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Li, Gongjie. 2015. Long-Term Dynamics of High Mass Ratio
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Multiples. Doctoral dissertation, Harvard University, Graduate
School of Arts & Sciences.
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Long-term Dynamics of High Mass Ratio Multiples
A dissertation presented
by
Gongjie Li
to
The Department of Astronomy
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the subject of
Astronomy & Astrophysics
Harvard University
Cambridge, Massachusetts
April 2015
c 2015 — Gongjie Li
(cid:13)
All rights reserved.
Dissertation Advisor: Professor Avi Loeb Gongjie Li
Long-term Dynamics of High Mass Ratio Multiples
Abstract
This thesis presents a series of studies on the dynamics of high mass ratio multiples, with
applications to planetary systems orbiting stars and stellar systems orbiting supermassive
black holes (SMBHs).
Almost two thousand exoplanetary systems have recently been discovered, and their
configurations gave rise to new puzzles related to planetary formation theories. We
studied the dynamics of planetary systems aiming to understand how the configuration
of planetary system is sculptured and to probe the origin of planetary systems. First,
we discussed hierarchical three-body dynamics, which can be applied to planets that
are orbiting a star while perturbed by a planet or a star that is farther away. The
perturbation from the farther object can flip the planetary orbits and produce counter
orbiting hot Jupiters, which cannot be formed in the framework of classical planetary
formation theory. In addition, we have studied the stellar encounters with planetary
systems in star clusters, which produce eccentric and inclined planets. Moreover, we
investigated the obliquity variation of the Earth, and the developed formalism can be
applied to exoplanetary systems. We note that the obliquity variation is important to
the habitability of the exoplanets.
Long term dynamics is also central to understanding stellar systems orbiting
SMBHs. SMBHs are common in the centers of galaxies and lead to rich dynamical
interactions with nearby stars. At the same time, dynamical features of nearby stars
iii
reveal important properties of SMBHs. The aforementioned hierarchical three-body
dynamics can be applied to stars near SMBH binaries, which are natural consequences of
galaxy mergers. We found that the distribution of stars surrounding one of the SMBHs
results in the shape of a torus due to the perturbation from the other SMBH, and the
dynamical interactions contribute to an enhancement of tidal disruption rates, which
can help identify the SMBH binaries. In addition, we investigated the heating of stars
near SMBHs, where the heating of stars due to gravitational waves as the SMBHs merge
may mark the merger, and provide an electromagnetic counterpart for gravitational wave
detection. Moreover, the accumulated tidal heating of stars may cause the stars to be
more vulnerable for tidal disruptions, as the stars orbit around a SMBH in an eccentric
orbit.
iv
Contents
Abstract iii
Acknowledgments x
Dedication xii
1 Introduction 1
1.1 Dynamics of Planetary Systems . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Origin of Planetary Systems . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Spin-axis Dynamics of Planets . . . . . . . . . . . . . . . . . . . . . 9
1.2 Interactions between Stars and Supermassive Black Holes . . . . . . . . . . 11
1.2.1 Implications for Stars Surrounding Supermassive Black Hole Binaries 12
1.2.2 Heating of Stars Surrounding Supermassive Black Holes . . . . . . . 13
Section I. Planets around Stars 15
2 EccentricityGrowthandOrbitFlipinNear-coplanarHierarchicalThree-
body Systems 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Coplanar Flip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Analytical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Systematic Study of 180◦ Flips . . . . . . . . . . . . . . . . . . . . . . . . . 27
v
CONTENTS
2.4 Application to Exoplanets and Tidal Disruption Events . . . . . . . . . . . 28
2.4.1 Counter Orbiting Hot Jupiters . . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Tidal Disruption Events - Systematic Study . . . . . . . . . . . . . 32
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Chaos in the Test Particle Eccentric Kozai-Lidov Mechanism 49
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Overview of the Eccentric Kozai-Lidov Mechanism in the Test-particle Limit 53
3.3 Surfaces of Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4 The Maximum Eccentricity and the Flip Condition . . . . . . . . . . . . . 61
3.5 Chaotic Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4 The Dynamics of the Multi-planet System Orbiting Kepler-56 75
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 The Obliquity Distribution Function . . . . . . . . . . . . . . . . . . . . . 79
4.3 Obliquity and Inclination Evolution in the Presence of an Outer Perturber 82
4.3.1 Overview of the System Architecture . . . . . . . . . . . . . . . . . 82
4.3.2 Dynamics of Kepler 56 . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3.3 Inferring the Inclination Distribution Function from Observations . 88
4.4 Tidal and Stellar Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Cross Sections for Planetary Systems Interacting with Passing Stars and
Binaries 101
vi
CONTENTS
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 Results for the Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4 Analysis and Scaling Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.5 The Solar Birth Aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.6.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6 On the Spin-axis Dynamics of a Moonless Earth 176
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2 A Simplified Perturbative Model . . . . . . . . . . . . . . . . . . . . . . . . 179
6.2.1 Large Chaotic Regions: 0◦ 45◦ & 65◦ 85◦ . . . . . . . . . . . . . 180
− −
6.2.2 Bridge Region: 45◦ 65◦ . . . . . . . . . . . . . . . . . . . . . . . . 185
−
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.3.1 Analytical Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.3.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
References 196
6.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
6.6.1 Dynamics of the Unsimplified Hamiltonian . . . . . . . . . . . . . . 200
6.6.2 Double Resonances and Triple Resonances . . . . . . . . . . . . . . 201
7 Pre-LHB Evolution of the Earth’s Obliquity 205
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.2 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
vii
CONTENTS
7.3 Numerical Integrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Section II. Stars around Supermassive Black Holes 220
8 Implications of the Eccentric Kozai-Lidov Mechanism for Stars Sur-
rounding Supermassive Black Hole Binaries 221
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.2.1 Comparison of Timescales . . . . . . . . . . . . . . . . . . . . . . . 226
8.2.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 234
8.3 SMBH-binary System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8.4 SMBH-IMBH System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
9 Accumulated Tidal Heating of Stars Over Multiple Pericenter Passages
Near SgrA* 253
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
9.2 Heating of Stars by Tidal Excitation of Modes . . . . . . . . . . . . . . . . 256
9.2.1 Mode Excitation and Interference in Multiple Pericenter Passages . 256
9.2.2 Tidal Heating of Stars . . . . . . . . . . . . . . . . . . . . . . . . . 264
9.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
10 Gravitational Wave Heating of Stars and Accretion Disks 281
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
10.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
10.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
10.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
viii
CONTENTS
References 297
ix