling |S⊥| = sin θ̄e and analyzing Eq. (7.40), we see that θ̄e oscillates about its
initial value with semi-amplitude
∣∣ ∣∆θ̄e∣
∣∣
∼ ∣∣ ∣∣ I˙̄e ∣∣∣∣ . (7.41)Ωe
In the adiabatic limit [Eq. (7.37)], θ̄e is indeed conserved, as the right-hand side
of Eq. (7.41) goes to zero. The bottom panel of Fig. 7.5 shows ∆θ̄e for the fiducial
example. Note that θ̄e is indeed mostly constant where Eq. (7.41) predicts small
oscillations.
∣ ∣
If we denote ∣∆θ̄e∣f to be the net change in θ̄e over t ∈ [0, tf], we can give a loose
bound ∣∣ ∣∣ ∣≲ ∣∣∣∣ I˙̄e ∣∆θ̄e f ∣
∣∣
Ωe ∣ . (7.42)max
213
Inspection of Fig. 7.5 indicates that the spin dynamics are mostly uninter-
esting except near the peak of |I˙̄e|, which occurs where Ω̄SL ≃ |ΩL|. We present
a zoomed-in view of dynamical quantities near the peak of |I˙̄e| in Fig. 7.6. In
particular, in the bottom-rightmost panel, we see that the fluctuations in θ̄e are
dominated by a second contribution, the subject of the discussion in Section 7.5.
For comparison, we show in Fig. 7.7 a more rapid binary merger starting with
I0 = 90.2◦, for which∣∣ |∆θ∣∣e|f ≈ 2
◦. If we again examine the bottom-rightmost panel,
we see that the net ∆θ̄e f obeys Eq. (7.42).
7.4.3 Estimate of Deviation from Adiabaticity from Initial
Conditions
To estimate Eq. (7.42) as a function of initial conditions, we first differentiate
Eq. (7.30), ( )
˙̄ = A
˙ A sin Ī
Ie
A 1+ + . (7.43)2A cos Ī A 2
It also follows from Eq. (7.23) that
= ∣∣∣ ∣∣( 2)1/2Ωe ΩL∣ 1+2A cos Ī+A , (7.44)
from which we obtain ∣∣∣ ∣I˙̄ ∣e∣ ( ˙)
= A ∣∣∣ 1 ∣∣∣ ( A sin Ī )3/2 . (7.45)Ωe A ΩL 1+2A cos Ī+A 2
214
Figure 7.5: The same simulation as depicted in Fig. 7.1 but showing several calcu-
lated quantities relevant to the theory of the spin evolution. Top: the four charac-
teristic frequencies of the system and dĪe/dt. Middle: the frequency ratios between
the zeroth and first Fourier components of Ωe to the LK frequency ΩLK. Bottom:
Time evolution of θe [grey line; Eq. (7.27)], θ̄e [red dots; Eq. (7.36)], as well as
estimates of the deviations from perfect conservation of θ̄e due to nonadiabaticity
[green, Eq. (7.41)] and due to the resonance Ωe ≈ΩLK [blue, Eq. (7.81)].
215
Figure 7.6: The same simulation as Fig. 7.1 but zoomed in on the region around
A ≡ΩSL/ΩL ≃ 1 and showing a wide range of relevant quantities. The first three
panels in the upper row depict a, e, I and Ī as in Fig. 7.1, while the fourth shows
Īe [Eq. (7.30)] and Ie1. The bottom four panels depict θsl, the four characteristic
frequencies of the system and dĪe/dt [Eqs. 7.23 and (7.24)] (as in the top panel of
Fig. 7.5), the relevant frequency ratios (as in the middle panel of Fig. 7.5), and
the deviation of θ̄e from its initial value compared to the predictions of Eqs. (7.41)
and (7.81).
216
Figure 7.7: Same as Fig. 7.6 except for I0 = 90.2◦ (and all other parameters are
the same as in Fig. 7.1), corresponding to a faster coalescence. The total change in
θ̄e for this simulation is ≈ 2◦.
Moreover, if we assume the eccentricity is frozen around e≃ 1 and use cos2ω≃ 1/2
in |ΩL| = |d/dt|, we obtain the estim(ate + )= ΩSL ≃ 3Gn m2 µ/3 [ ]15cos Ī −1A
ΩL 2c
2a j2(e) 8tLK j(e)
≃ 4 G(m2+µ/3)m12ã
3
out , (7.46)
5 (c2)m3a4 j(e)co(s Ī )
A˙ =− ȧ e de4 + 2 . (7.47)A a GW j (e) dt GW
With these, we see that Eq. (7.45) is largest around A ≃ 1, and so we find that the
maximum |I˙̄e|/Ωe is give∣n by∣∣∣∣ I˙̄ ∣∣∣
∣ ∣
e ∣ ≃ ∣∣∣A˙∣∣
∣
Ωe A
∣
max ∣∣∣
1 ∣∣∣ ( sin Ī+ )3/2 . (7.48)ΩL 2 2cos Ī
To evaluate this, we make two assumptions: (i) Ī is approximately constant
(see the third panels of Figs. 7.6 and 7.7), and (ii) j(e) evaluated at A ≃ 1 can be
217
approximated as a constant multiple of th√e initial j(emax), i.e.
j⋆ ≡ 5j(e⋆)= f cos2 I0, (7.49)3
where the star subscript denotes evaluation at A ≃ 1 and f > 1 is a constant.
Eq. (7.49) assumes that I far enough from 90◦0 that the GR effect is unimportant
in determining emax. The value of f turns out to be relatively insensitive to I0.
Using Eq. (7.47) and appro[xim]ating e⋆ ≈ 1 in Eqs. (7.11) and (8.15) give
A˙ 3≃ G µm
2
12 595
4 7 . (7.50)A c5⋆ a⋆ j⋆ 3
To determine a⋆, we require Eq. (7.46) to give A = 1 for a⋆ and j⋆. Taking this
and Eq. (7.50∣), w∣e rewrite Eq. (7.45) as∣∣∣ ∣ ∣3/8 [ ]1/8∣ I˙̄ ∣e ∣∣Ωe ∣
595s(in Ī ∣cos Ī)∣ 8000G9m9 m3≈ 12 3µ83 . (7.51)
max 36 cos Ī+1 2 ã9 37out j⋆ c18(m2+µ/3)11
We can also calculate |I˙̄e|/Ωe from numerical simulations. Taking characteristic
Ī ≈ 120◦ (Figs. 7.6 and 7.7 show that this holds across a range of I0), we fit the last
remaining free parameter f [Eq. (7.49)] to the data from numerical simulations.
This yields f ≈∣2.72∣∣∣ ˙̄ ∣
, leading to
∣ I ∣∣
( )
∣∣ cos I −37/8
( )
ã −9/8e ≃ 0.98◦ 0 out
Ωe ( cos(90.3max )◦) ( 2.2 pc )3/8 m9 8/(m + /3)11 1/8× m3 12µ 2 µ× 7 . (7.52)3 10 M⊙ (28.64M⊙)6
Figure 7.8 shows that when the merger time Tm is much larger than the initial LK
timescale, Eq. (7.52) provides an accurate estimate for |I˙̄e/Ω̄e|max when compared
with numerical results.
In the above, we have assumed that the system evolves through A ≃ 1 when
the eccentricity is mostly frozen (see Fig. 7.1 for an indication of how accurate this
is for the parameter space explored in Fig. 7.8). It is also possible thatA ≃ 1 occurs
218
∣
Figure 7.8: Comparison of ∣∣I˙̄ / ∣∣e Ωe∣ obtained from simulations and from the an-
max
alytical expression Eq. (7.52), where we take f = 2.72 in Eq. (7.49). The merger
time Tm is shown along the top axis of the plot in units of the initial LK timescale
tLK,0. The agreement between the analytical and numerical results is excellent
for Tm ≫ tLK,0.
219
when the eccentricity is still undergoing substantial oscillations. In fact, Eq. (7.52)
remains accurate in this case when replacing e with emax, due to the following
analysis. Recall that when emin ≪ emax, the binary spends a fraction ∼ j(emax)
of the LK cycle near e ≃ emax. This fraction of the LK cycle dominates both GW
dissipation and Ωe precession. Thus, both I˙̄e and Ωe in the eccentricity-oscillating
regime can be evaluated by setting e ≈ emax and multiplying by a prefactor of
j(emax). This factor cancels when computing the quotient |I˙̄e|/Ωe.
∣ ∣
The accuracy of Eq. (7.52) in bounding ∣∆θ̄e∣f is shown in Fig. 7.9, where we
carry out simulations for a range of I0, and for each I0 we consider 100 different,
isotropically distributed initial orientations for S (thus sampling a wide range
of initial initial θ̄e). Note that conservation of θ̄e is generally much better than
Eq. (7.52) predicts. This is because cancellation of phases in Eq. (7.40) is gener-
ally more efficient than Eq. (7.52) assumes (recall that Eq. (7.41) only provides an
estimate for the amplitude of “local” oscillations∣ of θ̄∣e). Nevertheless, it is clear
that Eq. (7.52) provides a robust upper bound of ∣∆θ̄e∣f, and serves as a good indi-
cator for the breakdown of adiabatic invariance.
7.4.4 Origin of the θ ◦sl,f = 90 Attractor
Using the approximate adiabatic invariant, we can now understand the origin of
the θsl,f = 90◦ attractor as shown in Fig. 7.2.
Recall from Eq. (7.23)
Ωe = (ΩLL̂out+ΩSLL̂),
= ΩL+ΩSL cos Ī Ẑ+ΩSL sin ĪX̂, (7.53)
220
Figure 7.9: Net change in θ̄e over the binary inspiral as a function of initial in-
clination I0. For each I0, 100 simulation∣s are∣ ru∣n for S∣ on a uniform, isotropic
grid. Plotted for comparison is the bound ∣ ∣ ∣∆θ̄e∣f ≲ ∣I˙̄e/Ωe∣ , using the analyticalmax
expression given by Eq. (7.52). It is c∣lear∣ that the expression provides a robustupper bound for the non-conservation∣of θ̄∣e due to nonadiabatic effects. Note thatat the right of the plot, the numerical ∆θ̄e f saturates; this is because we compute
the initial Ω̄e (in order to evaluate the initial θ̄e) without GW dissipation, and such
a procedure inevitably introduces fuzziness in θ̄e.
221
where Ẑ= L̂out and X̂ is perpendicular to Ẑ in the L̂out–L̂ plane. Note that
∝ cos ĪΩ ∝ a3/2L cos Ī, (7.54)tLK
∝ 1ΩSL 5/2 . (7.55)a
Adiabatic invariance implies that θ̄e, the angle between S and Ωe is conserved
between t= 0 and t= tf, i.e.
∣ ∣ θ̄e,f ≃ θ̄e,0. (7.56)
At t= tf, ΩSL ≫ ∣∣ΩL∣∣ and the spin-orbit misalignment angle θsl is “frozen”, imply-
ing Ωe is parallel to L, and so θ̄e,f = θsl,f. Eq. (7.56) then gives
θsl,f ≃ θ̄e,0, (7.57)
i.e. the final θsl is determined by the initial angle between S and Ωe.
Now, first consider the case where the initial spin S0 is al∣igned wi∣th the initial
L0. This initial spin is inclined with respect to Ωe by θ̄e,0 = ∣I0− Īe,0∣, where I0 is
the initial inclination angle between L and Lout and Īe,0 is the initial value of Īe.
Thus, adiabatic invariance implies
∣ ∣
θsl,f ≃ ∣I0− Īe,0∣ . ∣ ∣ (7.58)
In the special case where the binary initially satisfies ∣ ∣ΩSL ≪ ∣ΩL∣ or |A0| ≪ 1,
we find that Ωe is nearly parallel to Lout (for I0 < 90◦) or antiparallel to Lout (for
I0 > 90◦). Thus, I0 I0 < 90
◦,
θsl,f = (7.59)180◦− I0 I0 > 90◦.
Since LK-induced mergers necessarily require I0 close to 90◦, we find that θsl,f is
“attracted” to 90◦.
Eq. (7.57) can be applied to more general initial spin orientations. For initial
|A0| ≪ 1 (as required for LK-induced mergers), Ωe,0 is parallel to ±Lout. Suppose
222
the initial inclination between S and Lout is θs,out,0, then θ̄ ◦e,0 = θs,out,0 or 180 −
θ ◦ ◦s,out,0 (depending on whether I0 < 90 or I0 > 90 ). Thus,θ
◦
≃ s,out,0
I0 < 90 ,
θsl,f  (7.60)180◦−θs,out,0 I0 > 90◦,
[see also Yu et al., 2020a].
So far, we have analyzed the θsl,f distribution for smooth mergers. Next, we
can consider rapid mergers, for which θ̄e conservation is imperfect. We expect
∣∣ ∣ ∣θsl,f− θ̄e,0∣≲ ∣ ∣∆θ̄e∣f . (7.61)
∣ ∣
where ∣∆θ̄e∣f is given by Eq. (7.52). This is shown as the black dotted line in
Fig. 7.2, and we see it predicts the maximum deviation of θ ◦sl,f from ∼ 90 except
very near I0 = 90◦. This is expected, as Eq. (7.52) is not very accurate very near
I0 = 90◦, where it diverges (see Fig. 7.9).
When I0 = 90◦ exactly, Fig. 7.2 shows that θsl,f = 0◦. This can be understood:
I = 90◦ gives dI/dt = 0 by Eq. (7.6), so I = 90◦0 for all time. This then yields
d/dt = 0 [Eq. (7.5)], implying that L is constant. Thus, L is fixed as S precesses
around it, and θsl can never change. In Fig. 7.2, we take θsl,0 = 0, so θsl,f = 0.
Finally, Fig. 7.2 shows that the actual θsl,f are oscillatory within the envelope
bounded by Eq. (7.61) above. This can also be understood: Eq. (7.42) only bounds
the maximum of the absolute value of the change in θ̄e, while the actual change
depends on the initial and final complex phases of S⊥ in Eq. (7.40), denoted Φ(0)
and Φ(tf). When θsl,0 = 0, we have Φ(0) = 0, as S starts in the x̂-ẑ plane. Then,
as I0 is smoothly varied, the final phase Φ (tf) must also vary smoothly [since Ωe
in Eq. (7.39) is a continuous function, Φ (t) must be as well], so the total phase
difference between the initial and final values of S⊥ varies smoothly. This means
223
∣ ∣
the total change in θ̄e will fluctuate smoothly between ± ∣∆θ̄e∣f as I0 is changed,
giving rise to the sinusoidal shape seen in Fig. 7.2.
7.5 Analysis: Effect of Resonances
In the previous section, we have shown that the θsl,f distribution in Fig. 7.2 and the
“90◦ attractor” can be understood when neglecting the N ≥ 1 Fourier harmonics in
Eq. (7.25). In this section, we study the effects of these neglected terms and show
that they have a negligible effect in LK-induced mergers. Separately, we will also
consider the LK-enhanced regime and show that these Fourier harmonics play a
dominant role in shaping the θsl,f distribution.
For simplicity, we ignore the effects of GW dissipation in this section and as-
sume the system is exactly periodic (so I˙̄e = 0). The scalar spin evolution equation
Eq. (7.34) is then
dS⊥ ∞ [= ∑iΩeS + cos(∆Idt ⊥ ] eN)S⊥N=1
− i cosθe sin(∆IeN) ΩeN cos(NΩLKt). (7.62)
7.5.1 Intuitive Analysis
We first restrict our attention to the effect of just the N-th Fourier harmonic, and
Eq. (7.62) further simplifies to
dS⊥ [ ]= iΩe+ΩeN cos(∆IeN)cos(NΩdt LKt) S⊥
− iΩeN cos θ̄e sin(∆IeN)cos(NΩLKt). (7.63)
224
There are two time-dependent perturbations, a modulation of the oscillation fre-
quency (the term proportional to S⊥ on the right-hand side) and a driving term.
We can begin to understand the dynamics of S⊥ by considering the effect of each
of these two terms separately.
First, we consider the effect of frequency modulation alone. The equation of
motion is
dS⊥ [ ]≈ iΩe+ΩeN cos(∆IeN)cos(NΩLKt) S⊥. (7.64)dt
The exact solution is [ ]
ΩeN cos(∆IS eN
)
⊥(t)= S⊥(0)exp iΩet+ sin(NΩLKt) . (7.65)NΩLK
There is no combination of parameters for which the magnitude of S⊥ diverges, so
the modulation of the oscillation frequency does not cause any resonant behavior.
When considering only the time-dependent driving term instead, the equation
of motion is
dS⊥ ≈ iΩeS⊥− iΩeN cosθe sin(∆IeN)cos(NΩLKt) . (7.66)dt
We can approximate cos(NΩ iNΩ tLKt)≈ e LK /2, as the e−iNΩLKt component is always
farther from resonance (as ΩLK,Ωe > 0). Then the equation of motion has the
solution ∣t ∫t
e−i t
∣
Ω ∣∣ =− iΩeN sin(∆Ie S eN)⊥ e−iΩet+iNΩLKt cosθe dt. (7.67)
0 20
Since |S⊥| = sinθe, the instantaneous oscillation amplitude |∆θe| can be bound by
| 1∆θe| ∼ ∣∣∣∣ ∣ΩeN sin(∆IeN) ∣∣∣ . (7.68)2 Ωe−NΩLK
Thus, we see that large oscillation amplitudes in θ̄e occur when Ωe ≈ NΩLK, the
frequency of the N-th Fourier harmonic.
225
7.5.2 General Solution
Single Fourier Harmonic
Above, we began the analysis of Eq. (7.63) by considering the two time-dependent
perturbations separately. However, this is not necessary. Eq. (7.63) can be solved
exactly:
∣∣∣∣t ∫
t
e−iΦS⊥ =−iΩeN cos θ̄e sin∆IeN cos(NΩτ) e−iΦ(τ) dτ,
0  0
= iA ∣e−i ∣Φ∣ ∫t

∣t + iΩ e−iΦ(τ)e dτ . (7.69)
0 0
where A =−tan∆IeN cos θ̄∫e, andt
iΦ(t)≡ iΩe+ΩeN cos(NΩLKτ)cos(∆IeN) dτ,
≡ iΩet+ηsin(NΩLKt) , (7.70)
where η ≡ (ΩeN cos∆IeN) / (NΩLK). Eq. (7.69) shows that |S⊥| is bounded for all t
unless the integral I (x), given by ∫x
I (x)= e−iξ−ηsin(βξ) dξ, (7.71)
0
grows without bound as x→∞, where x=Ωet and β=NΩLK/Ωe.
To see where I (x) grows w∫ithout bound, w(e rewrite∑∞ x − ( ))k= ηsin βξI (x) (cosξ+ isinξ) dξ. (7.72)
k ( ) k=0
k!
0
These sin βξ terms can be expanded using the general trigonometric power-
reduction identities [Zwillin(ger,)2003]: ( )
2n = 1 2n + (−1)
n n∑−1 − l 2nsin y ( 1) cos[2(n− l) y] , (7.73)
22n n 22n−(1 l=0 ) l
n
sin2n+
n
1 = (−1) ∑ − 2n+1y n ( 1)l sin[(2n+1−2l) y] . (7.74)4 l=0 l
226
Due to the orthogonality relations among the trigonometric functions, I (x) only
grows without b(ou)nd if sink(βξ) contains a cosξ or sinξ term. Eqs. (7.73) and (7.74)
show that sink βξ only contains a term with unit frequency if β= 1/q for some in-
teger q≥ 1. When this is the case, all terms with k≥( q)in Eq. (7.72) contain a term
with unit frequency. Among these terms, the sinq βξ term has the largest pref-
actor. Neglecting the other terms with k > q, we can evaluate I for any integer
multiple m of its period 2πq to be ( q )
|I (2πmq)| ≈ η2πmq q . (7.75)2 q!
Within each period, I (x) has additional oscillatory behavior due to the other, off-
resonance terms in Eq. (7.72). However, these oscillations are periodic and van-
ish at every x = 2πmq, so they are bounded and do not affect the divergence in
Eq. (7.75). We conclude that I (x) grows without bound when β= 1/q, or
Ωe =NqΩLK. (7.76)
We see that this differs from the result of the intuitive analysis [Eq. (7.68)], as the
N-th Fourier harmonic generates infinitely many resonances indexed by q≥ 1.
∣ ∣
Instead, if β is near but not on a resonance, i.e. 0< ∣1− qβ∣≪ 1, the amplitude
of oscillation of I (x) is large but bounded. If we take q to be even, then the
maximum value of |I (x)| is dom∣ inated by the near-resonan∣ ce term in Eq. (7.72),∣∣∣∣∫
x q ∣
| | ≃ ∣ η 1 ( )I (x) cosξ cos qβξ dξ∣∣∣∣0 ∣ q! 2
q−1 ∣∣
q
≤ η
∣ ∣ 1 ∣ . (7.77)
2qq! ∣1− qβ∣
If q is instead odd, we use Eq. (7.74) instead of Eq. (7.73) and integrate against
isinξ instead of cosξ in Eq. (7.72), which results in the same bound on the os-
cillation amplitude. Returning to Eq. (7.69), we neglect the first, bounded term
227
(≲ eη ≃ 1) on the right-hand side and obtain the total oscillation amplitude due to
a q-th order resonance w∣ ith the N-th Fourier harmonic∣∣ ∣∆θ̄e∣ 1Nq ∼ q ∣∣∣ [ ]
( )∣
∣ Ω cos(∆I ) q Ω ∣tan I eN eN e ∣∆ eN ∣ . (7.78)2 q! NΩLK Ωe− qNΩ ∣LK
Since Ωe/NΩLK ≈ q, this reduces to Eq. (7.68) when q= 1, as expected.
Generalization to Multiple Fourier Harmonics
After having understood the effect of a single Fourier harmonic, we now return to
the spin evolution equation containing all of the Fourier harmonics [Eq. (7.62)]. If
Ωe/ΩLK ≈∣∣M, t∣∣here can now be multiple N and q satisfying Nq =M. By linearity,the total ∆θ̄e is given by the sum over all of the resonances, so that
∣∣ ∣∆θ̄e∣≈ ∑ ∣∣ ∣∆θ̄e∣Nq , (7.79)
∣ N,q|Nq=M
where ∣ ∣∆θ̄e∣Nq is given by Eq. (7.78).
We next attempt to understand whether particular combinations of N and q
dominate this sum. We make a few simplifying assumptions: (i) all N harmonics
are approximately equal2, ∣∣(ii) Ω∣∣ eN ∼ Ωe and cos∆IeN ≃ 1 (e.g. Figs. 7.5 and 7.6).Under these assumptions, ∆θ̄e Nq with fixed Nq=M scales with respect to q as
∣∣ ∣∆θ̄e∣ cosq (∆I ) qqeNNq ∝ q . (7.80)
∣ ∣ 2 q! p
Stirling’s formula then suggests that ∣∆θ̄ ∣ ∝ (cos(∆I ) e/2)qe Nq eN / q∼ 1. Thus, we
conclude that all combinations of N and q satisfying Nq=M result in comparable
oscillation amplitudes. For simplicity, we evaluate this amplitude for q = 1 and
N = M. If we denote the number of pairs of N and q satisfying Nq = M by d(M)
2This approximation is suitable for the problem studied in the main text because the only char-
acteristic frequency scale is j−1min ≫ 1, so all Fourier harmonics ΩeN for N ≲ j−1min are similar.
228
(the number of positive di∣visor∣s of M),∣we c∣an approximate Eq. (7.79) by:∣∆θ̄e∣≈ d(M) ∣∣∆θ̄e∣M1 ∣
∼ d(M) ∣∣ΩeM sin(∆IeM) ∣∣ , (7.81)
2 ∣ Ωe−M ∣ΩLK
Note that this agrees with Eq. (7.68) except for the factor of d(M). Appendix F.1.3
demonstrates that Eq. (7.81) is in good agreement with detailed numerical simu-
lations when M = 1 or M = 2, the two most relevant cases for our study.
7.5.3 Effect of Resonances in LK-Induced Mergers
We first consider the effect of these resonances (Ω̄e = MΩLK) on the LK-induced
regime, using the fiducial parameters (as in Figs. 7.1 and 7.2). Numerically, we
find thatΩe <ΩLK for the region of parameter space relevant to LK-induced merg-
ers (see Fig. 7.10), so we focus on the effect of the M = 1 resonance. If, for the entire
inspiral, Ωe <ΩLK by a sufficient margin that Eq. (7.81) remains small, then the
conservation of θ̄e cannot be significantly affected by this resonance. For the fidu-
cial simulation, Fig. 7.5 shows the ratio Ωe/ΩLK in the middle panel (black) and
the amplitude of oscillation of θ̄e due to the M = 1 resonance [Eq. (7.81)] in the
bottom panel (blue). We see that the system is never close to the resonant condi-
tion Ωe/ΩLK = 1, and as a result the net effect of the resonance never exceeds a
few degrees.
For∣∣ a m∣∣ ore precise comparison, the bottom-rightmost panel of Fig. 7.6 com-pares ∆θ̄e in the fiducial simulation to the expected contributions from nonadi-
abatic [Eq. (7.41)] and resonant [Eq. (7.81) for M = 1] effects in the regime where
A ≃ 1. We see that Eq. (7.81) for M = 1 describes the oscillations in θ̄e very well.
The agreement is poorer in the bottom-rightmost panel of Fig. 7.7, as the nonadi-
abatic effect is comparatively stronger.
229
It is s∣∣ome∣∣what surprising that the contribution of the resonances to the instan-taneous ∆θ̄e when A ≃ 1 is dominant over that of the nonadiabatic contribution.
In Section 7.4, we have shown that neglecting resonant terms still allows for an
accurate prediction of the final θ̄e deviation [∣Eq. ∣(7.52)]. This implies that, while
the resonances have a large∣r con∣ tribution to ∣ ∣∣ ∣ ∆θ̄e , the nonadi∣abat∣ic effect is moreimportant in determining ∆θ̄e f. This al∣so r∣equires that a ∣∆θ̄e∣ of up to a few
degrees due to resonant effects not affect ∣∆∣ ∣∣θ̄e f ∣∣by more ∣∣than∣∣ ∼ 0.01
◦. This differs
from the nonadiabatic case, where we find ∆θ̄e f ∼max ∆θ̄e . The origin of these
differences in behaviors may be due to the complex phases cancelling differently
in Eqs. (7.40) and (7.69) as the BH binary coalesces.
7.5.4 Effect of Resonances in LK-Enhanced Mergers
We turn now to the case of LK-enhanced mergers, as was studied in LL17, where
the inner binary is sufficiently compact (∼ 0.1 AU) that it can merge in isolation
via GW radiation. We consider a set of parameters that has the same tLK,0 as
the system studied in LL17 but has a tertiary SMBH: m1 = m2 = 30M⊙, a0 =
0.1 AU, e = 10−30 , m3 = 3×107M⊙, ãout = 300 AU, and eout = 0. We show that the
resonances studied above play an important role in shaping the θsl,f distribution
for this regime.
First, we illustrate the θsl,f distribution obtained via numerical simulation,
shown as the blue dots in Fig. 7.11. The prediction assuming adiabatic invariance
(i.e. the conservation of θ̄e) is shown in the red solid line. Good agreement is
observed both when no eccentricity excitation occurs (I ≲ 50◦ and I ≳ 130◦0 0 ) and
when |A |≫ 1 (80◦≲ I0 ≲ 100◦). However, we see in Fig. 7.11 that for intermediate
inclinations, I0 ∈ [50,80] and I0 ∈ [100,130], θsl,f varies over a large range and
230
Figure 7.10: emax and Ωe/ΩLK as a function of I(emin), the inclination of the inner
binary at eccentricity minimum, for varying values of emin (different colors as
labeled) for a LK-induced merger (with the parameters the same as in Figs. 7.1
and 7.2). In this case, only systems with I0 close to 90◦ will merge within a Hubble
time, I(emin)∼ 90◦ for most of the evolution (see Fig. 7.1) until emin ≈ 1 is satisfied.
This plot shows that Ωe ≲ 0.5ΩLK is a general feature of LK-induced mergers, as
is the case for the fiducial simulation (see Fig. 7.5).
231
Figure 7.11: The merger time (top), the magnitude of the initial adiabaticity
parameter |A | ≡ ΩSL/|ΩL| (middle), and the final spin-orbit misalignment angle
θsl,f (bottom) for LK-enhanced mergers, with m1 = m2 = 30M⊙, m3 = 3×107M⊙,
a0 = 0.1 AU, e = 10−30 , ãout = 300 AU, and eout = 0. In the middle panel, the
horizontal dashed line indicates |A | = 1. In the bottom panel, the blue dots de-
note results from numerical simulations with θsl,0 = 0 [these are symmetric about
I0 = 90◦, as the equations of motion (7.3–7.7) are as well]. The prediction for θsl,f
assuming conservation of θ̄e is shown as the red line, which agrees well with the
data both when there is no eccentricity excitation (I0 ≲ 50◦ and I0 ≳ 130◦) and
when |A |≫ 1. For a substantial range of intermediate inclinations (I0 ∈ [50◦,80◦]
and I ∈ [100◦,130◦0 ]), θsl,f is significantly affected by the resonances as they evolve
(see Fig. 7.12). As such, these initial inclinations are expected to give rise to a wide
range of θsl,f, and we denote this with broad red shaded regions.
232
does not agree with the prediction of θ̄e conservation. Note that these inclinations
correspond to neither the fastest nor slowest merging systems.
We attribute the origin of this wide scatter to resonance interactions. Fig-
ure 7.12 illustrates that for these intermediate inclinations, the condition Ωe ∼
ΩLK is satisfied. In addition, as the inner binary coalesces under GW radiation,
emin becomes larger (e.g. see Fig. 7.1). This causes the locations of the resonances
at inclinations less than (greater than) I = 90◦ to evolve to smaller (larger) incli-
nations. As the location of the resonances is a sensitive function of emin, resonance
passage is nonadiabatic. Thus, we expect that all systems that encounter the res-
onance, i.e. all systems with intermediate initial inclinations, will experience an
impulsive kick to θ̄e, resulting in poor θ̄e conservation. This result is denoted by
the two red shaded regions in Fig. 7.11.
While the outer edges of the red shaded regions described above are located
at the critical I0 required for nonzero eccentricity excitation, the inner edges are
harder to characterize. Systems with initial inclinations close to 90◦ start at the
edge of the M = 1 resonance and quickly evolve away from it (as emin increases
and a decreases). As such, they only interact briefly and weakly with the reso-
nances, and the cumulative effect of the resonance interaction can be estimated
by evaluating Eq. (7.81) for M = 1 at the initial conditions. We empirically choose
the transition between such “weakly” and “strongly” resonant systems, i.e. the
inner edge ∣∣of th∣∣e broad red shaded region in Fig. 7.11, when the oscillation semi-amplitude ∆θ̄e predicted by Eq. (7.81) exceeds 3◦.
To understand the general characteristics of systems that interact strongly
with resonances, we examine the quantities in Eq. (7.81):
• sin(∆IeN) is small unless A ≃ 1. Otherwise, Ωe does not nutate substan-
233
Figure 7.12: Same as Fig. 7.10, but for a LK-enhanced merger (with the parame-
ters of Fig. 7.11). In the LK-enhanced regime, all initial inclinations merge within
a Hubble time, and it is clear that Ωe ≈ΩLK can be satisfied for a wide range of
initial inclinations when the initial eccentricity is sufficiently small.
234
Figure 7.13: Definition of angles in the case where L/Lout is nonzero. Similar to
before, we choose the convention where Itot,e ∈ [0◦,90◦] when ΩL > 0 and Itot,e ∈
[90◦,180◦] when ΩL < 0. Here, Lout is not fixed, but Ltot ≡ L+Lout is. Note that
the coordinate system is now oriented with Ẑ∝Ltot.
tially within a LK cycle, and all the ΩeN are aligned with Ωe which implies
that the ∆IeN ≈ 0 for all N ≥ 1.
• Smaller values of emin increase Ωe/ΩLK, as shown in Fig. 7.12.
However, the timescales over which A increases and emin decreases are compa-
rable (see Fig. 7.1). This implies that, if A ≪ 1 initially, which is the case for
LK-induced mergers, then emin will be very close to unity when A grows to be
≃ 1, and the contribution predicted by Eq. (7.81) will remain small throughout the
entire evolution. On the other hand, only if A ≃ 1 and emin ≪ 1 initially, as is the
case for the intermediate inclinations in the LK-enhanced regime, are resonant
interactions likely to be significant.
7.6 Stellar Mass Black Hole Triples
In this section, we extend our predictions for the final spin-orbit misalignment
angle θsl,f to systems where all three masses are comparable and the ratio of the
235
angular momenta of the two bi∣naries, given b[y ]
≡ L ∣∣∣ µ m 1/2= 12aη , (7.82)Lout e=eout=0 µout m123aout
where m123 =m12+m3 and µout =m12m3/m123, is not negligible. When η ̸= 0, Lout
is no longer fixed, but the total angular momentum Ltot ≡ L+Lout is fixed. We
choose the coordinate system with Ẑ= L̂tot, shown in Fig. 7.13.
To analyze this system, we still assume eout ≪ 1 so that the octupole-order
effects are negligible. To calculate the evolution of L, it is only necessary to evolve
the orbital elements of the inner binary [a, e, , Itot (its inclination relative to
L̂tot), and ω] and a single orbital element for the outer binary, its inclination Itot,out
relative to L̂tot. The equa(tion)s of motion are given by [Liu et al., 2015a]:
da = da , (7.83)
dt dt GW ( )
de = 15 2 + dee j(e)sin2ωsin I , (7.84)
dt 8tLK ( dt GW )
d = Ltot 3 cos I 5e
2 cos2ω−4e2−1
, (7.85)
dt Lout 4tLK j(e)
dI 2tot = − 15 e sin2ωsin2I , (7.86)
dt 16tLK j(e)
dItot,out = − 15 ( )η { e
2 sin2ωsin I , (7.87)
dt 8tLK
dω = 3 4cos
2 I+ (5cos(2ω)−1)(1− e2−cos2 I)
dt tLK 8 j(e) }
+ ηcos I [ ]2+ e2(3−5cos(2ω)) +Ω
8 GR
, (7.88)
where I = Itot + Itot,out is the relative inclination between the two angular mo-
menta. The spin evolution of one of the inner BHs is then described in the frame
corotating with L about Ltot by t(he e)quation of motion
dS =Ωe×S, (7.89)dt rot
where
Ωe ≡ΩSLL̂+ΩLL̂tot. (7.90)
236
where ΩL = −d/dt [Eq. (7.85)] is the rate of precession of L about Ltot. As in
Section 7.4, we consider the LK-a(ver)aged Ωe and neglect the harmonic terms:
dS =Ωe×S, (7.91)dt
rot
where
( )
Ωe =ΩSL sin ItotX̂+ (ΩL+ΩSL cos Itot) Ẑ
≡ΩSL sin ĪtotX̂+ ΩL+ΩSL cos Ītot Ẑ. (7.92)
The results of Section 7.4 suggest that the angle θ̄e is an adiabatic invariant,
where θ̄e is given by
Ω
cos eθ̄e ≡ ·S, (7.93)
Ωe
where S is the spin vector averaged over a LK cycle. The orientation of Ωe is
described by the inclination angle Ītot,e (Fig. 7.13), which can be expressed using
Eq. (7.92)
A sin Ī
tan Ī tottot,e = + , (7.94)1 A cos Ītot
where A ≡ΩSL/ΩL is the adiabaticity parameter.
At t= tf, the inner binary is sufficiently compact that θsl is frozen (see bottom
right panel of Fig. 7.1), and the system satisfies A ≫ 1 (Ω −5/2SL ∝ a while ΩL ∝
a3/2). When this is the case, Ωe ∥ L, and so θ̄e,f = θsl,f. Then, since adiabatic
invariance implies θ̄e,f = θ̄e,0,
θsl,f = θ̄e,0. (7.95)
∣ ∣
We first consider the case where S0 ∝ L0. Then θ̄e,0 = ∣Itot,0− Ītot,e,0∣ (see
Fig. 7.13), and so
θsl,f =
∣∣ ∣Itot,0− Īe,0∣ . (7.96)
237
Suppose additionally that the binary initially satisfies |ΩL| ≫ ΩSL, then Ωe is
either parallel or anti-parallel to Ltot depending on whether ΩL is positive or neg-
ative. We denote the starting mutual inclination for which ΩL changes sign by
Ic. Note that I > 90◦c : even though ΩL changes sign at I0 = 90◦, the inclination
decreases over a LK cycle for I < Ilim (where I ◦lim > 90 is the starting mutual in-
clination that maximizes emax [Liu and Lai, 2018]), so the sign of ΩL changes over
a LK cycle for some I ◦c ∈ (90 , Ilim). We then obtain thatI
= tot,0
, I0 < Ic,
θsl,f  (7.97)180◦− Itot,0, I0 > Ic.
More generally, so long as A ≪ 1 initially, we can specify the initial spin orien-
tation by θs,tot,0, the initial anglebetween S and Ltot, givingθ= s,tot,0
, I < Ic,
θsl,f  (7.98)180◦−θs,tot,0, I > Ic.
We first compare these results to numerical simulations by considering a stellar-
mass BH triple that is in the LK-induced regime: we use the same inner binary pa-
rameters as the example of Fig. 7.1, but for a tertiary companion with m3 = 30M⊙,
aout = 4500 AU, and eout = 0. Figure 7.14 shows that Eq. (7.96) accurately predicts
θsl,f when θ ◦sl,0 = 0 for this parameter regime when conservation of θ̄e is good. Fur-
thermore, deviations from exact θ̄e conservation are well described by Eq. (7.61),
the prediction of the theory in Section 7.4. Unlike the η = 0 case (Fig. 7.2), θsl,f
is not symmetric about I ≈ 92.14◦c . This is because |ΩL| is not exactly equal on
either side of Ic. Additionally, unlike in the η = 0 case, the minimum θsl,f is not
exactly zero. We showed in Section 7.4.4 that when η= 0, L is fixed when I = 90◦0 ,
as d/dt= dI/dt= 0. For nonzero η, neither d/dt nor dItot/dt is zero at I0 = Ic.
Eq. (7.96) also gives good agreement in the LK-enhanced merger regime. We
238
Figure 7.14: Similar to Fig. 7.2 but for stellar-mass tertiary m3 = 30M⊙ and
ã3 = 4500 AU. The vertical black line denotes Ic ≈ 92.14◦, the initial inclination
for which ΩL changes signs, and the two horizontal lines denote the predictions of
Eq. (7.96). The dotted black lines bound the deviation due to non-adiabatic evolu-
tion, given by Eq. (7.61).
consider the same inner binary parameters as in Fig. 7.11 but use a tertiary com-
panion with m3 = 30M⊙, aout = 3 AU, and eout = 0. The results are shown in
Fig. 7.15.
7.7 Conclusion and Discussion
In this paper, we have carried out a theoretical study on the evolution of spin-orbit
misalignments in tertiary-induced black-hole (BH) binary mergers. Recent nu-
239
Figure 7.15: Similar to Fig. 7.11 except for a stellar mass tertiary m3 = 30M⊙ and
ãout = 3 AU.
240
merical works [Liu and Lai, 2017, 2018, Liu et al., 2019a, Antonini et al., 2018, Yu
et al., 2020a] have revealed that when binary BHs undergo mergers due to Lidov-
Kozai (LK) oscillations driven by a tertiary companion, the BH spin may evolve
toward a perpendicular state where the final spin-orbit misalignment angle θsl is
close to 90◦. Our theoretical analysis in this paper provides an understanding of
this “90◦ attractor” and characterizes its regime of validity and various spin evo-
lution behaviors during such LK-induced mergers. We focus on hierachical triple
systems where the inner BH binary experiences the “standard” quadupole LK os-
cillations and eventually merges, with the octuple effects playing a negigible role
[ϵoct ≪ 1; see Eq. (7.14)]. For such systems, the spin vectors of the inner BHs obey a
simple evolution equation, Eq. (7.22) or Eq. (7.91), where the “effective” precssion
rateΩe varies quasi-periodically due to the combined effects of LK oscillations and
gravitational radiation. Analysis of this equation yields the following conclusions:
• For BH binaries that have too large initial separations to merge in iso-
lation, LK-induced mergers require large/extreme eccentricity excitations
in the binary driven by a highly inclined tertiary companion. For such
systems, the BH spin evolution behavior can be generally captured by re-
placing Ωe with its LK-average Ωe [thus neglecting the Fourier harmonic
terms in Eq. (7.25)]. If the orbital decay is sufficiently gradual, the angle
θ̄e [Eq. (7.36)] between the spin axis and Ωe is an adiabatic invariant. This
naturally explains the “90◦ attractor” for the final spin-orbit misalignment
angle when the initial tertiary inclination I0 is not too close to 90◦ and the
initial BH spin axis is aligned with the orbital angular momentum axis (see
Fig. 7.2). We show that the deviation from perfect adiabaticity can be pre-
dicted from initial conditions [see Eq. 7.52 and Fig. 7.9].
• When the resonant condition Ωe ≈ MΩLK for integer M is satisfied, sig-
241
nificant variations in θ̄e can arise. We derive an analytic estimate of this
variation amplitude [Eq. (7.81)]. This estimate demonstrates that the reso-
nances are unimportant for “LK-induced” mergers (as depicted in Figs. 7.1–
7.2 and 7.14), but become important for “LK-enhanced” mergers, where the
BH binaries exhibit only modest eccentricity excitations. Our analysis of
the resonance effects qualitatively explain the behavior in θsl,f as seen in
LK-enhanced mergers (see Figs. 7.11 and 7.15).
• For LK-induced mergers of BH binaries with general tertiary companions,
we provide an analytic prescription for calculating the final spin-orbit mis-
alignment angle for arbitrary initial spin orientations (Section 7.6). This
prescirption is based on the approximate adiabatic invariance of θ̄e, and
produces results that are in agreement with numerical simulations (see
Fig. 7.14) in the appropriate regime.
There are several simplificatons in our theoretical analysis that are worth
mentioning. (i) We have neglected the octupole effects in the LK oscillations. This
is appropriate for systems where the tertiary orbit is circular and/or the semi-
major axis aout is much larger than the inner binary (as in the case when the
tertiary is a SMBH) and/or the inner binary BHs have nearly equal masses [see
Eq. (7.14)]. The octupole effects are known to significantly broaden the inclination
window for extreme eccentricity excitations, and therefore enhance the efficiency
of LK-induced mergers [Liu and Lai, 2018]. When the octupole effects are signifi-
cant, the LK orbital evolution is not integrable, and the eccentricity excitations are
no longer regular. As a result, Ωe has neither consistent direction nor magnitude,
and our theory cannot be applied. In fact, the resulting θsl,f distribution is largely
unrelated to the initial θ̄e,0 and the “90◦ attractor” is significantly “erased” [Liu
and Lai, 2018]. (ii) If the system is not sufficiently hierarchical (aout is too small),
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the double averaging approximation for the dynamics of the triple breaks down
[Liu and Lai, 2018]. In this case, there is little reason to expect any relation be-
tween θsl,f and θ̄e,0. However, Liu and Lai [2018] found that the double averaged
orbital equations predict the correct merger window and merger fractions even
beyond their regime of validity if the octupole effect is weak, so it is possible that
our results concerning θsl,f are also somewhat robust even when the double aver-
aged equations formally break down. (iii) In this work, we only consider spin-orbit
coupling, and follow the evolution of θsl until the orbital separation is sufficiently
small such that the inner binary is gravitationally decoupled from the tertiary
and the spin-orbit misalignment angle is frozen. To leading post-Newtonian (PN)
order, θsl is constant for small separations until the spin-spin interaction (2 PN)
becomes important. This interaction is non-negligible only when binary enters the
LIGO band and when the spin magnitude of each BH is appreciable [Liu et al.,
2019a, Yu et al., 2020a].
As noted in Section 7.1, the merging BH binaries detected by LIGO/VIRGO
in O1 and O2 have χeff ∼ 0 [Abbott et al., 2016, 2019]. One possible explanation
for this is that BHs are born slowly rotating [e.g. Fuller and Ma, 2019]. But our
“90◦ attractor” provides an alternative explanation with no assumptions on the
BH spin magnitudes if the mergers are “LK-induced” and θ ≈ 0◦sl,0 . In the O3
event GW190521, each BH has a significant spin magnitude and a large spin-
orbit misalignment angle [Abbott et al., 2020b]. If the evolution history of the
system resembled our LK-induced scenario [see Liu and Lai, 2021], a primordial
θsl,0 ≈ 0◦ would be consistent with the observed outcome of θ ◦sl,f ∼ 90 .
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CHAPTER 8
THE MASS RATIO DISTRIBUTION OF TERTIARY INDUCED BINARY
BLACK HOLE MERGERS
Originally published in:
Yubo Su, Bin Liu, and Dong Lai. The mass-ratio distribution of tertiary-induced
binary black hole mergers. MNRAS, 505(3):3681–3697, 2021b
8.1 Introduction
The 50 or so black hole (BH) binary mergers detected by the LIGO/VIRGO col-
laboration to date [Abbott et al., 2021] continue to motivate theoretical studies of
their formation channels. These range from the traditional isolated binary evolu-
tion, in which mass transfer and friction in the common envelope phase cause the
binary orbit to decay sufficiently that it subsequently merges via emission of grav-
itational waves (GWs) [e.g., Lipunov et al., 1997, 2017, Podsiadlowski et al., 2003,
Belczynski et al., 2010, 2016, Dominik et al., 2012, 2013, 2015], to various flavors
of dynamical formation channels that involve either strong gravitational scatter-
ings in dense clusters [e.g., Zwart and McMillan, 1999, O’leary et al., 2006, Miller
and Lauburg, 2009, Banerjee et al., 2010, Downing et al., 2010, Ziosi et al., 2014,
Rodriguez et al., 2015, Samsing and Ramirez-Ruiz, 2017, Samsing and D’Orazio,
2018, Rodriguez et al., 2018, Gondán et al., 2018] or mergers in isolated triple
and quadruple systems induced by distant companions [e.g., Miller and Hamilton,
2002, Wen, 2003, Antonini and Perets, 2012, Antonini et al., 2017, Silsbee and
Tremaine, 2017b, Liu and Lai, 2017, 2018, Randall and Xianyu, 2018b,a, Hoang
et al., 2018, Fragione and Kocsis, 2019, Fragione and Loeb, 2019, Liu and Lai,
2019, Liu et al., 2019a,b, Liu and Lai, 2020, Liu and Lai, 2021].
Given the large number of merger events to be detected in the coming years,
244
it is important to search for observational signatures to distinguish various BH
binary formation channels. The masses of merging BHs obviously carry impor-
tant information. The recent detection of BH binary systems with component
masses in the mass gap (such in GW190521) suggests that some kinds of “hierar-
chical mergers” may be needed to explain these exceptional events (Abbott et al.,
2020b; see Liu and Lai, 2021 for examples of such “hierarchical mergers” in stel-
lar multiples). Another possible indicator is merger eccentricity: previous studies
find that dynamical binary-single interactions in dense clusters [e.g., Samsing and
Ramirez-Ruiz, 2017, Rodriguez et al., 2018, Samsing and D’Orazio, 2018, Fragione
and Bromberg, 2019] or in galactic triples [Silsbee and Tremaine, 2017b, Antonini
et al., 2017, Fragione and Loeb, 2019, Liu et al., 2019a] may lead to BH binaries
that enter the LIGO band with modest eccentricities. The third possible indica-
tor is the spin-orbit misalignment of the binary. In particular, the mass-weighted
projection of the BH spins,
χ = m1χ1+m2χ2eff + · L̂, (8.1)m1 m2
can be measured through the binary inspiral waveform [here, m1,2 is the BH mass,
χ 21,2 = cS1,2/(Gm1,2) is the dimensionless BH spin, and L̂ is the unit orbital angu-
lar momentum vector of the binary]. Different formation histories yield different
distributions of χeff [Liu and Lai, 2017, 2018, Antonini et al., 2018, Rodriguez
et al., 2018, Gerosa et al., 2018, Liu et al., 2019a, Su et al., 2021a].
The fourth possible indicator of BH binary formation mechanisms is the dis-
tribution of masses and mass ratios of merging BHs. In Fig. 8.1, we show the
distribution of the mass ratio q≡m2/m1, where m1 ≥m2, for all LIGO/VIRGO bi-
naries detected as of the O3a data release [Abbott et al., 2021]1. The distribution
1Note that Fig. 8.1 should not be interpreted as directly reflecting the distribution of merging
BH binaries, as there are many selection effects and observational biases, e.g. systems with smaller
q are harder to detect for the same Mchirp or m12. For a detailed statistical analysis, see Abbott
et al. [2021].
245
Figure 8.1: Histogram of the mass ratios q ≡m2/m1 of binary BH mergers in the
O3a data release, excluding the two NS-NS mergers but including GW190814,
whose 2.5M⊙ secondary may be a BH [Abbott et al., 2021].
distinctly peaks around q∼ 0.7. BH binaries formed via isolated binary evolution
are generally expected to have q ≳ 0.5 [Belczynski et al., 2016, Olejak et al., 2020].
On the other hand, dynamical formation channels may produce a larger variety
of distributions for the binary mass ratio [e.g., Rodriguez et al., 2016, Silsbee and
Tremaine, 2017b, Fragione and Kocsis, 2019].
In this paper, we study in detail the mass ratio distribution for BH mergers
induced by tertiary companions in isolated triple systems. In this scenario, a ter-
tiary BH on a sufficiently inclined (outer) orbit induces phases of extreme eccen-
tricity in the inner binary via the von Zeipel-Lidov-Kozai (ZLK;von Zeipel, 1910,
Lidov, 1962, Kozai, 1962) effect, leading to efficient gravitational radiation and or-
bital decay. While the original ZLK effect relies on the leading-order, quadrupolar
gravitational perturbation from the tertiary on the inner binary, the octupole or-
der terms can become important [sometimes known as the eccentric Kozai mech-
246
anism, e.g. Naoz, 2016] when the triple system is mildly hierarchical, the outer
orbit is eccentric (eout ̸= 0) and the inner binary BHs have unequal masses [e.g.,
Ford et al., 2000, Blaes et al., 2002, Lithwick and Naoz, 2011, Liu et al., 2015a].
The strength of the octupole effect depends on the dimensionless parameter
m1−m2 a eout
ϵoct = + − 2 . (8.2)m1 m2 aout 1 eout
where a,aout are the semi-major axes of the inner and outer binaries, respectively.
Previous studies have shown that the octupole terms generally increase the incli-
nation window for extreme eccentricity excitation, and thus enhance the rate of
successful binary mergers [Liu and Lai, 2018]. As ϵoct ∝ (1− q)/(1+ q) increases
with decreasing q, we expect that ZLK-induced BH mergers favor binaries with
smaller mass ratios. The main goal of this paper is to quantify the dependence
of the merger fraction/probability on q, using a combination of analytical and nu-
merical calculations. We focus on the cases where the tertiary mass is compara-
ble to the binary BH masses. When the tertiary mass m3 is much larger than
m12 =m1+m2 (as in the case of a supermassive BH tertiary), dynamical stability
of the triple requires aout(1− eout)/[a(1+ e)] ≳ 3.7(m3/m )1/312 ≫ 1 [Kiseleva et al.,
1996], which implies that the octupole effect is negligible.
This paper is organized as follows. In Section 8.2, we review some analytical
results of ZLK oscillations and examine how the octupole terms affect the incli-
nation window and probability for extreme eccentricity excitation. In Section 8.3,
we study tertiary-induced BH mergers using a combination of numerical and an-
alytical approaches. We propose new semi-analytical criteria (Section 8.3.2) that
allow us to determine, without full numerical integration, whether an initial BH
binary can undergo a “one-shot merger” or a more gradual merger induced by the
octupole effect of an tertiary. In Section 8.4, we calculate the merger fraction as a
function of mass ratio for some representative triple systems. In Section 8.5, we
247
study the mass ratio distribution of the initial BH binaries based on the properties
of main-sequence (MS) stellar binaries and the MS mass to BH mass mapping. Us-
ing the result of Section 8.4, we illustrate how the final merging BH binary mass
distribution may be influenced by the octupole effect for tertiary-induced mergers.
We summarize our results and their implications in Section 8.6.
8.2 Von Zeipel-Lidov-Kozai (ZLK) Oscillations: Analytical Re-
sults
Consider two BHs orbiting each other with masses m1 and m2 on a orbit with
semi-major axis a, eccentricity e, and angular momentum L. An external, tertiary
BH of mass m3 orbits this inner binary with semi-major axis aout, eccentricity
eout, and angular momentum Lout. The reduced masses of the inner and outer
binaries are µ≡m1m2/m12 and µout ≡m12m3/m123 respectively, where m12 =m1+
m2 and m123 = m12+m3. These two binary orbits are further described by three
angles: the inclinations i and iout, the arguments of pericenters ω and ωout, and
the longitudes of the ascending nodes Ω and Ωout. These angles are defined in a
coordinate system where the z axis is aligned with the total angular momentum
J=L+Lout (i.e., the invariant plane is perpendicular to J). The mutual inclination
between the two orbits is denoted I ≡ i+ iout. Note that Ωout =Ω+180◦.
To study the evolution of the inner binary under the influence of the tertiary
BH, we use the double-averaged secular equations of motion, including the inter-
actions between the inner binary and the tertiary up to the octupole level of ap-
proximation as given by Liu et al. [2015a]. Throughout this paper, we restrict to
hierarchical triple systems where the double-averaged secular equations are valid
248
– systems with relatively small aout/a may require solving the single-averaged
equations of motion or direct N-body integration [see Antonini and Perets, 2012,
Antonini et al., 2014, Luo et al., 2016, Lei et al., 2018, Liu and Lai, 2019, Liu et al.,
2019a, Hamers, 2020a]2. For the remainder of this section, we include general rel-
ativistic apsidal precession of the inner binary, a first order post-Newtonian (1PN)
effect, but omit the emission of GWs, a 2.5PN effect – this will be considered in
Section 8.3. We group the results by increasing order of approximation, starting
by ignoring the octupole-order effects entirely.
8.2.1 Quadrupole Order
At the quadrupole order, the tertiary induces eccentricity oscillations in the inner
binary on the characteristic timescale
= 1 m
( )
t 12
aout,eff 3
ZLK , (8.3)
√ n m3 a √
where n≡ Gm12/a3 is the mean motion of the inner binary, and a 2out,eff ≡ aout 1− eout.
During these oscillations, there are two conserved quantities, the total energy and
the total orbital angular momentum. Through some manipulation, the total an-
gular momentum can be written in terms of the conserved quantity K given by
K ≡ j(e)cos I− η e2. (8.4)
2
p
Here, j(e)≡ 1− e2 and η is the ratio of the magnitudes of the angular momenta
at zero inner binary ec(centri)city: [ ]1/2
≡ L = µ m12aη . (8.5)
Lout = µout m a (1− e2e 0 123 out out)
2Although we do not study such systems in this paper, we expect that a qualitatively similar
dependence of the merger probability on the mass ratio remains, since the strength of the octupole
effect in the single-averaged secular equations is also proportional to (1− q)/(1+ q) [see Eq. 25 of
Liu and Lai, 2019].
249
Note that when η= 0, K reduces to the classical “Kozai constant”, K = j(e)cos I.
The maximum eccentricity emax attained in these ZLK oscillations can be com-
puted analytically at the quadrupolar order. It depends on the “competition” be-
tween the 1PN apsidal precession rate ω̇GR and the ZLK rate t−1ZLK. The relevant
dimensionless parameter is
3
≡ = 3Gmt 12 m12
aout,eff
ϵGR (ω̇GR ZLK)e=0 2 4 . (8.6)c m3 a
It can then be shown that, for an initially circular inner binary, emax is related to
the initial mutual inclinat[io(n I0 by [Li)u et(al., 2015a, Anderson et al., 2016]:3 j2(emax)− )1 η 2 95 cos I + − 3+[4ηcos I + ]η2 j2(e )8 j2(e ) 0 2 ] 0 4 maxmax
+η2 j4(emax) +ϵGR 1−
1 = 0. (8.7)
j(emax)
In the limit η→ 0 and ϵGR → 0, we r√ecover the well-known result
emax = 1− (5/3)cos2 I0. (8.8)
For general η, emax attains its limiting value elim when I0 = I0,lim, where [see also
Hamers, 2020b] [ ]
η 4
cos I 20,lim = j (elim)−1 . (8.9)2 5
Note that I0,lim ≥ 90◦ with equality only when η = 0. Substituting Eq. (8.9) into
Eq. (8.7), we find that elim satisfies
3 [ 2 − ][ ( )]− + η2 4j (elim) 1 3 [ j2(e8 4 5 lim])−1
+ − 1ϵGR 1 = 0. (8.10)j(elim)
On the other hand, eccentricity excitation (emax ≥ 0) is only possible when (cos I0)− ≤
cos I0 ≤ (cos I0)+ where  √ 
= 1(cos I0)± −η± η2+60− 80ϵGR . (8.11)10 3
250
Figure 8.2: The maximum eccentricity achieved for an inner binary in the test-
particle limit as a function of the initial inclination angle I0. The triple system
parameters are: a = 100 AU, aout,eff = 3600 AU, m12 = 50M⊙, m3 = 30M⊙, and
eout = 0.6; the corresponding octupole strength parameter is ϵoct = 0.02 and η≃ 0.
The octupole-level secular equations of motion are integrated for 2000tZLK (see
Eq. 8.3), and the maximum eccentricity attained during this time is recorded and
shown as a blue dot for each initial condition. We consider 1000 initial inclinations
in the range 50◦ ≤ I0 ≤ 130◦, and each I0 is simulated five times, with the initial
orbital elements ω, ωout, and Ω = Ωout −π chosen randomly ∈ [0,2π) shows the
quadrupole-level result (Eq. 8.7 with η = 0), and elim (Eq. 8.10) is shown as the
horizontal red line. The vertical purple lines denote the boundary of the octupole-
active inclination window, based on the fitting formula from Muñoz et al. [2016]
(Eq. 8.13).
For I0 outside of this range, no eccentricity excitation is possible. This condition
reduces to the well-known cos2 I0 ≤ 3/5 when η= ϵGR = 0.
8.2.2 Octupole Order: Test-particle Limit
The relative strength of the octupole-order potential to the quadrupole-order po-
tential is determined by the dimensionless parameter ϵoct (Eq. 8.2). When ϵoct is
251
non-negligible, K is no longer conserved, and the system evolution becomes chaotic
[Ford et al., 2000, Katz et al., 2011, Lithwick and Naoz, 2011, Li et al., 2014, Liu
et al., 2015a]. As a result, analytical (and semi-analytical) results have only been
given for the test-particle limit, where m2 = η= 0. We briefly review these results
below.
Due to the non-conservation of K , emax evolves irregularly ZLK cycles, and
the orbit may even flip between prograde (I < 90◦) and retrograde (I > 90◦) if K
changes sign (in the test-particle limit, K = j(e)cos I). During these orbit flips, the
eccentricity maxima reach their largest values but do not exceed elim [Lithwick
and Naoz, 2011, Liu et al., 2015a, Anderson et al., 2016]. These orbit flips occur
on characteristic timescale tZLK,oct, given by [Antognini, 2015]
p
= 128 10tZLK,oct tZLK p . (8.12)15π ϵoct
The octupole potential tends to widen the inclination range for which the eccen-
tricity can reach elim; we refer to this widened range as the octupole-active win-
dow. Figure 8.2 shows the maximum eccentricity attained by an inner binary
orbited by a tertiary companion with inclination I0. The octupole-active window
is visible as a range of inclinations centered on I0 = 90◦ that attain elim (the red
horizontal dashed line in Fig. 8.2). Katz et al. [2011] show that this window can be
approximated using analytical arguments when ϵoct ≪ 1. Muñoz et al. [2016] give
a more general numerical fitting formula describing the octupole-active window
for arbitrary ϵoct. They find that orbit flips and extreme eccentricity excitation
occur for Iflip,−≲ I0 ≲ Iflip,+ where ( ) ( )
0.26 ϵoct −0.536 ϵoct 20.1 0.1
cos2 Iflip,± =
 +
( ) (
12.05 ϵoct 3 ϵ
)
oct 4 (8.13)
 0.1
−16.78 0.1 ϵoct ≲ 0.05,
0.45 ϵoct ≳ 0.05.
252
Figure 8.3: An example of the triple evolution for a system with significant oc-
tupole effects and finite η (see Eq. 8.5). We use the same system parameters as
in Fig. 8.2 except for q = 0.2, corresponding to η ≈ 0.087 and ϵoct ≈ 0.007, and
I0 = 93.5◦. The three panels show the inner orbit eccentricity, the mutual inclina-
tion, and the generalized “Kozai constant” K (Eq. 8.4). In the first panel, elim is
denoted by the black dashed line. By comparing the second and third panels, we
see that orbit flips occur when K crosses the dotted line, given by K =Kc ≡−η/2.
In Fig. 8.2, we see that with the octupole effect included, emax indeed attains elim
when I0 is within the broad octupole-active window given by Eq. (8.13) (denoted
by the vertical purple lines in Fig. 8.2).
253
8.2.3 Octupole Order: General Masses
For general inner binary masses, when the angular momentum ratio η is non-
negligible, the octupole-level ZLK behavior is less well-studied [see Liu et al.,
2015a]. Figure 8.3 shows an example of the evolution of a triple system with
significant η and ϵoct. Many aspects of the evolution discussed in Section 8.2.2 are
still observed: the ZLK eccentricity maxima and K evolve over timescales ≫ tZLK;
the eccentricity never exceeds elim; when K crosses Kc ≡−η/2, an orbit flip occurs
(this follows by inspection of Eq. 8.4).
However, Eq. (8.13) no longer describes the octupole-active window as η is non-
negligible [see also Rodet et al., 2021]. In the top panel of Fig. 8.4, the blue dots
show the maximum achieved eccentricity of a system with the same parameters
as Fig. 8.2 except with q = 0.5 (so ϵoct = 0.007 and η= 0.087). Here, it can be seen
that no prograde systems can attain elim, and only a small range of retrograde
inclinations ≥ I0,lim (see Eq. 8.9) are able to reach elim. In fact, there is even a
clear double valued feature around I ≈ 75◦ in the top panel of Fig. 8.4 that is not
present in Fig. 8.2. If q is decreased to 0.3 (Fig. 8.5) or further to 0.2 (Fig. 8.6), ϵoct
increases while η decreases. This permits a larger number of prograde systems
to reach elim, though a small range of inclinations near I0 = 90◦ still do not reach
elim; we call this range of inclinations the “octupole-inactive gap”. On the other
hand, if q is held at 0.5 as in Fig. 8.4 and eout is increased to 0.9 while holding
aout,eff = 3600 AU constant, both ϵoct and η increase; the top panel of Fig. 8.7 shows
that prograde systems still fail to reach elim for these parameters, despite the
increase in ϵoct. The top panel of Fig. 8.8 illustrates the behavior when the inner
binary is substantially more compact (a = 10 AU): even though ϵoct is larger than
it is in any of Figs. 8.4–8.7, we see that prograde perturbers fail to attain elim.
All of these examples (top panels of Figs. 8.4–8.8) illustrate importance of η in
254
determining the range of inclinations for the system to be able to reach elim.
In general, we find that a symmetric octupole-active window (as in Eq. 8.13)
can be realized for sufficiently small η. Rodet et al. [2021] considered some ex-
amples of triple systems (consisting of MS stars with planetary companions and
tertiaries, for which the short-range forces is dominated by tidal interaction) and
found that η ≲ 0.1 is sufficient for a symmetric octupole-active window. In the
cases considered in this paper, a smaller η is necessary (e.g., η≃ 0.054 in Fig. 8.6).
Thus, the critical η above which the symmetry of the octupole-active window is sig-
nificantly broken likely depends on the dominant short-range forces and elim [in
Rodet et al. [2021], 1− e −3lim ∼ 10 , while in Figs. 8.2 and 8.4–8.8, 1− e ≲ 10−5lim ].
In general, when η is non-negligible, there are up to two octupole-active windows:
a prograde window whose existence depends on the specific values of η and ϵoct,
and a retrograde window that always exists.
8.3 Tertiary-Induced Black Hole Mergers
Emission of gravitational waves (GWs) affects the evolution of the inner binary,
which can be incorporated into the secular equations of motion for the triple [e.g.,
Peters, 1964, Liu and Lai, 2018]. The associated orbital and eccentricity decay
rates are [Peters, 1964]:∣
1 da ∣∣ ≡− 1
a dt ∣GW tGW
64 G3µm2
( )
=− 12 + 73 37∣ 5 4 7 1 e
2+ e4 , (8.14)
∣ 5 c a j (e) 24 96de ∣∣ 3 ( )=−304 G µm212 1 + 121 25 4 5 1 e . (8.15)dt GW 15 c a j (e) 304
GW emission can cause the orbit to decay significantly when extreme eccentric-
ities are reached during the ZLK cycles described in the previous section. This
255
Figure 8.4: Eccentricity excitation and merger windows for the fiducial BH triple
system (a= 100 AU, aout,eff = 3600 AU, m12 = 50M⊙, m3 = 30M⊙) with q = 0.5 and
eout = 0.6, corresponding to η ≈ 0.087 and ϵoct ≈ 0.007. In the top panel, for each
of 1000 initial inclinations, we choose 5 different random ω, ωout, and Ω as initial
conditions and evolve the system for 2000tZLK without GW radiation. The effec-
tive eccentricity eeff (Eq. 8.22; green dots) as well as the maximum eccentricity
emax (blue dots) over this period are displayed. For comparison, eeff,c (Eq. 8.23) is
given by the horizontal green dashed line, eos (Eq. 8.19) is shown as the horizontal
blue line, and elim (Eq. 8.10) is shown as the horizontal red dashed line. The ver-
tical purple lines denote the test-mass octupole-active window and are given by
the fitting formula of Muñoz et al. [2016]; they do not longer accurately describe
the elim-attaining inclination window because η is finite. The black dashed line is
is the quadrupole-level result as given by Eq. (8.7). In the middle panel, we show
the binary merger times when including GW radiation and using the same range
of initial conditions. Numerical integrations are terminated when Tm > 10 Gyr
and marked as unsuccessful mergers. The horizontal dashed line denotes tZLK
(Eq. 8.3) while the horizontal dash-dotted line indicates tZLK,oct (Eq. 8.12). Here,
each I0 is run 20 times with uniform distributions of ω, ωout, and Ω, so we can
estimate the merger probability Pmerger (Eq. 8.17) for each I0 – Pmerger is shown
as the black line in the bottom panel. As described in Section 8.3.2, the merger
probability can be predicted semi-analytically using the results of the top panel
and Eq. (8.25), and is denoted by Panmerger. In the bottom panel, the thick green
line shows Panmerger when using an integration time of 2000tZLK ≈ 3 Gyr for the
non-dissipative simulations, and thin green line shows the prediction using an
integration time of 500tZLK. The agreement of Panmerger with Pmerger is good and
improves when using the longer integration time.
256
Figure 8.5: Same as Fig. 8.4 but for q = 0.3, corresponding to η ≈ 0.07 and ϵoct ≈
0.011.
257
Figure 8.6: Same as Fig. 8.4 but for q = 0.2, corresponding to η≈ 0.054 and ϵoct ≈
0.014.
258
Figure 8.7: Same as Fig. 8.4 but for eout = 0.9 while holding aout,eff the same,
corresponding to η= 0.118 and ϵoct = 0.019.
259
Figure 8.8: Same as Fig. 8.4 but for a more compact inner binary; the parameters
are a0 = 10 AU, aout,eff = 700 AU, m12 = 50M⊙, m3 = 30M⊙, eout = 0.9, and q = 0.4,
corresponding to η= 0.118 and ϵoct = 0.029. Here, Pmerger is computed with only 5
integrations (for random ω, ωout, and Ω) for each I0.
260
allows even wide binaries (∼ 100 AU) to merge efficiently within a Hubble time.
While various numerical examples of such tertiary-induced mergers have been
given before (e.g., Liu and Lai [2018]; see also Liu et al. [2019a] for “population
synthesis”), in this section we examine the dynamical process in detail in order
to develop an analytical understanding. Our fiducial system parameters are as in
Fig. 8.3: aout,eff = 4500 AU, eout = 0.6, m12 = 50M⊙ (with varying q), m3 = 30M⊙,
and the inner binary has initial a0 = 100 AU and e0 = 10−3.
8.3.1 Merger Windows and Probability: Numerical Results
To understand what initial conditions lead to successful mergers within a Hubble
time, we integrate the double-averaged octupole-order ZLK equations including
GW radiation. We terminate each integration if either a = 0.005a0 (a successful
merger) or the system age reaches 10 Gyr. We can verify that the inner binary
is effectively decoupled from the tertiary for this orbital separation by evaluating
ϵGR (Eq. 8.6): ( )
m 2 ( a ) ( )−1 ( )= 1.8×106 12 out,eff 3 m3 a −4ϵGR . (8.16)50M⊙ 3600 AU 30M⊙ 0.5 AU
The middle panel of Fig. 8.4 shows the merger time Tm as a function of I0 for our
fiducial parameters with q = 0.5. We note that only retrograde inclinations lead
to successful mergers, and almost all successful mergers are rapid, with Tm ∼
tZLK,oct. These are the result of a system merging by emitting a single large burst
of GW radiation during an extreme-eccentricity ZLK cycle, which we term a “one-
shot merger”3. In Fig. 8.5, q is decreased to 0.3, and some prograde systems are
3It is important to note that these “one-shot mergers” are distinct from the “fast” mergers pre-
viously discussed in the literature [e.g. Wen, 2003, Randall and Xianyu, 2018a, Su et al., 2021a]:
The one-shot mergers discussed here occur when the maximum eccentricity attained by the inner
binary over an octupole cycle (i.e. within the first ∼ tZLK,oct) is sufficiently large to produce a prompt
merger, while the references cited above neglect octupole-order effects and study the scenario when
261
also able to merge successfully. However, these prograde systems exhibit a broad
range of merger times, with Tm ≳ tZLK,oct. These occur when a system gradually
emits a small amount of GW radiation at every eccentricity maximum – we term
this a “smooth merger”. Additionally, the octupole-inactive gap near I0 = 90◦ is
visible in the merger time plot (middle panel of Fig. 8.5). The middle panels of
Figs. 8.6–8.8 show the behavior of Tm for the other parameter regimes and also
exhibit these two categories of mergers and the octupole-inactive gap.
Due to the chaotic nature of the octupole-order ZLK effect, the initial inclina-
tion I0 alone is not sufficient to determine with certainty whether a system can
merge within a Hubble time. Instead, for a given I0, we can use numerical in-
tegrations with various ω, ωout, and Ω to compute a merger probability, denoted
by
Pmerger (I0; q, eout)= P (Tm < 10 Gyr) , (8.17)
where the notation Pmerger (I0; q, eout) highlights the dependence of Pmerger on q
and eout, two of the key factors that determine the strength of the octupole effect
(of course Pmerger depends on other system parameters such as m12, a0, aout, etc.).
The bottom panels of Figs. 8.4–8.8 show our numerical results. In all of these plots,
there is a retrograde inclination window for which successful merger is guaran-
teed. In Fig. 8.5, it can be seen that a large range of prograde inclinations have
a probabilistic outcome. In Fig. 8.6, while the enhanced octupole strength allows
for most of the prograde inclinations to merge with certainty, there is still a region
around I0 ≈ 80◦ where Pmerger < 1.
the maximum eccentricity attained in a quadrupole ZLK cycle (i.e. within the first ∼ tZLK) is suffi-
ciently large to produce a prompt merger. When the octupole effect is non-negligible, it can drive
systems to much more extreme eccentricities than can the quadrupole-order effects alone (compare
the blue dots and black dashed line in Fig. 8.4), and thus our “one-shot mergers” occur for a larger
range of I0 than do quadrupole-order “fast” mergers.
262
8.3.2 Merger Probability: Semi-analytic Criteria
By comparing the top and bottom panels of Figs. 8.4–8.8, it is clear that their
features are correlated: in all five cases, the retrograde merger window occupies
the same inclination range as the retrograde octupole-active window, while Pmerger
is only nonzero for prograde inclinations where emax nearly attains elim. Here, we
further develop this connection and show that the non-dissipative simulations can
be used to predict the outcomes of simulations with GW dissipation rather reliably.
In Section 8.3.1, we identified both one-shot and smooth mergers in our simu-
lations. Towards understanding the one-shot mergers, we first define eos to be the
emax required to dissipate an order-unity fraction of the binary’s orbital energy via
GW emission in a single ZLK cycle. Since a binary spends a fraction ∼ j(emax) of
each ZLK cycle near emax [e.g., Anderson et al., 2016], we set
dlna
j (eos) ∣∣∣∣ =− 1 , (8.18)dt e=eos tZLK
where d(lna)/dt is given by Eq. (8.14). This yields
425t 3 3 ( )
j6(e )≡ ZLK = 170 G µm12 aout,eff
3
os 5 4 , (8.19)96tGW,0 3 m3c a n a
where tGW,0 = (tGW)e=0 (see Eq. 8.14) is given by
64 G3µm2
t−1 = 12GW,0 5 , (8.20)5 c a4
we have approximated eos ≈ 1. (Eq. (8.1)9) i(s equivalent to
m 7/6 q/(1+ q)2 )1/3 ( )− ≈ × −6 12 m −1/31 e 3os 3 (10 50M)(⊙ ) 1/4 30M⊙
× aout,eff a −11/6 . (8.21)
3600 AU 100 AU
Then, if a system satisfies emax > eos with emax based on non-dissipative integra-
tion, it is expected attain a sufficiently large eccentricity to undergo a one-shot
merger.
263
Towards understanding smooth mergers, we seek a characteristic eccentricity
that captures GW emission over many ZLK cycles. We define eeff as an effective
ZLK maximum ecce〈ntricity〉, i.e. 〈
dlna =− 1 1+
〉
73e2/24+37e4/96
dt tGW,0 j7(e)
≡−425/96 1 , (8.22)
tGW,0 j6(eeff)
where the angle brackets denote averaging over many tZLK,oct in order to capture
the characteristic eccentricity behavior over many octupole cycles. In the second
line of Eq. (8.22), we have essentially replaced the ZLK-averaged orbital decay
rate by d(lna)/dt evaluated at eeff multiplied by j(eeff). In practice (see Figs. 8.4–
8.8), we typically average over 2000tZLK of the non-dissipative simulations to com-
pute eeff.
With eeff computed using Eq. (8.22), we can define the critical effective ec-
centricity eeff,c such that the ZLK-averaged inspiral time is a Hubble time, i.e.
〈d(lna)/dt〉 ≡− (10 Gyr)−1. This gives
( )
j6 eeff,c ≡
425 10 Gyr
, (8.23)
96 tGW,0
or equivalently ( )(
m 2
)1/3 ( )
1− e ≈ 10−4 12 q/(1+ q) a −4/3eff,c . (8.24)50M⊙ 1/4 100 AU
Thus, if a system is evolved using the non-dissipative equations of motion and
satisfies eeff > eeff,c, then it is expected to successfully undergo a smooth merger
within a Hubble time.
Therefore, a system can be predicted to merge successfully if it satisfies either
the one-shot or smooth merger criteria. The semi-analytical merger probability
(as a function of I0 and other parameters) is:
Pan
( )
merger (I0; q, eout)= P eeff > eeff,c or emax > eos . (8.25)
264
Although not fully analytical (since numerical integrations of non-dissipative sys-
tems are needed to obtain eeff and emax in general), Eq. (8.25) provides efficient
computation of the merger probability without full numerical integrations includ-
ing GW radiation.
The top panels of Figs. 8.4–8.8 show eeff and emax, and their critical values,
eeff,c and eos. Using these, we compute the semi-analytical merger probability,
shown as the thick green lines in the bottom panels of Figs. 8.4–8.8. We gener-
ally observe good agreement with the numerical Pmerger. However, Panmerger slightly
but systematically underpredicts Pmerger for some configurations, such as the pro-
grade inclinations in Figs. 8.5 and 8.8. These regions coincide with the inclina-
tions for which the merger outcome is uncertain. This underprediction is due to
the restricted integration time of 2000tZLK ≈ 3 Gyr used for the non-dissipative
simulations. To illustrate this, we also calculate Panmerger using a shorter integra-
tion time of 500tZLK for our non-dissipative simulations. The results are shown
as the light green lines in the bottom panels of Figs. 8.4–8.8, performing visibly
worse. A more detailed discussion of this issue can be found in Section 8.4.4.
A few observations about Eq. (8.25) can be made. First, it explains why some
prograde systems merge probabilistically (0< Pmerger < 1): for the prograde incli-
nations in Fig. 8.5, the eeff values scatter widely around eeff,c [or more precisely,
j(eeff) scatters around j(eeff,c)], even for a given I0, so the detailed merger outcome
depends on the initial conditions. For the prograde inclinations in Fig. 8.6, the
double-valued feature in the emax plot (the top panel) pointed out in Section 8.2.3
represents a sub-population of systems that do not satisfy Eq. (8.25). Second,
emax > eos often ensures eeff > eeff,c in practice, as the averaging in Eq. (8.22) is
heavily weighted towards extreme eccentricities. As such, eeff > eeff,c alone is of-
ten a sufficient condition in Eq. (8.25).
265
The one-shot merger criterion (emax > eos) can also be used to distinguish two
different types of system architectures: if elim ≳ eos for a particular architecture,
then all initial conditions leading to orbit flips (i.e., in an octupole-active window)
also execute one-shot mergers. For elim ≈ 1(, Eq. (8).10) reduces to
≈ 8ϵ
2 −1
j(e ) GR
η
lim 1+ . (8.26)9 12
which lets us rewr(ite the )constraint elim ≳ eos as ( )
a ( a ) 17/37
0.0186 out,eff
−7/37 m
≳ 12
aout,eff ( 36)00 AU( 50M⊙
30M⊙ 10/37 q/(1+ q)2 )−2/37× . (8.27)
m3 1/4
For the system architecture considered in Figs. 8.4–8.7, this condition is satisfied,
and we see indeed that wherever the top panel suggests orbit flipping (emax = elim),
the bottom panel shows Pmerger ≈ 1. When the condition (Eq. 8.27) is not satisfied,
one-shot mergers are not possible, and Pmerger is generally only nonzero for a small
range about I0,lim.
8.4 Merger Fraction as a Function of Mass Ratio
Having developed an semi-analytical understanding of the binary merger window
and probability in the last section (particularly Section 8.3.2), we now study the
fraction of BH binaries in triples that successfully merge under various conditions
– we call this the merger fraction.
8.4.1 Merger Fraction for Fixed Tertiary Eccentricity
We first consider the simple case where eout is fixed at a few specific values and
compute the merger fraction as a function of the mass ratio q. We consider
266
isotropic mutual orientations between the inner and outer binaries, i.e. we draw
cos I0 from a uniform grid over the range [−1,1] (recall that ω, ωout, and Ω are
drawn uniformly from the range [0,2π) when computing the merger probability
Pmerger at a given I0). The merger fraction is then given by:∫1
≡ 1fmerger (q, eout) dcos I0 Pmerger (I0; q, eout) . (8.28)2
−1
This is proportional to the integral of the black lines (weighted by sin I0) in the bot-
tom panels of Figs. 8.4–8.7. We can also use semi-analytical criteria introduced in
Section 8.3.2 to predict the outcome and merger fraction. This is computed by us-
ing Panmerger as the integrand in Eq. (8.28), or by evaluating the integral of the thick
green lines (weighted by sin I0) in the bottom panels of Figs. 8.4–8.7. Figure 8.9
shows the resulting fmerger and the analytical estimates for all combinations of
q ∈ {0.2,0.3,0.4,0.5,0.7,1.0} and eout ∈ {0.6,0.8,0.9}. It is clear that the numeri-
cal fmerger and the analytical estimate agree well, and that the merger fraction
increases steeply for smaller q.
To explore the impact of our choice of isotropic mutual orientations between the
two binaries, we also consider a wedge-shaped distribution of cos I0 as was found
in the population synthesis studies of Antonini et al. [2017]. We still use the same
uniform grid of cos I0 as before, but weight each eccentricity by its probability
probability density following the distribution:
= 1 + |cos I |P (cos I ) 00 . (8.29)4 2
The resulting fmerger for a tertiary with cos I0 distributed like Eq. (8.29) is shown
as the dashed lines in Fig. 8.9. While the total merger fractions decrease, the
strong enhancement of the merger fraction at smaller q is unaffected.
In the right panel of Fig. 8.9, we see that the merger fractions for the three
eout values overlap for small ϵoct. This implies that fmerger depends only on ϵoct
267
in this regime, and not on the values of q and eout independently. From Fig. 8.4
(which has ϵoct = 0.007), we see that this suggests that the size of the retrograde
merger window only depends on ϵoct, much like what Eq. (8.13) shows for the test-
particle limit. However, once ϵoct is increased sufficiently, the three curves in the
right panel of Fig. 8.9 cease to overlap. This can be attributed to their different
η values: for sufficiently small ϵoct, no prograde initial inclinations successfully
merge (e.g., Fig. 8.4), and the merger fraction is solely determined by the size of
the retrograde octupole-active window. But once ϵoct is sufficiently large, prograde
mergers become possible, and the merger fraction is also affected by the size of the
octupole-inactive gap, which depends on η. This again illustrates the importance
of the octupole-inactive gap, which we comment on in Appendix G.1.
Figure 8.10 depicts the merger fractions for systems with a0 = 50 AU (the other
parameters are the same as in Fig. 8.9). According to Eq. (8.27), these systems no
longer satisfy elim ≳ eos, so the merger fraction is expected to diminish strongly
and vary much more weakly with q, as one-shot mergers are no longer possible.
This is indeed observed, particularly for the eout = 0.6 curve in Fig. 8.10. We
also remark that the semi-analytical prediction accuracy is poorer in this case
than in Fig. 8.9. This is because the only mergers in this regime are smooth
mergers. As can be seen for the prograde I0 in Figs. 8.5 and 8.8, smooth mergers
occur over a wide range of merger times Tm, and the specific Tm that a system
experiences depends sensitively on its chaotic evolution. Thus, Eq. (8.22) is a
rather approximate estimate of the amount of GW emission that a real system
emits during a smooth merger; indeed, the prograde regions of Figs. 8.5 and 8.8
show that the merger times for smooth mergers are systematically underpredicted
by the semi-analytic merger criterion (see discussion in Section 8.4.4). The non-
monotonicity of the semi-analytic merger fraction for eout = 0.6 from q = 0.2 to
q= 0.3 is due to small sample sizes and finite grid spacing in cos I0.
268
Figure 8.9: Merger fraction (Eq. 8.17) of BH binaries in triples as a function of
mass ratio q (left panel) for several values of outer binary eccentricities. The
other system parameters are the same as in Figs. 8.4–8.7. The right panel shows
the same merger fraction, but plotted against the octupole parameter ϵoct. The
filled circles joined by the solid lines are numerical results (based on integrations
for full triple system evolution including GW emission; see the black solid lines in
the bottom panels of Figs. 8.4–8.7) assuming random mutual inclinations between
the inner and outer binaries (uniform in cos I0), and the dashed lines denote the
merger fractions if the mutual inclinations are distributed according to Eq. (8.29).
The crosses are semi-analytical results using an integration time of 2000tZLK (see
the thick green lines in the bottom panels of Figs. 8.4–8.7).
Figure 8.10: Same as Fig. 8.9 but for a0 = 50 AU.
269
8.4.2 Merger Fraction for a Distribution of Tertiary Eccen-
tricities
For a distribution of tertiary eccentricities, denoted P (eout), the merger fraction is
given by ∫
ηmerger(q)= deout P (eout) f∫ ∫merger
(q, eout) ,
1
= P (ede out)out dcos I0 Pmerger (I0; q, eout) . (8.30)2
−1
We consider two possible P(eout) with eout ∈ [0,0.9]: (i) a uniform distribution,
P (eout)= constant, and (ii) a thermal distribution, P (eout)∝ eout.
The top panel of Fig. 8.11 shows ηmerger (black dots) for the fiducial triple sys-
tems (with the same parameters as in Figs. 8.4–8.7). For each q, the integral in
Eq. (8.30) is computed using 1000 realizations of random eout, cos I0, ω, ωout, and
Ω. Not surprisingly, we see ηmerger increases with decreasing q. When q is small,
a thermal distribution of eout tends to yield higher ηmerger than does a uniform dis-
tribution. We also compute the merger fraction using the semi-analytical merger
probability of Eq. (8.25) on a dense grid of initial conditions uniformly sampled in
eout and cos I0; the result is shown as the blue dotted line in Fig. 8.11, which is in
good agreement with the uniform-eout simulation result (black).
To characterize the properties of merging binaries, the middle and bottom pan-
els of Fig. 8.11 show the distributions of merger times and merger eccentricities
(at both the LISA and LIGO bands) for different mass ratios. To obtain the LISA
and LIGO band eccentricities (with GW frequency equal to 0.1 Hz and 10 Hz re-
spectively), the inner binaries are evolved from when they reach 0.005a0 (at which
point we terminate the integration of the triple system evolution as the inner bi-
nary’s evolution is decoupled from the tertiary; see Eq. 8.16) to physical merger
270
using Eqs. (8.14–8.15). While the LIGO band eccentricities are all quite small
(≲ 10−3), the LISA band eccentricities (at 0.1 Hz) are significant, with median
≳ 0.2 for q ≲ 0.5. We note that these eccentricities are generally smaller than
those found in the population studies of Liu et al. [2019a]. This is because in
this paper we consider only sufficiently hierarchical systems for which double-
averaged evolution equations are valid, whereas Liu et al. [2019a] included a
wider range of triple hierarchies and had to use N-body integrations to evolve
some of the systems.
For comparison, Figure 8.12 shows the results when aout,eff = 5500 AU (instead
of aout,eff = 3600 AU for Fig. 8.11) with all other parameters unchanged. While
ηmerger is lower than it is for aout,eff = 3600 AU, there is still a large increase of
ηmerger with decreasing q. Since Eq. (8.27) is still satisfied, this is expected.
8.4.3 q≪ 1 Limit
For fixed m12 (and other parameters), even though the octupole strength ϵoct in-
creases as q decreases, the efficiency of GW radiation also decreases. It is there-
fore natural to ask at what q these competing effects become comparable and the
merger fraction is maximized. We show that this does not happen until q is ex-
tremely small.
We see from Figs. 8.4–8.7 that elim > eos for our fiducial triple systems. Indeed,
from Eq. (8.27), we see that even for q as small as 10−5, the condition elim >
eos is satisfied. This implies that most binaties execute one-shot mergers when
undergoing an orbit flip. In addition, recall that the characteristic time for the
binary to approach elim can be estimated by Eq. (8.12), which, for our fiducial
271
Figure 8.11: Upper panel: Binary BH merger fraction as a function of mass ratio q
for the fiducial triple systems (with parameters the same as in Figs. 8.4–8.7), as-
suming random mutual inclinations (uniform in cos I0), and either uniform (black
dots) or thermal distribution (red dots) for the tertiary eccentricity distribution
[with eout ∈ [0,0.9]]. These are obtained numerically using Eq. (8.30) by sampling
1000 combinations of eout, cos I0, ω, ωout, and Ω. The blue dotted line is the semi-
analytical result obtained by applying Eq. (8.25) in Eq. (8.30) (evaluated using a
dense uniform grid of cos I0 and eout). The thick green line is a power-law fit to the
analytical ηmerger with a power law index of −2.5. Middle panel: Merger times of
successful mergers for a uniform eout distribution (the median is denoted with the
large black dot). Bottom panel: Merging binary eccentricities (again, for a uniform
eout distribution) in the LISA band (0.1 Hz; green) and in the LIGO band (10 Hz;
black), with medians marked with large dots.
272
Figure 8.12: Same as Fig. 8.11 but for aout,eff = 5500 AU. The power law index of
the fit to the analytical ηmerger is −1.15.
273
triple systems, is given by ( )
m 1/2 (≃ 8 12 a ) ( )t 10 out,eff 7/2 a −2ZLK,oct ( 50M)⊙ [ 3600 AU ]100 AU
× m
−1
3 1− q
+ √ e
−1/2
out yr. (8.31)
30M⊙ 1 q 1− e2out
Since tZLK,oct ≪ 10 Gyr, this implies that the octupole-ZLK-induced binary merger
fractions are primarily determined by what initial conditions would lead to ex-
treme eccentricity excitation and only weakly depend on the GW radiation rate.
Indeed, Eq. (8.27) shows that, while elim > eos is indeed violated if q is decreased
sufficiently, the dependence is extremely weak. Thus, ηmerger is expected to be very
nearly constant for all physically relevant values of q, as can be seen in Fig. 8.13.
8.4.4 Limitations of semi-analytic Calculation
It can be seen in Fig. 8.9 that the semi-analytical merger fractions are system-
atically lower than the values obtained from the direct simulations. One reason
that this discrepancy arises is because the non-dissipative simulations used to
compute eeff and emax are only run for 2000tLK ≈ 3 Gyr, while the full simula-
tions including GW dissipation are run for 10 Gyr. Owing to the chaotic nature of
the octupole-order ZLK effect, this means that, if an initial condition leads to ex-
treme eccentricities only after many Gyrs, then eeff and emax are underpredicted
by the non-dissipative simulations. Additionally, there are times when eccentric-
ity vector of the inner binary is librating, during which orbit flips are strongly
suppressed [Katz et al., 2011]. Since the librating phase can last an unpredictable
amount of time, this suggests that the semi-analytical merger criteria can become
more complete as the integration time is increased.
We quantify the “completeness” of the semi-analytical merger fraction via the
274
Figure 8.13: Same as blue dashed line of the top panel of Fig. 8.11 but extended to
very small q. Due to the very weak q dependence in Eq. (8.27), fmerger is expected
to depend very weakly on q when q≪ 1 (such that ϵoct is approximately constant),
which agrees with the simulation results.
275
Figure 8.14: Completeness of the semi-analytical merger fraction, defined as
f anmerger/ fmerger, as a function of the integration time used for the non-dissipative
simulations, in the fiducial parameter regime while eout is fixed at a few val-
ues. The thin grey lines indicate the completeness for particular combinations
of (q, eout), and the thick black line denotes their average. We see that complete-
ness is still increasing as the integration time approaches 2000tZLK ≈ 3 Gyr.
ratio f anmerger/ fmerger as a function of non-dissipative integration time. We focus on
the fiducial triple systems for demonstrative purposes and compute the complete-
ness for each of the q and eout combinations shown in Fig. 8.9. Figure 8.14 shows
the completeness for each of these simulations in light grey lines and their mean
in the thick black line. We see that the completeness is still increasing even as the
non-dissipative simulation time is increased to 2000tZLK, so we expect that even
longer integration times would give even better agreement with the dissipative
simulations.
276
8.5 Mass Ratio Distribution of Merging BH Binaries
In Section 4, we have calculated the binary BH merger fractions fmerger and ηmerger
as a function of the mass ratio q for some representative triple systems. To de-
termine the distribution in q and m12 (total mass) of the merging binaries, we
would need to know both the initial distribution in q, m12 and a0 of the inner BH
binaries and the distribution in m3, aout and eout of the outer binaries, denoted
by:
dF dF
, out . (8.32)
dqdm12da0 dm3daoutdeout
The distribution in q an∫d m12 of the merging binaries is thendFmerger = dFda
dqdm 0
dmoutdaoutdeout
12 dqdm12da0
× dFout f (q, e
dm da de merger out
;m12,a0,m3,aout, eout), (8.33)
3 out out
where fmerger is given by Eq. (8.28) (assuming random mutual inclinations be-
tween the inner and outer binaries), and we have spelled out its dependence on
various system parameters. Some examples of fmerger are shown in Figs. 8.9–8.10.
If we further specify the ecc∫entricity distribution of the outer binaries, we havedFmerger = dFda
dqdm 0
dm3daout
12 dqdm12da0
× dFout ηmerger(q;m12,a0,m3,aout,eff), (8.34)dm3daout,eff
where ηmerger is given by Eq. (8.30). Some examples of ηmerger are shown in the
top panels of Figs. 8.11–8.12.
Clearly, to properly evaluate Eq. (8.33) or (8.34) would require large population
synthesis calculations and in any case would involve significant uncertainties, a
task beyond the scope of this paper. For illustrative purposes, we consider the
277
fiducial triple systems as studied in Section 8.4, and estimate the mass-ratio dis-
tribution of BH mergers as
dFmerger ∼ dF ηmerger(q). (8.35)dq dq
8.5.1 Initial q-distribution of BH Binaries
The initial mass-ratio distribution of BH binaries, dF/dq, is uncertain. It can
be derived from the the mass distributions of of main-sequence (MS) binaries,
together with the MS mass (mms) to BH mass (m) relation.
For the distribution of MS binary masses, we assume that each MS component
mass is drawn from a Salpeter-like initial mass function (IMF) independently,
with
dFms ∝m−α, (8.36)
dm msms
in the range mmin ≤ mms ≤ mmax. Note in this case the MS binary mass-ratio
distribution is (for q≤ 1) [ ( ) ]
dF 2−2αms ∝ qα−2 1− q , (8.37)
dq qmin
where qmin = mmin/mmax is the minimum possible binary mass ratio [this is a
generalization of the result of Tout, 1991]. We consider two representative values
of α: (i) α = 2.35, the canonical Salpeter IMF [Salpeter, 1955], and (ii) α = 2,
resulting in a uniform q distribution (for q ≳ 2qmin). The latter case is consistent
with observational studies of the mass ratio of high-mass MS binaries [Sana et al.,
2012, Duchêne and Kraus, 2013, Kobulnicky et al., 2014, Moe and Di Stefano,
2017].
To obtain dF/dq, we compute the BH binary mass ratio when each main se-
quence mass mms is mapped to its corresponding BH mass m. This mapping is
278
taken from Spera and Mapelli [2017] for the mass range 25M⊙ ≤ mms ≤ 117M⊙.
We consider both the case where Z = 0.02 (“high Z”) and where Z = 2.0×10−4 (“low
Z”), the two limiting metallicities used in Spera and Mapelli [2017]. We can then
numerically compute dF/dq by sampling masses for stellar binaries from the IMF,
translating these into BH masses, then calculating the resulting BH mass ratios
for each binary. The upper panel of Fig. 8.15 shows the dF/dq obtained via this
procedure for a Salpeter IMF (α= 2.35) when sampling 105 MS binaries for each
metallicity. In the lower four panels, we also show dF/dq restricted to particu-
lar ranges of m12. Note that the distributions differ significantly among the m12
ranges and also between the two metallicities. Figure 8.16 shows the case when
α= 2, which mostly resembles Fig. 8.15.
8.5.2 q-distribution of Merging BH Binaries
Using the results of Section 8.5.1, we can also estimate the mass ratio distribution
of merging BHs using Eq. (8.35). We consider representative triple systems con-
sidered in Section 8.4: for ηmerger, we use a simple approximation that lies roughly
between the two cases shown in Figs. 8.11–8.12:
ηmerger(q)≈ 0.2× [max(q,0.2)]−2 . (8.38)
The results for dFmerger/dq are displayed as the dotted curves in Figs. 8.15–8.16 in
each panel. Broadly speaking, dFmerger/dq peaks around q∼ 0.3 for low-Z systems,
and around q∼ 0.4 for high-Z systems, the latter reflecting the peak in the initial
BH binary q-distribution. Also note that dFmerger/dq can be quite different for
different m12 ranges. For example, merging BH binaries with m12 > 42M⊙ are
only produced in low-Z systems, and dFmerger/dq peaks around q ∼ 0.3 for m12 ∈
[42,67]M⊙, and is roughly uniform between q∼ 0.2 to 1 for m12 ≳ 67M⊙.
279
We emphasize that these results for dFmerger/dq refer to the representative
triple systems studied in Sections 8.2–8.4, and thus should be considered for il-
lustrative purposes only. As noted above, the merger fraction ηmerger depends on
various parameters of the triple systems. While we have not attempted to quan-
tify ηmerger for all possible triple system parameters, it is clear that the principal
finding of Section 8.4 (i.e., ηmerger increases with decreasing q) applies only for
systems with sufficiently strong octupole effects. In fact, from Figs. 8.9 and 8.10
we can estimate that the octupole-induced feature in ηmerger becomes prominent
only when ϵoct ≳ 0.005, or equivalently( ) √ 2
a 1+ q 1− eout 0.01
≳ 0.005 ≃ , (8.39)
aout,eff 1− q eout eout
where in the second step we have used q ∼ 0.5 and eout ∼ 0.6. When this condi-
tion is satisfied, the inner binary can usually also undergo a one-shot merger (see
Eq. 8.27), leading to strong dependence of the merger fraction on q. For triple sys-
tems with a/aout,eff ≲ 0.01 (such as the case when the tertiary is a supermassive
BH with m3 ≳ 106m12), the octupole effect is unimportant (see the discussion fol-
lowing Eq. 8.2), and we expect the merger fraction to be almost independent of q.
Indeed, an analytical fitting formula for BH mergers induced by pure quadrupole-
ZLK effect shows η ∝µ0.16merger ∝ q0.16/(1+q)0.32 (see Eq. 53 of Liu and Lai, 2018,
or Eq. 26 of Liu and Lai, 2021). For such systems, we expect dFmerger/dq to be
mainly determined by the initial q-distribution of BH binaries at their formation.
8.6 Summary and Discussion
We have studied the dynamical formation of merging BH binaries induced by a
tertiary companion via the von Zeipel-Lidov-Kozai (ZLK) effect, focusing on the
expected mass ratio distribution of merging binaries. The octupole potential of the
280
Figure 8.15: Mass ratio distributions of the initial BH binaries (solid lines) and
merging BH binaries (dotted lines) when using α= 2.35 for the MS stellar initial
mass function (see Sections 8.5.1 and 8.5.2). Top panel: Distribution of binary
mass ratio at formation and merger for all possible total binary BH masses. Each
BH mass is obtained from the MS mass using the fitting formula of Spera and
Mapelli [2017] for metallicities of 2×10−4 (Low Z) and 0.02 (High Z), while the
merger fraction of BH binaries is given by Eq. (8.38). To produce these distri-
butions, 105 initial MS binaries are used for each metallicity, and the number of
merging BH binaries has been scaled up by a factor of 10 for visibility. The counts
refer to the number per ∆q= 0.05 bin. Bottom four panels: Same as the top panel
but with specific ranges of m12, the total BH mass of the binary (as labeled). Note
that low-m12 systems are mainly produced from high-Z MS binaries, while high-
m12 systems are mainly produced in low-Z MS binaries.
Figure 8.16: Same as Fig. 8.15 but for α = 2, i.e. a nearly uniform distribution of
the main sequence binary mass ratio. The results are very similar to Fig. 8.15.
281
tertiary, when sufficiently strong, can increase the inclination window and prob-
ability of extreme eccentricity excitation, and thus enhance the rate of successful
binary mergers. Since the octupole strength ϵoct ∝ (1− q)/(1+ q) (see Eq. 8.2) in-
creases with decreasing binary mass ratio q, it is expected that ZLK-induced BH
mergers favor binaries with smaller mass ratios. We quantify the dependence
of the merger fraction/probability on q using a combination of numerical inte-
grations and analytical calculations, based on the secular evolution equations for
hierarchical triples. We develop new analytical criteria (Section 8.3.2) that al-
low us to determine, without full numerical integrations, whether an initial BH
binary can undergo a “one-shot merger” or a more gradual merger under the in-
fluence of a tertiary companion. These allow us to compute the merger probability
semi-analytically by only studying non-dissipative (i.e. no GWs) triple systems
(see Eq. 8.25). We show that for hierarchical triples with semi-major axis ra-
tio a/aout ≳ 0.01−0.02 (see Eq. 8.39), the BH binary merger fraction ( fmerger or
ηmerger) can increase by a larger factor (up to ∼ 20) as q decreases from unity to
0.2 (see Figs. 8.9–8.13). When combined with a reasonable estimate of the mass
ratio distribution of the initial BH binaries (Section 8.5.1), our results for the
merger fraction suggest that the final merging BH binaries have an overall mass
ratio distribution that peaks around q = 0.3 or 0.4, although very different distri-
butions can be produced when restricting to specific ranges of total binary masses
(see Figs. 8.15 and 8.16).
Taking our final results (Figs. 8.15 and 8.16) at face value, we tentatively con-
clude that the mass-ratio distribution dFmerger/dq of BH binary mergers induced
by a comparable-mass companion is inconsistent with the current LIGO/VIRGO
result (see Fig. 8.1), suggesting that such tertiary-induced mergers may not be the
dominant formation channel for the majority of the detected LIGO/VIRGO events.
However, there are at least two important issues/caveats to keep in mind:
282
(i) dFmerger/dq depends strongly on the initial mass-ratio distribution of BH bi-
naries at their formation (dF/dq), which is uncertain and depends sensitively on
the metalicity of the binary formation environment (see Section 8.5.1). It is also
possible that the initial BH binary mass ratio distribution is much more skewed
towards equal masses than what we found in Section 8.5.1 (e.g. if stellar binaries
with significantly asymmetric masses become unbound due to mass loss and su-
pernova kicks as their components become BHs). Such a distribution was found by
population synthesis studies that include octupole-order ZLK effects and models of
stellar evolution [e.g. Hamers et al., 2013, Toonen et al., 2018]. These studies find
that ZLK oscillations in stellar binaries with small q can experience mass trans-
fer and merge without forming a compact object binary; as a result, most compact
object binaries form with large mass ratios. The prevalence of this phenomenon
likely depends on the initial semimajor axes of the inner binaries. Further study
would be required to understand the competition between this primordial large-q
enhancement and the elevated merger fractions for small q found in the present
study in an astrophysically realistic population.
(ii) When the tertiary mass m3 is much larger than the BH binary mass m12, as
in the case of a supermassive BH tertiary, dynamical stability of the triple requires
aout ≫ a, which implies that the octupole effect is negligible (ϵoct ≪ 1). For such
triple systems, we expect the merger fraction to depend very weakly on the mass
ratio, and the final dFmerger/dq to depend entirely on the initial dF/dq. Although
the merger fraction of such “pure quadrupole” triples is small (≲ 6%; see Eq. 53 of
Liu and Lai, 2018), additional “external” effects can enhance the merger efficiency
significantly [e.g., when the outer orbit experiences quasi-periodic torques from
the galactic potential (Petrovich and Antonini, 2017; see also Hamers and Lai,
2017), or from the spin of a supermassive BH [Liu et al., 2019b]].
283
Near the completion of this paper, we became aware of the simultaneous work
by Martinez et al. [2021], who study a similar topic using a population synthesis
approach.
284
APPENDIX A
APPENDICIES TO DYNAMICS OF COLOMBO’S TOP: DISSIPATING
DISK
A.1 Cassini State Local Dynamics
In this appendix, we linearize the equations of motion near each CS and determine
its stability. We derive the local libration frequency or growth rate for perturba-
tions around each CS.
A.1.1 Canonical Equations of Motion and Solutions
We adopt spherical coordinate system where l̂ = ẑ and θ,φ are the polar and az-
imuthal angle of ŝ. We choose l̂d at coordinates θ = I,φ= π (see Figs. 2.1 and 3.1).
We use the convention 0≤ θ <π and 0≤φ< 2π.
( )
The equations of motion in φ,cosθ follow by applying Hamilton’s equations
to the Hamiltonian [Eq. (3.8)]:
dφ = ∂H =− ( )cosθ+η cos I+sin I cotθ cosφ , (A.1a)
dt ∂(cosθ)
d(cosθ) =−∂H =−ηsin I sinθsinφ. (A.1b)
dt ∂φ
These agree with Eq. (2.9).
The CSs satisfy φ̇= θ̇ = 0. For convenience, we give approximate solutions for
the CSs in the limits η≪ 1 and η≫ 1. For η≪ 1:
• CS1: φ1 = 0, θ1 ≃ ηsin I.
285
• CS2: φ2 =π, θ2 ≃π/2−ηcos I.
• CS3: φ3 = 0, θ3 ≃π−ηsin I.
• CS4: φ4 = 0, θ4 ≃π/2−ηcos I.
For η≫ 1, only CS2 and CS3 exist and are given by:
• CS2: φ2 =π, θ2 ≃ I+η−1 sin I cos I.
• CS3: φ3 = 0, θ3 ≃π− I+η−1 sin I cos I.
Note that in the convention of Fig. 3.1, CS1, CS3 and CS4 have negative θ values
since φ= 0.
A.1.2 Stability and Frequency of Local Oscillations
To examine stability of each CS, we linearize Eqs. (A.1) about an equilibrium lo-
cated at φcs = 0 (CS 1, 3, 4) or π (CS2) but arbitrary θcs. Setting φ = φcs+δφ,θ =
θcs+δθ yields
dδφ = sin Isinθ
dt cs
δθ∓η
sin2
δθ, (A.2a)
θcs
dδθ =±ηsin Iδφ, (A.2b)
dt
where the upper sign corresponds to φcs = 0. Eliminating δθ gives
d2δφ ≡λ22 δφ, (A.3)dt
where
λ2 ≡ ( )( )sinθ 2cs∓ηsin I csc θ ±ηsin I . (A.4)
A plot of λ2 for each of the CSs is given in Fig. A.1. It is clear that CS4 is unstable
while the other three are stable. The local libration frequency for these stable CSs
p
is simply ωlib = −λ2.
286
Figure A.1: λ2, given by Eq. (A.3), evaluated at each of the Cassini States. The ver-
tical axis is rescaled for clarity. Note that CS4 is unstable (λ2 > 0) when it exists
while all others are stable (λ2 < 0). The thin horizontal dashed line is the instabil-
ity boundary λ2 = 0 while the thin vertical dashed line labels η= ηc [Eq. (2.14)].
A.2 Approximate Adiabatic Evolution
In this appendix, we will use approximations valid for small η to derive the ex-
plicit analytic expressions for the final obliquities at small θsd,i and the associated
probabilities for the II → I and II → III tracks. These are the only possible tracks
for small η.
We first seek a simple parameterization for the separatrix, the level curve
of the Hamiltonian intersectin( g the un)stable equili(brium CS)4. Poi(nts alo)ng the
separatrix, parameterized by φ,θsep(φ) , satisfyH φ,θsep(φ) =H φ4,θ4 where
φ4 and θ4 are given in Appendix A.1.1. We obtain two solutions for θsep, given to
leading order in η by:
√ ( )
cosθsep(φ)≈ cosθ4± 2ηsin I 1−cosφ . (A.5)
287
These two solutions parameterize the two legs of the separatrix. Integration of
the phase area enclosed by the separatrix yields then
√
AII(η)≈ 16 ηsin I. (A.6)
We can now compute the final obliquities and their associated probabilities for
each track as follows:
1. For a given θsd,i, we know that if (η→∞ then) the trajectory executes simple
libration about l̂d, and so A = 2π 1−cosθsd,i ≈ πθ2sd,i. This then implies η⋆
must be the solution to A( II(η⋆)= A, or( ))2 (πθ2 )2
η⋆ ≈
2π 1−cosθsd,i 1 ≈ sd,i 1 . (A.7)
16 sin I 16 sin I
2. Upon separatrix encounter, a transition to either zone I or zone III occurs.
These can be calculated to have the associated probabilities [using the ap-
proximate area Eq. (A.6) and Eqs. (2.25)] √
2πη cos I+4 η sin I
Pr(II→ I)≈ ⋆ √ ⋆√ , (A.8a)8 η⋆ sin I
→ ≈ −2πη⋆ co√s I+4 η⋆ sin IPr(II III) . (A.8b)
8 η⋆ sin I
3. Upon a transition to zone I or zone III, the final obliquity can be predicted
by observing the final adiabatic invariant Af = −AI(η⋆) in the zone I case
and Af =AI(η⋆)+AII(η⋆) in the zone III case. As η→ 0, these correspond to
obliquities ( )2
πθ2 2
cos ≈ sd,i
θ
( θf)II→I cot I+ sd,i , (A.9a)( 16 ) 42
πθ2 θ2
(cos
sd,i
θf)II→III ≈ cot I− sd,i . (A.9b)16 4
These are the black dotted lines overplotted in Fig. 2.5.
288
APPENDIX B
APPENDICIES TO COLOMBO’S TOP: WEAK TIDAL FRICTION
B.1 Convergence of Initial Conditions Inside the Separatrix
to CS2
In Section ??, we studied the stability of the CSs under of tidal alignment torque
given by Eq. (4.7), finding that CS2 is locally stable. Later, in Section 3.3.3, we
found that all initial conditions within the separatrix converge to CS2, which is not
guaranteed by local stability of CS2. In this section, we give an analytic demon-
stration that all points inside the separatrix indeed converge to CS2, focusing on
the case where η≪ 1.
Similarly to the analytic calculation in Section 3.3.4, we seek to compute the
change in the unperturbed Hamiltonian over a single libration cycle. To calculate
the evolution of H, we first parameterize the unperturbed trajectory (similarly
to Eq. 3.27). For initial conditions inside the separatrix, the value of H can be
written H =Hsep+∆H where ∆H > 0, and the two legs of the libration trajectory
can be written: √
≈ ± [ ( ) ]cosθ± ηcos I 2η sin I 1−cosφ −∆H . (B.1)
We have taken sinθ ≈ 1, a good approximation in zone II when η≪ 1. Note that
there are some values of φ for which no solutions of θ exist, reflecting the fact
that the libration cycle does not extend over the full interval φ ∈ [0,2π]. During a
libration cycle, θ− [θ+] is traversed while φ′ > 0 [φ′ < 0], i.e. the trajectory librates
counterclockwise in (cosθ,φ) phase space (see Fig. 3.2).
The leading order change to H over a single libration cycle can then computed
289
by integrati∮ng dH/dt a∮lon(g this tr)ajectory, yielding:
dH = d(cosθ)dt dφ,
dt ∫ dt tideφmax
= 1 ( )sin2θ−−sin2θ+ dφts
φmin
φ∫max
≈ 1
√ [ ( ) ]
4ηcos I 2η sin I 1−cosφ −∆H dφ> 0. (B.2)
ts
φmin
Her[e, φmin > 0]and φmax < 2π are defined such that the trajectory librates over
φ ∈ φmin,φmax . Thus, H is strictly increasing for all initial conditions inside the
separatrix, and they all converge to CS2.
B.2 Approximate TCE2 Probability for Small ηsync
In this appendix, we seek a tentative analytic understanding for the probability
of convergence to tCE2 when ηsync is small, i.e. the left extremes of Figs. 3.20
and 3.21. In this regime, following the discussions in Sections 3.3.4 and 3.4.3, we
understand that initial conditions (ICs) in zone I always converge to tCE1, ICs
in zone II always converge to tCE2, and ICs in zone III experience separatrix en-
counter and probabilistically converge to either one of the tCE. To further proceed,
we will assume an isotropic distribution of initial spin orientations; different dis-
tributions again will only change the quantitative but not qualitative character
of the discussion. Then the tCE2 probability, which we denote by PtCE2, can be
expressed as the sum of: (i) the probability that an IC is in zone II, and (ii) the
probability that an IC is both in zone III and undergoes a III → II transition. To
simplify the discussion, we will approximate that PtCE2 can be calculated as
∼ AP II + AIIItCE2 〈P4 4 III→II〉 , (B.3)π π
290
where AII and AIII are the phase space areas of zones II and III respectively, and
〈PIII→II〉 is the average III → II transition probability for a random IC in zone III.
Next, we evaluate each of the expressions in Eq. (B.3).
We first consider AII and AIII. Exact analytic forms for both AII and AIII is
known (Ward and Hamilton, 2004, Paper I), but an accurate approximation can
be obtained using Eq. (3.27) since ηi ≪ 1. We obtain that:
AII = 4
√
ηi sin I, (B.4)4π π
AIII = 1+ηi cos I
√
− 2 ηi sin I. (B.5)4π 2 π
Next, we need to evaluate 〈PIII→II〉, for which we must understand the out-
comes of the separatrix encounters that ICs in zone III experience. We proceed by
analytically calculating ∆K± (Eq. 3.45) for use in Eq. (3.46) to obtain the probabil-
ities of the outcomes of s∮eparatrix encounter. We first rewrite Eq. (3.45) as:
= dH − dHK sep∆ ± ∮ dtC± (dt d)t ( )
= d(cosθ) + Ω̇s ∂H − ∂Hsep dφ. (B.6)
C± dt tide φ̇ ∂Ωs ∂Ωs
Then, using the full equations of motion for the planet’s spin including weak tidal
friction in component form, given by Eqs. (3.34–3.36), we can evaluate ∆K± by
integrating along the two legs of the separatrix C± (see Fig. 3.2). Note that we
must use the value of η at the moment of separatrix encounter, which we denote
ηcross, as the evolution of Ωs changes the spin-orbit precession frequency α and
291
thus η itself:
η2
[ (
cross √ )ts∆K± ≈ −2cos I ±2πηcross cos I+8 ηcross s]in I ∓4πsin Iηsync √ √
− + 4η8cos I sin I sync(ηcross sin I/η( ) crossηcross )+ 2ηcross p √∓2π 1−2ηcross sin I +16cos Iη3/2cross sin I +8 ηcross sin Iηsync
± 64 ( )2 3/2πηcross cos I− η3 cross sin I . (B.7)
The resulting PIII→II obtained using this analytic ∆K± in Eq. (3.46) is shown as
the green dashed line in the top panel of Fig. 3.18, where it can be seen that
agreement is reasonable for ηcross ≲ 0.05. For the purposes of this section, we
drop all but the leading order terms in both the numerator and denominator of
Eq. (3.46) and obtain: √
≃ 6ηsync sin IPIII→II . (B.8)
π ηcross
However, ηcross cannot be expressed in closed form as a function of the ICs.
Based on the bottom panel of Fig. 3.18, we make the crude approximation that
ηcross is uniformly distributed between ηi and ηsync. Note that if Ωs ≃ n, then this
approximation is invalid: since nearly anti-aligned spins (θi ≈ 180◦) will undergo
significant spin-down before tidal friction can realign the spin orientation, Ωs,i
being too close to n results in ηcross ≪ ηsync.∫We thus obtain:ηsync
〈PIII→II〉 ∼
1
− PIII→II dηcrossηsy√nc ηi ηi
= 12( η√sync sin I+ ) . (B.9)π 1 n/Ωs,i
With this resu√lt, we can fi[n√ally express Eq. (B.3) as:]
≃ 4 ηsync sin I 3 ( )PtCE2 n/Ωs,i+ ( +√ ) +O ηsync . (B.10)π 2 1 n/Ωs,i
This is exactly Eq. (3.47). We remark again that this is valid in the regime where
ηsync ≪ 1 and Ωs ≳ n.
292
APPENDIX C
APPENDICIES TO COLOMBO’S TOP: MULTIPLANETARY SYSTEMS
C.1 Inclination Modes of 3-Planet Systems
In the linear regime, the evolution of the orbital inclinations in a multi-planet
system is described by the Laplace-Lagrange theory [Murray and Dermott, 1999,
Pu and Lai, 2018]. In this section, we consider 3-planet systems. We denote the
magnitude of the angular momentum of each planet by L j and the inclination
relative to the total angular momentum axis Ĵ by I j, and we define the complex
inclination I j =I j exp[iΩi]. The evolution equations for I1, I2, and I3 are
I 1
   
 −ω12−ω ω  13 12
ω13 I1d = i 
dt I2  ω21 −ω21−ω23 ω23 I2 , (C.1)
I3 ω31 ω32 −ω31−ω32, I3
where ω jk is the precession rate of the j-th planet induced by the k-th planet, and
is given by
= mk a ja<ω (1)
( ) jk
n
(4M ) 2 j
b3/2(α), (C.2)
⋆ a>
where a< =min a j,ak , a> =max a( j,ak , n j = (GM /a
3 1/2
⋆ j) ) , α= a</a>, and
15 175
b(1)3/2(α)= 3α 1+ α2+ α4+ . . . (C.3)8 64
is the Laplace coefficient. Eq. (C.1) can be solved in general, giving two non-trivial
eigenmodes. In the limit L1 ≪ L2,L3, the two eigenmodes have simple solutions
and interpretations:
• Mode I has frequency
g(I) =− (ω12+ω13) . (C.4)
293
It corresponds to free precession of L̂1 around the total angular momentum
J=L2+L3. The amplitude of oscillation of L̂1, I(I), is simply the inclination
of L̂1 with respect to Ĵ.
• Mode II has frequency
J
g(II) =− (ω23+ω32)=− ω23, (C.5)L3
which is simply the precession frequency of L̂2 (or L̂3) about Ĵ. The forced
oscillation of L̂1 has an amplitude
= ω(12L3−ωI 13L2(II) g(II)− ) Ig 23, (C.6)(I) J
where I23 is the mutual inclination between the two outer planets and is
constant.
We consider two archetypal 3-planet configurations, systems with three super
Earths (SEs) and systems with two inner SEs and an exterior cold Jupiter (CJ).
In both cases, we take the inner two planets to have m1 = M⊕, m2 = 3M⊕, a1 =
0.1 AU, and we consider three values of a2 = {0.15,0.2,0.25} AU. For the 3SE case,
we take m3 = 3M⊕ and the characteristic inclinations I ◦1 ≃ I23 ≃ 2 [corresponding
to three nearly-coplanar SEs; see Fabrycky et al., 2014, Dai et al., 2018]. For
the 2SE + CJ case, we take m3 = 0.5MJ and the characteristic inclinations I1 ≃
I ≃ 10◦23 [corresponding to a mildly inclined CJ; see Masuda et al., 2020]. In both
cases, we compute the mode precession frequencies g(I,II) and characteristic mode
amplitudes I(I,II) for a range of a3. Figures C.1–C.3 show examples of our results.
294
Figure C.1: Inclination mode frequencies and amplitudes for the 3SE (left) and
2SE + CJ (right) systems. In both systems, the inner planets’ parameters are
m1 = M⊕, m2 = 3M⊕, a1 = 0.1 AU, a2 = 0.15 AU. In the 3SE case, m3 = 3M⊕ and
I1 = I ◦ ◦23 = 2 , while in the 2SE + CJ case, m3 = 0.5MJ and I1 = I23 = 10 , In the top
panels, the black dashed lines show the mode frequencies from the exact solution
of Eq. (C.1), while the solid, colored lines are given by Eqs. (C.4, C.5).
295
Figure C.2: Same as Fig. C.1 but for a2 = 0.2 AU.
Figure C.3: Same as Fig. C.1 but for a2 = 0.25 AU.
296
C.2 Additional Comments and Results on Mixed Mode Equi-
libria
In the main text, we provided an example of the spin evolution into a mi∣xed-∣mode
equilibrium in Fig. 4.2 for the parameters I(I) = 10◦, I(II) = 1◦, α = 10 ∣g(I)∣, and
g(II) = 10g(I). In Fig. C.4, we provide several further examples of the evolution into
other mixed-mode equilibria with different values of g when I = 3◦res (II) is used.
We find that their average equilibrium obliquities θeq are still well-described by
Eq. (4.15). Furthermore, we find that, if gres/∆g = p/q for integers p and q, then
the steady-state oscillations of θsl are periodic with period 2πq/ |∆g| (see bottom
example in Fig. 4.2 for the case of gres =∆g/2).
Figure C.5 shows the equilibrium obliquity θeq as a function of gres for a system
with I ◦(II) = 3 and g(II) = 8.5g(I). We can see for gres = ∆g that there are three
distinct equilibrium values of θsl. The largest-obliquity equilibrium is CS2(II),
and the equilibrium with an intermediate obliquity is a mixed-mode equilibrium
with gres = ∆g, as it directly intersects the green line (Eq. 4.15). The existence
and stability of this equilibrium is responsible for the extra dot a few degrees
below the CS2(II) curve in the bottom panels of Figs. 4.6, 4.9, and 4.11 (e.g. most
visible for g(II)/g(I) = 6.5, 7.5, and 8.5 in Fig. 4.9). The origin of the lowest-obliquity
equilibrium at gres = ∆g in Fig. C.5 is distinct, though it is within the range of
oscillation of θsl of the mixed-mode steady state.
In Fig. 4.4, we presented the numerical stability analysis of initial conditions
in the neighborhood of the mixed-mode equilibrium for the I ◦(II) = 1 case. When
I(II) is increased to 3◦, the amplitudes of oscillations of the mixed-mode equilibria
begin to overlap (see Fig. 4.8), and so we might expect that the basins of attraction
for the resonances overlap in θsl,0 space. Figure C.6 shows that this is indeed the
297
Figure C.4: Same as Fig. 4.2 but for g(II) = 10g(I) and I ◦(II) = 3 . The three exam-
ples correspond to capture into mixed-mode equilibria with resonant angles cor-
responding to gres = ∆g/3, gres = 3∆g/5, and gres = 3∆g/4 respectively. The three
pairs of vertical red lines are separated by 6π/ |∆g|, 10π/ |∆g|, and 8π/∆g in the
three examples respectively.
298
Figure C.5: Same as Figs. 4.5 and 4.8 but for I(II) = 3◦ and g(II) = 8.5g(I), where
ranges of oscillation in θsl have been suppressed for clarity.
case, and that the basins of attraction of the mixed modes are very distorted (likely
due to interactions among the resonances) compared to those seen in the I ◦(II) = 1
case.
Finally, in the main text, we have focused on the regime where I(II) ≪ I(I).
In Fig. C.7, we show the effect of choosing I(II) = 9◦ (with the same that I(I) =
10◦). I∣∣t can∣∣ be ∣∣seen∣∣ that the gres =∆g/2 mixed mode is the most common outcomewhen g(II) ≫ g(I) , as it is the preferred low-obliquity equilibrium (θeq ≲ 20◦) for
g(II) = 15g(I). The degraded agreement of the θeq values with Eq. (4.15) is because
our theoretical results assume I(II) ≪ I(I). A broad range of final obliquities is
observed when g(II) = 6.5g(I), very close to the critical value of g(II) (denoted by the
vertical dashed line) where the number of mode II CSs changes from 4 to 2 [Su
and Lai, 2021]. This is likely due to the unusual phase space structure near this
bifurcation.
299
Figure C.6: Same as Fig. 4.4 but for I(II) = 3◦. The range of the vertical axis is
chosen to include the ranges of oscillation of the 1/2, 3/5, 2/3, and 3/4 mixed mode
resonances (see Fig. 4.8). The decreasing density of points as θsl,0 decreases is
because the grid of initial conditions is uniform in cosθsl,0 rather than θsl,0 itself.
Figure C.7: Same as Fig. 4.6 but for I ◦(II) = 9 .
300
APPENDIX D
APPENDICIES TO PHYSICS OF TIDAL DISSIPATION
D.1 Derivation of Fluid Equations
We aim to model wave dynamics over multiple density and pressure scaleheights.
Start with the compressible Euler equations.
∂tρ+ (u ·∇)ρ+ρ∇·u=0, (D.1)
∂ts+ (u ·∇) s=0, (D.2)
1
∂tu+ (u ·∇)u+ ∇p=g, (D.3)
ρ
where s is the entropy. The pressure is calculated from the entropy and density
using the equation of state
s 1/γ= plog , (D.4)
cp ρ
where cp is the specific heat at constant pressure, and γ is the ratio of specific
heats. For computational ease, it is convenient to filter out the fast sound waves
from these equations. One approach is to assume pressure perturbations are
small [yielding the “pseudo-incompressible” equations, Vasil et al., 2013], or that
all thermodynamic perturbations are small [yielding the “anelastic” equations,
Brown et al., 2012]. In these approximation(s, one)of the thermodynamic equations
is r(epla)ced by a constraint equation: ∇ · p1/γ0 u = 0 for pseudo-incompressible;∇ · ρ0u = 0 for anelastic. Here p0 and ρ0 are the background density and pres-
sure profiles. The pressure in the momentum equation can be interpreted as a
Lagrange multiplier which enforces the constraint [Vasil et al., 2013]. Upon lin-
earization, both approximations conserve a wave energy
= 1
2
| |2+ 1 g (ρ
′)2 = 1 1 (s
′)2
E ρ u 2 2w 2 0 2 N2
ρ
2 0
|u| + ρ
2 0
N |∇ |2 , (D.5)ρ0 s0
301
where ρ′ and s′ represent the density and entropy perturbations.
Rather than assume thermodynamic perturbations are small, we instead filter
out sound waves by taking the limit γ→∞. Then the entropy and log density are
proportional to each other, and the entropy equation becomes
∂t logρ+ (u ·∇) logρ = 0. (D.6)
Together with mass conservation, this implies
∇·u= 0. (D.7)
We solve these equations together with the normal momentum equation.
Although these equations are non-standard, they have various desirable prop-
erties. They conserve mass and momentum, and the linearized equations conserve
the wave energy Ew [Eq. (D.5) above] similar to the pseudo-incompressible equa-
tions and anelastic equations. Our equations also satisfy the same linear disper-
sion relation as the fully compressible equations in the limit of large sound speed
[this is also true for the anelastic equations, but not pseudo-incompressible, Brown
et al., 2012, Vasil et al., 2013]. Thus, the vertical propagation of internal gravity
waves is similar to the pseudo-incompressible equations and anelastic equations.
In this work we are interested in waves which reach large amplitudes and
break. For breaking waves, Achatz et al. [2010] suggests that the anelastic equa-
tions may miss important effects. Although the pseudo-incompressible equations
may capture wave-breaking more accurately, they are more complicated, and do
not satisfy the correct dispersion relation to order (k H)−2z [Vasil et al., 2013]. In
the absence of a clear choice to study this wave breaking problem, we have elected
to use these simple equations derived in the γ→∞ limit.
302
D.2 Forcing Solution
To solve for the linear excited IGW amplitude due to bulk forcing (see Eq. (5.13)),
we consider the linearized system of equations, with all dynamical variables hav-
ing dependence uz(x, z, t) = ũ (z)eikxx−iωtz . Thus, ∂/∂t → −iωt, ∂x → ikx, and the∂
dynamical fluid equations become (see Eqs. (5.2) and (5.5)):
duz + ik u
dz x x
= 0,
−iωux+ ikxϖ+ gHikxΥ= 0,
− dϖiωuz+ + dΥ ϖgH − = 0,dz dz H [ ]
− − uz = − (z− z
2
i C exp 0
)
ωΥ ≡C (z).
H 2σ2
These can be recast solely in terms of uz as
d2uz − 2 − 1 duz + N
2k2 2
k u u x2 x z z 2 =−
gkx
dz H dz ω ω2
C (z)..
= [( 1 ± ) ]The homogeneous solutions are of form uz,±(z) exp 2H ikz (z− z0) where kz
satisfies the dispersion relation (Eq. (5.7)). We compute the solution to the inho-
mogeneous ODE by the meth∣ od of variation of p∣arameters. The Wronskian is∣∣∣∣ ∣∣ u ∣∣ z,+
uz,−
W ≡ ∣det ∣∣∣=−2ik ez/H∣ z . (D.8)duz,+/dz duz,−/dz∣
The general solu∫tion is th(en 2 ) ∫ ( 2 )
uz =− 1
gk 1 gk
uz,+ u x xW z,−
− 2 C (z) dz+uz,− uW z,+ −ω ω2 C (z) dz. (D.9)
303
Taking these integrals and applying the boundary conditions uz (z→∞) = uz,+,
uz (z→−∞)= uz,− give the( exact solup  )
tion:
 σ22 ± ik σ2
2
= πCσ gkx 2H
z [
u(z) 3/2 2 exp 2  −uz,+ (1+ ]erf(ξ+))+uz,− (erf(ξ−)−1) ,2 ikz ω 2σ
[ ] (D.10)
≡ pz− z0ξ± + σ ikzσ3/2 ± p (D.11)2σ2 2 H 2 ( p )∫z ( )
Here, the error function is defined erf(z) ≡ 2/ π exp −t2 dt. If we are con-
0
cerned with only z scales significantly larger than σ, then we may take erf(ξ±) ≈
Θ(z−z0) (the Heaviside step function). If we further assume |kzH|≫ 1 and restore
the eikxx−iωt factor, we recover Eq. (5.14) in themain text√  [( 1 + ) 2 ]C gk2 exp ik z− z + i kzσ for z> z ,2 2
uz(x, z, t)=− x e−kzσ /2 2πσ2eikxx−iωt×2ikz ω2  [(
2H z ( 0)) 2H2 ]
0
exp 12H − ik (z− z )− i kzσz 0 2H for z> z0.
(D.12)
Note that in the main text, this approximate form is used to compute uan, as it
is easier to work with and sufficiently accurate in the regions of interest (many σ
away from z0).
D.3 Equation Implementation
The system of equations we wish to simulate consists of Eqs. (5.2a), (5.5b), and (5.13).
The nonlinear terms in the these equations will transfer energy from lower wavenum-
bers to higher wavenumbers. Since spectral codes have no numerical diffusion, ex-
plicit diffusion must be added. To ensure the non-ideal system conserves horizon-
tal momentum exactly, we begin by adding diffusion terms to the flux-conservative
304
form of the Euler fluid equations (equivalent to Eqs. 5.2):
∇·u= 0, (D.13a)
∂tρ+∇· (ρu−ν∇(ρ−ρ))= 0, (D.13b)
∂t(ρu)+∇· (ρuu+diag(ρϖ)−νρ∇u)+ρgẑ= 0. (D.13c)
For simplicity, we use the same diffusivity ν for both the momentum and mass
diffusivities. Although mass diffusivity is not physical, we include it for numerical
stability. We choose the mass diffusion term to conserve mass, and not to affect
the background density profile.
It is necessary to mask out nonlinear terms in the forcing zone using a form
similar to Eq. (5.27). In the absence of this mask, a nonphysical mean flow local-
ized to the forcing zone deve[lops. We use the mask ]
= 1 + z− (z0+8σ) z− zΓNL(z) 2 tanh − tanh B . (D.14)2 σ σ
Including the damping zones and forcing terms as described in Section 5.4,
and again making change of variables to Υ,ϖ, we finally obtain the full system of
305
equations as simulated in Dedalus:
∇·u= 0, (D.15a)
u (z−z )2z
∂tΥ− = − F
0
Γ(z)Υ[ + e
−
2σ2( cos(kxx−ωt)H ρ(z)
+ 2 1−
)
e−Υ ]
ΓNL − (u ·∇)Υ+ν ∇2Υ+ (∇Υ) · (∇Υ)− ∂zΥ+ 2 ,H H
( (D.15b) )
∂ux + ∂ϖ
′
+ ∂Υ
[ −Υ
gH = −Γ(z()ux+Γ ν∇2NL ux−u 2x)ν ∇ Υ+ (∇Υ) · ∇
2 1− e
( Υ)− ∂ Υ+
∂t z∂x ∂x ] H H2
+ 1 ∂Υ2ν ((∇Υ) ·∇)ux− ∂zux − (u ·∇)ux−ϖ′ ,H ∂x
(D.15c)
∂u ∂ϖ′ ′ [ ( −Υ )z + + ∂ΥgH − ϖ = − 2 1− eΓ(z()uz+ΓNL ν∇2uz−uz)ν ∇2Υ+ (∇Υ) · (∇Υ)− ∂ Υ+∂t ∂z ∂z H ] H z H2
+ 1 ∂Υ2ν ((∇Υ) ·∇)uz− ∂zuz − (u ·∇)u ′H z−ϖ .∂z
(D.15d)
306
APPENDIX E
APPENDICIES TO DYNAMICAL TIDES IN ECCENTRIC BINARIES
E.1 Stellar Structure calculations with MESA
We attempt to reproduce the stellar structure parameters for the main-sequence
star in PSR J0045-7319 as reported from the so-called “Yale model” of [Kumar
and Quataert, 1998] using the stellar evolution code Modules for Experiments in
Stellar Astrophysics [MESA; Paxton et al., 2011, 2013, 2015, 2018, 2019]. In par-
ticular, the most relevant parameter to our results is the radius of the convective
core. To calculate this quantity, we perform a multi-step procedure, described be-
low:
• First, we pick a set of parameters for the stellar evolution (we explored dif-
ferent values of stellar rotation, metallicity, and convective overshoot param-
eter α). For the stellar rotation, we tried both non-rotating and critically-
rotating models. The non-rotating models are expected to be inaccurate
since B stars are typically rapidly rotating. The measured vsin i ≈ 110 km/s
at the surface [Kaspi et al., 1994], so we assume that vsurf ∼ 160 km/s,
which is a substantial fraction of the critical vc ≡ GM/R ∼ 500 km/s. Since
PSR J0045-7319 is in the SMC, its metallicity can be taken from measure-
ments of the typical SMC metallicity. We find that this value is in the range
Z = 0.1Z⊙–0.2Z⊙ [Piatti, 2012, Bolatto et al., 2011]. For the solar metallic-
ity, this value ranges from 0.012–0.02 [Vagnozzi, 2019]. Finally for convec-
tive overshoot, we used a value of αov = 0.6, satisfying the value required
to match the larger convective core masses reported in recent literature
[Tkachenko et al., 2020b].
307
• For each set of parameters, we initialize stellar models with a range of ZAMS
masses. We then evolve each of these models until core hydrogen depletion.
• For each model, we can find the age at which its luminosity and effective
temperature best match the observed values of Teff = 2.4× 104 K and L =
1.2×104L⊙. This gives us a single stellar model, described by its hyperpa-
rameters, mass, and age, that best matches the observed properties of the
MS star in PSR J0045-7319.
• For this best-fit stellar model, we calculate its core radius as the innermost
radius at which its Brunt-Väisälä frequency N2 > 0, i.e. the innermost ra-
dius at which it is convectively stable.
This procedure is illustrated for a few of these models in Figs. E.1–??. We see
that the core radii do not exceed ∼ R⊙, though larger convective overshoot and
metallicity improve agreement. The values of the convective core radius are in
disagreement with the conclusions shown in Fig. 6.8, which require a core radius
rc ≳ 1.3R⊙.
308
Figure E.1: Plots illustrating the procedure used to find the best-fitting stellar
model to the properties of PSR J0045-7319 using a near-critically rotating stellar
model with no convective overshoot, and metallicity Z = 0.1Z⊙ = 0.0012. In the
top-left panel, the plots show the fit to the observed stellar luminosity and surface
temperature (red dot) as a function of the ZAMS stellar mass (each line is an
evolutionary track for a star with mass denoted in the legend). In the top-right
column, the plots show the evolution of the best-fitting star as a function of stellar
age: the orange line shows the overall stellar radius, the blue line the radius of
the convective core, and the green line the central hydrogen fraction (right axis).
The vertical black line denotes the age at which the stellar model best matches
the observed luminosity and surface temperature (i.e. the age used for the third
column). In the bottom panel, the plots show the signed Brunt-Väisälä frequency
N2 as a function of the stellar radius for the best-fitting stellar mass and age
combination.
309
Figure E.2: Same as Fig. E.1 but with a convective overshoot parameter of 0.04
(see MESA documentation).
Figure E.3: Same as Fig. E.1 but with a convective overshoot parameter of 0.04
(see MESA documentation) and with Z = 0.2Z⊙ = 0.004
310
APPENDIX F
APPENDICIES TO SPIN DYNAMICS OF TERTIARY-INDUCED BLACK
HOLE MERGERS
F.1 Floquet Theory Analysis
In this appendix, we provide an alternative approach to analyzing the BH spin
dynamics based on Floquet theory [Chicone, 2006] that provides results comple-
mentary to those in the main text. Although the resulting equations cannot be
analytically solved, they place strong constraints on the allowed behavior of the
system. Additionally, Eq. (7.81) has a natural interpretation in this formulation,
and its accuracy is numerically tested in Appendix F.1.3. We again work in the
corotating frame and neglect GW dissipation, so the equation of motion is given
by Eq. (7.22). If we define the matrix operator à satisfying ÃS=Ωe×S, then the
equation of motion is ( )
dS = ÃS. (F.1)
dt rot
Here, Ã is periodic with period PLK [see Eq. (7.19)].
F.1.1 Without Nutation
First, for simplicity, let us assume that Ωe does not nutate, so its orientation is
fixed. In this case, Eq. (F.1) admits an exact conserved quantity:
d [
e−
]
ΦS = 0, (F.2)
dt
where ∫t
Φ(t)≡ Ã dt. (F.3)
311
Separately, since Eq. (F.1) is linear and has periodic coefficients, Floquet theory
tells us that S (t+PLK) is related to S(t) by the monodromy matrix M̃:
S (t+PLK)= M̃S(t). (F.4)
Comparing Eqs. (F.2) and (F.4), we immediately find that
M̃= exp P∫

LK
à dt ,
[ 0 ]
= exp PLKÃ , (F.5)
where again the overbar denotes time averaging. Note that M̃ is a rotation matrix,
so it must have exactly one eigenvector with eigenvalue 1; call this eigenvector R.
But R= Ω̂e, so S ·Ω̂e must be constant for every t=NPLK.
This example is somewhat trivial: since Ωe does not nutate, S just precesses
around fixed R= Ω̂e at a variable rate, and S·Ω̂e is conserved. However, the equa-
tion of motion studied in Section 7.4 [Eq. (7.35)] neglects nutation yet provides a
good description of the evolution of θ̄e. For the fiducial LK-induced merger, Fig. 7.6
shows that Ωe nutates substantially within a LK period when A ≃ 1 (nutation is
equivalent to Ωe1 ̸= 0 and ∆I ≠ 0◦e1 ). We infer that, even when Ωe does nutate
appreciably, the nutation can sometimes be neglected to good approximation.
In Section 7.5, we showed that being close to a resonanceΩe ≈MΩLK results in
non-conservation of θ̄e. However, the converse is not obviously true: our approx-
imate analysis does not prove that being far from these resonances guarantees
good conservation of θ̄e. In the next section, we argue that, for the dynamics stud-
ied in this paper, this converse is likely true as well.
312
F.1.2 With Nutation
When Ωe is allowed to nutate within PLK, the quantity given by Eq. (F.2) is no
longer conserved, as [ ]
d [ − ]Φ = dS dSe S e−Φ − Ãe−ΦS ̸= e−Φ − ÃS = 0. (F.6)
dt dt dt
Instead, we define two new quantities Φ′ and B̃ via
∫t
′ (
[ Φ (t])≡ Ã+
)
B̃ dt, (F.7)
d
e−Φ
′
S = 0. (F.8)
dt
This requires [ ′ ]
B̃= e−Φ ,Ã eΦ′ , (F.9)
where the square brackets denote thecommutator. The monodromy matrix is then P∫LK= (M̃ exp Ã+ )B̃ dt . (F.10)
0
We next want to understand when Eq. (F.10) can be well approximated by
Eq. (F.5). We first expand the matrix exponential using the Zassenhaus formula
[the inverse of the well-known Baker-Campbell-Hausdorff formula, see e.g. Mag-
nus, 1954] [ [ ]P2 Ã,B̃ ]
M̃= ePLKÃePLKB̃ LKexp − + . . . . [ ] (F.11)2
Note that since M̃ is a rotation matrix (see Section 7.3.1), and exp PLKÃ is also a
rotation matrix (Ã is skew-symmetric), the remainder of the right-hand side above
must also be a rotation matrix (as the rotation matrices are closed under matrix
multiplication). For convenience, define
[ [ ]P2 ]
≡ P Ã ≡ P B̃ − LK
Ã,B̃
R̃A e LK R̃B e LK exp + . . . , (F.12)2
313
where R̃A and R̃B are rotation matrices that effect rotations by angles θA and θB
about their respective axes.
Wh[en ca]n R̃B be neglected? From Eq. (F.9), we see that B̃= 0 vanishes (θB = 0)
when Ã,Φ = 0, which occurs when Ωe does not nutate, and we recover Eq. (F.5).
In fact, we will argue later that θB is generally small for the spin dynamics studied
in the main text. However, a small θB alone is not sufficient to guarantee M̃≈ R̃A.
To see this, note if θB is small, then R̃B ≈ 1 where 1 is the 3×3 identity matrix. On
the other hand, θA =ΩeTLK. Then, if θA is not too near an integer multiple of 2π,
M̃≈ R̃A and M̃≈ R̃A as before. However, if ΩeTLK ≈ 2πM for integer M, then R̃A
itself is near the identity as well, and R̃B cannot be neglected when calculating M̃.
The criterion for neglecting R̃B is then clear: θB must be much closer to an integer
multiple of 2πM than θA.
To complete this picture, we argue that, for the spin dynamics studied in the
main text, θB is small, so generally R̃B ≈ 1, and Ωe ≈ MΩLK is a necessary condi-
tion for R to differ significantly from Ω̂e. To do this, we recall that ΩePLK ≲ 2π
(Figs. 7.10 and 7.12), and we seek to show that θB must generally be small com-
′
pared to θA, which would imply θB ≪ 2π. We first approximate that eΦ ≈ eΦ in
Eq. (F.9) (requiring θB ≪ θA, which we will verify retroactively, and being far from
resonance, θA ≠ 2π), which gives
B̃≈ [e− ]Φ,Ã eΦ. (F.13)
Next, recall that Φ is the integral of −Ã, and so the magnitude of B̃ [B̃ primarily
affects M̃ via its average, see∫ Eq. (F.10)] depends on the average misalignment
between the vectors tΩe and Ωe dt ≃Ωe. There are a few possible regimes to
consider: (i) if A ≫ 1, then LK oscillations are frozen, and Ωe does not nutate;
(ii) if A ≪ 1, then Ω̂e ≈ L̂tot ≈ Ω̂e for almost all of TLk; and (iii) if A ≃ 1, then
emax cannot be too large [Eq. (7.18)], and so ΩL and ΩSL cannot vary too much
314
within an LK cycle and the nutation of Ωe is limited. This analysis suggests that,
at least far from resonance, the nutation of Ωe is limited, and the commutator in
Eq. (F.13) is small in the sense that θB ≪ θA, justifying our earlier claim. While
this analysis is not rigorous, it suggests that the only resonances present in the
system are near Ωe = MΩLK and that otherwise R ∥Ωe, in agreement with the
results of the main text.
F.1.3 Quantitative Effect
Above, we have given a qualitative analysis of the exact solution for the mon-
odromy matrix M̃. In this section, we aim to reconcile this with the quantitative,
approximate results in the text and suggest that the results in the text constitute
a complete characterization of the spin dynamics.
In Section 7.5, we found that one effect of the Fourier harmonics in Eq. (7.25)
are fluctuations in θ̄e when Ωe ≈MΩLK for some integer M with amplitude given
by Eq. (7.81). On the other hand, in Appendix F.1.2, we found that R is aligned
with Ωe except when Ωe is sufficiently close to MΩLK that the nutation of Ωe
becomes important, but we were not able to determine a closed-form expression
for the misalignment. In this section, we show numerically that the formulas
given in the main text give good predictions for the orientation of R.
To validate the analytic prediction given by Eq. (7.81), we numerically compute
M̃. We study the η ̸= 0, LK-enhanced regime (m1 =m2 =m3 = 30M⊙, a0 = 0.1 AU,
e −30 = 10 ,ãout = 3 AU). We still neglect GW dissipation in order for Floquet anal-
ysis to be applicable. For 2000 different I0 of the inner binary, we construct M̃
by evolving the spin equation of motion Eq. (7.22) starting with the three initial
conditions S0 = x̂, S0 = ŷ, and S0 = z (see Fig. F.1) over a single LK period, then
315
Figure F.1: Definition of angles for numerical study of the monodromy matrix
rotation axis. R is the eigenvector of the monodromy matrix M̃ with eigenvalue 1.
using
M̃=Φ (PLK)Φ−1(0)=Φ (PLK) , (F.14)
where Φ(t) is the principal fundamental matrix solution whose columns are so-
lutions to Eq. (7.22) and Φ(0) is the identity. R is then the eigenvector that has
eigenvalue 1. Note that if v is an eigenvector, so too∣is −v; w∣ e choose convention
that R points in the same direction as Ωe, i.e. ∆Im ≡ ∣Im− Ī ∣< 90◦e .
∣ ∣
The orientation of R is related to the ∣∆θ̄e∣ predicted by Eq. (7.81): if R andΩe
are misaligned by angle ∆Im, then θ̄e oscillates with semi-amplitude ∆Im. Thus,
we can infer ∆Im in the vicinity of eachΩe =MΩLK resonance from Eq. (7.81), and
we obtain by linearity that
∑∞ ∣ ∣∼ d(M) ∣∣∣ΩeM sin∆I ∣∆I eMm ∣ . (F.15)
M=1 2 Ωe−M ∣ΩLK
Figure F.2 shows that this calculation predicts the numeric ∆Im well. In par-
ticular, the amplitude of both resonances are well predicted, suggesting that the
approximate factor d(M) introduced in Eq. (7.81) is accurate. This supports our
316
assertion that R deviates from Ω̂e at resonances Ωe ≈ MΩLK, and the misalign-
ment is captured by Eq. (7.81).
317
Figure F.2: Comparison of the orientation R obtained from numerical simulations
of Eq. (7.22) in the η ̸= 0, LK-enhanced parameter regime (m1 =m2 =m3 = 30M⊙,
a0 = 0.1 AU, e −30 = 10 ,ãout = 3 AU) with the analytic resonance formula given
by Eq. (F.15) as a function of initial inclination, in the absence of GW dissipa-
tion. The top panel shows the ratio Ωe/ΩLK is shown as the solid black line, while
the horizontal dashed lines denote Ωe = ΩLK and Ωe = 2ΩLK. The bottom panel
shows the misalignment angle between the numerically-computed R and Ωe as
the black line. Separately, the predicted misalignment ∆Im due to interaction
with the Fourier harmonics is given by Eq. (F.15). We see that the scaling of the
misalignment angle near resonances is well captured by our analytic formula, but
the numerical misalignment angle crosses 0 within the M = 1 resonance, which is
not predicted by our simple theory.
318
APPENDIX G
APPENDICIES TO MASS RATIO DISTRIBUTION OF
TERTIARY-INDUCED BLACK HOLE MERGERS
G.1 Origin of Octupole-Inactive Gap
We investigate the origin of the “octupole-inactive gap”, an inclination range near
I0 ≈ 90◦ for which emax does not attain elim despite being in between two octupole-
active windows. This gap was first identified in Section 8.2.3, and is seen in both
the non-dissipative and full simulations with GW dissipation (see Figs. 8.4–8.8).
To better understand this gap, we first review the mechanism by which ex-
treme eccentricity excitation occurs. In the test-particle limit, Katz et al. [2011]
showed that K (Eq. 8.4) oscillates over long timescales when ω, the argument of
pericenter of the inner orbit, is circulating. This then leads to orbit flips (and ex-
treme eccentricity excitation) between prograde and retrograde inclinations when
K changes signs: since j(e) is nonnegative, the sign of K determines the sign of
cos I. Katz et al. [2011] obtained coupled oscillation equations in K andΩe, the az-
imuthal angle of the inner eccentricity vector in the inertial reference frame. The
amplitude of oscillation of K can then be analytically computed, and the octupole-
active window (the range of I0 over which orbit flips occur) is the region for which
the range of these oscillations encompasses K = 0 [Katz et al., 2011]. When ω is
librating instead, Ω ◦e jumps by ∼ 180 every ZLK cycle, and the oscillations in K
are suppressed.
In the finite-η case, we commented in Section 8.2.3 that the relation between
K oscillations and extreme eccentricity excitation (and orbit flipping) can be gen-
eralized even when η is nonzero. K still oscillates over timescales ≫ tZLK when
319
Figure G.1: Octupole-active windows and amplitude of oscillation of K (Eq. 8.4).
Top panel: Maximum eccentricity emax attained by the inner binary with initial
tertiary inclination I0 when integrated for 2000tZLK (blue dots), reproduced from
the top panel of Fig. 8.6. Also shown are elim (Eq. 8.10, horizontal red dashed line),
the quadrupole-level result for emax (Eq. 8.7, dashed black line), the empirically-
determined center of the gap, located at I ◦0 ≈ 88.32 (vertical black line), and the
inclinations that can lead to extreme eccentricities (shaded purple regions). Bot-
tom panel: Minimum and maximum values of K , denoted Kmin and Kmax, attained
by the systems. Also shown are the initial K for a given I0 (black dashed line) and
the critical Kc =−η/2 for orbit flipping (horizontal red dashed line). The center of
the octupole-inactive gap and the octupole-active windows are labeled as in the
top panel.
320
ω is circulating, and if its range of oscillation contains Kc ≡ −η/2, then the in-
ner orbit flips, in the process attaining extreme eccentricities. To be precise,
orbit flips are defined to be when the range of inclination oscillations changes
from (cos I0)− < cos I < cos I0,lim to cos I0,lim < cos I < (cos I0)+ or vice versa, where
(cos I0)± are given by Eq. (8.11) and I0,lim satisfies Eq. (8.9).
However, the range of oscillation of K is more complex than it is in the test-
particle limit. Figure G.1 compares the behavior of emax in the non-dissipative
simulations (top panel; reproduced from the top panel of Fig. 8.6) to the range of
oscillations in K (bottom panel). Denote the center of the gap I0,gap (shown as
the vertical black line in both panels of Fig. G.1). Near I0,gap, K oscillates about
K(I0,gap), which is positive, and the oscillation amplitude goes to zero at I0,gap.
On the other hand, orbit flips (and extreme eccentricity excitation) are possible
when the range of oscillation of K encloses Kc (i.e., Kmin <Kc <Kmax). The purple
shaded regions in both panels of Fig. G.1 illustrate this equivalence, as they show
both the elim-attaining inclinations in the top pane( l and) the inclinations where
Kmin <Kc <Kmax in the bottom panel. But since K I0,gap > 0 while Kc < 0, there
will always be a( rang)e of I0 about I0,gap for which the oscillation amplitude is
smaller than K I0,gap −Kc, and orbit flips are impossible in this range. This
range then corresponds to the octupole-inactive gap.
This analysis has simply pushed our lack of understanding onto a new quan-
tity: why are K oscillations suppressed in the neighborhood of I0,gap? A quantita-
tive answer to this question is beyond the scope of this paper, but for a qualitative
understanding, we can examine the evolution of a system in the octupole-inactive
gap. The left panel of Fig. G.2 shows the same simulation as Fig. 8.3 but with an
additional panel showing Ωe, while the right panel shows a simulation with the
same parameters except I = 88◦0 , which is near I0,gap (see Fig. G.1). The oscilla-
321
Figure G.2: The left panel is the same as Fig. 8.3 but includes the evolution of
the azimuthal angle of the eccentricity vector, Ωe. The right panel is the same as
the left but for I0 = 88◦. For both of these examples, we have used ω0 = 0, but the
evolution is similar for all ω0.
tions in K (third panels) are much smaller for I = 88◦0 than for I0 = 93.5◦, and no
orbit flips occur. Most interestingly, the fourth panel shows that the evolution of
Ωe is much less smooth than in Fig. 8.3, jumping at almost every other eccentric-
ity maximum. Katz et al. [2011] have already pointed out that jumps in Ωe occur
when ω is librating, rather than circulating.
When the octupole-order terms are neglected, the circulation-libration bound-
ary is a boundary in e-ω space: as long as the ZLK separatrix exists in the e-ω
plane and e0 > 0, then an initial ω0 = 0 causes ω to circulate, while an initial
ω0 = π/2 causes ω to librate [e.g., Kinoshita and Nakai, 1999, Shevchenko, 2016].
However, when including octupole-order terms, this picture breaks down. To il-
lustrate this, for a range of I0 and both ω0 = 0 and ω0 = π, we evolve the fiducial
system parameters for a single ZLK cycle, using q = 0.2 as is used for Figs. G.1
and G.2, and consider both the dynamics with and without the octupole-order
terms. Figure G.3 gives the resulting changes in Ωe over a single ZLK period
322
when the octupole-order effects are neglected (top) and when they are not (bot-
tom). Two observations can be made: (i) I0,gap is approximately where ∆Ωe = 0 for
circulating initial conditions when neglecting octupole-order terms, and (ii) the
inclusion of the octupole-order terms seem to cause Ωe to exclusively vary slowly
(|∆Ωe| ≪ 180◦) except for I0,gap < I0 < I0,lim. The former is plausible: if K(I0,gap)
is the location of an equilibrium in K-Ωe space, then it must satisfy ∆Ωe = 0. The
latter suggests that the assumption of circulation of ω in Katz et al. [2011] may
be satisfied for many more initial conditions than the quadrupole-level analysis
suggests, as long as they are not in octupole-inactive gap.
Finally, examination of the bottom panel of Fig.∣G.1 sugge∣sts that the oscil-
lation amplitude in K grows roughly linearly with ∣I0− I0,gap∣ in the vicinity of
I0,gap [this may be because, in the test-particle limit, librating ω give oscillation
amplitudes in K that are higher-order in K and Ωe, as pointed out by Katz et al.,
2011]. Assuming this, the gap width can then be given by
( )
Gap Width= 2 I0,lim− I0,gap . (G.1)
This explains why the gap does not exist in the test-particle regime, as I0,lim =
I0,gap = 90◦ by symmetry of the equations of motion.
It is clear from the preceding discussion and Fig. G.3 that the octupole-order,
finite-η dynamics are complex, and our discussion can only be considered heuristic.
Nevertheless, in the absence of a closed form solution to the octupole-order ZLK
equations of motion or a full generalization of the work of Katz et al. [2011], they
provide a preliminary understanding of the octupole-inactive gap.
323
Figure G.3: Plot of ∆Ωe, the change in Ωe over a single ZLK cycle, for q = 0.2
and the fiducial parameters using different initial conditions. In the top panel,
octupole-order terms are neglected, while in the lower panel, they are not. The
solid and dashed vertical black lines denote I0,gap and I0,lim respectively.
324
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