Adventures in the Kozai-Lidov Mechanism

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Joseph Michael Antognini

Graduate Program in Astronomy

The Ohio State University 2016

Dissertation Committee: Professor Todd A. Thompson, Advisor Professor Chris S. Kochanek Professor Marc H. Pinsonneault Copyright by

Joseph Michael Antognini

2016 Abstract

Hierarchical triple systems are ubiquitous in our Galaxy. The population of

Sun-like stars in triple systems is 10% and this fraction rises for high mass stars. ∼ The most powerful dynamical phenomenon exhibited by hierarchical triples is the

Kozai-Lidov (KL) mechanism. If the orbit of the tertiary is at high inclination relative to the orbit of the inner binary, it can drive the inner binary to high eccentricity over timescales long compared to the orbital periods. In principle a tertiary at some critical inclination (close to, though not generally exactly equal to

90◦) can drive the inner binary to arbitrarily high eccentricity.

The equations of motion of the KL mechanism can be derived by taking a multipole expansion of the gravitational potential of the triple system and employing the von Zeipel transformation to average over each binary orbit. From the equations of motion I derive the period of KL oscillations exactly using the quadrupole term and in the test particle limit. I then explore the variation of the period of KL oscillations over the parameter space of all possible triples in the test particle limit.

The KL period does not vary by more than a factor of a few except for a narrow band around the separatrix between circulation and libration where the KL period diverges.

ii When the next term of the multipole expansion, the octupole term, is considered, the maximum inclination achieved by the triple system varies from one

KL oscillation to the next (the “eccentric KL mechanism” or EKM). If the EKM causes the inclination of the triple to pass through 90◦ (a “flip”), the individual KL oscillations can be extremely strong, often pushing the inner binary to eccentricities of e 0.99999. I derive the period of oscillations induced by the octupole term by ≈ averaging the equations of motion over individual KL oscillations. I demonstrate that when constants of motion are held fixed, the period of EKM oscillations varies

1/2 as ǫoct− .

I demonstrate using N-body integrations that the orbital motion of the tertiary induces fluctuations in the angular momentum of the inner binary, violating the double orbit averaged approximation. I also derive the equations of motion of a hierarchical by averaging only over the inner binary of the triple and find excellent agreement between the N-body and semi-secular calculations. I show that these

fluctuations are of constant magnitude, and so when the inner binary has an angular momentum less than the magnitude of the fluctuations (as during the maximum of a

KL oscillation), they can lead to large changes in the eccentricity. These fluctuations can then push the inner binary to much higher eccentricities than predicted in secular theory. I show that for mergers of supermassive black holes in triples these

fluctuations can lead the binary to merge with substantial eccentricity (e > 0.8), ∼ even when relativistic corrections are included.

iii Scattering events involving triples are common, particularly in cluster environments, and are a potential mechanism to form multiple systems. I calculate the cross section of various outcomes from binary-binary, triple-single, and triple-binary scattering events using a suite of N-body integrations. I explore the dependence of the cross sections on the semi-major axis ratio, mass ratio, eccentricities, and incoming velocity. I derive the velocity dependence of the cross section for new triple formation from triple-single scattering and show that it is shallower at high velocities than the cross section for triple formation from binary-binary scattering. I derive the cumulative cross section for changes to the orbital parameters from flybys and show that the cross section for large changes to the orbital parameters is similar to the cross section for ionization.

I apply these results to the Type Ia supernova progenitor problem. While the

KL-accelerated WD-WD merger model has a number of attractive features, it has the difficulty that KL oscillations would drive the stars to strong tidal interaction on the main sequence, thereby quenching future KL oscillations. Scattering can push triple systems from low inclination to high inclination after the stars have evolved into WDs. However, I derive the rate of new triple formation in the field to be many orders of magnitude smaller than the SN Ia rate. In open clusters the new triple formation rate is smaller by a factor of 10, but the uncertainties in the estimate ∼ are too large to rule out scattering in open clusters. In globular clusters the new

iv triple formation rate is comparable to the Galactic SN Ia rate scaled by mass, but it is unclear if the specific SN Ia rate is enhanced in globular clusters.

I estimate the rate of scattering-induced stellar mergers and find it to be lower

1 than the observed rate of 0.5 yr− by an order of magnitude. I calculate the cross section for planets to be ejected due to scattering with an interloping system and show that most free floating planets are not ejected in this way. Finally, I show that in all environments the timescale of KL oscillations is short compared to the timescale for disruption due to scattering.

v Dedication

To my grandpa

vi Acknowledgments

Only rarely is research performed in a vacuum, and the work presented in this dissertation is no exception. Without the help of many people this work could not have been done (or, at the very least, it would have been done worse and more slowly).

I am very grateful to my thesis advisor, Todd Thompson, for the many ways in which he has helped me to grow as a scientist. A few in particular stand out in my mind. Todd has a remarkable capability to approach any physical problem without fear and use whatever resources are available to come up with an order-of-magnitude solution. But he is quick to state that this capability is not an innate talent, but a skill that can (and should) be instilled in any enterprising scientist through enough practice. Our meetings together have been valuable experiences for me to learn what makes a good scientific question and how to approach a solution. Todd has also been a great source of life advice and I know that I will be applying his advice for many years in the future.

I would like to thank Paul Martini for being my first-year advisor. Paul worked patiently with me on my first research project in graduate school and helped me

vii write my first publication. I learned a great deal about scientific writing from Paul through his many comments and corrections on my drafts and I am thankful that he was so thorough.

I am thankful for the friendship of my officemates Scott Adams, Dale Mudd, and Ben Shappee over the years. I am also thankful for the graduate student population in the astronomy department for their friendship. The entering class one year above me was very accepting of me, being the only student entering the program my year. I am particularly thankful to Calen Henderson for organizing our chamber music parties and for hosting many parties besides. I am also thankful to

Calen and Brett Andrews for letting me sleep on their couch for a few weeks while I was transitioning between my apartment and condo.

I am grateful for the astronomy department for letting me use their spare

CPU cycles through the Condor network. I am especially grateful to David Will for helping to set up Condor and humoring my requests for unusual software. I am also grateful to Michael Savage for keeping the networks in the department running smoothly. I am very grateful to the Ohio Supercomputing Center for their grant of computer time and for being very responsive with some of the issues I had. I would also like to thank Annika Peter for being instrumental in setting up the CCAPP cluster at the Ohio Supercomputing Center, on which I used a disproportionate amount of time.

viii Finally, I would like to thank my parents for the education and support that they gave me. Most importantly, I would like to thank Valerie for her support and encouragement, particularly in these last few months as this work was being completed.

ix Vita

July 11, 1988 ...... Born – Sacramento, California

2010 ...... B.S. Astrophysics, California Institute of Technology

2010 – 2015 ...... Graduate Teaching and Research Associate, The Ohio State University

2010, 2015 ...... Distinguished University Fellowship, The Ohio State University

Publications

Research Publications

1. R. Stoll, J. L. Prieto, K. Z. Stanek, R. W. Pogge, D. M. SzczygieÃl, G. Pojma´nski, J. Antognini, H. Yan, “SN 2010jl in UGC 5189: Yet Another Luminous Type IIn Supernova in a Metal-poor Galaxy”, ApJ, 730, 34, (2011).

2. J. Antognini, J. Bird, P. Martini, “The Lifetime and Powers of FR IIs in Galaxy Clusters”, ApJ, 756, 116, (2012).

3. O. Pejcha, J. M. Antognini, B. J. Shappee, T. A. Thompson, “Greatly enhanced eccentricity oscillations in quadruple systems composed of two binaries: implications for stars, planets and transients”, MNRAS, 435, 943, (2013).

4. J. M. Antognini, B. J. Shappee, T. A. Thompson, P. Amaro-Seoane, “Rapid eccentricity oscillations and the mergers of compact objects in hierarchical triples”, MNRAS, 439, 1079, (2014).

x 5. J. M. Antognini, “Timescales of Kozai-Lidov oscillations at quadrupole and octupole order in the test particle limit”, MNRAS, 452, 3610 (2015).

Fields of Study

Major Field: Astronomy

xi Table of Contents

Abstract ...... ii

Dedication ...... vi

Acknowledgments ...... vii

Vita ...... x

List of Tables ...... xvi

List of Figures ...... xvii

Chapter 1: Introduction ...... 1

1.1 Historical development of the Kozai-Lidov mechanism ...... 1

1.1.1 Limitations of the standard KL formalism ...... 5

1.1.2 Higher order terms in the KL expansion and convergence . . . 7

1.2 Applications of the Kozai-Lidov mechanism ...... 8

1.3 ScopeoftheDissertation...... 9

Chapter 2: Timescales of Kozai-Lidov oscillations ...... 10

2.1 Introduction...... 10

2.2 Basicequations ...... 11

2.2.1 Notation...... 11

2.2.2 TheHamiltonian ...... 12

2.2.3 Integrals of motion ...... 13

2.2.4 Equationsofmotion ...... 14

xii 2.2.5 Librationvs.rotation...... 15

2.3 Derivation of the period of KL oscillations ...... 16

2.4 Abriefsurveyofparameterspace ...... 18

2.5 Approximations...... 19

2.5.1 The timescale of KL oscillations ...... 19

2.5.2 High inclination, low eccentricity triples ...... 21

2.6 TheeccentricKLmechanism...... 22

2.6.1 Equations of motion and integrals of motion ...... 24

2.6.2 The period of EKM oscillations ...... 26

2.6.3 ParameterspaceoftheEKM ...... 27

2.6.4 The dependence on ǫoct ...... 28

2.7 Conclusions ...... 34

Chapter 3: Rapid Eccentricity Oscillations in Hierarchical Triples ...... 36

3.1 Introduction...... 36

3.2 Numericalmethods&Setup ...... 37

3.2.1 Energyconservation ...... 39

3.2.2 Inspiraltime...... 40

3.2.3 Comparison to the secular approximation ...... 43

3.3 Rapid eccentricity oscillations ...... 45

3.4 Effectonmergertimes ...... 52

3.4.1 Testcase...... 52

3.4.2 Populationstudy ...... 56

3.5 Discussionandconclusions...... 61

3.5.1 Implications for extreme-mass-ratio inspirals ...... 63

xiii 3.5.2 Implications for gravitational wave emission ...... 63

3.5.3 Tides and implications for stars and planets ...... 66

Chapter 4: Dynamical formation & scattering of triples ...... 69

4.1 Introduction...... 69

4.2 Numericalmethods ...... 73

4.2.1 Notation...... 73

4.2.2 Crosssections...... 74

4.2.3 Initial conditions and halting criteria ...... 76

4.3 Scatteringexperiments ...... 79

4.3.1 Cross sections of model systems ...... 79

4.3.2 Dependence on initial parameters ...... 86

4.4 Analyticapproximations ...... 95

4.4.1 Three-bodyscattering ...... 97

4.4.2 Four-bodyscattering ...... 101

4.5 Orbital parameter distributions after scattering ...... 106

4.5.1 Changes to the orbital parameters from flybys ...... 106

4.5.2 Distribution of orbital parameters in dynamically formed triples 112

4.5.3 Populationstudy ...... 118

4.6 Discussion...... 123

4.6.1 Application to Type Ia Supernovae ...... 123

4.6.2 Application to free-floating planets ...... 132

4.6.3 Stellar collisions during scattering events ...... 136

4.6.4 How long do high inclination triples survive? ...... 139

4.7 Conclusions ...... 141

xiv Chapter 5: Future directions ...... 145

5.1 Modelling non-secular effects with single-orbit averaging ...... 145

5.1.1 Motivation...... 145

5.1.2 Vectorial equations of motion ...... 145

5.2 HighmasstriplesasLIGOsources ...... 150

5.2.1 Motivation...... 150

5.2.2 Methods...... 152

5.2.3 Results...... 153

5.3 Conclusions ...... 156

References ...... 157

xv List of Tables

3.1 Initial conditions for a system that undergoes weak Kozai-Lidov oscillations...... 49

3.2 Initial conditions for triple systems studied in this chapter...... 49

4.1 Normalized cross sections for the outcomes of triple-single scattering. 82

4.2 Normalized cross sections for the outcomes of triple-binary scattering. 83

4.3 Normalized cross sections for the outcomes of binary-binary scattering. 84

4.4 Cross sections for outcome classes in triple-single scattering atv ˆ = 1. 87

4.5 Cross sections for outcome classes in triple-binary scattering atv ˆ = 1. 88

4.6 Cross sections for outcome classes in binary-binary scattering atv ˆ = 1. 92

4.7 Best-fitting parameters of the eccentricity cross sections in Fig. 4.4 to theinverseGompertzfunction...... 92

4.8 Cross sections for planet ionization. We include both the normalized cross section,σ ˆ, and the physical cross section, σ. Multi-planet systems consist of two planets, circumbinary systems consist of a planet orbiting a binary, and S-type planets consist of a planet orbiting a star in a binarysystem...... 135

xvi List of Figures

2.1 Variation in the period of KL oscillations over all of parameter space. 20

2.2 Residuals for the approximation in equation (2.48) to the period of KL oscillations in the high inclination, low eccentricity limit...... 23

2.3 Parameter space where EKM oscillations with flips are possible for two choices of ǫoct...... 29

2.4 The period of EKM oscillations with flips as a function of φq for three choices of χ...... 30

2.5 The period of the EKM relative to the period of KL oscillations as a function of ǫoct calculated analytically...... 33

3.1 Energy conservation in Fewbody for orbits over a range of eccentricities in the Newtonian case and including non-radiative post- Newtoniantermsuptoorder3...... 41

3.2 The evolution of the semi-major axis of a binary of two 107 M SMBHs ⊙ with an initial semi-major axis of 1 pc and an initial eccentricity of 3 1 e 10− ...... 44 − ∼ 3.3 The evolution in eccentricity of a system undergoing Kozai-Lidov oscillations...... 46

3.4 The final moments of the evolution in eccentricity of the system presentedinFigure3.3...... 47

3.5 SystemsexhibitingREOs...... 50

3.6 The impact of REOs on the evolution of the inner binary of a hierarchicaltriple...... 53

3.7 Timescales of the system presented in Figure 3.6 calculated with direct integration and in the secular approximation...... 55

xvii 3.8 The fraction of time that the system presented in Figure 3.6 spends at eccentricities greater than any given eccentricity in the direct integration and in the secular approximation...... 58

3.9 The time required for the inner binary of triple systems to merge as a function of the eccentricity of the orbit of the tertiary using direct three-body integrations and using the secular approximation...... 60

3.10 Merger time distribution for fixed e2...... 62

3.11 REOs in an EMRI calculated with direct three-body integration and inthesecularapproximation...... 64

3.12 The eccentricity distribution of the inner orbit when the inner two components of the systems calculated in Figure 3.9 come within 10 RSch ofeachother...... 67

4.1 Convergence of the scattering cross sections with bmax for triple-single scattering...... 78

4.2 A sample binary-binary, triple-single, and triple-binary scattering experiment projected onto the xy-plane...... 80

4.3 Cross sections of various classes of outcomes as a function of semi- major axis ratio, α, for binary-binary, triple-single, and triple-binary scattering...... 90

4.4 Cross sections of outcome classes for triple-single, triple-binary, and binary-binary scattering as a function of eccentricity...... 94

4.5 Cross sections of outcome classes for binary-binary, triple-single and triple-binary scattering as a function of the mass ratio...... 96

4.6 Cross sections of outcome classes for binary-binary, triple-single and triple-binary scattering as a function of the incoming velocity. . . . . 96

4.7 Normalized cross sections for exchange and ionization in equal-mass binary-single scattering as a function of incoming velocity...... 99

4.8 The cross section for a collision to occur in binary-single scattering as a function of the normalized incoming velocity...... 102

4.9 Cross sections for outcomes of equal mass binary-binary scattering with a semi-major axis ratio of 100 and circular orbits as a function of incomingvelocity...... 104

xviii 4.10 The velocity dependence of the cross section for new triple formation for binary-binary and triple-single scattering of our model system. . . 107

4.11 The cumulative cross section for changes to the orbital parameters from triple-single flybys and triple-binary flybys with incoming velocities of vˆ =1...... 109

4.12 The relationship between changes to the mutual inclination and changes to eout fromflybyevents...... 111

4.13 The cumulative cross section for changes to the semi-major axis ratio, α, for a variety of choices of bmax...... 113

4.14 The semi-major axis ratio, α over time for an isolated triple...... 114

4.15 Orbital parameters of dynamically formed triples from binary-binary, triple-single, and triple-binary scattering...... 116

4.16 The distribution of the timescale of KL oscillations of dynamically formedtriplesinasetofmodelexperiments...... 119

4.17 Orbital parameters of new triples resulting from scattering experiments inourpopulationstudy...... 121

4.18 Distribution of the timescales of KL oscillations and the eccentric KL mechanism in dynamically formed triples in our population study. . . 122

4.19 The lifetime of high-inclination triple systems in globular clusters and inthefield...... 140

5.1 A comparison of a calculation of a KL oscillation performed by code using double orbit averaging (dotted line), a calculation using the N- body code rebound (solid line), and the single-orbit averaged equations of motion (dashed line). The single-orbit averaged equations of motion reproduce the oscillations in the eccentricity of the inner orbit on the timescale of the outer orbit that are missed by double-orbit averaging. Note that the single-orbit averaged code and the N-body code predict a substantially higher eccentricity than the double-orbit averaged code. The lines have been slightly offset in time for clarity...... 151

5.2 The distribution of semi-major axes of the inner binaries of triples that survive two SNe. The distribution is bimodal with a peak at 1 AU, and a secondary peak at R ...... ∼ 155 ∼ ⊙ xix 5.3 The distribution of the mutual inclination of the surviving triples. The distribution is roughly uniform in the cosine of the inclination. . . . . 155

5.4 The distribution of the parameter ǫoct (see equation 2.50). The vast majority of surviving triples are very compact and are near the boundaryofstability...... 158

xx Chapter 1: Introduction

1.1. Historical development of the Kozai-Lidov mechanism

After deducing the Law of Universal Gravitation, a young Isaac Newton applied it to the motion of two bodies in orbit around each other and was able to derive the general version of Kepler’s laws for the first time in history. Newton then turned to the next logical problem — the motion of three bodies in orbit around each other. In particular, Newton was interested in understanding the motion of the Moon around the Earth and the Sun. While Newton made a great deal of progress in understanding the Earth-Moon-Sun system (he devoted more space to this problem in the Principia Mathematica than to the two-body problem), he was unable to derive a general equation for the motion of the Moon under the gravitational influence of the Earth and the Sun in the same way that he was able to derive Kepler’s laws from the two-body problem. Indeed, Newton’s calculation of the motion of the lunar perigee disagreed with observations by a factor of two and after over fifty years of effort other physicists were unable to resolve the discrepancy. These difficulties even cast doubt on the Law of Universal Gravitation, with the 4 French astronomer Alexis Clairaut proposing to add a small r− term to the Law of Universal Gravitation (for a historical overview see Gutzwiller 1998).

As the techniques for understanding the three-body problem advanced, astronomers were better able to find approximate solutions and were able to bring the predicted motion of the lunar perigee into agreement with observations. Nevertheless, a general analytic solution to the three-body problem eluded physicists for the next three centuries. It is no exaggeration to say that work on the three-body problem directly led to most fundamental developments in the theory of classical mechanics.

1 By the end of the nineteenth century the three-body problem had attached to it a major prize — in celebration of his sixtieth birthday, King Oscar II of Sweden decreed that a medal and a reward of 2500 crowns (the equivalent of about a third of a professor’s yearly salary) be awarded to anyone who provided a solution to the three-body problem. The French mathematician Henri Poincar´eclaimed the prize shortly after it was announced. Though he did not provide a solution to the three-body problem, he published a treatise (Poincar´e1892) containing innovative approaches to understanding the three-body problem, many of which are still in use today (for a thorough historical treatment see Green 1997). Poincar´edemonstrated that in general trajectories in three-body dynamics are sensitive to the initial conditions — even arbitrarily small changes to the initial conditions will generally lead to qualitatively different long-term dynamics. Poincar´e’s work marked the founding of chaos theory and today the three-body problem is understood to be the archtypical chaotic system.

Despite the notoriety of the three-body problem even today, it is unfortunate that the contributions of the Finnish mathematician Fritiof Sundman remain relatively neglected. Sundman (1907) derived a series in powers of t1/3 that was convergent for all initial conditions (except the degenerate orbits of zero total angular momentum), thereby finding the solution that eluded the best physicists of the past three centuries. But Sundman’s solution was a Pyrrhic victory — the series converges quite slowly. Beloriszky (1930) estimated that to use Sundman’s series for astronomical calculations of orbits in the Solar System would require some 108,000,000 terms.

After Poincar´e’s result, study of the three-body problem underwent a lull throughout the first half of the twentieth century until the launch of artificial satellites and the development of computers in the late 1950’s spurred a renewal of interest in the problem. By averaging over the motions of the orbit of a satellite and the orbit of the Moon around the Earth, in 1961 the Soviet physicist Michael Lidov derived a set of partial differential equations that describe the orbital evolution of a satellite due to the perturbative influence of the Moon (Lidov 1961). Lidov studied these equations and found that if the inclination of the satellite’s orbit exceeds

2 arccos 3/5 39.2◦ with respect to the orbit of the Moon, the satellite’s eccentricity ≈ will oscillate.p Lidov further showed that a mutual inclination of 90◦ will lead to the satellite’s orbit being pushed to arbitrarily high eccentricity.

The Japanese astronomer Yoshihide Kozai introduced this phenomenon to the West in a study of the orbit of an asteroid under the gravitational influence of the Sun and Jupiter (Kozai 1962), though Kozai was aware of Lidov’s work.1 It is remarkable to note that the techniques Lidov and Kozai used to derive the KL mechanism had already existed for nearly two centuries and Pierre Laplace or Joseph-Louis Lagrange (for example) could certainly have discovered it in the late 1700s had either of them been so inclined as to study the problem that interested Lidov and Kozai.

Lidov and Kozai analyzed only hierarchical triples, which are a subset of all possible triples. A hierarchical triple is one for which the maximum distance between two particles in the system is much less than the minimum distance between either of those two particles and the third. Hierarchical triples lend themselves more easily to analytic techniques because, to zeroth order, the triple behaves like two independent binaries. As these techniques fail for more compact triples, other techniques must be used to study stable non-hierarchical triple systems such as the famous figure-eight orbit of Chenciner & Montgomery (2000). Astrophysical triple systems, however, do not span the range of all possible stable triple systems because the probability of forming a non-hierarchical stable triple systems from scattering events is very small (Heggie 2000). Even determining general conditions for the stability of a three-body system can only be done numerically (Mardling & Aarseth 2001; Petrovich 2015),

1A note on nomenclature: Because Lidov’s work was hidden behind the iron curtain, Western scholars have historically referenced only Kozai’s work. Due to the fact that Lidov discovered the mechanism first and Kozai was aware of Lidov’s result, it is now generally accepted to refer to the mechanism as the “Kozai-Lidov” mechanism. Indeed some authors have even begun to refer to the mechanism as the “Lidov-Kozai” mechanism in recognition of Lidov’s greater contribution. Unfortunately, there is so much literature already referring to the Kozai mechanism that this author believes it would be a disservice to future astronomers to muddle the nomenclature further and “Kozai-Lidov” may be here to stay. This designation has the further advantage of bringing the KL mechanism into agreement with Stigler’s Law of Eponymy, which states that no scientific law is named after the person who originally discovered it Merton (1968).

3 Lidov (1962) and Kozai (1962) both examined the physics of hierarchical triples in more detail by expanding the Hamiltonian in a power series of the ratio, α, between the inner binary semi-major axis to the outer binary semi-major axis. The α0 term is stationary and describes two independent binary systems and the α1 term vanishes. Lidov (1962) and Kozai (1962) derived the first non-trivial term of this series, the α2, or quadrupole, term. If α is small, Lidov (1962) and Kozai (1962) assumed that the Hamiltonian can be well approximated by neglecting higher order terms of the series. They then performed a von Zeipel transformation, which eliminates explicit reference to the instantaneous positions of the particles of the system or their mean anomalies. Hence, rather than representing the complete motion of three point masses, the Hamiltonian after the von Zeipel transformation represents the gravitational interaction between two hoops of matter. Implicit in this transformation is the “secular” or “double-orbit-averaged” approximation, in which it is assumed that all changes to the orbital elements are slow compared to the orbital timescales of the problem.

From this Hamiltonian Lidov (1962) and Kozai (1962) were able to derive a set of coupled non-linear differential equations which describe the evolution of the orbital elements of the inner and outer binaries. Surprisingly, these equations revealed that if the two binaries were initially circular2 with mutual inclination i, the eccentricity of the inner binary, e1, will oscillate between its initial value and some maximum eccentricity given by

5 2 e1, max = 1 cos i (1.1) r − 3 as long as the inclination is larger than a critical value of 3 cos icrit = , (1.2) r5 or i 39.2◦. Kozai (1962) found that for inclinations less than i the system is crit ≈ crit stationary. This calculation was generalized by Harrington (1968), who provided the equations of motion for the case of a non-circular outer orbit (though see Soderhjelm 1982, for corrections). 2Strictly speaking, a perfectly circular inner binary is a stationary, though unstable, solution to the equations of motion. Hence Eq. 1.1 is valid if the inner binary has some very small, but non-zero eccentricity.

4 The Hamiltonian was first expanded to its next order α3, or octupole, term by Soderhjelm (1984) in the limit of low eccentricities and inclinations. While the quadrupole term vanishes for inclinations less than icrit, Soderhjelm (1984) found that the octupole remains nonzero and therefore dominates the evolution, producing eccentricity oscillations even for coplanar systems. These oscillations were further studied by Li et al. (2014b), who demonstrated that such systems can be driven to the high eccentricities characteristic of high inclination triple systems and can lead the inner binary to flip by 180◦. ∼ The general octupole-order Hamiltonian and their corresponding equations of motion were derived by Krymolowski & Mazeh (1999) and Ford et al. (2000). These authors found that the introduction of the octupole term breaks one of the symmetries of the quadrupole-order Hamiltonian and therefore leads to much more complicated behavior. Whereas in quadrupole theory all trajectories are closed, only a measure-zero subset of trajectories are closed in octupole theory. Krymolowski & Mazeh (1999) and Ford et al. (2000) found that the octupole term therefore leads to qualitatively different behavior, characterized by a long-term oscillation of the maximum eccentricity of individual KL cycles. Indeed, like Soderhjelm (1984), Ford et al. (2000) found that in certain regimes the octupole term can become the dominant term of the Hamiltonian. The importance of the octupole order term was more systematically explored in a pair of papers by Lithwick & Naoz (2011) and Katz et al. (2011). Katz et al. (2011) provided an analytic study of the octupole-order equations of motion and derived the conditions under which the inner orbit flips. Lithwick & Naoz (2011) complemented this work by numerically surveying the behavior of the equations of motion over a broad subset of parameter space, in particular examining where the inner orbit flips.

1.1.1. Limitations of the standard KL formalism

There are several important regimes in which the standard KL theory fails. A critical assumption of KL theory is that α is small and high-order terms can be neglected. If the triple is insufficiently hierarchical this assumption will not be valid. At its extreme, this clearly implies that KL theory will be unable to account for unstable

5 triples, but it is unclear whether KL theory can correctly model triples which are sufficiently hierarchical to be stable, but which are close to the boundary of stability (e.g., Mardling & Aarseth 2001), and if so, how many terms in the expansion are required.

The standard KL theory must also be modified if other forces are present in the system. These forces manifest themselves as additional sources of precession of the inner or outer binary. Since the strength of a KL oscillation depends sensitively on the evolution of the difference between the arguments of periapsis between the inner and outer orbits, additional sources of periapsis precession serve to weaken KL oscillations. Indeed, if the precession timescale becomes much shorter than tKL, inst, KL oscillations will become “de-tuned” and will halt entirely (Antognini et al. 2014). There are three major sources of precession in astrophysical triple systems: non-Keplerian stellar potentials, tidal forces, and relativistic precession. Precession due to non-Keplerian potentials is generally only important for triples close to the stellar cusp of a potential (Merritt et al. 2011). Tidal forces are ordinarily negligible, but become extremely important if the two objects of the inner binary come within a few radii of each other. While some prescriptions exist to incorporate these forces into the secular calculation (e.g., Fabrycky & Tremaine 2007), the theory of dynamical tides is not currently well understood, so while the general principle that tides serve to de-tune KL oscillations is well established, the details of this de-tuning process are only known approximately. Precession from general relativity (GR) is typically (though not always) important only for inner binaries consisting of two compact objects since more extended objects usually come into tidal contact before GR precession becomes important.

Relativistic effects were first incorporated into the octupole equations of motion by Blaes et al. (2002), who included the first post-Newtonian (PN) term to model precession and a correction for gravitational radiation using the calculations of Peters (1964). Recently Naoz et al. (2013b) derived an additional post-Newtonian “cross term” in the relativistic three body Hamiltonian and found that when the GR precession timescale is comparable to the KL timescale, GR precession can excite resonant eccentricity oscillations and thereby strengthen the KL oscillations

6 rather than de-tune them. Will (2014) explicitly derived this cross term in the full post-Newtonian expansion of the metric without making the secular approximation. While the calculations of Naoz et al. (2013b) and Will (2014) do not agree, the general result that this additional cross term can change the dynamics of a hierarchical triple system over a GR precession timescale appears to be robust.

A further limitation to the standard KL mechanism is the phenomenon of rapid eccentricity oscillations (REOs, Seto 2013; Antonini et al. 2014; Antognini et al. 2014). The changing tidal force on the inner binary from the tertiary due to its motion through its orbit induces an oscillation in the eccentricity of the inner binary with a frequency twice that of the orbital frequency of the tertiary, thereby violating the secular approximation. The amplitude of the change in angular momentum was first calculated by Ivanov et al. (2005). Because the change in angular momentum is constant over the course of a KL cycle, REOs are ordinarily negligible when the inner orbit has low eccentricity (i.e., large angular momentum). But during the high-eccentricity phase of a KL cycle, the fluctuations in angular momentum of the inner orbit due to REOs can become comparable to or larger than the total amount of angular momentum in the inner orbit. When this occurs the oscillations can become stochastic and can drive the components of the inner binary to collide (Katz & Dong 2012).

1.1.2. Higher order terms in the KL expansion and convergence

The octupole term is known to have an important long-term effect on the evolution of hierarchical triple systems in which the inner binary has a large mass ratio by changing the maximum eccentricity from one KL cycle to the next in the so-called “eccentric KL mechanism” (Ford et al. 2000; Katz et al. 2011; Lithwick & Naoz 2011). However, the octupole term vanishes if the two components of the inner binary are of equal mass. Although Marchal (1990) derived an “interaction” term of order α7/2, and Laskar & Bou´e(2010) presented the Hamiltonian to the next order, the equations of motion were not derived. The next term in the multipole expansion, the hexadecapole term, was first derived in the literature along with the equations

7 of motion by Hamers et al. (2015), but it appears to affect the long-term dynamics only negligibly in every region of parameter space where it has been checked.

1.2. Applications of the Kozai-Lidov mechanism

The KL mechanism has therefore been invoked in a wide range of contexts. KL oscillations may play an important role in the formation of hot Jupiters by bringing gas giants formed beyond the snow line into tidal contact with their host star (Holman et al. 1997; Fabrycky & Tremaine 2007; Naoz et al. 2012). KL oscillations may also be involved in the formation of blue stragglers (Perets & Fabrycky 2009). A particularly interesting application of the KL mechanism was proposed by Thompson (2011) in white dwarf-white dwarf (WD-WD) binaries. If a tertiary drives the WD-WD binary to high eccentricities, the gravitational wave (GW) radiation timescale is reduced by many orders of magnitude. The WD-WD binary could then merge in less than a Hubble time, possibly creating a Type Ia supernova. Thompson (2011) further showed that the number of WD-WD binaries in high-inclination triple systems is potentially large enough to explain the observed Type Ia SN rate. Katz & Dong (2012) further demonstrated that these mergers could be head-on collisions from highly eccentric orbits rather than gradual coalescences from circular orbits. By similar arguments, Thompson (2011) proposed that KL oscillations may be involved in the formation of gamma-ray bursts by driving neutron star-neutron star binaries to rapid merger. On the largest scales, KL oscillations can drive supermassive black holes (SMBHs) to rapid merger (Blaes et al. 2002; Antognini et al. 2014). KL oscillations may alse be a dynamical feature of extreme mass ratio inspirals (Antonini et al. 2014; Antognini et al. 2014).

KL oscillations additionally have important implications for GW observations. Because binaries can be driven to extremely high eccentricities, they may retain some residual eccentricity when they emit GWs in the frequency ranges of LIGO or VIRGO (Antonini et al. 2014; Antognini et al. 2014). These detectors require the use of waveform templates to detect the faint signals expected in the noise background, but if waveforms from circular orbits are used, even orbits with residual eccentricities as modest as e 0.1 will be undectable by such waveforms (Brown & Zimmerman 1 ∼

8 2010). These signals from eccentric WD-WD binaries may even overwhelm many other sources in space-based detectors (Gould 2011).

1.3. Scope of the Dissertation

I begin this dissertation with a study of orbital evolution over very long timescales and move to progressively shorter timescales in each chapter. An outline is as follows: In Chapter 2 I derive the equations of motion of a hierarchical triple system using both the quadrupole and octupole terms of the Hamiltonian and explore the periods of oscillations in the orbital elements of the triple due to these terms. In Chapter 3 I demonstrate the breakdown of double-orbit averaging and show that non-secular changes to the orbital parameters can lead to greatly enhanced gravitational wave emission and much shorter merger times of compact object binaries. In Chapter 4 I provide a numerical study of the gravitational scattering of hierarchical triple systems and the dynamical formation of hierarchical triple systems via scattering. Finally, in Chapter 5 I conclude with preliminary results from ongonig work studying triple population synthesis.

9 Chapter 2: Timescales of Kozai-Lidov oscillations

2.1. Introduction

KL oscillations are a secular phenomenon, occurring on timescales much longer than the orbital periods. It is therefore possible to average the motions of the individual stars over their orbits and study only the secular changes to the orbital elements. If there is a large mass ratio in the inner binary then on even longer timescales the strength of the KL oscillations (i.e., the maximum eccentricity reached) will vary (Ford et al. 2000; Katz et al. 2011; Lithwick & Naoz 2011; Naoz et al. 2013a). These variations have been termed the ‘eccentric KL mechanism’ (EKM), and in some cases can cause the inner binary to pass through an inclination of 90◦ with respect to the outer binary in a ‘flip’ from prograde to retrograde or vice versa. During a flip the eccentricity of the inner binary can be driven to extremely large values because the strength of KL oscillations is very sensitive to the mutual inclination, with arbitrarily strong oscillations occurring as the inclination approaches 90◦ exactly in the test particle limit. Although EKM oscillations do not occur when the two stars of the inner binary are of equal mass, mass loss from one of the stars in the course of stellar evolution can induce EKM oscillations (Shappee & Thompson 2013; Michaely & Perets 2014). EKM oscillations and flips have generally been studied in the context of hierarchical triples, but flips occur over a wider range of parameter space in both 2+2 quadruples (Pejcha et al. 2013) and 3+1 quadruples (Hamers et al. 2015).

Because the extreme eccentricity oscillations that occur during a flip can affect the evolution of the objects in the inner binary, the timescale for EKM oscillations is another important quantity. Yet no derivation of the timescale for EKM oscillations has appeared in the literature, although several studies have asserted that t t /ǫ is a plausible timescale (e.g., Katz et al. 2011; Naoz EKM ∼ KL oct 10 et al. 2013b; Li et al. 2015), where ǫoct measures the strength of the octupole order term relative to the quadrupole order term of the Hamiltonian (see equation 2.50 for a definition). I show that t t /√ǫ . EKM ∼ KL oct In Section 2.2, I present the basic parameters and equations that govern a hierarchical three-body system. In Section 2.3, I then derive the period of KL oscillations. In Section 2.4, I explore how the period varies over the parameter space and in Section 2.5, I provide an approximation to the exact period. In Section 2.6, I treat the period of EKM oscillations and derive the corrected timescale. I conclude in Section 2.7.

To perform the calculations in this chapter I wrote the kozai Python module. This module can evolve hierarchical triple systems in the secular approximation up to hexadecapole order using either the Delaunay orbital elements or the eccentricity and angular momentum vectors. I have released this code under the MIT license and it is available at https://github.com/joe-antognini/kozai.

2.2. Basic equations

2.2.1. Notation

Throughout this chapter orbital properties referring to the inner and outer binary are labelled with a subscript 1 and 2, respectively. The masses of the two components of the inner binary are m1 and m2, and the mass of the tertiary is m3.

We will often refer to the orbital parameters using Delaunay’s elements: the mean anomalies, lx; the arguments of periapsis, gx, and the longitudes of ascending nodes, hx, where x = 1 or 2 and refers to the inner or outer binary, respectively. Their conjugate momenta are

m1m2 L1 = G(m1 + m2)a1, (2.1) m1 + m2 p m3(m1 + m2) L2 = G(m1 + m2 + m3)a2, (2.2) m1 + m2 + m3 p G = L 1 e2 , (2.3) x x − x p 11 and

Hx = Gx cos ix. (2.4)

Delaunay’s elements form a set of canonical variables. Note that G1 and G2 are the angular momenta of the inner and outer binaries, respectively. We furthermore define the reduced angular momentum

j2 1 e2 . (2.5) x ≡ − x

The angular momentum of an orbit may thus be written Gx = Lxjx.

2.2.2. The Hamiltonian

If a three-body system is sufficiently hierarchical, its Hamiltonian may be considered to be that of two isolated binaries (the inner binary, consisting of the two closest bodies, and the outer binary, consisting of the distant body plus the inner binary taken as a point mass) plus a perturbative interaction term:

Gm1m2 G(m1 + m2)m3 = + + pert. (2.6) H 2a1 2a2 H This interaction term captures the change in the orbital motion of each binary in the tidal field of the other. Because we are assuming that the triple is hierarchical, the semi-major axis ratio, α = a1/a2, is a small parameter that we can use to expand the perturbative component of the Hamiltonian in a multipole expansion (Harrington 1968),

j j+1 G ∞ r a = αj 1 2 P (cos Φ), (2.7) Hpert a Mj a r j 2 j=2 1 2 X µ ¶ µ ¶ th where Pj is the j Legendre polynomial, rx is the distance between the two components of the xth binary, Φ is the angle between r and r and is a mass 2 1 Mj parameter defined by

j 1 j 1 m1− ( m2) − j = m1m2m3 − − j . (2.8) M (m1 + m2)

If we are only interested in changes to the orbital elements that occur on timescales much longer than the orbital periods (so-called ‘secular’ changes), we must

12 average the Hamiltonian over both mean anomalies. To do this while maintaining the canonical structure of the Hamiltonian requires a technique known as von Zeipel averaging. The general case for three massive bodies is quite complicated even at quadrupole order as one must be careful to include the longitudes of ascending nodes (Naoz et al. 2013a). However, the Hamiltonian simplifies considerably if one component of the inner binary is taken to be a test particle as this allows one to fix the longitudes of ascending nodes and eliminate them from the Hamiltonian. The resulting double-averaged Hamiltonian at quadrupole order in the test particle limit is

= C 2 + 3e2 1 3 cos2 i 15e2 1 cos2 i cos 2g , (2.9) Hq 2 1 − − 1 − 1 £¡ ¢ ¡ ¢ ¡ ¢ ¤ where C2 is a constant parameterizing the strength of the quadrupole term given by Gm m m C = 1 2 3 α2. (2.10) 2 16(m + m )a (1 e2)3/2 1 2 2 − 2 The semi-major axes and e2 do not change at quadrupole order, so C2 is also constant. We will henceforth refer to the dimensionless Hamiltonian,

q ˆq H . (2.11) H ≡ C2

2.2.3. Integrals of motion

There are no dissipative forces in the problem, so the total energy, ˆ, remains H constant. Moreover, because no energy is transferred between the two binaries at quadrupole order, each term of the Hamiltonian is conserved separately, so ˆ Hq remains constant as well.

The total angular momentum is also conserved and may be expressed in the form of the geometrical relation, G2 G2 G2 cos i = tot − 1 − 2 . (2.12) 2G1G2 This relation is valid in the general case of three massive bodies. In the test particle limit the geometrical relation may be approximated by

G G + G cos i. (2.13) tot ≃ 2 1 13 Since Gtot and G2 are constant, we must have that G1 cos i is constant as well.

Furthermore, G1 = L1j1, and L1 is also constant, so this requires that j1 cos i be constant as well. We notate this constant of motion as

Θ (j cos i)2. (2.14) ≡ 1 In this form, the constant of motion is known as ‘Kozai’s integral’ and is equal to the square of the z-component of the reduced angular momentum, jz (Holman et al. 1997). Kozai’s integral implies that the component of angular momentum perpendicular to the plane of the outer binary is constant. However, the test particle assumption is crucial to its derivation. In the general case of three massive bodies this component of angular momentum is not conserved, although it is possible to derive a generalized version which is conserved (e.g., Wen 2003). The generalized Kozai integral is due to the fact that, as Lidov & Ziglin (1976) showed, ˆ is Hq independent of g2, thereby implying that G2 (and hence also e2) must be constant.

Because ˆ only depends on e , cos i, and g , and there are two integrals Hq 1 1 of motion, ˆ and Θ, there is only one degree of freedom and so the system is Hq integrable. Moreover, because these variables are all bounded, the motion is periodic (with the exception of a locus of stationary points of measure zero). The Hamiltonian to quadrupole order may be expressed as

ˆ 1 2 2 2 2 q = 2 (5 3j1 )(j1 3Θ) 15(1 j1 )(j1 Θ) cos 2g1 (2.15) H j1 − − − − − £ ¤ in terms of j1 and Θ.

2.2.4. Equations of motion

We are interested in the time evolution of the variables j1, cos i, and g1. Of these, only g1 is a canonical variable so its time evolution follows directly from Hamilton’s equations: dg ∂ C ∂ ˆ 1 = Hq = 2 Hq (2.16) dt ∂G1 L1 ∂j1 Carrying out the differentiation of equation (2.15) we find

dg1 6C2 1 4 4 = 3 5 Θ j1 (1 cos 2g1) + 4j1 (2.17) dt L1 j1 − − £ ¡ ¢ ¤ 14 The variable j1 is related to a canonical variable, G1, by a constant, so we find its time evolution to be dj 1 ∂ C ∂ ˆ 1 = Hq = 2 Hq . (2.18) dt L1 ∂g1 L1 ∂g1 Again carrying out the differentiation of equation (2.15) we find

dj1 30C2 1 2 2 = 2 1 j1 j1 Θ sin 2g1. (2.19) dt L1 j1 − − ¡ ¢ ¡ ¢ The time evolution of the inclination is complicated by the elimination of nodes from the Hamiltonian. Due to this procedure the time evolution of the inclination cannot be recovered from Hamilton’s equations directly. Instead, the inclination must be derived by calculating j1 and solving the geometrical relation given in equation (2.12).

2.2.5. Libration vs. rotation

During a KL oscillation, the argument of periapsis of the inner binary may either rotate or librate. This is to say, g1 may sweep through the full range of angles from 0 to 2π (rotation) or it may be restricted to just a subset of them (libration). In the case of libration, the set of librating trajectories must librate about a fixed point of g1 and j1. Inspection of equation (2.19) reveals that j1 is stationary only when 2 g1 takes half- or whole-integer multiples of π (recall that Θ < j1 ). Now, inspection of equation (2.17) reveals that g1 cannot be stationary at integer multiples of π. This implies that trajectories can only librate about half-integer multiples of π, so g = π/2 and j2 = 5Θ/3. 1,fix ± 1,fix p To determine whether a particular system (i.e., a given ˆ and Θ) librates or Hq rotates we must see whether there exists a physical solution of equation (2.15) for j1 2 when g1 = 0. Setting g1 = 0 in equation (2.15) and solving for j1 we find 1 j2 = (10 + ˆ + 6Θ). (2.20) 1 12 Hq The critical system on the boundary between libration and rotation will have a solution for j1 exactly equal to unity and libration will occur if the only solution for j1 exceeds unity. Defining the libration constant as 1 C 2 ˆ 6Θ , (2.21) KL ≡ 12 − Hq − ³ ´ 15 we will have libration if CKL < 0 and rotation if CKL > 0. This constant was first presented in Lidov (1962) and may be calculated equivalently by

5 C = e2 1 sin2 i sin2 g . (2.22) KL − 2 1 µ ¶ Note that the condition for rotation then becomes

2 1 sin g1 . (2.23) ≤ r5 sin i

Because CKL naturally parameterizes a dynamical property of the triple, it is often convenient to work with it instead of ˆ where possible. Hq 2.3. Derivation of the period of KL oscillations

Because the Hamiltonian at quadrupole order is integrable, the period of KL oscillations, tKL, may be determined exactly. The period may be written dt t = dt = dj . (2.24) KL dj 1 I I 1

Solving equation (2.15) for cos 2g1 and rewriting in terms of sin 2g1, we have

1 2 2 4 2 ˆ 3j1 + j1 q 9Θ 5 + 15Θ sin 2g = 1 H − − . (2.25) 1  −  15³ (1 j2)(j2 Θ)´   − 1 1 −       Substituting equation (2.25) into equation (2.24) and substituting the result into equation (2.24) we find

2 L1 j1 tKL = 2 2 30C2 (1 j1 )(j1 Θ) I − − 1 2 − 2 3j4 + j2( ˆ 9Θ 5) + 15Θ 1 1 1 Hq − − dj . (2.26) ×  − 15(1 j2)(j2 Θ)  1 Ã − 1 1 − !   We note that this integral may be rewritten in terms of incomplete elliptic integrals of the first kind, but we do not do so here because it complicates the expression considerably.

16 The integral in equation (2.26) proceeds from the maximum value of j1 to the minimum value of j1 and back again to the maximum value of j1, so we may instead integrate from j to j and multiply by two. Eliminating ˆ in favor of C by min max Hq KL making use of equation (2.21), and rearranging, we have

jmax L1 1 tKL = 2 15C2 jmin (1 j1 ) Z − 1 2 Θ 2 1 Θ 4 C 2 − 1 + KL dj . (2.27) × − j2 − 5 − j2 5 1 j2 1 "µ 1 ¶ µ 1 − 1 ¶ #

Eccentricity maxima (j ) occur for g = π/2. Eccentricity minima (j ) also min 1 ± max occur at g = π/2 in the case of libration but occur at g =0 or π in the case of 1 ± 1 rotation. We may therefore solve for jmin and jmax by substituting the appropriate values of g1 into equation (2.15) and solving for j1. We therefore have

1 2 jmin = ζ ζ 60Θ (2.28) r6 − − ³ p ´

1 2 jmax = ζ + ζ 60Θ , CKL < 0 (2.29) r6 − ³ p ´

j = 1 C , C > 0. (2.30) max − KL KL p where we have defined

ζ 3 + 5Θ + 2C . (2.31) ≡ KL

For convenience, we define the integral in equation (2.27) to be f(CKL, Θ) such that

15tKLC2 f(CKL, Θ) . (2.32) ≡ L1

Having calculated the limits of integration, we can now use equation (2.27) to calculate the period of KL oscillations to quadrupole order in the test particle limit for any hierarchical triple.

17 2.4. A brief survey of parameter space

We now turn to a brief exploration of the range of values that the integral in equation (2.27) may take. The overall timescale for KL oscillations is determined by the coefficient before the integral, which we present in more detail in Section 2.5.1. The integral, however, depends on only two parameters describing the triple: ˆ Hq and Θ, or equivalently, CKL and Θ. Thus, once the timescale of KL oscillations has been set, only two degrees of freedom remain.

What values may ˆ , C , and Θ take? It is easy to see from equation (2.14) Hq KL that 0 Θ 1 since both j and cos i are bounded by 0 and 1. Moreover, it is clear ≤ ≤ 1 from equation (2.22) that 3 C 1 (2.33) −2 ≤ KL ≤ since all the terms are bounded by 0 and 1. From the bounds on Θ and CKL, we can conclude from equation (2.21) that the bounds on ˆ are Hq 10 ˆ 20. (2.34) − ≤ Hq ≤ However, the limits on ˆ and Θ are not independent. In the case of g = 0, the Hq 1 requirement that Θ j2 implies that ≤ 1 10 + 6Θ ˆ 20. (2.35) − ≤ Hq ≤

This, in turn, translates to the requirement in CKL that

C 1 Θ. (2.36) KL ≤ −

In order for equation (2.27) to have a solution, the square roots in equations (2.28), (2.29), and (2.30) must exist. The existence of the inner square root in equation (2.28) requires that 1 C 5Θ 2√15Θ + 3 . (2.37) KL ≥ −2 − ³ ´ This requirement is always satisfied in the case of rotation (CKL > 0). In the case of libration (CKL < 0), this requirement may instead be written in terms of CKL as 1 Θ 3 2 6C 2C , C 0. (2.38) ≤ 5 − − KL − KL KL ≤ ³ p ´ 18 In the case of libration, the square root in equation (2.29) exists everywhere that the square root in equation (2.28) does, so the existence of this square root adds no new constraints. In the case of rotation, the requirement that the square root in equation (2.30) exist is satisfied by the same condition set in equation (2.36).

Equation (2.38) implies that there is a critical inclination, below which librating

KL oscillations do not occur. Taking CKL = 0 we recover the usual critical inclination of cos i 3/5. Although we do not provide an explicit derivation, we note that ≥ the criterionp in equation (2.38) can also be arrived at by requiring that d2j 1 < 0 (2.39) dt2 when j1 is at a maximum. In other words, KL oscillations occur when the minimum eccentricity is an unstable equilibrium.

Knowing now the region of parameter space in which KL oscillations occur, we can numerically integrate the integral in equation (2.27) over the entire parameter space. The results of this procedure are presented in Figure 2.1. Except for a narrow strip of parameter space centered around the boundary between rotation and libration (CKL = 0) the integral only varies by a factor of a few. Near the rotation-libration boundary the integral diverges and KL oscillations have arbitrarily large periods. Figure 2.1 also indicates that the period of KL oscillations depends most strongly on CKL and only weakly on Θ.

2.5. Approximations

2.5.1. The timescale of KL oscillations

So long as the integral in equation (2.27) is of order unity, the period of KL oscillations will be given by the coefficient before the integral to within an order of magnitude:

L1 tKL . (2.40) ≃ 15C2 Substituting equations (2.1) and (2.10) and noting that we are working in the test particle limit so m 0, we have the timescale in terms of the semi-major axes, 2 → 19 Fig. 2.1.— Variation in the period of KL oscillations over all of parameter space.

The contours (dotted lines) show different values of f(CKL, Θ) (i.e., the integral in equation 2.27). The period varies only by a factor of a few except very near the boundary between rotation and libration (dashed line), where it diverges. The gray regions indicate where KL oscillations are not possible. The large dot at CKL = 0, Θ = 3/5 marks the largest value of Θ for which libration is possible.

20 masses, and eccentricities:

3 16 a m 3/2 t 2 1 1 e2 . (2.41) KL ≃ 15 3/2 Gm2 − 2 Ãa1 ! r 3 ¡ ¢ This timescale may be expressed more elegantly in terms of the periods of the inner and outer orbits, Pin and Pout, respectively, by making use of Kepler’s law:

2 8 m P 3/2 t 1+ 1 out 1 e2 . (2.42) KL ≃ 15π m P − 2 µ 3 ¶ µ in ¶ ¡ ¢ This is the form of the KL period that typically appears in the literature, but with an additional mass term and numerical coefficient. The mass term implies that KL oscillations lengthen indefinitely as the tertiary approaches zero mass. In the case of a massive tertiary but a test particle primary and secondary (e.g., a WD-WD binary orbiting a SMBH), the period of KL oscillations approaches a constant value. Note that in the case of an equal mass primary and tertiary, neglecting the numerical coefficient will lead to an overestimate of the period of KL oscillations by a factor of nearly three.

2.5.2. High inclination, low eccentricity triples

In most cases of interest in astronomy, the inner binary of a hierarchical triple starts with a low to moderate eccentricity. Moreover, KL oscillations are strongest (and therefore most interesting) when the tertiary is at high inclination. It is therefore worth finding an approximation to tKL in the high inclination, low initial eccentricity limit. In this limit, we have Θ 0 and C 0 and equation (2.27) may be solved → KL → exactly:

jmax 5 1+ j1 f(CKL, Θ) ln (2.43) ≃ 4√6 1 j1 µ ¶¯jmin − ¯ We also have in this limit that j ¯ 1 and 1 j 1 so that min ≪¯ − max ≪ 5 2 f(CKL, Θ) ln . (2.44) ≃ 4√6 1 jmax µ − ¶ where C j 1+ KL (2.45) max ≃ 3 21 for libration (CKL < 0), and C j 1 KL . (2.46) max ≃ − 2 for rotation (CKL > 0).

The dependence of f(CKL, Θ) on Θ is non-trivial to approximate from first principles. After experimenting with several functional forms, we found that f(C , Θ) varies most closely with (1 Θ). If there is a Θ dependence both inside KL − and outside the logarithm, we then expect f(CKL, Θ) to take the form

b 5 a(1 Θ) c f(CKL, Θ) ln − (1 Θ) , (2.47) ≈ 4√6 CKL − µ ¶ where a, b, and c are fitting parameters. We fit numerical integrations of f(CKL, Θ) to this form over the range 0 Θ 0.25, 0.1 C 0.1 and find the remarkably ≤ ≤ − ≤ KL ≤ good fit,

2 1 2 m1 Pout 2 3/2 tKL 1+ 1 e2 ≈ 3π r3 m3 Pin − µ ¶ µ ¶ 2.36 ¡ ¢9.42(1 Θ) 1.53 ln − (1 Θ)− . (2.48) × C − µ KL ¶ We attempted to add several auxiliary parameters but found that they did not substantially improve the fit.

The approximation provided in equation (2.48) fits the true value of f(CKL, Θ) to within 2% over the range sampled, and over the vast majority of the range sampled the residuals are less than 0.3%. This is therefore an appropriate formula to use for triples in which the inner binary has an eccentricity e1 < 0.3 and an ∼ inclination 60◦

2.6. The eccentric KL mechanism

If the two masses of the inner binary are not equal and the outer orbit has non-zero eccentricity, the next term in the expansion of the Hamiltonian, the octupole order

22 Fig. 2.2.— Residuals for the approximation in equation (2.48) to the period of KL oscillations in the high inclination, low eccentricity limit. The approximation is correct to within 2% at all points in this range and typically does much better.

This range of CKL and Θ corresponds to triples in which the inner binary has an eccentricity e1 < 0.3 and the inclination is i > 60◦. ∼ ∼

23 term, becomes dynamically significant. This term leads to changes to the orbital parameters of the outer orbit that are slow relative to individual KL oscillations. These long-term changes can cause the inner orbit to eventually pass through an inclination of 90◦. During these orbital flips, the nearly perpendicular inclination leads to strong KL oscillations which drive the inner binary to extremely large eccentricities. For this reason, the dynamical effect of the octupole term has been called the ‘eccentric KL mechanism’ (EKM) (e.g., Lithwick & Naoz 2011).

The introduction of the octupole term breaks the integrability of the

Hamiltonian. Consequently, neither CKL or Θ remain constants of the motion. Furthermore, in the test particle limit at quadrupole order it is possible to eliminate the longitude of ascending node, Ω, from the Hamiltonian. At octupole order either this parameter or g2 necessarily enters into the equations of motion. In this section we will follow the analysis of Katz et al. (2011) and work in terms of the longitude of ascending node of the eccentricity vector, Ωe, defined such that e = e(sin ie cos Ωe, sin ie sin Ωe, cos ie), and e points toward periapsis of the inner binary.

In the case of rotation (CKL > 0), the parameters Ωe, CKL, and Θ all change on a timescale which is long compared to individual KL cycles. It is therefore possible to assume that Ωe, CKL, and Θ are all approximately constant over individual KL oscillations and only examine the long-term changes to these parameters. In this approximation the system remains integrable with new integrals of motion.

Due to the integrability of the system the variations in CKL, Θ, and Ωe are all strictly periodic. In this section we derive the period of these EKM oscillations. We note that there is a related octupole-order dynamical phenomenon in which nearly coplanar orbits at high eccentricity can undergo a flip by rolling over its major axis. An analysis of this phenomenon, including a derivation of the timescale, can be found in Li et al. (2014b).

2.6.1. Equations of motion and integrals of motion

Since energy is conserved, the quadrupole order term of the Hamiltonian, , is Hq also conserved in the time-averaged behavior of the system. This implies that the

24 relationship between C , Θ, and ˆ in equation (2.21) remains valid and that the KL Hq quantity 1 φ C + Θ (2.49) q ≡ KL 2 is a constant of motion.

It is convenient to work with the parameter ǫoct, which measures the relative size of the octupole order term of the Hamiltonian to the quadrupole order term.

The parameter ǫoct is conventionally defined as e a ǫ 2 1 . (2.50) oct ≡ 1 e2 a − 2 2 Some authors have added a mass term (e.g., Naoz et al. 2013a) to capture the fact that the octupole term is zero and EKM oscillations do not occur for an equal mass inner binary. However, because we are working exclusively in the test particle limit we do not do so here.

Following Katz et al. (2011), the long-term evolution in Ωe and Θ are given by

dΩ 6E(x) 3K(x) e = Θ − , (2.51) dτ 4K(x) µ ¶

dΘ 15πǫoct √Θ sin Ωe = (4 11CKL) 6 + 4CKL, (2.52) dτ − 64√10 K(x) − p where K(x) and E(x) are complete elliptic functions of the first and second kind, respectively,

3(1 CKL) x(CKL) − , (2.53) ≡ 3 + 2CKL and the time coordinate has been scaled to the secular timescale:

3 t m3 Ga1 τ = = 3 2 3/2 t. (2.54) tsec a2(1 e2) s m1 − Katz et al. (2011) also derive another integral of motion,

χ F (C ) ǫ cos Ω , (2.55) ≡ KL − oct e 25 where the function F (CKL) is defined to be 32√3 1 K(η) 2E(η) F (C ) − dη. (2.56) KL ≡ π (41η 21)√2η + 3 Zx(CKL) −

Although there are two integrals of motion, φq and χ, they are not sufficient to completely describe the dynamical behavior of the triple. This is because ǫoct carries dynamical information as well, most importantly whether or not flips are possible.

The dynamical significance of ǫoct can be seen from the fact that ǫoct enters into the definition of χ. Thus, in the octupole case there are three independent parameters describing the system as opposed to the case of quadrupole-order KL oscillations in which there are only two.

2.6.2. The period of EKM oscillations

In the case of EKM oscillations it is easier to derive their period directly from the equations of motion rather than from action angle variables. We have from equation (2.49) that dC 1 dΘ KL = , (2.57) dτ −2 dτ so the period may be written

dCKL τEKM = dτ = . (2.58) C˙ I I KL Substituting equation (2.52) we find 128√10 K(x) 1 τEKM = dCKL. (2.59) 15πǫoct 2(φ C ) sin Ω (4 11CKL)√6 + 4CKL I q − KL e µ − ¶ To write Ωe in terms ofpCKL, we note that equation (2.55) implies that χ F (C ) 2 sin Ω = 1 − KL . (2.60) e − ǫ s µ oct ¶ Substituting equation (2.60) into equation (2.59) and explicitly writing the limits of the integral yields 256√10 CKL,max K(x) τEKM = 15πǫoct CKL,min 2(φq CKL) (4 11CKL) Z − − 1 2 2 p (χ F (CKL)) − 1 − (6+4C ) dC . (2.61) × − ǫ2 KL KL ·µ oct ¶ ¸

26 The upper limit of the integral can be deduced by noting that CKL is maximized when Θ is minimized and that Θ = 0 during a flip. We therefore have

CKL,max = φq. (2.62)

The lower limit is more subtle. It is clear from equation (2.52) that Θ is maximized when sin Ωe = 0. This implies from equation (2.55) that

F (C )= χ ǫ . (2.63) KL,min ± oct

To decide whether to take the plus or minus sign, we must solve both for CKL and then take the value of CKL which is less than CKL,max. This equation can then be used to solve for CKL,min numerically. Together, equations (2.61), (2.62), and (2.63) can be used to solve for the period of EKM oscillations exactly.

2.6.3. Parameter space of the EKM

As in the quadrupole case we first explore over what region of parameter space EKM oscillations with flips may occur. We then determine the variation in tEKM over this range of parameter space. Unfortunately, the parameter space cannot be mapped quite as easily as in the case of quadrupole KL oscillations because there are now three parameters describing the system instead of two: φq, χ, and ǫoct. As such, we 3 2 explore parameter space for two choices of ǫoct: ǫoct = 10− and ǫoct = 10− . Strong 2 octupole-order effects occur in many triple systems with ǫoct = 10− , but these effects 3 are much weaker for most triples when ǫoct = 10− (e.g., Lithwick & Naoz 2011).

To determine the boundaries of the parameter space of spin flips we first recall that 0 Θ 1, and for rotation 0 C 1 (which is the only case we are ≤ ≤ ≤ KL ≤ considering to octupole order). The occurrence of a spin flip is equivalent to having Θ = 0, and hence during a flip C = φ . Since cos Ω is bounded by 1, we then KL q e ± have the following constraint:

F (φ ) ǫ χ F (φ )+ ǫ . (2.64) q − oct ≤ ≤ q oct The parameter space can be divided into two regions based on the maximum of the function F (φq). This maximum can be found by solving K(xcrit) = 2E(xcrit) for xcrit,

27 which yields x 0.826, and then calculating crit ≈ 3(1 xcrit) φq,crit = − 0.112. (2.65) 3 + 2xcrit ≈

Now, φq cannot be arbitrarily large because F (φq) diverges at φq = 4/11. Thus we have 4 φ < . (2.66) q 11

Since, for φq < φq,crit, F (φq) cannot be less than zero, this then implies a constraint on χ:

χ ǫ (φ < φ ). (2.67) ≥ oct q q,crit Finally, the above relation implies that

F (φq,min)= ǫoct. (2.68)

Taken together, these relations bound the parameter space over which flips are 3 2 possible. The resulting maps of parameter space for ǫoct = 10− and ǫoct = 10− are shown in Figure 2.3. Because the parameter space over which flips occur is somewhat narrow and the dependence of τEKM on φq is fairly complicated we do not show contours as we did at quadrupole order in Fig. 2.1. Instead, we show τEKM as a function of φ with the choice of χ = F (φ ) and χ = F (φ ) ǫ /2 in Fig. 2.4. q q q ± oct The timescale for EKM oscillations depends most sensitively on φq. The timescale has two singularities: one at the maximum value of φq of 4/11, and another which is dependent on the choice of χ, but is near φq,crit. Except very close to these singularities, the period of EKM oscillations does not vary by more than a factor of a few. Thus, over a broad range of parameter space EKM oscillations have similar timescales. The existence of these singularities, however, does imply that tEKM has some dependence on, e.g., the initial inclination as was found by Teyssandier et al. (2013).

2.6.4. The dependence on ǫoct

If the constants φq and χ are held fixed and ǫoct is varied, how does the period of 1 EKM oscillations vary? Equation (2.61) exhibits a ǫoct− dependence in the coefficient

28 Fig. 2.3.— Parameter space where EKM oscillations with flips are possible for two choices of ǫoct. We only explore the parameter space where individual KL cycles are rotating instead of librating (i.e., CKL > 0), as librating cycles cannot be correctly analyzed using this technique of averaging over individual KL oscillations. At smaller values of ǫoct the area of parameter space where rotating flips are possible shrinks about the line χ = F (φ).

29 Fig. 2.4.— The period of EKM oscillations with flips as a function of φq for three choices of χ. The solid line is given by the choice χ = F (φq), the dashed line by χ = F (φ ) ǫ /2, and the dotted line by χ = F (φ )+ ǫ /2. Except very near q − oct q oct the two singularities, the period of EKM oscillations does not vary by more than a factor of a few. Over a broad range of parameter space EKM oscillations have similar timescales.

30 before the integral, so it is tempting to conclude that the timescale for EKM 1 oscillations scales as ǫoct− . This conclusion has been asserted in several studies in the literature, but we show here that it is incorrect. The integral in equation (2.61) in 1/2 fact exhibits a ǫoct− dependence.

To determine this dependence we first note that for EKM oscillations to occur, in general C 1. This then implies that x is very close to unity, so we may write KL ≪ x = 1 ε, where ε 1. For values of x very close to unity, the complete elliptic − ≪ integral of the first kind may be approximated 1 K(1 ε) ln ε, (2.69) − ≃ −2 and the complete elliptic integral of the second kind is approximated by E(1 ε) 1. − ≃ We note that the coefficient in equation (2.69) is off by several tens of percent for realistic values of ε, but the important feature of this approximation is that it carries the correct dependence on ε. The function F (CKL) can then be approximated as

8 3 ε 1 F (C ) ln (ε′) + 2 dε′. (2.70) KL ≃ −5π 5 2 r Z0 µ ¶ Now, because we are integrating over a small range, the integral can then be approximated as

8 3 1 ε F (C ) ln ′ + 2 ε (2.71) KL ≃ −5π 5 2 2 r µ µ ¶ ¶ Furthermore, we note that 2 ε C (2.72) ≃ 3 KL so we finally have

16 3 1 C F (C ) C ln KL + 2 . (2.73) KL ≃ −15π 5 KL 2 3 r µ µ ¶ ¶

Let us now consider the lower limit of the integral in equation (2.61). For simplicity, let us for the time being restrict ourselves to the locus χ = F (φq) since here flips occur for arbitrarily small values of ǫoct. We then have

F (C )= F (φ ) ǫ . (2.74) KL,min q − oct 31 Now, the approximation in equation (2.73) may be written more simply as F (C ) kC , where k is a parameter that has only a sub-linear dependence on KL ∼ KL CKL. For small CKL, then, the function F is nearly linear in CKL. This then implies that for points on the locus we are considering ǫ φ C oct . (2.75) q − KL,min ∼ k

This then means that the width over which we are integrating, ∆CKL is proportional to ǫoct since

∆C C C = φ C ǫ . (2.76) KL ≡ KL,max − KL,min q − KL,min ∼ oct

Let us now consider the various terms of the integrand of equation (2.61). We have already seen that because x is close to unity, K(x) ln(C /3). This term ∼ KL is sublinear so we ignore it. The (4 11C ) term reduces to 4, and similarly the − KL √6 + 4C term reduces to √6. The term 2(φ C ) reduces by equation (2.75) KL q − KL to √2ǫ . This leaves only the sin Ω term. Now, if φ C ǫ and F is ∼ oct e p q − KL ∼ oct approximately linear in this limit, it must be the case that

F (φ ) C = χ C ǫ . (2.77) q − KL − KL ∼ oct

Comparing this to equation (2.60), we find that to lowest order, sin Ωe does not exhibit any dependence on ǫoct. It is straightforward to verify this claim numerically.

Putting these results together, we find that the only dependencies on ǫoct in the integral come from the width of integration (which yields a dependence of ǫoct), 1/2 and from the term 1/ 2(φ C ) (which yields a dependence of ǫ− ). Since the q − KL oct 1 integral has a coefficientp of ǫoct− , this then implies that the overall dependence of the period of the EKM is 1 τEKM . (2.78) ∼ √ǫoct

We demonstrate this dependence explicitly in Fig. 2.5 by numerically calculating the period using equation (2.61) for fixed values of φq and χ but over a range of ǫoct. We have compared these values with the periods obtained by integrating the secular equations of motion directly and find excellent agreement.

32 Fig. 2.5.— The period of the EKM relative to the period of KL oscillations as a function of ǫoct calculated analytically using equation (2.61) (lines) and by integrating the secular equations of motion (points). The timescale for the EKM is almost exactly 1/2 proportional to ǫoct− and there is excellent agreement between the secular and analytic calculations. We show this relationship for an arbitrary choice of φq = 0.015 and two 4 choices of χ = F (φ ) (black line and points), and χ = F (φ ) + 9 10− (gray dotted q q × line and points). For χ = F (φq) flips are possible at arbitrarily small values of ǫoct, 4 4 whereas for χ = F (φ ) + 9 10− flips are only possible for values of ǫ > 9 10− . q × oct × The relationship between tEKM and ǫoct becomes slightly shallower near this critical value of ǫoct. Note that ǫoct cannot exceed χ. Although flips occur at larger values of

ǫoct, the evolution is no longer integrable because the inner binary switches between rotation and libration. In this regime the timescale for flips steepens as a function of

ǫoct, although there is no longer a simple relationship between the two because the evolution becomes essentially chaotic.

33 By combining this result with the numerical coefficient of equation (2.61) we find that 256√10 tEKM tsec. (2.79) ∼ 15π√ǫoct During a single EKM cycle the inner binary will undergo two flips, so the flip timescale is half this value. The flip timescale can then be obtained by substituting for tsec, 128a3 10 m (1 e2)3 t 2 1 − 2 . (2.80) flip ∼ 3/2 ǫ m Gm 15πa1 s oct µ 3 ¶ 3 Over most of parameter space this expression is valid to within a factor of a few. For extremely large values of ǫ (ǫ 0.1) our numerical experiments demonstrate oct oct ∼ that the dependence of tEKM on ǫoct steepens and this expression overpredicts the timescale for flips, but in this limit non-secular effects become important, so the above analysis does not apply (e.g., Antonini & Perets 2012; Katz & Dong 2012; Seto 2013; Antonini et al. 2014; Bode & Wegg 2014; Antognini et al. 2014). Moreover, the above analysis also requires individual KL oscillations to be short relative to the EKM cycle. If this is not the case, then resonances between the quadrupole and octupole order terms can induce chaotic variation of the orbital eccentricity (Li et al. 2014a).

2.7. Conclusions

Using action angle variables we have derived the period of KL oscillations at quadrupole order and in the test particle limit (equation 2.27). From the exact period we have derived the timescale for KL oscillations. We have explored the full range of parameter space over which KL oscillations are possible and found that except very near the boundary between rotation and libration( C 1) the period | KL| ≪ of KL oscillations does not vary by more than a factor of a few from the derived timescale (Fig. 2.1). By employing several approximations in the high-inclination, low initial eccentricity limit we have found a function that matches the true KL period to within 2% for triples for which e1 < 0.3 and i > 60◦ (equation 2.48). ∼ ∼ The strength of KL oscillations varies due to the octupole term of the Hamiltonian. We average over individual KL cycles to calculate the period of EKM

34 oscillations, and hence, the timescale for spin flips to occur. We map the parameter space over which spin flips occur (Fig. 2.3) and show that apart from near two singularities where spin flips do not occur, the timescale for EKM oscillations does not vary by more than a factor of a few (Fig. 2.4). Finally, we show numerically and analytically that the dependence of ǫoct on the timescale for EKM oscillations 1/2 is ǫoct− (Fig. 2.5) in contrast to previous studies. We provide the EKM timescale in equation (2.79) and the timescale for flips in equation (2.80).

35 Chapter 3: Rapid Eccentricity Oscillations in Hierarchical Triples

3.1. Introduction

It is becoming increasingly evident that the secular approximation can fail in certain circumstances. Antonini & Perets (2012) found that in extreme-mass-ratio systems, eccentricities change rapidly compared to the period of the tertiary if the tertiary is in an eccentric orbit (this behavior can also be seen in Antonini et al. 2010). Bode & Wegg (2014) found that in a more general set of systems, the eccentricity of the inner binary varies on the timescale of the orbit of the tertiary. Recently, Katz & Dong (2012) found that these rapid variations can lead to collisions of WD-WD binaries if the tertiary is at very high inclination.1 Finally, Seto (2013) examined the impact of these rapid fluctuations on gravitational wave astronomy.

In this chapter we revisit earlier calculations of the merger times of compact objects by Blaes et al. (2002) and Hoffman & Loeb (2007). We extend these works by directly integrating the equations of motion of the three-body system and including post-Newtonian (PN) force terms up to order 3.5 to account for GR effects. We show that motion of the tertiary on its orbit (even in relatively low eccentricity orbits) leads to rapid eccentricity oscillations (REOs) in the inner binary and we quantify the importance of these oscillations. Our goal is to better understand the effect of the eccentric Kozai-Lidov mechanism and non-secular effects on the merger time distribution and dynamics of compact object binaries. In systems with tertiaries in

1Katz & Dong (2012) distinguish between “head-on collisions,” in which two objects merge without substantial tidal interaction, and “collisions,” in which two objects merge with or without previous tidal interaction. We use “collision” to refer exclusively to mergers without tidal interaction. Any event in which the two objects undergo substantial tidal interaction before combining is termed a “merger” in this chapter.

36 low eccentricity orbits we find that the double-orbit-averaged secular approximation fails by predicting merger times many orders of magnitude longer than those of the direct three-body integration.

This chapter is structured as follows. In 3.2 we describe our numerical § methods and characterize the accuracy of our integration. In 3.3 we describe the § breakdown of the secular approximation in calculating the eccentricity of the inner binary. In 3.4 we demonstrate one regime in which this breakdown of the secular § approximation leads to catastrophic failure, namely in predicting the merger times of compact objects. We conclude and discuss a number of applications in 3.5. § 3.2. Numerical methods & Setup

We numerically evolve triple systems with the open source Fewbody suite (Fregeau et al. 2004). Fewbody is designed to compute the dynamics of hierarchical systems of small numbers of objects (N < 10) either in scattering experiments or in bound ∼ systems. The underlying integrator for the Fewbody suite is the GNU Scientific Library ordinary differential equations library (Gough 2009). By default Fewbody uses eighth-order Runge-Kutta Prince-Dormand integration with adaptive time steps. It is straightforward to modify Fewbody to use any of the other roughly half-dozen integration algorithms supported by GSL.2 In our experience the choice of integration algorithm does not affect the results since the adaptive steps force the size of the error to be within the same target value regardless of the algorithm used. All results in this chapter were obtained using the default eighth-order Runge-Kutta Prince-Dormand algorithm. We have extended Fewbody to include post-Newtonian (PN) force terms up to order 3.5, presently the state of the art. Due to their length, we do not reproduce the terms here (the third-order term alone spans more than a page), but instead refer the reader to Equations 182, 183, 185, and 186 of Blanchet (2006). These terms are conjectured to accurately reproduce general relativistic effects to within several Schwarzschild radii. Though analytic error estimates do not exist in the literature, comparisons of PN calculations with direct integration of the

2GSL also supports an additional five integration algorithms, but these require the calculation of the Jacobian. When post-Newtonian terms are included this becomes nontrivial to implement.

37 Einstein field equations support the consensus that the PN terms are effective to within this range (Will 2011).

The PN terms are stronger functions of velocity and radius than the Newtonian term. Consequently, the inclusion of the PN terms makes integration of the orbits much more difficult, particularly for highly eccentric orbits where the radial distance and velocity are both changing very rapidly. Thus, while the PN terms might in principle be accurate down to several Schwarzschild radii, in practice the efficient computation of the orbit may limit the regime of applicability.

These difficulties are compounded by the roundoff error introduced by integrating close encounters far from the origin (see, e.g., Mikkola & Merritt 2008, for further discussion). If the positions of two nearby objects are represented with respect to a distant origin, the numerical precision is reduced by roughly the ratio of the distance of the two objects from the origin to their separation. (For example, if a computer has only four digits of precision and two objects are 3 separated by 1.234 10− and are a distance of 1 from the origin, their positions × must be represented by 1.001, leading to a loss of three digits.) In practice this can lead to a loss of six or seven orders of magnitude of precision and can render the evolution of high eccentricity systems intractable. In general, roundoff error in N-body dynamical simulations can be avoided with some variation of algorithmic regularization (see, e.g., Mikkola & Tanikawa 1999; Aarseth 2003). However, because we are only concerned with the special case of hierarchical triple systems, we have modified Fewbody to avoid roundoff error by automatically recentering the triple on every step so that the center of mass of the inner binary is placed at the origin.

To test the correctness of our implementation and to characterize its regime of accuracy we run several numerical tests. We first examine the degree of energy conservation in orbits at several eccentricities when no PN terms are included and when non-radiative PN terms are included. We then show that the orbital decay due to the 2.5 order PN term closely matches analytic calculations. Finally we compare the results from Fewbody to an octupole-order secular model in several simple cases to demonstrate that systems in which the approximations of the secular model are valid produce similar behavior as direct three-body integration.

38 3.2.1. Energy conservation

The most straightforward way to determine the numerical accuracy of an N-body integrator is to determine how well energy is conserved. Once the change in energy becomes non-negligible compared to the total energy it is certain that the calculated dynamics are qualitatively incorrect. Typical energy conservation tolerances are set at least several orders of magnitude below this point. The largest tolerance often 5 invoked is of order 10− .

In ordinary integration (i.e., without taking a Kustannheimo-Stiefel or similar transformation), energy conservation is worst during close encounters of extremely eccentric orbits. Due to the steep 1/r2 profile of the gravitational force and the rapid change in r near the periapsis of a highly eccentric orbit, such orbits are difficult to calculate accurately. This problem is exacerbated with the introduction of PN terms since the PN terms are even stronger functions of distance and include strong velocity terms which vary rapidly as well. For an orbit with a given semi-major axis, there is thus a maximum eccentricity to which we can accurately integrate.

To estimate Fewbody’s numerical accuracy and determine this maximum eccentricity, we integrate 3 106 orbits (approximately one Hubble time) of two 107 × M SMBHs with a semi-major axis of 1 pc and eccentricities ranging from 1 e = 1 ⊙ − 5 to 10− . This system is the inner binary of the systems we integrate in 3.4. Since § the triple systems we later integrate consist of a tertiary with a mass of 107 M at ⊙ a distance of 20 pc, we offset the binary in these energy tests to a distance of 10 pc so as to account for roundoff error that will be present when we integrate the triple systems. This initial offset has only a negligible effect on the calculation, however, because our code automatically recenters the triple system on the center of mass of the inner binary on every step so as to eliminate this roundoff error.

We perform these calculations both with and without the PN terms. In calculations with the PN terms we exclude the odd-order 2.5 and 3.5 terms since these serve to describe the effects of gravitational radiation. These force terms are not conservative and are therefore not appropriate in our checks for energy conservation. The accuracy of these odd terms is characterized in 3.2.2. We further §

39 note that when PN force terms are included in the integration, the expression for the energy changes. The energy term including PN terms up to third-order is lengthy, so like the force terms, we do not reproduce it here, but instead refer the reader to Equation 2.11 of Mora & Will (2004). (The energy term is also provided in Equation 170 of Blanchet 2006, but is represented in a different gauge that contains an undesirable logarithm.)

3 For the low-eccentricity systems (1 e 10− ), gravitational radiation is weak − ≥ and so we integrate them for a Hubble time. The high-eccentricity systems will merge in under a Hubble time, however, so we simply integrate them for as long as it takes them to merge via gravitational radiation. The results of these integrations are presented in Figure 3.1. The results from including the PN 1 term alone and from including just the PN 1 and PN 2 terms are very close to the results from including all three PN terms. They are therefore not displayed in Figure 3.1. As expected, Fewbody conserves energy best at low eccentricities. At high eccentricities energy conservation is also quite good because only a small number of orbits need to be integrated. At all eccentricities, however, Fewbody performs the integration for 5 the necessary number of orbits and conserves energy to well under 10− (and often much better).

3.2.2. Inspiral time

To test the accuracy of the radiation reaction terms we compare the orbital decay of a highly eccentric orbit to the analytic expressions of Peters (1964). From Peters (1964), the semi-major axis and eccentricity evolution of the orbit are described by the following two differential equations:

da 64 G3m m (m + m ) 73 37 = 1 2 1 2 1+ e2 + e4 , dt − 5 c5a3(1 e2)7/2 24 96 ¿ À − µ ¶ de 304 G3m m (m + m ) 121 = e 1 2 1 2 1+ e2 . dt − 15 c5a4(1 e2)5/2 304 ¿ À − µ ¶ The orbital decay time of the system is given by the integral

12 c4 e0 e29/19 [1 + (121/304)e2]1181/2299 T (a ,e )= 0 de, 0 0 19 β (1 e2)3/2 Z0 − 40 Fig. 3.1.— Energy conservation in Fewbody for orbits over a range of eccentricities in the Newtonian case (red squares) and including non-radiative post-Newtonian terms up to order 3 (black dots). We integrate the orbits for a Hubble time or for the gravitational radiation inspiral time, whichever is less. In all cases energy 5 conservation is better than 10− .

41 where

64 G3m m (m + m ) β = 1 2 1 2 , 5 c5 and

870 a (1 e2) 121 − 2299 c = 0 − 0 1+ e2 . 0 e12/19 304 µ ¶

To compare Fewbody to the analytic results, we calculate the evolution of the orbital parameters of the orbit of two 107 M SMBHs with a semi-major axis ⊙ 4 of one parsec and an initial eccentricity of 1 e = 10− . For the purposes of this − comparison, we perform the Fewbody calculation with the 2.5 PN term alone. This is because Peters (1964) assumes that the orbits are Keplerian and calculates the gravitational wave power in the quadrupole approximation. For very eccentric orbits, the deviation from a perfect ellipse manifests itself as a longer dwelling time at periapsis. Since most of the gravitational radiation is emitted near periapsis, a fully relativistic orbit results in more radiation emitted than Peters (1964) predicts. As the 2.5 PN term is the only term that captures quadrupole radiation emission, this is the only term consistent with the assumptions of Peters (1964).

We find that the difference in the overall merger time between Fewbody 3 and Peters (1964) is less than 10− . We believe this discrepancy is due to the fact that Fewbody treats the energy loss more realistically by emitting most of the orbital energy during passage through periapsis. Peters (1964), however, assumes that energy loss is continuous throughout the orbit. For very eccentric orbits like the ones we are modelling, Fewbody’s treatment leads to stepwise changes in the orbital parameters, whereas Peters (1964) assumes that these orbital parameters vary continuously across the entire orbit. Over many orbits, this difference manifests itself in small discrepancies in the orbital parameters between the two calculations.

During most of the orbit the semi-major axis calculated by Fewbody is within 2 10− of the semi-major axis predicted by Peters (1964). At the end of the inspiral the discrepancy is much worse, but this is simply because the overall merger time of the orbit is slightly different between Fewbody and Peters (1964). Although

42 we therefore cannot adequately test Fewbody in this regime, however, we are not interested in the precise dynamics prior to merger, only the overall merger time.

The orbital decay including only the 2.5 PN term is presented in Figure 3.2. The effect of adding the additional PN terms is a 0.5% change in the overall merger time. At higher eccentricities the other PN terms become stronger and yield even larger discrepancies.

3.2.3. Comparison to the secular approximation

If any changes to the orbital parameters in a hierarchical triple are slow compared to the outer orbital period, the orbits need not be integrated directly, but instead can be calculated from a time-averaged Hamiltonian (e.g., Blaes et al. 2002). Although we show in this chapter that this approximation breaks down in important regions of parameter space, there are broad regimes in which this approximation works well. In particular, the secular approximation works very well when the Kozai-Lidov mechanism does not excite extremely high eccentricities.

We here show the agreement between the orbital evolution in the secular approximation with the direct three-body integration. For the secular approximation we use the formalism of Blaes et al. (2002), which is an octupole-order calculation that includes general relativistic precession and gravitational radiation. (Note that Blaes et al. 2002 use the equations of general relativistic orbital decay from Peters 1964. As discussed in 3.2.2, this slightly underpredicts the merger time for highly § eccentric orbits.) The orbital evolution is compared to the explicit orbit integration using Fewbody for a slowly-varying hierarchical triple undergoing Kozai-Lidov oscillations. Because the Hamiltonian in Blaes et al. (2002) uses the results from Peters (1964) to account for gravitational radiation, the only radiation term we include is PN 2.5. Similarly, the formalism for handling apsidal precession in Blaes et al. (2002) is equivalent to the first PN term. Thus the only PN terms we include in this comparison are PN 1 and PN 2.5.

We calculate the evolution of a triple system in which the Kozai-Lidov mechanism is present, but does not excite extremely high eccentricities. Properties

43 Fig. 3.2.— The evolution of the semi-major axis of a binary of two 107 M SMBHs ⊙ 3 with an initial semi-major axis of 1 pc and an initial eccentricity of 1 e 10− . − ∼ For purposes of comparison with Peters (1964), the calculation shown includes only the 2.5 PN term. The difference between our results and Peters (1964) is much less than the thickness of the line. We also calculate the evolution using all PN terms up to and including PN 3.5. In this experiment the difference in the merger time between this calculation and the full PN calculation is 0.5%. For clarity we omit displaying the evolution in the full PN approximation. At higher initial eccentricities the discrepancy between the full PN calculation and the 2.5 PN term alone is larger. The inner binaries that we examine in this chapter begin to suffer copious energy loss 4 due to gravitational radiation at an eccentricity of 1 e 10− . At this eccentricity − ∼ the difference between the full PN calculation and the 2.5 PN approximation in Peters (1964) is 5%. ∼

44 of this system are listed in Table 3.1. We evolve the system for 1010 yr, or about 15.5 Kozai-Lidov cycles. The eccentricity evolution of both calculations are presented in Figure 3.3. The difference between the two systems is equivalent to a 0.1% scaling in time. This small difference is due to the fact that Fewbody begins the integration with each object at a random point along its orbit and at random longitudes of ascending node. These different starting conditions yield slightly different orbits. The initial phases of the orbits and longitudes of ascending node do not impact the orbit-averaged evolution of Blaes et al. (2002). The variation due to these random initial conditions from one realization to the next is consistent with the difference between any particular realization and the orbit-averaged calculation.

To more clearly illustrate the differences between the secular and Fewbody calculations, we run the Fewbody calculation 100 times with random initial mean anomalies for each run. The variation in the evolution of the eccentricity of the inner orbit is shown in Figure 3.4. The magnitude of this variation is a 0.15% ∼ scaling in the time, which amounts to an offset of 15 Myr after 1010 yr. The ∼ evolution predicted by the secular calculation is consistent with the range calculated by Fewbody.

Throughout this chapter we use m1 and m2 to refer to the masses of the objects in the inner binary, and m3 to refer to the mass of the tertiary. For other quantities the subscript ‘1’ refers to the inner binary and the subscript ‘2’ refers to the outer binary. The semi-major axis is represented by a, the eccentricity by e, the argument of periapsis by g, and the mutual inclination by i.

3.3. Rapid eccentricity oscillations

Most studies of three-body dynamics have employed the secular approximation in which any changes to the orbital parameters of either orbit are assumed to be slow compared to the orbital period of both orbits. Such models cannot account for any changes that occur on more rapid timescales, and it is implicitly assumed that if such changes do occur, their effect would be negligible.

45 Fig. 3.3.— The evolution in eccentricity of a system undergoing Kozai-Lidov oscillations. This system does not exhibit oscillations to extremely high eccentricities, so it is in the regime in which the secular approximation is valid. Properties of this system are listed in Table 3.1. We evolved this system for 1010 years using both the secular model of Blaes et al. (2002) and by performing the direct three-body integration using Fewbody. The difference between the two techniques is much less than the thickness of the line and amounts to a 0.1% offset in time at the end of ∼ the calculation. The difference between the calculation in the secular approximation and the direct three-body integration is explored further in Figure 3.4.

46 Fig. 3.4.— The final moments of the evolution in eccentricity of the system presented in Figure 3.3. Properties of this system are listed in Table 3.1. The secular calculation (dotted line) is shown with the results of 100 runs using Fewbody (gray region). The orbits of the triple system in each run were given random initial mean anomalies. The variation in the orbital evolution due to these random initial mean anomalies results in a 0.15% offset in time. The difference between the secular calculation and any ∼ given calculation using Fewbody is consistent with this variation.

47 We find that over a broad region of parameter space, the inner binaries in triple systems undergo oscillations in eccentricity (or, equivalently, angular momentum) on the timescale of the outer orbital period (“rapid eccentricity oscillations,” REOs). REOs are typically small and do not affect the dynamics of the triple system for almost all of its evolution. But when the inner binary is already at high eccentricity, as during an eccentric Kozai cycle, the magnitude of the oscillations in angular momentum becomes comparable to the total angular momentum of the inner binary. REOs can then drive the inner binary to rapid merger.

The existence of REOs was predicted by Ivanov et al. (2005), who found that the amplitude of the change in angular momentum during an oscillation is 2 ∆L 15 m3 a1 = cos imin Gm3a2, (3.1) µ 4 m1 + m2 a2 µ ¶ p where µ is the reduced mass of the inner binary and imin is the minimum mutual inclination between the two orbits during a Kozai-Lidov cycle.3 Equation 3.1 can also be written as a change in eccentricity, although this form of the equation is somewhat more cumbersome:

3 3 2 15 m 2 a 2 ∆e = e + 1 1 e2 + 3 cos i 1 . (3.2) 1 1 v 1 min − u − " − 4 m1 + m2 a2 # u q µ ¶ µ ¶ These equations aret only accurate near the eccentricity maximum of a Kozai-Lidov cycle.

Our numerical experiments are in agreement with Ivanov et al. (2005). We show in Figure 3.5 the evolution of two example systems exhibiting REOs (see Table 3.2). The two systems are identical except that the system in the left panel begins with g = g = 0◦ and the right panel begins with g g = 90◦. To demonstrate 1 2 1 − 2 that REOs are a non-relativistic phenomenon, we have suppressed PN terms in this figure.

Intuitively, REOs can be understood as similar to a Kozai-Lidov oscillation in miniature. In the double-orbit-averaged approximation, the Kozai-Lidov mechanism 3See Appendix B of Ivanov et al. 2005 for the complete derivation. Note that in Ivanov et al. 2005, ∆L refers to the change in the specific angular momentum from the average value to the maximum value. This quantity therefore differs from our ∆L by a factor of µ/2.

48 Parameter Value 7 m1 10 M ⊙ 7 m2 10 M ⊙ 6 m3 3 10 M × ⊙ a1 1 pc

a2 20 pc

e1 0.1

e2 0.2

g1 0

g2 0 cos i 0.5

Table 3.1: Initial conditions for a system that undergoes weak Kozai-Lidov oscillations. See Figure 3.3 for the evolution of this system.

m1 m2 m3 a1 a2 e1 e2 g1 g2 i 7 5 7 10 M 10 M 10 M 1 pc 20 pc 0.1 0.1–0.8 0–360◦ 0–360◦ 80◦ ⊙ ⊙ ⊙ Table 3.2: Initial conditions for triple systems studied in this chapter. Throughout this chapter g refers to the argument of periapsis and i refers to the mutual inclination.

49 0.1 N-body 0.1

Secular Predicted envelope 1 1

e 0.05 e 0.05 - - 1 1

0.02 0.02

6.´107 6.5 ´107 7.´107 6.5 ´107 7.´107 7.5 ´107 t HyrL t HyrL

Fig. 3.5.— Systems exhibiting REOs. The eccentricity of the inner binary during a Kozai-Lidov cycle as calculated by direct three-body integration (black solid line) and as calculated in the secular approximation (red dashed line) for g1 = g2 = 0◦

(left panel) and g g = 90◦ (right panel). PN terms are not included. The 1 − 2 secular and three-body calculations match on average in the left panel, but the three-body calculation exhibits oscillations in e1. In the right panel, the secular calculation correctly predicts the minimum eccentricity, but the REOs in the three- body calculation push the binary exclusively to higher eccentricities. Blue dotted lines show the amplitude of the REOs predicted by Equation 3.1. The period of the REOs is twice the period of the outer binary. The asymmetry in the period of the oscillations is due to fact that the tertiary is on an eccentric orbit (e2 = 0.2). The initial conditions of the system are presented in Table 3.2 but with g1 and g2 fixed as stated above.

50 occurs due to the fact that the outer orbit exerts a stronger force on the inner orbit at the line of nodes than at other regions of the outer orbit. But because in reality the outer orbit is a point mass in motion rather than a continuous hoop of matter, this force is strongest along the line of nodes as the tertiary is actually passing through the line of nodes. For weak Kozai-Lidov oscillations, the driving force contributed during any single orbit is small, so there is only a gradual change in the eccentricity of the inner orbit and any rapid eccentricity oscillations are negligible. However, during a strong eccentricity maximum, the inner orbit has lost nearly all of its angular momentum and is therefore extremely sensitive to torquing.

This implies that the arguments of periapsis of the inner and outer orbits determine the direction of the oscillation. If the apsides are aligned with the line of nodes (as in the left panel of Figure 3.5), the eccentricity will be driven to higher values relative to the secular calculation when the tertiary passes through periapsis and to lower values when the tertiary passes through apoapsis. If the apsides are 90◦ from the line of nodes, however, the eccentricity will be exclusively driven to higher values relative to the secular calculation while the amplitude of the oscillations will remain fixed (as in the right panel of Figure 3.5).

Although oscillations in the orbital elements on the timescale of the period of the outer orbit were first predicted by Soderhjelm (1975), an explicit formula for the change in angular momentum was first derived by Ivanov et al. (2005). Moreover, these oscillations were not confirmed by three-body integrations until Bode & Wegg (2014), who found them in the test-particle limit, and Antonini & Perets (2012), who found them in the equal-mass case. Katz & Dong (2012) further explored these oscillations in the context of WD-WD collisions. They argued that these oscillations are fundamentally a stochastic phenomenon, but only examined systems in which the inclination of the tertiary was near the Kozai “pole” of i 93◦, where certain terms ∼ in the Hamiltonian formally diverge at quadrupole order and Kozai-Lidov oscillations become extremely strong (Miller & Hamilton 2002). Although a complete analytic treatment of REOs is beyond the scope of this chapter, our results suggest that at lower inclinations they could be modelled analytically. As we discuss in Section 3.4.2, we only examine REOs in inner binaries on prograde orbits.

51 3.4. Effect on merger times

REOs are important when 1 e 0 and nearly all of the angular momentum in − 1 ∼ the inner orbit has been transferred to the outer orbit. Here, fluctuations in the angular momentum given by Equation 3.1 become comparable to the total angular momentum in the inner orbit itself. REOs then can have a substantial impact on the long-term dynamics. This is especially true if relativistic effects are important because the PN terms are strong functions of distance and thus a small change in the distance at periapsis dramatically changes their strength. A secular calculation does not account for these effects and will overpredict the merger time, in some cases by many orders of magnitude. There are therefore certain regions of parameter space in which the double-orbit-averaging approximation fails.

3.4.1. Test case

The importance of REOs for the long-term evolution of an example system is illustrated in Figure 3.6. The secular calculation (blue dashed line) closely matches the three-body integration performed by Fewbody (black line) during the first Kozai-Lidov cycle, but afterwards they begin to diverge. In this particular case, the eccentricity of the inner binary in the three-body integration increases to 4 3 1 e < 10− , whereas in the secular calculation it only reaches 1 e 10− . As a − 1 − 1 ∼ consequence, the three-body integration predicts the inner binary to merge within one eccentric-Kozai-Lidov timescale whereas the secular calculation predicts that the inner binary will effectively never merge. To demonstrate the importance of resonant post-Newtonian eccentricity excitation (Naoz et al. 2013b) we additionally show the same three-body calculation without any PN terms (blue dotted line). We find that without relativistic effects the inner binary gets excited to much lower eccentricities, in agreement with Naoz et al. (2013b).

During the eccentric phase of the Kozai-Lidov cycles the GR precession timescale, tGR, shortens since e1 approaches unity (e.g., Blaes et al. 2002):

3/2 5/2 6 m1 + m2 − a1 2 tGR 2.3 10 yr 6 2 1 e1 . (3.3) ∼ × 2 10 M 10− pc − µ × ⊙ ¶ µ ¶ ¡ ¢ 52 1.

10-1

10-2 1 e - 1 10-3

N-body Secular + GR 10-4 Secular + No GR

10-5 0. 1.´109 2.´109 3.´109 t HyrL

Fig. 3.6.— The impact of REOs on the evolution of the inner binary of a hierarchical triple. We show the direct three-body integration (solid black line) and the calculation in the secular approximation (red dashed line). To illustrate the importance of relativistic terms at high eccentricities we also show the direct three-body integration without any PN terms (blue dotted line). Although the direct integration matches closely with the two secular calculations during the first Kozai-Lidov cycle, the calculations diverge thereafter. The secular calculation predicts that the system only 3 reaches a maximum eccentricity of 1 e 10− in the time period shown, whereas − 1 ∼ the direct integration predicts that the inner binary is driven to sufficiently high eccentricities to merge after 2 109 yr. ∼ ×

53 If the eccentricity becomes sufficiently large, as in the final Kozai-Lidov cycles of the system presented in Figure 3.6, tGR can become shorter than the Kozai-Lidov timescale, tKL. This will ordinarily not suppress the Kozai-Lidov mechanism because at high eccentricity the inner binary requires only very small torques to change its eccentricity. Bode & Wegg (2014) therefore introduce the instantaneous

Kozai-Lidov timescale, tKL,inst as the timescale for the inner binary to change its angular momentum by order unity. tKL,inst is related to tKL by

t 1 e2 t (3.4) KL,inst ∼ − 1 KL q up to constant factors of order unity. If tKL,inst exceeds tGR the Kozai-Lidov cycles are “detuned,” and the Kozai-Lidov mechanism will be suppressed (Holman et al. 1997). This kind of detuning occurs in the regime in which the secular approximation is valid. The detuning which occurs in the system presented in 3.6 does not occur in this regime, however. In this system tGR becomes shorter than P2 before it becomes shorter than tKL,inst. When this occurs, it is impossible for the outer binary to exert any secular influence on the inner binary. At this point, the inner binary decouples from the outer binary and gravitational radiation drives the inner binary to rapid merger because the inner binary is at high eccentricity. Hence the Kozai-Lidov mechanism is detuned as a consequence of the breakdown of the secular approximation. To emphasize that tKL is not relevant for determining if Kozai-Lidov cycles will be detuned in the middle of a Kozai-Lidov cycle, we show in the left panel of Figure 3.7 the ratio between tKL and tGR for the system presented in Figure 3.6.

During the final Kozai-Lidov cycles tGR becomes much shorter than tKL, but is driven back to longer timescales by REOs before the inner binary can merge by gravitational radiation. The right panel shows the ratio between P2 and tGR for the same system. Once this ratio reaches unity, the inner binary decouples from the outer binary and merges via gravitational radiation.

We emphasize that REOs are only important when the eccentricity is large. REOs therefore only affect the dynamics during a small fraction of the system’s lifetime. We illustrate in Figure 3.8 the fraction of time that the system spends at high eccentricity. The fluctuations in the angular momentum of the inner binary become 10% of the total angular momentum of the inner binary when the

54 Fig. 3.7.— Timescales of the system presented in Figure 3.6 calculated with direct integration (black line) and in the secular approximation (dotted red line). Left panel. The ratio between the Kozai-Lidov timescale and the GR precession timescale (Eq. 3.3) of the inner binary. During the final several Kozai-Lidov cycles the GR precession timescale becomes shorter than the Kozai-Lidov timescale. Although the Kozai-Lidov mechanism is detuned, REOs are sufficient to restore the system to lower eccentricities and continue the Kozai-Lidov cycle. Right panel. The ratio between the outer period and the GR precession timescale. Once the GR precession timescale of the inner binary becomes comparable to the outer orbital period, the two orbits completely decouple and gravitational radiation drives the inner binary to merge.

55 inner binary reaches an eccentricity of 0.9. From Figure 3.8, REOs are therefore ∼ non-negligible for only 10% of the system’s lifetime. The secular approximation ∼ is therefore valid 90% of the time. Nevertheless, as illustrated in Figure 3.6 and ∼ in the next subsection, these short periods in which the secular approximation fails dramatically influence the evolution of the system and lead to a sharp divergence from the secular predictions because of the strong eccentricity dependence of the GR terms.

3.4.2. Population study

Here we compare the merger times of a variety systems calculated in both the secular approximation and in the full three-body integration. We fix the masses of the 7 5 7 SMBHs (m1 = 10 M , m2 = 10 M , m3 = 10 M ), the semi-major axes (a1 = 1 ⊙ ⊙ ⊙ pc, a2 = 20 pc), the initial eccentricity of the inner binary (e1 = 0.1), the inclination of the tertiary (i = 80◦), and the arguments of periapsis (g1 = 0◦, g2 = 90◦). The initial mean anomalies are chosen randomly. The eccentricity of the tertiary, e2, is systematically varied from e2 = 0.1 to 0.85 in steps of 0.001. (Systems at e2 > 0.85 ∼ are unstable and few systems with e2 < 0.1 ever merge.) Note that we do not ∼ choose our masses to model any specific physical system (the REO phenomenon is not specific to any particular mass range), but instead choose them for ease of comparison to Blaes et al. (2002), and because we wish to study this phenomenon in the test-particle case.

At each choice of e2 we calculate the merger time. We define a merger as occurring when the two components of the inner binary come within 10 Schwarzschild radii (RSch) of each other (where RSch hereafter refers to the Schwarzschild radius of the larger BH). We are forced to integrate only to 10 RSch rather than down to 1 or 2 RSch because the PN terms begin diverging when the relative velocity exceeds 0.2c (see Section 9.6 of Blanchet 2006). In the systems we examine the relative ∼ velocities start to become close to 0.2c when the two components come within 10 ∼

56 RSch of each other. In practice, when the inner objects come within 10 RSch of each other, the orbital decay timescale is short compared to the overall merger time.4

The results of these calculations appear in Figure 3.9. Because these orbits spend the vast majority of time in the Newtonian regime, they can be rescaled to other masses, distances, and times until the small fraction of time prior to merger that the eccentricity becomes large enough that relativistic effects become important. For this reason we run each calculation to completion even if the merger time exceeds a Hubble time for the particular case that we analyze.

There is substantial scatter in tmerge. This scatter is a result of the slightly different choices of e2 from point to point, but also the different mean anomalies. Two systems with identical starting conditions but different initial mean anomalies can have merger times that differ by up to two orders of magnitude. The most rapidly merging systems all merge within one eccentric Kozai-Lidov timescale. This timescale is given by Katz et al. (2011) and Naoz et al. (2013a) as

tKL tEKM , ∼ ǫoct where ǫoct is the strength of the octupole-order term in the expansion of the three-body Hamiltonian. The eccentric Kozai-Lidov timescale can be written as

1/2 3/2 m + m − a t 2.1 109 yr 1 2 1 EKM ∼ × 2 106 M 1 pc µ × ⊙ ¶ µ ¶ m + m a /a 4 (1 e2)5/2 1 2 2 1 − 2 . (3.5) × 2m 20 e µ 3 ¶ µ ¶ 2 This function matches the lower envelope of the merger time distribution very closely. Systems above this line fail to merge within a single eccentric Kozai-Lidov cycle, but merge after several. Usually, however, the first eccentric Kozai-Lidov cycle so disturbs the system that future eccentric Kozai-Lidov cycles operate on different timescales. This is primarily due to changes in the argument of periapsis of the inner binary. The merger times are therefore not integer multiples of the first eccentric Kozai-Lidov timescale. 4To verify this we reran the system displayed in Figure 3.6 and set the merger criterion to 100, 50, 20, 10, and 5 Schwarzschild radii. The overall merger times are all within 0.01% of each other. ∼ 57 Fig. 3.8.— The fraction of time that the system presented in Figure 3.6 spends at eccentricities greater than any given eccentricity in the direct integration (solid line) and in the secular approximation (red dashed line). For comparison we also present the line y = x and y = 3x (dotted lines). At low eccentricities, the fraction of time that a hierarchical triple undergoing Kozai-Lidov cycles spends at eccentricities greater than e is approximately 1 e . REOs drive the system to spend more time 1 − 1 at higher eccentricities. If eccentric Kozai-Lidov oscillations are also present, as in this case, the fraction of time spent at higher eccentricities is slightly larger, but is always within a factor of a few of 1 e . Although REOs are only important at high − 1 eccentricities, their effect during these brief periods drastically changes the overall evolution of the system.

58 Of the systems we study, approximately one-quarter merge within one eccentric Kozai-Lidov cycle. This is an overestimate of the true fraction of systems that merge within tEKM since we study only a small region of parameter space. Although our choice of inclination (i = 80◦) is not finely tuned, our choice of arguments of periapsis (g1 = 0◦,g2 = 90◦) is tuned to encourage strong eccentric KL resonances. We do this for two reasons. First, it is computationally expensive to integrate a sufficient number of systems to marginalize over the arguments of periapsis and obtain good statistics. Second, it demonstrates more clearly that the lower envelope of the tmerge distribution is set by tEKM because both Kozai-Lidov resonances and

REOs are stronger when the arguments of periapsis are different by 90◦.

To examine the effect of a more realistic distribution of initial arguments of periapsis on the merger time distribution, we pick two choices of e2 (0.2 and 0.6), and calculate the evolution of 100 systems with uniform distributions of g1 and g2. We compare this distribution of tmerge (black line) to the distribution when the arguments of periapsis are fixed to g1 = 0◦ and g2 = 90◦ (blue dashed line) in Figure 3.10. As expected, the distribution shifts to longer merger times when the arguments of periapsis are chosen randomly. Nevertheless, about 15% of systems ∼ with e = 0.2 and about 30% of systems with e = 0.6 merge within a few t . 2 ∼ 2 × EKM We additionally compare this result to that calculated in the secular approximation and find that it is shifted to much longer tmerge than either calculation performed using direct three-body integration.

Double-orbit averaging fails to correctly predict the merger times most drastically for triple systems in which the tertiary is at low eccentricity. At best the secular calculation overpredicts the merger time by two orders of magnitude, and at worst it overpredicts the merger time by nearly four. The catastrophic failure of double-orbit averaging is due to the fact that it cannot account for REOs. When the orbit of the tertiary has a low eccentricity, Kozai-Lidov resonances (including eccentric Kozai-Lidov resonances) are weakened. Consequently, when the outer orbit is at sufficiently low eccentricities, the Kozai-Lidov resonance is not strong enough to drive the inner binary to merger on its own. Kozai-Lidov resonances nevertheless drive the inner binary to sufficiently high eccentricities that REOs become important.

59 Fig. 3.9.— The time required for the inner binary of triple systems to merge as a function of the eccentricity of the orbit of the tertiary (see Table 3.2 for the system parameters) using direct three-body integrations (points) and using the secular approximation (red dashed line). The scatter in both the direct three-body integration and in the double-orbit averaged calculation is due to the fact that these systems are chaotic. Slight changes in e2 or the initial mean anomalies can change tmerge by over an order of magnitude. Approximately 25% of the systems we study merge in tEKM (Eq. 3.5, solid line). The shaded region depicts merger times within

1 2 tEKM. At e2 < 0.3 the eccentric Kozai-Lidov mechanism weakens and cannot − × ∼ drive systems to merger as shown by the large difference between the secular and three-body calculations. As a consequence, REOs become an important mechanism to drive binaries to merger. Because REOs are fundamentally non-secular, the secular calculations overpredict the merger times by many orders of magnitude at low e2.

60 REOs drive the inner binary to higher eccentricities, thereby causing relativistic effects to become much more important. In particular, gravitational wave radiation is much more efficient when the inner binary is at higher eccentricities.

3.5. Discussion and conclusions

We have performed an exploration of a dynamical effect in hierarchical triple systems that is not captured by secular double orbit averaging. By directly integrating the orbits of the three bodies and including post-Newtonian terms up to order 3.5, we show that the eccentricity of the inner binary oscillates on the timescale of the period of the outer binary with amplitude given by Eqs. 3.1 and 3.2 (see Figure 3.6 and Ivanov et al. 2005; Bode & Wegg 2014). During most of the evolution of the triple system these oscillations are negligible and secular calculations are valid. However, when the eccentricity of the inner binary is close to unity, fluctuations in the angular momentum of the inner binary due to REOs become comparable to the total angular momentum in the inner binary itself. This is because the system spends more time at higher e1 (see Figure 3.8). We find that the time spent at eccentricities greater than any given eccentricity e′ is few 1 e′ . As a consequence of this, 1 ∼ × − 1 REOs can substantially affect the dynamics of the triple system. Though we have limited our analysis in this chapter to triples of SMBHs for concreteness, our results apply generally to any triple system for which the inner binary consists of black holes or neutron stars. Because relativistic effects are extremely strong functions of eccentricity, REOs can drive binaries to merge more rapidly by many orders of magnitude. As this discrepancy occurs over a broad range of parameter space, REOs will drive many systems to merge which otherwise would not merge within a Hubble time. Though a complete treatment of the delay time distribution of compact object mergers across a broad range of parameter space is beyond the scope of this chapter, REOs may be an important correction to calculations of the merger rate and delay time distribution of compact object binaries in triple systems (e.g., Thompson 2011; Katz & Dong 2012) and perhaps systems like stars and planets that may be strongly affected by tides (see Figure 3.9 and Section 3.5.3 below).

61 e2 = 0.2 e2 = 0.6 80. 80.

N-body + random g2 - g1

Secular + random g2 - g1 60. 60. N-body + g2 - g1 = 90°

N 40. N 40.

20. 20.

0. 0. 109 1010 1011 1012 108 109 1010 1011 1012

tm erge HyrL tm erge HyrL

Fig. 3.10.— Merger time distribution for fixed e2. We compare the distribution when the arguments of periapsis are chosen randomly from a uniform distribution

(solid line) to when the arguments of periapsis are fixed at g1 = 0◦ and g2 = 90◦ calculated using direct three-body integration (blue dashed line) and in the secular approximation with a uniform distribution of arguments of periapsis (red dotted line) for 100 systems. Since Kozai-Lidov oscillations and REOs are stronger when the arguments of periapsis differ by 90◦, the merger time distribution shifts to longer merger times when the arguments of periapsis are chosen randomly. About 15 30% − of systems still merge rapidly when the arguments of periapsis are chosen randomly. The last bin of the secular calculation is a lower bound. The distribution is shifted to larger merger times when the secular approximation is employed.

62 3.5.1. Implications for extreme-mass-ratio inspirals

Since we have examined hierarchical triples with extreme mass ratios, a potential application of our results is to extreme-mass-ratio inspirals (EMRIs). EMRIs consist of a binary of stellar-mass objects in orbit around a SMBH (see, e.g., Amaro-Seoane et al. 2007; Amaro-Seoane 2012; Amaro-Seoane et al. 2013a). In such cases there is an extremely large mass ratio between the tertiary and the inner binary. Although we ignore many important features of real EMRIs (i.e., a stellar background, which causes important different phenomena, such as the Schwarzschild barrier, discussed in Amaro-Seoane 2012, but also the role of the spin of the central MBH, Amaro-Seoane et al. 2013b), we here discuss the potential impacts of REOs on EMRIs as a motivation to future works. Because the amplitude of the eccentricity oscillations given by Equation 3.1 is proportional to the mass ratio, arbitrarily large mass ratios can lead to arbitrarily large eccentricity oscillations. If the oscillations are too large they can unbind the inner binary. But encounters at large enough distances that the binary system does not become unbound could therefore lead to REOs with amplitudes comparable to the amplitude of the Kozai-Lidov resonance itself. Eccentricity oscillations for a fiducial EMRI are shown in Figure 3.11. At the peak of the Kozai-Lidov cycle the oscillations reduce the distance at periapsis of the inner binary by over a factor of five. Since the gravitational wave merger timescale for a very eccentric orbit is proportional to (1 e2)7/2 (Peters 1964), this reduction − in the distance at periapsis due to REOs could lead to a significant reduction in the merger time and increase gravitational wave luminosity if the dynamical features of realistic EMRIs do not suppress this effect. These results should be studied with more detail in the context of secular effects in semi-Keplerian systems with relativistic corrections, such as in the works of Merritt et al. (2011); Brem et al. (2012).

3.5.2. Implications for gravitational wave emission

In this subsection we discuss an important application of our findings that will be very soon expanded significantly in a dedicated statistical study of the dynamics and the implications for ground-based gravitational wave detectors such as Advanced

63 Fig. 3.11.— REOs in an EMRI calculated with direct three-body integration (solid line) and in the secular approximation (red dashed line). Because the mass ratio between the inner binary and outer binary is very large, the amplitude of the fluctuations in the angular momentum in the inner binary due to REOs becomes comparable to the angular momentum in the inner binary itself. Parameters of the 8 system are provided in Table 3.2, but with m3 = 3 10 M , g1 = 0◦, g2 = 90◦, and × ⊙ e2 = 0.2.

64 LIGO/VIRGO, along with the detailed description of the modification of the integrator to incorporate relativistic effects.

It is uncertain what the dynamics immediately prior to merger will be in systems dominated by REOs. Although the orbit will have substantially circularized by the time the two objects of the inner binary come within 10 RSch of each other, the orbit nevertheless retains a non-negligible eccentricity (e 0.1) in most 1 ∼ systems (Wen 2003; Gould 2011). Whether or not typical compact object binaries retain non-negligible eccentricity immediately prior to merger is of key importance to gravitational wave detectors like LIGO and LISA. Because these experiments require gravitational wave templates to find gravitational wave signals in their data, accurate a priori predictions of the waveform shapes are crucial for the success of these experiments. Gravitational wave searches like LIGO have generally assumed that by the time a merging binary is emitting gravitational waves at frequencies to which they are sensitive, it has completely circularized. But if a substantial number of compact object binaries are driven to merger due to Kozai-Lidov resonances, then the assumption of perfectly circular inspirals will be mistaken. Since gravitational radiation is a very strong function of distance, even a modest residual eccentricity

(e1 > 0.1) would suffice to bury a gravitational wave signal in the data if a circular ∼ orbit template is used (Brown & Zimmerman 2010).

We calculate the eccentricity distribution of the inner binaries of the triple systems investigated in Section 3.4.2. Figure 3.12 presents the distribution of e1 as the two components of the inner binary come within 10 RSch of each other calculated using direct three body integration (solid line) and in the secular approximation

(red dashed line). At 10 RSch a Keplerian orbit becomes an increasingly poor approximation to the true orbit of the inner binary. As such, we define the eccentricity of the orbit to be

1 L 2 e 1 1 , (3.6) 1 ≡ − G(m + m )a µ s 0 1 1 µ 1 ¶ where L1 is the angular momentum of the inner binary and µ1 is the reduced mass of the inner binary. We add PN corrections to L1 up to second order (e.g., Iyer & Will 1995). To increase the computational efficiency, in the secular approximation

65 we calculate systems until a1 has decreased to 1% of its initial value. At this point the inner binary has decoupled from the outer binary and can be calculated independently. We then calculate the eccentricity of the orbit when the two components come within 10 RSch of each other using the adiabatic calculation of Peters (1964).

The eccentricity distribution calculated using direct integration predicts that 10% of systems merge at high eccentricity (e > 0.8). In the secular approximation, ∼ 1 however, nearly all systems merge at low eccentricity (e1 < 0.2). We also present the ∼ distribution of e1 at 10 RSch as a function of e2 in the right panel of Figure 3.12.

Triples with larger e2 have a greater chance of merging at high eccentricity. Since we assume a uniform distribution of e2 in the left panel of Figure 3.12, a more realistic thermal distribution will lead to a larger fraction of systems merging at high eccentricity. Population synthesis studies of hierarchical triple systems which employ the secular approximation will therefore miss an important source of unique gravitational waveforms. If an important residual eccentricity was indeed typical in these situations, gravitational wave experiments would possibly have to take this into account in the preparation of the waveform banks.

One interesting possible outcome of a NS-NS merger in a triple system would be a head-on collision similar to those between white dwarfs described in Katz & Dong (2012). In the case of binaries consisting of objects more compact than white dwarfs, however, the chance of a collision is much smaller. This is due to two factors. Firstly, the objects themselves have a smaller cross section than do white 6 7 dwarfs by a factor of 10 − . Secondly, close encounters will lead to circularization ∼ of the orbit at larger distances relative to the object’s radius due to the stronger relativistic effects. Head-on collisions between pairs of neutron stars or black holes should therefore be rarer than collisions between WD-WD binaries by a large factor.

3.5.3. Tides and implications for stars and planets

Another well known class of triple systems with high mass ratios is that of planets in binary star systems. The formation of hot Jupiters is a long-standing problem in the theory of planet formation and Kozai-Lidov cycles have been proposed as a

66 1.

0.8

N-body 0.8 Secular 0.6 0.6 Sch R

0.4 at 10 1

e 0.4 Fraction of systems

0.2 0.2

0. 0. 0. 0.2 0.4 0.6 0.8 1. 0.2 0.4 0.6 0.8

e1 at 10 RSch e2

Fig. 3.12.— The eccentricity distribution of the inner orbit when the inner two components of the systems calculated in Figure 3.9 come within 10 RSch of each other. Left panel: The secular approximation (red dashed line) underpredicts the number of binaries which merge at high eccentricities (e > 0.2) relative to the direct ∼ integration (black line). In particular, the secular approximation predicts that no binaries will come within 10 RSch at e1 > 0.4, whereas the direct integration predicts that 20% of hierarchical triples do. Right panel: The distribution of e as a ∼ final function of e2. A larger fraction of systems merge at high efinal when e2 is large.

Note that we have assumed a uniform distribution in e2; if e2 is distributed thermally more systems will merge at high e1. Gravitational wave detectors will need to employ templates of eccentric binaries to detect such systems.

67 mechanism to drive planets formed far from the host star into tight orbits (Wu et al. 2007). Tidal effects are very important for stellar and planetary systems and while a complete treatment is beyond the scope of this chapter (though see Naoz et al. 2012, for a discussion of the effect of tides on Kozai-Lidov cycles), we nevertheless make some qualitative statements about the impact of tides on our results and the implications for stars and planets.

The overall effect of tides on eccentric orbits is to circularize them and reduce the semi-major axis (Hut 1981) on a characteristic tidal friction timescale tTF. If the 3/2 eccentricity is very close to unity, t (1 e )− (Hut 1982). Tides therefore TF ∝ − 1 prevent stars and planets from remaining on high eccentricity orbits for long periods of time and will disrupt sufficiently strong Kozai-Lidov cycles. However, REOs occur on a shorter timescale than the Kozai-Lidov cycle by a factor of P2/P1. Tides may not have enough time to circularize the orbit at a relatively low eccentricity before REOs drive the orbit to higher eccentricities. Because REOs can reduce (1 e ) by − 1 a factor of 5, this results in a reduction in t by an order of magnitude. The orbit ∼ TF will thus circularize more rapidly and be brought into a closer orbit than it would by Kozai-Lidov oscillations calculated in the secular approximation.

68 Chapter 4: Dynamical formation & scattering of triples

4.1. Introduction

Gravitational scattering events are common in globular clusters and are also dynamically important for many systems in open clusters and the field (Hills & Day 1976). Although two-body scattering can be studied analytically, three-body scattering is too complex to permit general, practical analytic results (Poincar´e 1892; Sundman 1907). The three-body scattering problem was therefore not studied in detail until the development of computers (e.g., Saslaw et al. 1974; Heggie 1975; Hut & Bahcall 1983). Since then binary-single and binary-binary scattering events have been studied extensively (e.g., Hut 1983; Mikkola 1983; Hills 1991; Valtonen & Mikkola 1991; Sigurdsson & Phinney 1993; Bacon et al. 1996; Fregeau et al. 2004, and the references therein), but little attention has been devoted to the scattering of triple systems or to the orbital characteristics of triple systems formed from binary-binary scattering (though see Ivanova 2008; Ivanova et al. 2008; Leigh et al. 2011; Leigh & Geller 2012; Moeckel & Bonnell 2013; Leigh & Geller 2013, 2015). Yet due to the Kozai-Lidov (KL) mechanism (Lidov 1962; Kozai 1962), it is now appreciated that the dynamics of triples may play an important role in a wide variety of astrophysical phenomena (e.g., Holman et al. 1997; Ford et al. 2000; Miller & Hamilton 2002; Blaes et al. 2002; Wu & Murray 2003; Wen 2003; Fabrycky & Tremaine 2007; Wu et al. 2007; Ivanova et al. 2008; Perets & Fabrycky 2009; Naoz et al. 2011; Thompson 2011; Naoz et al. 2012; Katz & Dong 2012; Shappee & Thompson 2013; Antonini et al. 2014; Antognini et al. 2014). Furthermore, it is now known that triple systems are not rare in our Galaxy; demographic surveys of the field have revealed that triples constitute 10% of all stellar systems (Duquennoy & Mayor 1991; Raghavan et al. 2010; Tokovinin 2014; Sana et al. 2014).

69 Given the prevalence of triple systems it is important to understand in detail how triples interact with other stellar systems. Observations of the Tarantula Nebula have demonstrated that binary interactions strongly affect the observed multiplicity fraction (Sana et al. 2013) so triple interactions may contribute to the observed multiplicity fraction as well. Moreover, the large multiplicity fraction of high mass stars indicates that triple dynamics may be even more important for these systems (Leigh & Geller 2013); for example, scattering of triples with binaries may be an important formation mechanism for quadruple systems. Raghavan et al. (2010) showed that quadruple systems are nearly as common as triple systems and many even higher-order systems have been discovered (e.g., Koo et al. 2014), but it is unclear what fraction of such systems are formed in situ versus dynamically (Goodwin & Kroupa 2005). Quadruple and higher-order systems may be even more dynamically important than triples since numerical and semi-analytic experiments have demonstrated that they exhibit stronger KL oscillations than triples over a wider region of parameter space (Pejcha et al. 2013; Hamers et al. 2015).

The perturbative influence of interloping stars on hierarchical triple systems may have an important effect on the long-term evolution of triples. The KL mechanism can drive the inner binary to very high eccentricities, but the maximum eccentricity reached is sensitive to the orbital parameters, in particular the mutual inclination and the outer eccentricity. In the standard quadrupole order KL mechanism the tertiary can drive the inner binary to arbitrarily large eccentricities over a narrow range of inclinations near 90◦ (the exact value depends on the particular system). However, at octupole order, in the so-called “eccentric KL mechanism,” the range of inclinations over which the tertiary can drive the inner binary to extreme eccentricities is significantly increased, especially if the eccentricity of the orbit of the tertiary is large. Thus, perturbations to the inclination or eccentricity of the tertiary may produce much stronger KL oscillations for star-star or star-planet systems. If these KL oscillations drive the inner binary to sufficiently high eccentricities, the components of the inner binary will tidally interact (e.g., Mazeh & Shaham 1979; Wu & Murray 2003; Fabrycky & Tremaine 2007; Perets & Fabrycky 2009; Naoz et al. 2011, 2012; Naoz & Fabrycky 2014), potentially explaining the observation that nearly all close binaries (Tokovinin et al. 2006) and certain subsets of warm and

70 hot Jupiters (Wu & Murray 2003; Wu et al. 2007; Naoz et al. 2011, 2012; Socrates et al. 2012; Dong et al. 2014) have tertiary companions.

Tidal interactions may not always occur in highly inclined triple systems; Li et al. (2014b) showed that some coplanar systems can be driven to very large eccentricities. Moreover, certain systems undergo changes in eccentricity so rapidly that the angular momentum of the inner orbit can change by an order of magnitude in a single orbit (Antonini & Perets 2012; Katz & Dong 2012; Seto 2013; Antonini et al. 2014; Antognini et al. 2014). These rapid eccentricity oscillations present the possibility that the components of the inner orbit may be driven to merger more rapidly than the tidal circularization process can circularize the orbit and quench KL oscillations. The components of the inner binaries of such systems will then not merge gently, but will instead collide head-on. These head-on collisions may produce a variety of unusual astrophysical phenomena depending on the objects comprising the inner binary. Most notably, if the inner binary components are two white dwarfs, these collisions may produce a Type Ia supernova (SN Ia; Thompson 2011; Katz & Dong 2012; Hamers et al. 2013; Kushnir et al. 2013; Prodan et al. 2013).

The possible connection between triple dynamics and SNe Ia should be explored because, despite their crucial role in constraining cosmological parameters (Riess et al. 1998; Perlmutter et al. 1999), it is unknown whether the progenitor systems consist of one white dwarf (the single degenerate model; Whelan & Iben 1973; Nomoto 1982) or two (the double degenerate model; Iben & Tutukov 1984; Webbink 1984). Although observational evidence currently favors the double degenerate model (Howell 2011; Maoz & Mannucci 2012; Maoz et al. 2014), it is unclear how to drive white dwarf binaries to merge at a rate large enough to be consistent with the observed SN Ia rate (Ruiter et al. 2009, 2011). One approach is by driving the binary to high eccentricity through the perturbative influence of a tertiary via the KL mechanism. At high eccentricities the inner binary would then emit much more gravitational radiation (Blaes et al. 2002; Miller & Hamilton 2002; Wen 2003), thereby leading to more rapid coalescence of the WD-WD binary (Thompson 2011). However, it is unclear how to prevent the inner binary from merging or colliding while still on the main sequence. KL oscillations would bring the two stars into tidal

71 contact and tidal circularization would then shrink the orbit, greatly increasing the semi-major axis ratio and “freezing” the inner binary at an inclination close to the critical Kozai angle (Fabrycky & Tremaine 2007). These two effects would effectively shut off further KL oscillations and prevent the inner binary from merging when the stars evolve into white dwarfs. The dynamical formation of high-inclination triples from scattering may be one means to circumvent this difficulty.

Scattering of high-multiplicity systems may also be one channel to produce free-floating planets. Microlensing studies have revealed that there may be as many as two free-floating planets for every bound planet in the Galaxy (Zapatero Osorio et al. 2000; Sumi et al. 2011). Moreover, there is some evidence that it is difficult for planet-planet scattering to produce free floating planets in the required numbers (Veras & Raymond 2012). Dynamical scattering of field stars off of binary stars may be an important component to the rate of planet ejection.

Scattering events involving high-order stellar systems could also be a source of 1 stellar collisions. Stellar collisions are estimated to occur at a rate of 0.5 yr− in ∼ the Galaxy (Kochanek et al. 2014) and it is unknown whether these collisions are due to KL oscillations, scattering events, or a combination of the two. Even when scattering does not lead to a stellar collision the scattering event will change the orbital parameters of the system. This will lead to evolution in the distribution of orbital parameters of the triples in the population. This evolution would be strongest in globular clusters due to their high stellar densities and large ages.

This chapter has three goals. The first is to perform idealized binary-binary, triple-single, and triple-binary scattering experiments in order to derive general relationships between the cross sections of various outcomes with the initial orbital parameters and the incoming velocity. In this way our chapter is similar to Hut & Bahcall (1983) and Mikkola (1983) except that we study higher-order scattering. The second goal of our chapter is to determine the orbital parameters of triple systems after scattering events. The final goal of this chapter is to apply the cross sections we derive in several contexts. In particular we estimate the rate of formation of WD-WD binaries in highly inclined triples, the ejection rate of planets due to

72 scattering events, the stellar collision rate, and the lifetime of high inclination triples. To meet these goals we have performed over 400 million scattering experiments.

Details of the numerical methods of this chapter are presented in Section 4.2. We study triple scattering in detail in Section 4.3 by performing numerical experiments and comparing them with analytic approximations in Section 4.4. We then discuss the distribution of orbital parameters of dynamically formed triples in Section 4.5. We discuss the implications of these results for SNe Ia, the longevity of triple systems undergoing KL oscillations, and provide estimates for the rate of planetary ejection and stellar collisions in Section 4.6. We conclude in Section 4.7.

4.2. Numerical methods

We use the open source Fewbody suite to perform our scattering experiments (Fregeau et al. 2004). Fewbody is optimized to numerically compute the dynamics of systems with small numbers of components (N < 10). Fewbody uses the ∼ ordinary differential equations library of the GNU Scientific Library (GSL) for its underlying integrator (Gough 2009). The GSL ordinary differential equations library supports six integration algorithms with adaptive time steps, from which we use eighth-order Runge-Kutta Prince-Dormand integration. We find, however, that the choice of integration algorithm makes little difference to the results of the calculations because GSL’s adaptive time steps are chosen to target a specified 14 relative and absolute accuracy (10− in our experiments) regardless of the algorithm used. Fewbody also supports the use of Kustaanheimo-Stiefel (KS) regularization (Kustaanheimo & Stiefel 1965), a coordinate transformation which removes the singularities in the gravitational force present in ordinary N-body integration. Throughout this chapter we use KS regularization as it has the particular advantage of making eccentric orbits much easier to compute.

4.2.1. Notation

Throughout this chapter we use the same notation as Fregeau et al. (2004) and Fewbody in which each star in a system with n stars is labelled with a unique index running from 0 to n 1. Bound pairs are denoted by square brackets surrounding −

73 the pair and collisions between two stars are denoted by colons between the pair. For example, a hierarchical triple with an unbound interloping star is notated [[0 1] 2] 3 (stars 0 and 1 form the inner binary, star 2 is the tertiary, and star 3 is the interloping star), and a hierarchical triple in which the two stars of the inner binary have collided is notated [0:1 2].

Orbital parameters (e.g., semi-major axis or eccentricity) of the component binaries of the system are notated by subscripts in a top-down fashion. Leftmost indices in subscripts represent the outermost binaries of the separate hierarchies, and rightward indices represent inner binaries in those hierarchies. For example, in a triple-binary scattering event, the semi-major axis of the outer binary of the triple is a1, the semi-major axis of the inner binary of the triple is a11, and the semi-major axis of the interloping binary is a2. Similarly, the mass of the second star in the innermost binary of the triple is m112, the mass of the tertiary is m12, and the mass of the first star of the interloping binary is m21. We also combine masses in the subscripts. Thus the total mass in the inner binary of the triple is m11 = m111 + m112 and the total mass of the triple is m1 = m11 + m12.

For clarity we occasionally refer to the orbital parameters of hierarchical triples using the subscripts “in” and “out.” Thus the eccentricity of the inner orbit of a triple may be referred to equivalently by e11 or ein and the eccentricity of the outer orbit by e1 or eout.

4.2.2. Cross sections

Throughout this chapter we provide cross sections for the outcomes of scattering events involving triple or higher-order hierarchical systems. Our treatment of these cross sections and their uncertainties follows that of Hut & Bahcall (1983). We use the usual definition of the cross section for a particular outcome, X:

2 nX σX = πbmax , (4.1) ntot where bmax is the maximum impact parameter in a set of experiments, nX is the number of experiments with outcome X, and ntot is the total number of experiments performed. Throughout this chapter we present our results using normalized

74 cross sections,σ ˆ. The cross sections from triple-single scattering experiments are normalized to the area of the outer binary of the triple: σ σˆ 2 (triple-single). (4.2) ≡ πa1 The cross sections for binary-binary experiments are normalized to the sum of the areas of the two binaries and similarly in triple-binary experiments the cross sections are normalized to the sum of the areas of the outer binary of the triple and the interloping binary: σ σˆ 2 2 (binary-binary, triple-binary). (4.3) ≡ π(a1 + a2)

There are two uncertainties in the calculation of the cross sections. The first is the statistical uncertainty due to the finite number of experiments performed:1

σX ∆statσX = . (4.4) √nX The second source of uncertainty is due to the fact that certain systems require prohibitively long computation times to resolve. For example, while it is impossible for a single star scattering off of a triple system to produce a stable quadruple system (Chazy 1929; Littlewood 1952; Heggie 1975; Heggie & Hut 2003), certain systems can enter into a “metastable” state where a quadruple system is produced that takes an exceedingly long time to dissociate. Such unresolved systems produce a separate systematic uncertainty in the calculation of the cross sections given by

2 nunres ∆sysσX = πbmax , (4.5) ntot where nunres is the number of experiments with unresolved outcomes.

It is important to note that the systematic uncertainty is completely asymmetric and only serves to increase the cross section of any outcome. That is to say, our estimates of the cross sections are lower bounds on the true cross sections. 1In cases where the fraction of outcomes with the outcome X is close to unity or zero it is better to use a confidence interval like the Wilson score interval rather than the normal approximation in order to capture the asymmetry of the statistical uncertainty (Wilson 1927). However, we perform enough experiments that the difference between the Wilson score interval and the normal approximation is almost always negligible. Where it is not, we use the Wilson score interval.

75 4.2.3. Initial conditions and halting criteria

Each scattering experiment begins with the interloping system at a large, but finite, distance from the target system. We choose the initial separation between the two systems to be the distance at which the tidal force on the outer binary of the target system is some small fraction, δ, of the relative force between the two components of the outer binary when at apocenter. Hence, the initial separation, r, is given by F tid = δ, (4.6) Frel where 2G(m + m )m F = 11 12 2 a(1 + e), (4.7) tid r3 and

Gm11m12 Frel = , (4.8) [a(1 + e)]2 where a and e refer to the semi-major axis and eccentricity, respectively, of the outer 5 binary of the target system. Throughout this chapter we use δ = 10− , the same choice as Fregeau et al. (2004). Smaller choices of δ do not change the results but increase the running time. The interloping system is then analytically brought along a hyperbolic orbit from infinity with the bmax and velocity at infinity, v , that has ∞ been fixed for that particular experiment.

It is also essential to choose the appropriate bmax when computing cross sections.

Too small a choice of bmax will result in in an underestimate of the cross section, whereas too large a choice of bmax will result in few experiments producing outcomes of interest, thereby leading to large statistical uncertainties. We adopt the choice of bmax similar to that of Hut & Bahcall (1983) of: 4v b = crit + 3 a , (4.9) max v 1 µ inf ¶ where vinf is the incoming velocity at infinity and vcrit is the critical velocity at which the total energy of the system is zero. For binary-binary scattering, vcrit is G(m + m ) m m m m v2 = 1 2 11 12 + 21 22 , (4.10) crit, bin-bin m m a a 1 2 µ 1 2 ¶ 76 for triple-single scattering, vcrit is

2 G(a1m111m112 + a11m11m12)(m1 + m2) vcrit, trip-sing = , (4.11) a1a11m1m2 and for triple-binary scattering vcrit is

2 G(m1 + m2) vcrit, trip-bin = a1a11a2m1m2 [a a m m + a (a m m + a m m )] . (4.12) × 11 2 11 12 1 2 111 112 11 21 22 We will frequently refer to velocities in terms of the normalized velocity,v ˆ, which is the velocity, v, scaled to vcrit, v vˆ . (4.13) ≡ vcrit

We test our choice of bmax by computing the cross sections for four categories of outcome in triple-single scattering over a broad range of bmax. We demonstrate that the cross sections are well converged in Fig. 4.1.

The calculation halts when one of four conditions is met: (1) if the system consists of some number of stable hierarchical systems as determined by the Mardling stability criterion (Mardling & Aarseth 2001):

2/5 a1 m11 1+ e1 0.3 (1 e1) > 2.8 1+ 1 i (4.14) a11 − m12 √1 e − π ·µ ¶ − 1 ¸ µ ¶ (where the mutual inclination, i, is in radians), and all the systems are far enough apart that the ratios between their tidal forces to their relative forces is less than δ; (2) the system integrates for 106 times the initial orbital period of the outer binary of the target system; (3) the system integrates for one hour; or (4) the total energy or angular momentum of the system changes from its initial value by more than one part in 103. If conditions (2) or (3) are met the outcome is classified as unresolved and included in our systematic uncertainty.2

2Portegies Zwart & Boekholt (2014) have argued that typical energy conservation standards in N-body studies are extremely conservative and that maintaining energy conservation to better than one part in 10 is sufficient to preserve the statistical properties of dynamical experiments. These results have been confirmed by Boekholt & Portegies Zwart (2015). Since very few of our experiments violate our energy conservation standard, this finding does not change our results, but does imply that we may have slightly overestimated our systematic uncertainties.

77 Fig. 4.1.— Convergence of the scattering cross sections with bmax for triple-single 5 scattering. We performed 10 scattering experiments at each choice of bmax. All masses are equal point masses and the orbits of the initial triple were circular with a semi-major axis ratio of 10. The cross sections for four categories of outcomes are presented: exchanges (black circles), single ionization (blue triangles), double binary formation (red squares), and exchange-ionization (orange stars). (See Section 4.3.2 for a description of the categories.) The choice for bmax we adopt in this chapter (dashed line) is given by equation (4.9) and is similar to that of Hut & Bahcall (1983). Smaller bmax’s do not probe the full area over which non-flyby outcomes occur, whereas the statistical power of larger bmax’s is reduced because of the small fraction of non-flyby outcomes. The choice we adopt is converged but is small enough to have strong statistics.

78 4.3. Scattering experiments

Hierarchical triples can be formed through scattering in one of three ways: (1) binary-binary scattering, (2) scattering of triples, or (3) scattering of higher-order systems. Binary-binary scattering has been studied extensively, but only rarely have any studies addressed the formation of triple systems and, moreover, scattering of higher-order systems in general has received little attention at all. We study triple and binary scattering in detail in this section. We do not study the scattering of higher-order systems (e.g., triple-triple, quadruple-single) because such scattering events are rare in most environments (Leigh & Geller 2013).

We compute the cross sections for the outcomes of both single and binary stars scattering off of hierarchical triples along with binaries scattering off of binaries in the point mass limit using Newtonian gravity. The masses of all stars in the system are equal unless we explicitly vary the mass of one star of the inner binary of the triple. In an individual scattering event, the semi-major axis and eccentricity of all component binaries (both binaries of the triple and, in the case of triple-binary scattering, that of the interloping binary) in the system are fixed, but the argument of pericenter and the mean anomaly are chosen from a uniform distribution between 0 and 2π. The component binaries are then oriented randomly by pointing each binary’s angular momentum vector toward a randomly chosen point on a sphere. We present a sample binary-binary, triple-single, and triple-binary scattering experiment projected in the xy-plane in Fig. 4.2.

We first perform scattering experiments on a model system in Section 4.3.1 and then vary several of the initial orbital and physical parameters of this system one at a time to determine the dependence of the cross sections on these parameters in Section 4.3.2.

4.3.1. Cross sections of model systems

Our model system for triple-single scattering is a moderately hierarchical triple system with semi-major axis ratio, α a /a = 10. This choice of α is large enough ≡ 1 11 that the triple is stable (Mardling & Aarseth 2001), but is still small enough that

79 Fig. 4.2.— A sample binary-binary, triple-single, and triple-binary scattering experiment projected onto the xy-plane. The incoming velocity is 0.1vcrit in the case of triple scattering and 0.2vcrit in the case of binary-binary scattering. The semi- major axis ratio of the triple in the case of triple scattering is 10 and the semi-major axis of the incoming binary is equal to the semi-major axis of the outer binary in the case of triple-binary scattering. In the case of binary-binary scattering the semi-major axes are equal. In the case of triple scattering the triple approaches from the left and the interloping systems approaches from the right. Left panel: In this binary-binary scattering event one star from each binary combine to form a new binary, and the two remaining stars are ejected. Middle panel: In this triple-single scattering event one star from the inner binary of the triple is ejected and the interloping star is captured as a tertiary. Right panel: In this triple-binary scattering event one star from the inner binary is ejected, leaving two binaries and a single star.

80 there will be dynamical interactions between the inner and outer binaries. In this model system, the interloping star has an incoming velocity of v /vcrit = 1, which ∞ is approximately the velocity dispersion of stars in the thin disk of the Milky Way 1 ( 40 km s− ) relative to the critical velocity of a triple system consisting of three 1 ∼ M stars with a1 = 10 AU and a11 = 1 AU (Binney & Merrifield 1998, p. 656). We ⊙ take both orbits to be circular.

Our model system for triple-binary scattering is identical, but with the interloping star replaced by an interloping binary with semi-major axis equal to the outer semi-major axis of the triple, a2 = a1 = 10a11. The incoming velocity is again taken to be vcrit, but with vcrit now calculated using equation (4.12) instead of equation (4.11). Again, all stars are of equal mass, so now the mass ratio of the interloping system to the triple is 2/5 instead of 1/3.

Lastly, our model system for binary-binary scattering is identical to the case of triple-binary scattering but with the inner binary of the triple replaced by a point mass of 1 M . The incoming velocity is taken to be vcrit, but with vcrit calculated ⊙ using equation (4.10). In this case the mass ratio of the interloping system to the target system is 1.

We perform 106 scattering experiments for each model system. We present the cross sections for all possible outcomes with their statistical and total uncertainties in Table 4.3.1 for the triple-single case, Table 4.3.1 for the triple-binary case, and Table 4.3.1 for the binary-binary case.

Because there are a large number of possible outcomes in binary-binary, triple-single, and triple-binary scattering (22, 22, and 161, respectively), many of which are qualitatively similar, we group these outcomes into broad classes. In the case of triple-single scattering we define six classes: (1) flybys, in which the hierarchical structure of the system remains the same, although the orbital parameters may have changed; (2) exchanges, in which the interloping star replaces one of the stars in the triple; (3) double binary formation, in which one star of the triple is ionized and binds to the interloping star; (4) single ionization, in which one star from the triple is ionized, leaving a binary and two unbound stars; (5) full

81 Triple-single

Outcomeσ ˆ ∆statσˆ Outcome class [0 1] 3 2 1.309 0.013 Single ionization [0 2] 1 3 0.092 0.003 Single ionization 0 [1 2] 3 0.080 0.003 Single ionization 0 [1 3] 2 0.076 0.003 Exchange ionization [0 3] 1 2 0.074 0.003 Exchange ionization [[0 1] 3] 2 0.033 0.002 Exchange, new triple [[0 3] 2] 1 0.027 0.002 Exchange, new triple 0 [[1 3] 2] 0.021 0.002 Exchange, new triple [0 1] [2 3] 0.017 0.001 Double binary [0 2] [1 3] 0.011 0.001 Double binary 3 [[1 2] 0] 0.010 0.001 Scramble, new triple [0 3] [1 2] 0.010 0.001 Double binary [[0 2] 1] 3 0.008 1e-03 Scramble, new triple 0 1 [2 3] 0.005 8e-04 Exchange ionization

∆sysσˆ 0.032 0.002

Table 4.1: Normalized cross sections for the outcomes of triple-single scattering (see equation 4.2 for the normalization). The initial conditions are described in Section 4.3 and the initial hierarchy is [[0 1] 2] 3. We present both the statistical uncertainty and the systematic uncertainty. Note that the systematic uncertainty only represents an uncertainty toward larger cross sections. That is, the cross sections presented are lower limits and may be larger by the systematic uncertainty—see Section 4.2.2. We only present cross sections for which the statistical uncertainty is less than 10 per cent. The cross section for the outcome equal to the initial hierarchy (i.e., a flyby) is not well defined and so is not included. Such interactions can change the orbital parameters of the triple system, however, and are discussed in Section 4.5.1. The outcome classes are defined in Section 4.3.2. We also note if the scattering event produces a new triple with a hierarchy distinct from the original hierarchy.

82 Triple-binary

Outcomeσ ˆ ∆statσˆ Outcome class [0 1] 4 2 3 1.110 0.016 Double ionization [0 1] [3 4] 2 0.990 0.015 Double binary [[0 1] 2] 4 3 0.911 0.015 Binary disruption [0 1] 3 [2 4] 0.072 0.004 Double binary [0 1] 4 [2 3] 0.070 0.004 Double binary [0 2] 1 4 3 0.048 0.003 Double ionization 0 [1 2] [3 4] 0.048 0.003 Double binary 0 [1 2] 4 3 0.048 0.003 Double ionization [0 2] 1 [3 4] 0.045 0.003 Double binary [[0 1] 4] 3 2 0.025 0.002 Bin. disruption, new triple 0 [1 3] 2 4 0.025 0.002 Double ionization [0 4] 1 2 3 0.022 0.002 Double ionization [[0 1] 3] 4 2 0.021 0.002 Bin. disruption, new triple [0 3] 1 2 4 0.021 0.002 Double ionization 0 [1 4] 2 3 0.018 0.002 Double ionization [[0 1] 3] [2 4] 0.011 0.002 Exchange, new triple [[0 1] 4] [2 3] 0.011 0.002 Exchange, new triple

∆sysσˆ 0.045 0.003

Table 4.2: Normalized cross sections for the outcomes of triple-binary scattering (see equation 4.3 for the normalization). The initial conditions are described in Section 4.3 and the initial hierarchy is [[0 1] 2] [3 4]. We present both the statistical uncertainty and the systematic uncertainty. Note that the systematic uncertainty only represents an uncertainty toward larger cross sections. That is, the cross sections presented are lower limits and may be larger by the systematic uncertainty—see Section 4.2.2. We only present cross sections for which the statistical uncertainty is less than 10 per cent. The outcome classes are defined in Section 4.3.2. We also note if the scattering event produces a new triple with a hierarchy distinct from the original hierarchy.

83 Binary-binary

Outcomeσ ˆ ∆statσˆ Outcome class 0 1 [2 3] 1.335 0.016 Single ionization [0 1] 3 2 1.310 0.016 Single ionization [0 2] 1 3 0.829 0.013 Exchange ionization [0 3] 1 2 0.823 0.013 Exchange ionization 0 [1 3] 2 0.807 0.013 Exchange ionization 0 [1 2] 3 0.800 0.013 Exchange ionization [0 2] [1 3] 0.152 0.005 Exchange [0 3] [1 2] 0.150 0.005 Exchange [[0 3] 2] 1 0.003 8e-04 Triple formation [[0 1] 2] 3 0.002 6e-04 Triple formation [[0 3] 1] 2 0.002 6e-04 Triple formation 0 [[1 2] 3] 0.002 6e-04 Triple formation 0 [[1 3] 2] 0.002 6e-04 Triple formation [[2 3] 0] 1 0.002 6e-04 Triple formation [[0 2] 1] 3 0.002 6e-04 Triple formation 2 [[1 3] 0] 0.002 6e-04 Triple formation

∆sysσˆ 0.002 0.001

Table 4.3: Normalized cross sections for the outcomes of binary-binary scattering. The initial conditions are described in Section 4.3 and the initial hierarchy is [0 1] [2 3]. The cross sections have been normalized to the sum of the areas of the two binaries. We present both the statistical uncertainty and the systematic uncertainty. Note that the systematic uncertainty only represents an uncertainty toward larger cross sections. That is, the cross sections presented are lower limits and may be larger by the systematic uncertainty—see Section 4.2.2. We only present cross sections for which the statistical uncertainty is less than 50%.

84 ionization, in which all stars become unbound from each other; and (6) a scramble, in which the tertiary exchanges with one of the stars of the inner binary.

In the case of binary-binary scattering we define six classes: (1) flybys, defined as in the triple-single case; (2) exchanges, in which one star from each binary exchanges places with the other; (3) triple formation, in which a stable triple is formed, leaving a single unbound star; (4) single ionization, in which one star is ionized, leaving a binary and two unbound stars; (5) full ionization, defined as in the triple-single case; and (6) exchange + ionization, in which one star from each binary binds to form a new binary, and the two other stars remain unbound.

In the case of triple-binary scattering we define seven classes: (1) flybys, defined as in the triple-single case; (2) exchanges, in which one star from the interloping binary exchanges with one star from the interloping binary; (3) double binary formation, in which one star from the triple is ionized, resulting in two binaries and an unbound star; (4) binary disruption, in which the two stars of the interloping binary become unbound; (5) triple disruption, in which the three stars of the triple become unbound from each other; (6) quadruple formation, in which one star from the interloping binary becomes bound to the triple, forming a quadruple; (7) and a scramble, defined as in the triple-single case.

Note that in triple-single and triple-binary scattering several of these outcome classes produce a ‘new’ triple (i.e., the final hierarchy differs from the original hierarchy). (In binary-binary scattering triple formation is its own class and is mutually exclusive with the other classes.) The cross sections for these classes for triple-single, triple-binary, and binary-binary scattering are displayed in Tables 4.3.1, 4.3.1, and 4.3.1, respectively. We include there the cross section for new triple formation, though for triple-single and triple-binary scattering it is not independent of the cross sections for other outcome classes.

In the case of triple scattering the systematic uncertainty is dominated by marginally unstable triples (according to Equation 4.14) for which the integration time exceeded the one-hour CPU time limit; 0.0263 per cent of the triple-single scattering experiments failed to complete, and of these 81 per cent failed to complete

85 due to the CPU time limit (the rest failed because they violated the energy conservation limit). Because the Mardling stability criterion of equation (4.14) is not a hard boundary (Petrovich 2015) it is possible that these triples are, in fact, stable and should be classified as flybys. If, however, they are unstable on longer timescales than our integrations permit they should be classified as single ionization events since no other outcome is possible. Thus the systematic uncertainty of triple scattering should be considered as an uncertain contribution to the cross section for single ionization. In the case of binary-binary scattering the systematic uncertainty is dominated by systems that violated the energy conservation criterion during an extremely close passage. This is a much rarer occurrence than the formation of a marginally unstable triple during a triple scattering event, so the systematic uncertainty of binary-binary scattering is much lower than that of triple scattering (0.0012 per cent of systems failed to resolve, of which all were due to violations of the energy conservation limit).

4.3.2. Dependence on initial parameters

We next explore the dependence of the cross sections on the initial parameters of the system. In particular we separately vary the semi-major axis ratio, the incoming velocity, the eccentricities, and the masses.

Semi-major axis ratio

We first vary the semi-major axis ratio from α a /a = 100.6 102. At choices ≡ 2 1 − of α much below our lower limit the triples become unstable and at choices of α much larger than our upper limit the computational time becomes prohibitively long because the time step is set by the orbital period of the inner binary, but the crossing time is set by the size of the outer orbit. We hold all other parameters (e.g., eccentricities, masses) fixed to their values in the model system (Section 4.3.1). The normalized cross sections for binary-binary, triple-single, and triple-binary scattering are shown in Fig. 4.3. Single ionization is the dominant outcome of triple-single scattering for all α. At the low-α end this is because the triples are only marginally stable so minor perturbations from the interloper tend to ionize one member of the triple. At the high-α end this is because the tertiary becomes weakly bound to the

86 Triple-single

Outcome classσ ˆ ∆statσˆ Exchange 0.082 0.003 Single ionization 1.481 0.013 Full ionization 0.000 0.000 Double binary formation 0.038 0.002 Exchange + ionization 0.155 0.004 Scramble 0.018 0.001 New triple 0.100 0.003

∆sysσˆ 0.032 0.002

Table 4.4: Cross sections for outcome classes in triple-single scattering atv ˆ = 1. Cross sections for “Exchange + ionization” are not included in either “Exchange” or “Single ionization.” However, the cross sections for new triples are not independent of the other cross sections. (I.e., they are a sum of subsets from the other classes.) See Section 4.3.2 for definitions of the outcome classes.

87 Triple-binary

Outcome classσ ˆ ∆statσˆ Exchange 0.036 0.003 Quadruple formation 0.000 2e-04 Scramble 0.006 0.001 Binary disruption 0.996 0.016 Double binary formation 1.250 0.017 Triple disruption 1.303 0.018 Full ionization 0.000 2e-04 New triple 0.127 0.006

∆sysσˆ 0.045 0.003

Table 4.5: Cross sections for outcome classes in triple-binary scattering atv ˆ = 1. Cross sections for “Exchange + ionization” are not included in either “Exchange” or “Single ionization.” However, the cross sections for new triples are not independent of the other cross sections. (I.e., they are a sum of subsets from the other classes.) See Section 4.3.2 for definitions of the outcome classes.

88 inner binary. Weak perturbations from the interloper are therefore likely to ionize the tertiary. Other classes of outcomes generally require some interaction with the stars of the inner binary. The probability of such an interaction scales with the 2 orbital area of the inner binary, so we observe thatσ ˆ α− . ∝

For the case of triple-binary scattering we fix the initial value of a1 to be equal to a2 as we vary the semi-major axis, α. At large α, binary disruption is the dominant outcome whereas at small α double binary formation dominates. In the low-α limit the triple comes closer to instability, so single ionization from the triple is also likely to occur in addition to the disruption of the interloping binary, leading to double binary formation. In the large α case the scattering problem approaches that of binary-binary scattering in which one binary is more massive than the other. Earlier experiments have shown (e.g., Fregeau et al. 2004) that the less massive binary is more likely to be disrupted, in accordance with our results here. In this limit we might na¨ıvely expect no dependence on α since we fix a1 = a2. Instead 1 we observe a dependence of σ α− . This is because as α increases the incoming ∝ velocity (which, in the large-α limit is comparable to the orbital velocity of the inner binary of the triple) becomes larger relative to the orbital velocities of the outer 2 binary and interloping binary. In the limit of large incoming velocitiesσ ˆ vˆ− (Hut ∝ 1/2 1 & Bahcall 1983), and sincev ˆ a− , we have thatσ ˆ a α− . For the same ∝ ∝ 11 ∝ reason the scaling of the single ionization cross section in triple-single scattering exhibits the same dependence.

Eccentricity

We next vary the eccentricity of binaries in the systems, both individually and together while holding constant all other parameters (e.g., semi-major axis ratio and masses). To increase the range over which we can vary the outer eccentricity in triple scattering we set the outer semi-major axes to 20 AU (i.e., we use α of 20 instead of

10 as before). We vary the outer eccentricity of the triples (e1) from 0 to 0.7 and the eccentricity of the inner binary (e11) from 0 to 0.98. Even with the larger value of α at outer eccentricities larger than 0.7 the triple violates the Mardling stability ∼ criterion (equation 4.14). In the case of triple-binary scattering we fix e1 = e2. In the case of binary-binary scattering we vary the eccentricities of either one binary

89 Fig. 4.3.— Cross sections of various classes of outcomes as a function of semi-major axis ratio, α, for binary-binary (left panel), triple-single, (middle panel), and triple- binary (right panel) scattering. In the limit of large semi-major axis ratios for triple- single scattering, interactions involving the inner binary should scale like the area of the inner orbit (i.e., an inverse square dependence on the semi-major axis). Indeed, the cross sections for scrambles and exchange + ionization (both of which require the interloper interact with the inner binary) are consistent with the inverse square dependence shown (black dashed line). Single ionization is the dominant outcome of triple-single scattering at all semi-major axes. At small α the triple becomes unstable to ionization from weak perturbations due to the interloper and at large α the tertiary is weakly bound relative to the inner binary so weak perturbations from the interloper lead to ionization. See Section 4.3.2 for a discussion of the origin of the α dependence of different cross sections.

90 or both binaries from 0 to 0.98. The cross sections for triple-single, triple-binary, and binary-binary scattering are shown in Fig. 4.4. In the case of binary-binary scattering the cross sections are completely independent of the eccentricity. In the case of triple scattering the cross sections are independent of the inner eccentricity and are generally independent of the outer eccentricity, although there is a modest increase in the cross section for single ionization in the case of triple-scattering and in the cross section for double binary formation in the case of triple-binary scattering.

The physical mechanism behind these dependences is twofold. First, there is some baseline cross section for the outcome which is independent of any eccentricity. Hut & Bahcall (1983) argue that this is because the cross sections for a particular orbit are only dependent on the average velocity of the orbit, which is independent of the eccentricity as a result of the virial theorem. This baseline cross section applies to all outcomes. Second, in the case of triple scattering there is an additional contribution to the cross section of ionization-like outcomes (i.e., single ionization in the case of triple-single scattering and double binary formation in the case of triple-binary scattering) due to perturbations to the eccentricity of the outer orbit of the triple, e1. If e1 exceeds some critical eccentricity, ecrit, the triple will violate the Mardling stability criterion and will become unstable. As we show in Section 4.5.1, and in particular in panels b) of Fig. 4.11, the logarithm of the cross section for cumulative changes in the eccentricity follows the inverse of the Gompertz function:

∆e = exp ( c exp (c lnσ ˆ)) = exp( c σˆc2 ) , (4.15) 1 − 1 2 − 1 where c1 and c2 are positive constants. This implies that the cross section to undergo an ionization-like reaction takes the form

1 1/c2 σˆ =σ ˆ +( c− ln(e e )) , (4.16) 0 − 1 crit − 1 whereσ ˆ0 is the baseline cross section described above. Taking ecrit from the Mardling stability criterion (e 0.733), we find an excellent match between this functional crit ≈ form and the cross sections for ionization-like outcomes in Fig. 4.4. Best-fitting parameters are shown in Table 4.7.

Because ein does not influence the stability of the triple, we do not see a similar effect when ein is varied. This result that the cross sections are independent of

91 Binary-binary

Outcome classσ ˆ ∆statσˆ Exchange 0.302 0.008 Triple formation 0.020 0.002 Single ionization 2.646 0.023 Full ionization 0.000 0.000 Exchange + ionization 3.259 0.025

∆sysσˆ 0.002 0.001

Table 4.6: Cross sections for outcome classes in binary-binary scattering atv ˆ = 1. Cross sections for “Exchange + ionization” are not included in either “Exchange” or “Single ionization.” However, the cross sections for new triples are not independent of the other cross sections. (I.e., they are a sum of subsets from the other classes.) See Section 4.3.2 for definitions of the outcome classes.

Scattering type panelσ ˆ0 c1 c2 Triple-single b 0.35 6.21 0.57 Triple-single c 0.34 6.05 0.60 Triple-binary b 0.23 3.45 0.35 Triple-binary c 0.14 4.88 0.39

Table 4.7: Best-fitting parameters of the eccentricity cross sections in Fig. 4.4 to the inverse Gompertz function. See equation (4.16) for the definition of the parameters.

92 the eccentricities, except when related to the stability of the triple, is consistent with the results of binary-single scattering experiments, which have also found that cross sections are independent of eccentricity (e.g., Hut & Bahcall 1983). We note, however, that is only the cross sections of strong interactions that are independent of eccentricity. Heggie & Rasio (1996) found that the magnitude of small secular perturbations to the eccentricity of a binary from a distant flyby is proportional to e√1 e2. − Mass ratio

We show in Fig. 4.5 the effect of changing the mass of one component of the system in binary-binary, triple-single, and triple-binary scattering. In the case of triple scattering the mass of one component of the inner binary of the triple is varied. All other parameters (e.g., α, eccentricities, andv ˆ) are held fixed. The mass of the one star is varied from 0.03 M to 40 M . For small mass ratios (i.e., as the test particle ⊙ ⊙ limit is approached) the cross section for all outcomes in triple scattering increases by roughly an order of magnitude. In the high mass limit single ionization becomes the dominant outcome of binary-binary scattering, rising roughly with the square root of mass, and binary disruption becomes the dominant outcome of triple-binary scattering with roughly the same mass dependence. Conversely, although single ionization is the dominant outcome at high mass for triple-single scattering, the cross section decreases, nearly with the square of the mass.

Hut (1983) derived the mass dependence of the ionization cross section for 2 binary-single scattering in the largev ˆ limit and found it to be proportional to m− when the most massive body resides in the binary and proportional to m2 when the most massive body is the interloping star (see equation 4.17). In the large mass limit of binary-binary and triple-binary scattering, the mass dependence should be similar to the case of binary-single scattering with a massive interloping star. Likewise, the case of triple-single scattering should exhibit a similar mass dependence to binary-single scattering with a massive binary. However, we are not in the high velocity limit (ˆv = 1) so we expect the mass dependencies to soften, which is what we observe. Nevertheless, the trend of increased ionization cross sections for low-mass

93 Fig. 4.4.— Cross sections of outcome classes for triple-single (top row), triple-binary (middle row), and binary-binary (bottom row) scattering as a function of eccentricity. For triple scattering we vary the inner eccentricity alone (panels a), the outer eccentricity alone (panels b), and the inner and outer eccentricities simultaneously (panels c). In the case of triple-binary scattering we set the eccentricity of the incoming binary to be equal to that of the tertiary. In the case of binary-binary scattering we vary the eccentricity of one binary only (left panel) and both binaries simultaneously (right panel).

94 interloping binaries and decreased ionization cross sections for low-mass interloping single stars holds even in the moderate velocity regime.

Incoming velocity

We finally examine the effect of the velocity at infinity of the interloping system. We measure these velocities relative to the critical velocity of the system (see equations 4.11 and 4.12) and explore a range of two dex centered around vcrit. It is difficult to explore a much larger range because velocities much larger than 10vcrit yield very small cross sections and therefore have large statistical uncertainties.

Velocities much smaller than 0.1vcrit require an excessive amount of computing time because the low total energy of the system means that it takes a long time for the system to random walk to a region of phase space where one object has a sufficient amount of energy to escape and leave behind a stable system.

The cross sections are presented in Fig. 4.6. Hut (1983) found two high-velocity limits in binary-single scattering: the cross section for ionization, which has a 2 dependence ofσ ˆ vˆ− and the cross section for exchange, which has a dependence ∝ 6 ofσ ˆ vˆ− . Although there are many more possible outcomes in binary-binary, ∝ triple-single, and triple-binary scattering, the behavior of the cross sections is qualitatively similar to binary-single scattering, in that at high velocity the outcomes 2 may be grouped into those with av ˆ− dependence (“ionization-like” outcomes), and those with a steeper dependence (“exchange-like” outcomes). In some cases 6 exchange-like outcomes follow the samev ˆ− dependence as in binary-single scattering (e.g., triple formation in binary-binary scattering and quadruple formation in triple-binary scattering), but in others the velocity dependence is shallower (e.g., 4 exchanges in binary-binary scattering, which display av ˆ− dependence). We discuss these results in more depth in Section 4.4.2.

4.4. Analytic approximations

We here develop analytic approximations to the cross sections computed in Section 4.3. We first review binary-single scattering, and then generalize the theory of binary-single scattering to four-body scattering.

95 Fig. 4.5.— Cross sections of outcome classes for binary-binary (left panel), triple- single (middle panel) and triple-binary (right panel) scattering as a function of the mass ratio of one component of the binary (in the case of binary-binary scattering) or of one component of the inner binary (in the case of triple scattering) to all other components of the system. The masses of all other components of the system remain fixed and equal to each other. In the high mass limit the more massive system tends to disrupt the incoming system. This leads to an increase in the cross section for single ionization in binary-binary and triple-binary scattering. In the case of triple-single scattering all cross sections decrease because the incoming system cannot disrupt.

Fig. 4.6.— Cross sections of outcome classes for binary-binary (left panel), triple- single (middle panel) and triple-binary (right panel) scattering as a function of the incoming velocity. The velocities are written relative to the critical velocity of the system (see equations 4.11 and 4.12). Outcomes which require an ionization from 2 one system follow a v− dependence at high velocities, whereas those requiring an exchange follow a steeper dependence. The kink in the cross section for single ionization is explained in Section 4.4.2.

96 4.4.1. Three-body scattering

The analytic theory of binary-single scattering of point masses was developed by Heggie (1975) and Hut (1983) and has been found to agree with numerical scattering experiments quite well (Hut & Bahcall 1983). Although the derivation is complicated, the results are straightforward to summarize. In binary-single scattering of point masses only three classes of outcomes are possible: (1) a flyby, (2) an exchange, or (3) an ionization. It is not possible to form a stable triple from binary-single scattering (Chazy 1929; Littlewood 1952; Heggie 1975; Heggie & Hut 2003). The cross section of a flyby is not well defined, but Hut (1983) explicitly calculated the cross section for the other two classes in the high velocity limit (ˆv 1) and found them to be ≫ 3 40 m2 1 σˆion = 2 , vˆ 1 (4.17) 3 m11m12(m11 + m12 + m2) vˆ ≫ and

2 4 20 m (m + m2) 1 σˆex = 3 3 6 , vˆ 1 (4.18) 3 m2(2m + m2) vˆ ≫ where the latter is only valid for equal mass binaries, m m = m . Hut (1983) ≡ 11 12 found that both cross sections are independent of eccentricity.

In the low velocity limit (ˆv 1) the cross sections cannot be computed with ≪ precise numerical coefficients. Nevertheless, the velocity scaling in the low velocity case is simple because ionization is not possible and the only mechanism to change the cross section is gravitational focusing. This is because at low velocities the speed of the interloping star when it is close enough to interact strongly with the binary will be very close to the escape speed. Thus for any low velocity system, the incoming velocities when the system begins strong interactions will be nearly the same, and the cross section will scale as 1 σˆ , vˆ 1. (4.19) ∝ vˆ2 ≪

The cross sections may be combined by taking the reciprocal of the sum of the reciprocals of the cross sections in the two extreme limits. This is because

97 different physical effects limit the rate of production of particular outcomes and the combined effect of rate limiting processes is generally the reciprocal of the sum of the reciprocals. Because the mass terms are in general not known, we introduce normalization factors (ˆσa,σ ˆb, etc.), which can be determined empirically. The cross section for exchange is then 1 vˆ2 vˆ6 = + . (4.20) σˆex σˆa σˆb The validity of this functional form is demonstrated in Fig. 4.7 where we fit equation (4.20) to numerical exchange cross sections (blue dashed line) for binary-single scattering of equal mass stars.

The cross section for full ionization is necessarily zero forv

√13 1 (√13 1)/2 σˆε 1 ε − (ˆv 1) − . (4.21) ≪ ∝ ∝ − Therefore the general cross section for ionization is

(1 √13)/2 2 1 (ˆv 1) − vˆ = − + . (4.22) σˆion σˆc σˆd

Note, however, that the exponent on theσ ˆc term is only valid in the case of approximately equal masses. If one object contains more than 1/2 of the mass, ∼ the dominant orbital configuration leading to full ionization will change, leading to a different exponent (Sweatman 2007). Sweatman (2007) lists exponents for other configurations, but this exponent could also be determined empirically. The validity of the combined cross section for ionization is also illustrated in Fig. 4.7 (red dashed line).

The collision cross section

The dependence of the collision cross section in few-body scattering problems on the incoming velocity and stellar radius has been investigated by a number of studies (e.g., Fregeau et al. 2004; Leigh & Geller 2015), but an analytic fit to these dependences does not exist in the literature. We present such a fit here in the case of equal mass stars.

98 103 exchange 102 ionization 101 100 ˆ σ 10−1 10−2 10−3 10−4 10−1 100 101 vˆ∞

Fig. 4.7.— Normalized cross sections for exchange (blue circles) and ionization (red squares) in equal-mass binary-single scattering as a function of incoming velocity (cf. Fig. 5 of Hut & Bahcall 1983). The dashed lines are an empirical fit to equations (4.20) and (4.22). The analytic approximation is based on a combination of the low velocity limits (ˆv 1 for exchange andv ˆ 1 1 for ionization) and ≪ − ≪ the high velocity limits (ˆv 1). Values for the normalization parametersσ ˆ ,σ ˆ , ≫ a b etc. are estimated from the numerical data with best-fitting values of σ 9.76, a ≈ σ 28.9, σ 12.8, and σ 10.3. The good agreement between the data and the b ≈ c ≈ d ≈ approximation for allv ˆ implies that the two limits capture all the relevant physics of binary-single scattering.

99 Collisions can occur in two different ways: through a three-body interaction or through a two-body interaction. In a three-body interaction the interloping star exchanges momentum with one star of the binary before eventually colliding with the other. In a two-body interaction the interloping star collides into one of the stars of the binary without interacting with the other at all.

For three-body interactions, the collision cross section scales with velocity in the same way that the cross section for exchange does. That is, forvv ˆ crit the velocity scaling isv ˆ− . The low-velocity scaling is due to gravitational focusing, whereas the high-velocity scaling is likely because the mechanism by which collisions occur in three-body scattering events is similar to the mechanism by which exchanges occur. Exchanges occur when two stars undergo a close approach and exchange momenta (Hut 1983). This ejects one star formerly in the binary and replaces it with the interloping star on a similar orbit. A collision may occur if the momentum exchange places the interloping star on an orbit eccentric enough to lead to a collision. Since the mechanism for collision 6 is similar, we would then expect the collision cross section to display the samev ˆ− velocity dependence as the exchange cross section. Note, however, that this is simply a plausibility argument. We do not attempt here to show rigorously that this is the mechanism responsible for the observed velocity dependence. Nevertheless, the 6 numerical calculations indicate that the cross sections indeed show av ˆ− velocity dependence in this regime just like exchange.

We may combine these velocity dependences as we did in Section 4.4.1 to find the overall cross section: 1 vˆ2 vˆ6 = + , (4.23) σˆ3-body σˆa σˆb whereσ ˆa andσ ˆb are constants of order unity that must be fit to numerical data.

Let us consider now two-body interactions. Like three-body interactions, these exhibit two velocity regimes. However, in this case the two velocity regimes are separated by the escape velocity of the stars, v . Forv ˆ v , gravitational esc ≫ esc focusing from the individual stars is negligible and the cross section for collision will approach a constant value equal to the geometric cross section of the stars. For

100 velocities less than vesc, gravitational focusing from individual stars will lead to a 2 vˆ− dependence. For two-body interactions, then, the overall cross section will be σˆ σˆ =σ ˆ + c , (4.24) 2-body geometric vˆ2 where the geometric cross section is just given by

R 2 σˆ = 2π , (4.25) geometric a µ ¶ 2 andσ ˆc must be fit to numerical data, though it will be of orderσ ˆgeometricvˆesc.

The collision cross sections for two- and three-body interactions may then be combined additively to yield the overall collision cross section:

1 vˆ2 vˆ6 − σˆ σˆ =σ ˆ +σ ˆ = + + c +σ ˆ . (4.26) coll. 2-body 3-body σˆ σˆ vˆ2 geometric µ a b ¶ A fit of equation (4.26) to numerical data is shown in Fig. 4.8. There is an excellent match between the numerical data and the analytic fit.

4.4.2. Four-body scattering

Four-body scattering can occur through binary-binary or triple-single scattering.3 Because the number of objects interacting is the same in both cases, they generally may be studied in the same framework. Although there are 22 distinct possible outcomes of four-body scattering, we here examine the velocity dependences of three broad classes of outcomes: (1) ionization, including single and full ionization; (2) exchange; and (3) triple formation.

It is instructive to study the velocity dependence of ionization in the extreme semi-major axis ratio limit. If the semi-major axes between the two binaries are very different or the semi-major axis ratio of the triple is very large, then the problem can be considered to be a three-body scattering problem in two limits: (1) at low velocities the smaller binary can be considered a point mass and (2) at high

3In principle a simultaneous interaction between a binary and two independent single stars or a simultaneous interaction between four single stars are also possible, but such events are rare and so we do not study them here.

101 Fig. 4.8.— The cross section for a collision to occur in binary-single scattering as a function of the normalized incoming velocity. We show both numerical data (gray points) and a fit to equation (4.26) (black dashed line). This system consists of three 1 M stars each with a radius of 0.3 R . The binary orbit is circular with a semi- ⊙ ⊙ 2 major axis of 1 AU. The best-fitting parameters areσ ˆ 0.475,σ ˆ 2.00 10− , a ≈ b ≈ × 2 5 σˆ 2.25 10− , andσ ˆ 1.61 10− . c ≈ × geometric ≈ ×

102 velocities one component of the larger binary can be ignored. There are therefore two critical velocities in the four-body scattering problem: the critical velocity of the wide binary scattering off of interloping system,v ˆcrit,1, and the critical velocity of the interloping system scattering off of one component of the wide binary,v ˆcrit,2.

In the limit that the semi-major axis of the smaller binary approaches zero, the cross section for single ionization will appear like the cross section for ionization in three-body scattering given by equation (4.22) with the only difference being that one object in the three-body scattering event has a mass twice as great as the others because it is a binary. The effect of this is to change the exponent on the (ˆv 1) − term from √13 1 to (2+ √10)/8 (section 4.3 of Heggie & Sweatman 1991), and to − change the base from (ˆv 1) to (ˆv vˆ ). In the limit that the semi-major axis − − crit,1 of one binary approaches zero the cross section for single ionization approaches zero forv

1 (2+√10)/8 2 − σˆa (ˆv vˆcrit,1) vˆ σˆion = 2 + − + . (4.28) vˆ Ã σˆb σˆc ! In Fig. 4.9 we present the cross section for single ionization for binary-binary scattering with a semi-major axis ratio of 100, equal masses, and circular orbits. We fit the numerical data to the functional form of equation (4.28) and find an excellent match. (See the caption of Fig. 4.9 for the best-fitting parameters.)

To study exchange and triple formation we return to more moderate semi-major axis ratios: unity for binary-binary scattering, and 10 for triple-single scattering. As discussed in Section 4.4.1, Hut (1983) derived the high-velocity exchange cross

103 Fig. 4.9.— Cross sections for outcomes of equal mass binary-binary scattering with a semi-major axis ratio of 100 and circular orbits as a function of incoming velocity. We fit the cross sections for single ionization to equation (4.28) (dashed line). The 2 2 best-fitting parameters areσ ˆ 2.16 10− ,σ ˆ 3.47, andσ ˆ 8.08 10− . The a ≈ × b ≈ c ≈ × ionization cross section consists of two components: a component similar to the full ionization cross section in binary-single scattering (curved dot-dashed line), and a component due to hardening of the inner binary (straight dot-dashed line). When summed, these two components produce a kink at velocities intermediate between vˆcrit,1 andv ˆcrit,2 (dotted lines).

104 6 section to be proportional tov ˆ− for binary-single scattering. We find from our numerical experiments in Section 4.3.2 that the high-velocity exchange cross section 4 for exchange is shallower, being instead proportional tov ˆ− for binary-binary and triple-single scattering (Fig. 4.6, black points).

In binary-binary scattering new triple formation occurs via an exchange. We find empirically that the velocity dependence for the new triple cross section is 6 proportional tov ˆ− in the high velocity limit for binary-binary scattering (blue points, Fig. 4.10). In triple-single scattering, exchanges are the dominant channel for new triple formation at low velocities and velocities slightly larger than the critical velocity. In the high velocity limit the cross section for new triple formation via 6 exchange is likewise proportional tov ˆ− for triple-single scattering.

In the case of triple-single scattering new triple formation may also occur via a scramble. In the high velocity limit the cross section for scrambles is proportional to 2 vˆ− (red points, Fig. 4.10). This is because a scramble is, in a sense, an incomplete ionization. The interloping star imparts enough energy to one star of the inner binary such that its semi-major axis becomes much larger than that of the tertiary of the original system, but not enough energy to ionize it. At low velocities a scramble is a rare outcome, but due to the shallower velocity dependence, at sufficiently high velocities it becomes the dominant mechanism by which to form new triples in triple-single scattering.

We may combine these velocity dependences to obtain the general velocity dependence of the cross section for triple formation. For binary-binary scattering, the cross section has an identical form to the exchange cross section in binary-single scattering, except that the break occurs nearv ˆcrit,1 rather thanv ˆ = 1 (this is reflected in the smaller best-fitting value forσ ˆb): 1 vˆ2 vˆ6 = + . (4.29) σˆnew trip. σˆa σˆb For triple-single scattering the cross section is similar, except that an additional high 2 velocityv ˆ− term must be added to account for scrambles:

1 σˆ vˆ2 vˆ6 − σˆ = a + + . (4.30) new trip. vˆ2 σˆ σˆ µ b c ¶ 105 We present a fit of the new triple cross section to equations (4.29) and (4.30) in Figure 4.10. The match between equation (4.29) and the binary-binary scattering cross sections is excellent. The match between equation (4.30) and the triple-single scattering cross sections is good, but we note that at high velocities the scramble 3/2 cross section appears to be better fit with av ˆ− dependence. We present this alternative velocity dependence in Figure 4.10 in gray. At present we do not have an analytic understanding of this shallower slope.

While the normalization parametersσ ˆa,σ ˆb, etc., should be determined by fitting equations (4.28), (4.29), and (4.30), for systems not too dissimilar from those presented in Figures 4.9 and 4.10 the ratios between the normalization parameters should be approximately constant. The overall scale can then be estimated from the dependences of the cross sections in Figures 4.3–4.5. For systems far from the systems presented in Figures 4.9 and 4.10, the normalization parameter may be quite different from these estimates because they may depend on each other in a nonlinear way.

4.5. Orbital parameter distributions after scattering

Scattering events can affect the orbital parameter distribution of triples in two ways: by perturbing the orbital elements of an existing triple in a flyby, and by creating a new triple system. We are particularly interested in determining the distribution of orbital parameters that govern the strength and timescale of KL oscillations, namely cos i, α, and eout. We first examine the case of flybys in Section 4.5.1 and then examine newly formed triples in Section 4.5.2.

4.5.1. Changes to the orbital parameters from flybys

Cross sections for flybys are not well defined because they diverge for infinitesimally small perturbations. As such, we instead calculate the cumulative cross section, σ(X > x), defined to be the cross section for a change in parameter X by at least x. From our experiments involving the model system (see Section 4.3.1) we select those which result in a flyby. We then calculate the cumulative cross section for generating changes in the orbital parameters of a given magnitude or larger. These cross sections

106 Fig. 4.10.— The velocity dependence of the cross section for new triple formation for binary-binary and triple-single scattering of our model system (see Section 4.3.1). We fit the velocity dependence to equations (4.29) and (4.30) (black dashed lines). At high velocities the triple formation cross section from binary-binary scattering is 6 proportional tov ˆ− . At velocities slightly above the critical velocity, the new triple 6 formation cross section from triple-single scattering carries the samev ˆ− dependence, but at higher velocities, new triple formation is dominated by scrambles, which are 2 ionization-like and would therefore carry av ˆ− dependence. We note, however, that 3/2 the triple-single data are better fit by av ˆ− dependence (gray dashed line). For binary-binary scattering our best fitting parameters areσ ˆ 0.567 andσ ˆ 0.0279. a ≈ b ≈ 2 For triple-single scattering our best fitting parameters assuming av ˆ− dependence for 3/2 scrambles isσ ˆ 0.247,σ ˆ 0.0402, andσ ˆ 0.0133. Assuming av ˆ− dependence a ≈ b ≈ c ≈ 3 for scrambles the best fitting parameters areσ ˆ 0.257,σ ˆ 0.044,σ ˆ 8.89 10− . a ≈ b ≈ c ≈ ×

107 are shown in Fig 4.11. The normalized cross section for inducing a change in the semi-major axis of order unity isσ ˆ 0.54, which is very close to the cross section for ≈ single ionization of 1.48. That the two cross sections should be comparable is to be expected since there is only a small difference in energy in changing α by order unity and in ionizing the object. The logarithm of the cumulative cross section for changes 4 to e1 is well described by the inverse of a Gompertz function (cf. equation 4.15):

∆e = exp ( c exp (c lnσ ˆ)) = exp( c σˆc2 ) . (4.31) 1 − 1 2 − 1 We present a fit to the cumulative cross sections in Fig. 4.11, but the difference between the two is typically smaller than the width of the lines (the best-fitting parameters are presented in the caption to Fig. 4.11). The cross section for large changes to the eccentricity is small (ˆσ 1 for 1 ∆e 1) because to remain ≪ − 1 ≪ stable such systems need to gain a large amount of energy and are therefore more likely to be ionized than remain bound. The cross section for moderate changes to the eccentricity (∆e 0.5) is of order unity (ˆσ 1), similar to the case for a 1 ∼ ∼ change in α. The cumulative cross section for changes to eout drops more steeply for ∆eout > 0.7. This is due to the fact that the initial triple is unstable for ∼ eout 0.7 (see equation 4.14). To produce ∆e > 0.7 two things are required: (1) the ∼ ∼ semi-major axis ratio must increase by enough that at the final eout the triple is no longer unstable, and (2) eout must change by ∆eout. The addition of requirement

(1) induces the sharper drop in the cross section for ∆eout > 0.7. To check if ∼ the form of equation (4.31) is independent on the assumption of equal masses we repeated the calculations used to produce panel b) of the upper row of Figure 4.4 and panel b) of the upper row of Figure 4.11 using the following sets of masses: 6 (m ,m ,m ,m )in (4, 3, 2, 1), (1, 2, 3, 4), and (1, 10− , 1, 1) . We found 111 112 12 2 { } equation (4.31) to be an excellent fit in all cases. While this does not prove that equation (4.31) is necessarily valid for all possible mass ratios, it demonstrates that the form of equation (4.31) is not strongly dependent on the assumption of equal masses. 4A Gompertz function is defined to be any function of the form

−be−ct f(t) = ae , where a is the asymptote and b and c are positive constants.

108 Fig. 4.11.— The cumulative cross section for changes to the orbital parameters from triple-single flybys (top row) and triple-binary flybys (bottom row) with incoming velocities ofv ˆ = 1. We show here changes to the semi-major axis ratio (panels a), the outer eccentricity (panels b), and the mutual inclination (panels c). The cross sections presented are the cross sections for a change in the orbital parameter of that magnitude or greater. The 1-σ confidence interval is shown by the gray shaded region, although typically these uncertainties are smaller than the width of the lines. We fit the ∆e1 cumulative cross sections to a Gompertz function (panels b, dotted line) and find an extremely close match. Except for very small or large

∆e1 the difference between the numerical data and the fit is smaller than the width of the line. Since isolated triples can undergo changes to their inclination due to KL oscillations (an effect which is necessarily present when we perform our scattering experiments), we compare the magnitude of this effect to the cross sections due to scattering by calculating the evolution of an identical set of triples for an identical length of time (panels c, dashed line). The difference between the scattering cross section and the cross section from the isolated triples is shown by the thin black line with 1-σ uncertainties. KL oscillations are unable to produce large changes in the inclination because the flyby timescale is much shorter than the KL timescale.

109 The change in the inclination is particularly interesting due to its relevance for inducing KL oscillations in a triple system. However there is a confounding effect in the cumulative cross section for this orbital parameter. Since some triples will begin at high inclinations, their inclination can change in isolation due to KL oscillations. This is a difficult effect to disentangle from changes that are solely due to scattering or perturbations from interloping stars. To estimate the magnitude of this effect, we evolve a sample of triple systems identical to those used in our model systems but without any interloping stars for the same length of time as our flyby calculations. We then calculate the cumulative distribution of the change in inclination to compare its magnitude to the distribution when an interloping star is present. Small changes in inclination can be accounted for almost entirely by KL oscillations. Because these scattering calculations complete in 10% of the KL timescale, t , KL oscillations ∼ KL cannot produce large changes in inclination. Thus, the cumulative cross sections for large changes in inclination are entirely due to the interloping star. In more compact systems, however, the KL timescale may be comparable to the timescale of the scattering event, in which case either scattering or KL oscillations would be able to produce large changes in inclination. The distributions of positive and negative changes in cos i are equal to within the statistical uncertainty of our results.

We additionally show the relationship between changes to the mutual inclination and changes to eout in Figure 4.12. To demonstrate the magnitude of the effect of KL oscillations we also include the relationship between changes to the inclination 2 and eout from the isolated triples. Flyby events for which ∆ cos i < 10− do not ∼ exhibit a strong correlation between the change to the inclination and eout. In these systems the dynamics of the isolated triple therefore overwhelm the contributions 2 of the interloping star. Flyby events for which ∆ cos i > 10− do exhibit a strong ∼ correlation between the change in the inclination and eout, albeit with substantial scatter. This result is consistent with other dynamical studies (e.g., Li & Adams 2015).

A note on convergence

The issue of whether our calculations are converged in the maximum impact parameter, bmax, is subtler in flyby events than the convergence analysis presented in

110 Fig. 4.12.— The relationship between changes to the mutual inclination and changes to eout from flyby events. The darkness of the hexagonal bins corresponds to the number of events, binned logarithmically. To demonstrate the magnitude of KL oscillations we include this relationship for several isolated triples (green points). 2 For events in which there is a change to cos i less than 10− there is not a strong ∼ correlation between ∆ cos i and eout. Changes to the orbital parameters are due primarily to KL oscillations rather than the dynamical effect of the interloping star. 2 For events in which there is a change to cos i greater than 10− there is a strong ∼ correlation between ∆ cos i and ∆eout, though with substantial scatter.

111 Section 4.2.3, so we revisit it here. In Section 4.2.3 we considered the convergence of the cross sections of outcomes from “strong” interactions (i.e., interactions for which the hierarchy changed). In flyby events, however, arbitrarily distant encounters can have arbitrarily weak dynamical effects, so our calculations can never be truly converged in bmax. We can, however, determine the magnitude of the change of orbital parameters down to which our calculations are converged by calculating the cumulative cross sections using a variety of bmax’s. We present the results of this calculation for changes to the semi-major axis ratio, α, in Figure 4.13.

2 Our results are converged down to relative changes in α of 10− . The cross ∼ section fro smaller changes to α diverges, however. This is due to the fact that the secular approximation is not perfectly valid, and the semi-major axes of the inner and outer orbits of an isolated triple will exhibit small variability over time. We demonstrate the scale of this variability by integrating an isolated triple in Figure 4.14. This scale corresponds with the point at which the cross sections diverge in Figure 4.13. Note, however, that the relative size of this variability is dependent on the orbital parameters. In particular, as α , the relative size of → ∞ this variability will approach zero. Calculation of the cross sections of changes to the orbital parameters from flybys of triples with very large α may therefore require a larger bmax than given by equation (4.9).

4.5.2. Distribution of orbital parameters in dynamically formed triples

The scattering experiments presented in Section 4.3 demonstrated that some fraction of binary-binary, triple-single, and triple-binary scattering events produce new triple systems. Because this fraction is relatively small, we here run three larger suites of scattering experiments to obtain better statistics on the distribution of orbital parameters of dynamically formed triples. One suite consists of binary-binary scattering experiments, another of triple-single experiments, and the last of triple- binary experiments. In the case of triple-binary scattering the interloping binary has a semi-major axis equal to the outer binary of the triple. We set the semi-major axis ratio of the triples to be 10 and set a1 = a2 in the case of binary-binary scattering.

112 Fig. 4.13.— The cumulative cross section for changes to the semi-major axis ratio, α, 2 for a variety of choices of b . For changes larger than ∆α/α 10− our calculations max 0 ∼ 2 are converged. For changes to α smaller than 10− the cross sections diverge because ∼ α in an isolated triple will vary on this scale over time (see Figure 4.14).

113 Fig. 4.14.— The semi-major axis ratio, α over time for an isolated triple with initial orbital parameters of: a11 = 1 AU, a1 = 10 AU, e11 = e1 = 0, i = 70◦. The relative 3 variation of α is few 10− , in agreement with the point of divergence seen in ∼ × Figure 4.13.

114 We furthermore set all masses to be equal and set an incoming velocity of v = vcrit. We do not include the orbital parameters of triples resulting from flybys in these distributions as these were studied in Section 4.5.1. To ensure that the resulting triples are stable we run them in isolation for 100 outer orbital periods and reject those which do not maintain the same hierarchy. (E.g. we reject systems which begin in the configuration [[0 1] 2] but end in the configuration [[0 2] 1]). This is similar to the stability test used by Mardling & Aarseth (1999).

We must be careful when calculating the inclination distribution. Because the calculations can often run for times comparable to tKL, KL oscillations can bias the inclination distribution towards the Kozai angles of 39◦ and 141◦. To correct for this we run the triples in isolation for half the length of time that the scattering event was computed; this is approximately the length of time that the triple has existed for. We then use the inclination closest to 90◦ in the inclination distribution because the triple spends most of its time at this inclination. (For example, an equal mass triple with an initial inclination of 85◦ spends over 80 per cent of its time within 10◦ of its initial inclination.)

The distribution of orbital parameters is shown in Fig. 4.15. The most important feature of these distributions is that scattering produces triples which are extremely compact, in that the ratio of the distance at periapsis of the outer orbit, r , to a is quite small, typically 10. Indeed, the vast majority of these peri, out in ∼ triples are on the verge of instability.

Because these triples are so compact, they have extremely large ǫoct. This parameter is a measure of the strength of octupole-order contributions to the KL oscillations and is defined to be (Lithwick & Naoz 2011; Naoz et al. 2013a) 1 e ǫ out . (4.32) oct ≡ α 1 e2 µ − out ¶ Triples with larger ǫoct have stronger eccentric KL oscillations. In particular, 2 Lithwick & Naoz (2011) note that for ǫoct > 10− the parameter space for orbital flips (and hence arbitrarily large eccentricities in the inner binary) widens dramatically. 2 Our results indicate that nearly all dynamically formed triples have ǫoct > 10− . We note that the particular triples formed from these experiments do not undergo typical

115 Fig. 4.15.— Orbital parameters of dynamically formed triples from binary-binary (solid lines), triple-single (dashed lines), and triple-binary (dotted lines) scattering. The systems consist of 1 M stars in circular orbits with a semi-major axis of 1 AU ⊙ in the case of binary-binary scattering, and ain = 1 AU and aout = 10 AU in the case of triple scattering. We have excluded triples whose final hierarchy is identical to their initial hierarchy. Upper left: the inclination distribution for dynamically formed triples. We correct the inclinations for KL oscillations; this panel shows the minimum cos i that is achieved by KL oscillations. Upper middle: the distribution of inner eccentricities. Upper right: The distribution of outer eccentricities. Lower left: the distribution of semi-major axis ratios. Lower middle: the distribution of distances at periapsis relative to the inner semi-major axis. Lower right: the distribution of

ǫoct, a measure of the strength of the octupole order term in the Hamiltonian.

116 eccentric KL oscillations because they are of equal mass, so the octupole order terms of the Hamiltonian vanish (e.g., Ford et al. 2000). Nevertheless, in extremely compact triples non-secular effects can drive the inner binary to arbitrarily large eccentricities (e.g., Antonini et al. 2014; Bode & Wegg 2014; Antognini et al. 2014). Moreover, as we show in Section 4.5.3, these results for the dynamical formation of triples are generic and apply to unequal mass triples.

The inclination distributions appear to be approximately uniform, but are clearly biased towards inclinations less than 90◦, i.e., prograde orbits. This is curious because we began with a distribution of inclinations uniform in cos i (i.e., as many retrograde orbits as prograde), but many studies have found that retrograde orbits are generally more stable than prograde orbits (e.g., Harrington 1972; Mardling & Aarseth 2001; Morais & Giuppone 2012). It is possible that scattering is biased towards producing triples in prograde orbits and there remains an excess of prograde orbits after the unstable triples have dissociated.

The inner eccentricity distributions have two components, one of which is approximately thermal for high ein. In the case of triple scattering there is also a strong component peaked near e 0. This is an artifact of the fact that we in ≈ initialized triples with ein = 0 and many new triples were formed from an exchange between the tertiary and the interloping star. Since these exchange reactions do not strongly affect the inner binary there remains a peak at e 0. in ≈ It is clear from this orbital parameter distribution that a significant fraction of dynamically formed triples undergo strong KL oscillations. The period of these oscillations, tKL, for equal mass systems is given approximately by (Holman et al. 1997; Innanen et al. 1997; Antognini 2015)

2 8 Pout 2 3/2 tKL = 1 eout . (4.33) 15π Pin − ¡ ¢ Although our calculations are unitless, if we fix the initial semi-major axes of the binaries to 1 AU in the case of binary-binary scattering and in the case of triple scattering we fix the initial semi-major axis of the inner binary to 1 AU and set all masses to 1 M we can calculate the resulting distribution of tKL in years. This ⊙ distribution is presented in Fig. 4.16. There are two interesting features: the first is

117 that they are fairly narrow; the vast majority of dynamically formed triples span only one decade in tKL. The second is that tKL is comparable to the initial dynamical timescale of the systems. In the case of binary-binary scattering, the initial periods of the binaries are 0.7 yr and the KL timescales are peaked at approximately 3 yr. Likewise in the case of triple scattering the outer period is 18 yr and the KL timescales have a peak at around 20–30 yr. In the case of triple-single scattering there is an additional peak at even shorter timescales. Such short KL timescales are a consequence of two facts: (1) dynamically formed triples are extremely compact, and (2) at the critical velocity triples are more easily formed by hardening binaries than by softening them, thereby leading to smaller semi-major axes and shorter periods.

4.5.3. Population study

The above analysis reveals that triples produced from our model scattering experiments are extremely compact. It is possible that this result is an artifact from the fact that in binary-binary scattering we scattered binaries with equal semi-major axes and in triple scattering we scattered relatively compact triples. Binary and triple systems in the Galaxy span many decades of semi-major axis, so the components of typical scattering events will not be of comparable scales.

To examine the distribution of orbital parameters of triples produced from a more realistic population of binary and triple systems in the Galaxy we run 106 apiece of binary-binary, triple-single, and triple-binary scattering experiments. In each experiment we draw a primary mass from the IMF provided by Maschberger (2013) which is an analytic approximation to the IMFs found by Kroupa (2001) and Chabrier (2003) We then draw secondary and, if applicable, tertiary masses from a uniform distribution bounded by 0.08 M and the primary mass. We next draw ⊙ semi-major axes from a distribution uniform in ln a from 0.01 to 105 AU. In the case of a triple, we draw two semi-major axes and take the smaller to be the semi-major axis of the inner binary. We then draw eccentricities from a uniform distribution (e.g., Raghavan et al. 2010; Duchˆene & Kraus 2013). Lastly we determine the stability of the triple by applying the Mardling stability criterion (equation 4.14).

118 Fig. 4.16.— The distribution of the timescale of KL oscillations of dynamically formed triples in a set of model experiments. In the case of binary-binary scattering the initial semi-major axes were set to 1 AU, and in the case of triple scattering we set ain = 1

AU and aout = 10 AU. The masses are all 1 M . Triple scattering generally produces ⊙ larger triples and hence longer KL timescales because there is a larger scale in the problem, namely aout. The distributions of tKL do not extend much beyond a decade, except in the case of triple-binary scattering which has a longer tail towards large tKL.

119 If the triple is unstable we throw out the entire experiment and resample all 1 parameters. We set the incoming velocity to 40 km s− , which is approximately the velocity dispersion of older stars in the thin disk (Edvardsson et al. 1993; Dehnen & Binney 1998; Binney & Merrifield 1998). Because we are only interested in the orbital parameters here, and not the cross sections for collisions, we assume all stars to be point masses.

The distribution of orbital parameters of new triples is shown in Fig. 4.17.

Instead of ǫoct (equation 4.32) we calculate ǫoct,M , defined to be (Naoz et al. 2013a)

m m ǫ = 111 − 112 ǫ , (4.34) oct,M m + m oct µ 111 112 ¶ where m111 and m112 are the masses of the two components of the inner binary. This term accounts for the fact that the octupole term in the Hamiltonian has a mass term which vanishes in the case of an equal mass inner binary. Thus, ǫoct,M indicates the strength of eccentric KL oscillations. We estimate the timescale for eccentric KL oscillations (i.e., the time between orbital flips) to be (Antognini 2015)

128√10 tEKM = tKL. (4.35) 15π√ǫoct,M

These results indicate that dynamically formed triples in the field are, as in our model system, extremely compact. The peak of the ǫoct,M distribution is broader 2 than the ǫ distribution, but centered on 10− , indicating that the vast majority oct ∼ of dynamically formed triples at high inclination undergo strong eccentric KL oscillations. We additionally show the KL timescales and eccentric KL mechanism (EKM) timescales for these triples in Fig. 4.18. Despite the wide range of orbital periods in the initial systems (with outer periods up to 107 yr), the KL timescales ∼ are all quite short. Nearly all of them are less than 1000 yr. This is due to two facts: (1) the dynamically formed triples are all extremely compact, and so don’t have KL timescales much longer than the outer orbital period; and (2) above periods of 1000 yr the binaries become soft and so the cross section for triple formation ∼ drops dramatically (see Fig. 4.9). Furthermore, because ǫoct,M is so large, the EKM timescales are generally not substantially larger than the KL time.

120 Fig. 4.17.— Orbital parameters of new triples resulting from scattering experiments in our population study (Section 4.5.3). The panels are as in Fig. 4.15, except for the bottom right panel, where we have used ǫoct,M instead of ǫoct (see equation 4.34). As in the model case, triples formed from scattering are extremely compact and have large ǫoct.

121 Fig. 4.18.— Distribution of the timescales of KL oscillations (left panel) and the eccentric KL mechanism (right panel) in dynamically formed triples in our population study (Section 4.5.3). Dynamically formed triples have very short KL timescales, with nearly all KL timescales being less than 1000 yr. This is a consequence of the fact that the triples formed are compact and very wide systems are soft, and therefore have low interaction cross sections. Because the triples are so compact the timescale for the eccentric KL mechanism is also very short, nearly always less than 105 yr.

122 4.6. Discussion

Here we apply our results on binary-binary, triple-single and triple-binary scattering in four contexts. In Section 4.6.1 we compare the production rate of WD-WD binaries with highly inclined tertiaries via scattering to the SN Ia rate. In Section 4.6.2 we estimate the number of planets in multiple systems ejected due to scattering and compare to the estimated number of free-floating planets. We then refine our previous calculations by modeling stars with non-zero radii to estimate the rate of stellar collisions in Section 4.6.3. Finally, in Section 4.6.4 we address the question of how long KL oscillations persist for triple systems in globular clusters and in the field until they are disrupted by scattering events.

4.6.1. Application to Type Ia Supernovae

The progenitor systems of SNe Ia are currently unknown. The two most viable models are the single degenerate model, in which a WD accretes mass from a main sequence or giant star (Whelan & Iben 1973; Nomoto 1982), and the double degenerate model, in which two WDs merge (Iben & Tutukov 1984; Webbink 1984). In the past several years the double degenerate model has received much observational support (Bloom et al. 2012; Schaefer & Pagnotta 2012; Edwards et al. 2012; Gonz´alez Hern´andez et al. 2012; Maoz et al. 2014; Shappee et al. 2013), but population synthesis models demonstrate that while it may be possible to match the rate of sub-Chandrasekhar mass WD-WD mergers to the SN Ia rate (Ruiter et al. 2009, 2011; Maoz et al. 2012), hydrodynamical simulations suggest that a large fraction of these mergers may not produce SNe Ia (Nomoto et al. 1997; Maoz & Mannucci 2012; Maoz et al. 2014).

Thompson (2011) provided the first analysis of WD-WD mergers in triple systems and showed that the perturbative influence of the tertiary can decrease the merger time by several orders of magnitude (Miller & Hamilton 2002; Blaes et al. 2002). Thompson (2011) argued that most SNe Ia may occur in triple systems due to the relatively large triple fraction and the larger fraction of triple systems that merge rapidly relative to isolated WD-WD binaries. Moreover, Thompson (2011) speculated that some of these mergers may result in a collision. Katz & Dong (2012)

123 first showed that clean, head-on collisions of two WDs are possible in triple systems using direct N-body calculations and claimed that 5% of WD-WD binaries with ∼ tertiaries produce a direct collision as a result of non-secular dynamics. The triple scenario for SNe Ia is attractive because collisions potentially soften the requirement that the merger mass exceed the Chandrasekhar mass (Rosswog et al. 2009; Raskin et al. 2010; Kushnir et al. 2013). Tertiaries might also hide WD-WD progenitor systems (Thompson 2011). Since the rate of sub-Chandrasekhar mass mergers is comparable to the SN Ia rate (van Kerkwijk et al. 2010; Maoz & Mannucci 2012; Maoz et al. 2012, 2014), this model may solve several problems with the double degenerate model.

The triple scenario has an important obstacle, however. The same KL mechanism that drives the WD-WD binaries to rapidly merge or collide would drive the stars to tidal contact or merger on the main sequence. The main sequence stars would then have either tidally circularized at a few stellar radii or have undergone common envelope evolution. In both cases, the semi-major axis ratio between the inner binary and tertiary would likely increase significantly, rendering the KL mechanism potentially ineffective by the time the stars evolved into WDs. Furthermore, mass loss from one of the stars generally exacerbates this problem by decreasing the mass ratio of the inner binary and triggering the eccentric KL mechanism (Shappee & Thompson 2013). The eccentric KL mechanism can drive the inner binary to much higher eccentricities (Lithwick & Naoz 2011; Katz et al. 2011); thus, even stars that had escaped tidal circularization on the main sequence may be brought into tidal contact after the primary evolves into a WD. While this may be an important source of single degenerate systems, the resulting double degenerate system when the smaller mass star evolves into a WD may be too wide to merge within a Hubble time. It is, however, possible that some fraction of these systems might evolve into tight WD-WD binaries during common envelope evolution, though it is currently very difficult to make estimates of this fraction (Ivanova et al. 2013).

Na¨ıvely, for the triple scenario to work, it must be possible for a binary to begin its life in a system that does not undergo KL oscillations and then efficiently change into a system that undergoes KL oscillations after evolving into

124 a WD-WD binary. Scattering is a candidate by which this can occur. Although a detailed, comprehensive calculation of the rate is left for future work, we can present an estimate to determine if scattering is a viable channel for the production of high inclination triples. In this estimate we consider three environments: the field, open clusters and globular clusters. We study the field and open clusters separately because the SN Ia delay time distribution is often decomposed into two separate parts: a prompt component which explodes within a few hundred Myr of star formation, and a delayed component which explodes a Gyr or more after star formation (e.g., Scannapieco & Bildsten 2005; Mannucci et al. 2006; Maoz & Badenes 2010). Since the KL timescale is much shorter than the scattering timescale (see Section 4.6.4), any KL-induced collisions may occur rapidly after the scattering event (e.g., figure 1 of Katz & Dong 2012). The maximum lifetime of open clusters is generally only a few hundred Myr (de La Fuente Marcos 1997; Lamers et al. 2005), so it is plausible that scattering events that contribute to the prompt component occur in open clusters and scattering events that contribute to the delayed component occur in the field. However, WD-WD mergers due to KL oscillations do not necessarily occur instantaneously. Particularly in the case of gravitational wave driven mergers, KL oscillations can take more than a Gyr to drive the WD-WD binary to merger (Thompson 2011). It is therefore possible that scattering events in open clusters could produce SN Ia progenitors which eventually contribute to the delayed component of the SN Ia delay time distribution rather than the prompt component.

We additionally consider scattering in globular clusters. The SN Ia rate in globular clusters is poorly constrained. Based on a lack of detections of SNe Ia in globular clusters, Voss & Nelemans (2012) placed an upper limit on the globular cluster SN Ia rate, but this limit was larger than the Galactic SN Ia rate by a factor of 50. Graham et al. (2015) found one hostless SN Ia which they claimed was likely to be in a globular cluster. Based on this detection they estimated that the SN Ia rate might be enhanced in globular clusters by a factor of 25. ∼ 3 1 The Galactic SN Ia rate is Γ 5 10− yr− (Maoz & Mannucci 2012). If Ia ∼ × scattering in the field is the primary channel to form high-inclination triple SN Ia

125 progenitors, then the rate at which scattering events produce high-inclination triples must be at least as large as the SN Ia rate. The rate of dynamical formation of new triples may be written

Γnew triple = Nnσv, (4.36) where N is the total number of binaries in the case of binary-binary scattering or the total number of triples in the case of triple scattering, n is the number density of stars, and σ is the cross section to produce a new triple. If we consider triples or binaries in a small range of semi-major axes, da, then the differential rate is dN dΓ(a)= nvσ(a) da. (4.37) da For our calculation we assume Opik’s¨ law (Opik¨ 1924), dN N = 0 , (4.38) da a where N0 is a normalization constant set by the total number of binaries or triples, N: 1 a − N = N ln max . (4.39) 0 a · µ min ¶¸ Although a wide log-normal distribution is more accurate (e.g., Duquennoy & Mayor 1991), the flat distribution is simpler to work with analytically and provides an upper limit on the number of systems at the widest semi-major axes.

The cross section for collision can be written in terms ofσ ˆ by noting that

σ(a)= πa2σˆ(ˆv). (4.40)

We found in Section 4.4.2 that the normalized cross section for new triple formation 2 6 from exchange is proportional tov ˆ− forv ˆ 1 and is proportional tov ˆ− forv ˆ 1. ≪ ≫ We may therefore write the cross section as v2 πaσˆ η2 σ = πa2σˆ crit = 0 , vˆ < 1 (4.41) 0 v2 v2 and v6 πσˆ η6 σ = πa2σˆ crit = 0 , vˆ > 1 (4.42) 0 v6 av6 126 whereσ ˆ0 is the normalized cross section for high-inclination triple formation at the critical velocity and where we have defined η2 to be

η2 av2 (4.43) ≡ crit so as to separate out the dependence of vcrit on a. η is the same for binaries or triples of any size as long as the masses and semi-major axis ratio remain constant (cf. equations 4.10 and 4.11). For the triples we are considering (0.6 M and α = 10) ⊙ 2 3 2 2 we have η 10 AU km s− . ∼ We additionally found in Section 4.4.2 that at very high velocities the cross section for new triple formation from triple-single scattering is dominated by 2 scrambles, which have av ˆ− dependence. This contribution to the overall cross section for new triple formation may be written v2 πaσˆ η2 σ = πa2σˆ crit = 0,sc. , (4.44) 0,sc. v2 v2 whereσ ˆ0,sc. is the normalized cross section for scrambles at the critical velocity.

Substituting equations (4.38), (4.40), (4.41), and (4.42) into equation (4.37) and integrating with respect to a, we find that for binary-binary scattering πN nσˆ η2 Γ = 0 0 (a a ) , (4.45) aacrit v5 a − a µ crit max ¶ where we have defined acrit such that the incoming velocity is equal to vcrit, Gm m a = 11 12 . (4.47) crit µv2

For triple-single scattering Γa

2 4 πN0nη η 1 1 Γa>acrit = σˆ0 +σ ˆ0,sc.amax , (4.48) v v acrit − amax · ³ ´ µ ¶ ¸ where we have assumed that a a . We may then combine these to obtain the max ≫ min overall rate of new triple formation, given by πN nσˆ η2 η4 1 1 Γ = 0 0 (a a )+ (4.49) new triple v crit − min v4 a − a · µ ¶ µ crit max ¶¸ 127 for binary-binary scattering, and given by πN nσˆ η2 Γ = 0 0 new triple v η4 1 1 σˆ (a a )+ + 0,sc. a (4.50) × crit − min v4 a − a σˆ max · µ ¶ µ crit max ¶ µ 0 ¶ ¸ for triple-single scattering. In most cases we have a a a , so this may min ≪ crit ≪ max be further simplified to 2 4 πN0nσˆ0η η 1 Γnew triple = acrit + (4.51) v v acrit · ³ ´ ¸ for binary-binary scattering, and 2 4 πN0nσˆ0η η 1 σˆ0,sc. Γnew triple = acrit + + amax (4.52) v v acrit σˆ0 · ³ ´ µ ¶ ¸ for triple-single scattering. This implies that nearly all binary-binary scattering events which produce new triples occur for systems in which the outer semi-major axis is such that the critical velocity of the outer binary is close to the velocity dispersion of the surrounding stars. By contrast, new triple formation from triple-single scattering events is dominated by scrambles in systems near the maximum semi-major axis.

Using our scattering experiments from Section 4.3.2 we calculateσ ˆ0 for binary-binary scattering by calculating the cross section for outcomes in which a triple is formed with a mutual inclination of 80◦

We simply assume the inclination range of 80◦ 141.8◦). We find that for 3 triple scattering (both triple-single and triple-binary)σ ˆ = 2 10− . We likewise 0 × calculate the cross section for high-inclination new triple formation from scrambles 5 at the critical velocity in a similar way and findσ ˆ = 7 10− . 0,sc. × 128 Estimate for the field

To estimate Γnew triple in the field we make the following assumptions: (1) the total number of white dwarfs in the Galaxy is 1010 (Napiwotzki 2009); (2) the fraction of white dwarfs in WD-WD binaries is 10 per cent (Holberg 2009); (3) the fraction of WD-WD binaries with tertiary companions is the same as the fraction of stellar binaries with tertiary companions, i.e., 20 per cent (Raghavan et al. 2010); (4) the 3 volume density of stars in the field is 0.09 M pc− (Flynn et al. 2006); (5) the mean ⊙ star mass is 0.36 M (Maschberger 2013); and (6) the velocity dispersion of stars in ⊙ 1 the thin disk is 40 km s− (Edvardsson et al. 1993; Binney & Merrifield 1998, p. 656).

9 Under these assumptions the total number of WD-WD binaries is Nbin = 10 , the total number of WD-WD binaries with tertiary companions is N = 2 108, trip × 3 4 and the number density in the field is n = 0.25 pc− . We additionally take amax = 10 2 7 AU and a = 10− AU. From equation (4.39) we then have N = 7 10 , min 0,bin × 7 and N0,trip = 10 , where N0,bin is the normalization for WD-WD binaries and

N0,trip is the normalization for WD-WD binaries with tertiary companions. Using our data from Section 4.3.2 we calculate the normalized cross section for new 3 triple formation to beσ ˆ = 2 10− at the critical velocity of the tertiary for 0 × 5 triple scattering andσ ˆ = 2 10− for binary-binary scattering. For systems 0 × 1 consisting of solar mass stars with an incoming velocity of 40 km s− , acrit = 0.8 9 1 AU. With these numbers, equation (4.51) gives Γ 5 10− yr− for new triple ∼ × 13 1 triple scattering and Γ 2 10− yr− for binary-binary scattering in the new triple ∼ × Galaxy. The rates from triple scattering are several orders of magnitude larger than the rates from binary-binary scattering because the rate from triple scattering is dominated by scrambles at large semi-major axes, which are not a possible outcome of binary-binary scattering. However, rates from both binary-binary and triple scattering are far below ΓIa. In elliptical galaxies the rates are further depressed relative to the estimate in the Galactic field due to the lower number densities and higher velocity dispersions (and hence lower acrit).

129 Estimate for open clusters

To estimate Γnew triple for open clusters we make the following assumptions: (1) the 3 typical number density of an open cluster is 10 pc− (Piskunov et al. 2007); (2) the 1 typical velocity dispersion of an open cluster is 0.3 km s− ; (3) the binary fraction is 50 per cent and the triple fraction is 10 per cent (Raghavan et al. 2010); (4) the total number of open clusters in the Galaxy is 105 (Piskunov et al. 2006); and (5) the typical system is in an open cluster with 300 members (Porras et al. 2003).

Under these assumptions the total number of triples in open clusters is 3 ∼ 106 and the total number of binaries in open clusters is 2 107. From equation × ∼ × (4.39) we have N = 106 and N = 2 105. Note that with our assumed 0,bin 0,trip × velocity dispersion we have a 104 AU, which is comparable to the semi-major crit ∼ axes of the widest binaries observed. Thus in an open cluster environment we are in the low-velocity regime even for the widest systems. Since scrambles are a dominant channel to produce high inclination triples only at very high velocities, the contribution of scrambles can be neglected in open clusters.

5 1 With these numbers, equation (4.51) gives Γ 2 10− yr− for triple new triple ∼ × 4 1 scattering and Γ 5 10− yr− for binary scattering in all open clusters in new triple ∼ × the Galaxy. Thus, even in open cluster environments the new triple formation rate is below the SN Ia rate. These estimates suggest that it is difficult for scattering to produce high-inclination triples at a rate consistent with the SN Ia rate. Moreover, the estimated rate is dominated by scattering events near a , which is 104 AU. crit ∼ These separations are only smaller than the typical separations between stars in the open cluster by a factor of a few, so our implicit assumption that each scattering event proceeds in dynamical isolation is likely unwarranted at the largest semi-major axes (see Section 4.6.1).

Estimate for globular clusters

To estimate Γnew triple for globular clusters we make the following assumptions: (1) the typical mass of a globular cluster is 2 105 M (Harris 1996); (2) the typical × ⊙ half-mass radius is 3 pc (Harris 1996); (3) there are 150 globular clusters in the Galaxy (Harris 1996); (4) 0.7 per cent of the systems in a globular cluster are

130 WD-WD binaries (Shara & Hurley 2002); (5) the triple fraction relative to the binary fraction is the same as the field, i.e., 0.2 (Raghavan et al. 2010);5 and (6) the 1 typical velocity dispersion of a globular cluster is 6 km s− (Harris 1996; Binney & Tremaine 2008, p. 31).

Under these assumptions the average number density in the half-mass radius is 3 4 2500 pc− , and the typical separation between stars is 10 AU. To maintain our ∼ ∼ 3 assumption of dynamical isolation we therefore take amax = 10 AU. Given a median star mass of 0.36 M (Maschberger 2013), the total number of stars within the ⊙ half-mass radii of all globular clusters is 4 107, the number of WD-WD binaries ∼ × is 3 105, and the total number of WD-WD binaries with tertiaries is 6 104. ∼ × ∼ × This then implies that N 3 106 and N 7 105. We furthermore have 0,bin. ∼ × 0,trip. ∼ × a 40 AU. crit ∼ 7 1 With these numbers, equation (4.51) gives Γ 10− yr− for binary- new triple ∼ 6 1 binary scattering and Γ 4 10− yr− for triple scattering for all globular new triple ∼ × clusters in the Galaxy. To compare these rates with the Galactic SN Ia rate, we note that the total mass in globular clusters is 3 107 M whereas the total stellar ∼ × ⊙ mass in the Galactic disk is 5 1010 M (McMillan 2011). Thus the fraction of ∼ × ⊙ 4 stellar mass in globular clusters is 5 10− , so the expected SN Ia rate in globular ∼ × 6 1 clusters assuming no enhancement is 3 10− yr− . This is comparable to the ∼ × scattering rates that we find. Thus, if the SN Ia rate is not enhanced in globular clusters, scattering could be an important channel to form WD-WD binaries with high inclination tertiaries. If, however, the SN Ia rate is enhanced by a factor of 25 ∼ as claimed by Graham et al. (2015) scattering would be unlikely to be the primary channel by which to produce SN Ia progenitors in triples.

5In reality the triple fraction in globular clusters will be determined by the dynamical evolution of the cluster, so there is no reason that the triple fraction in globular clusters should be similar to that of the field. However, the triple fraction in globular clusters is poorly constrained, though it is unlikely to be very much higher than that of the field, so we assume here a triple fraction to binary fraction of 0.2 as a rough upper limit.

131 Summary of rate estimates

Although there are substantial uncertainties and simplifications in the analysis of this subsection, these results imply that if the triple scenario is correct, either scattering of very wide triples in open clusters leads to the formation of high-inclination triple systems or a mechanism other than scattering is responsible for the formation of high-inclination triple systems. It also may be that the full dynamics of open clusters beyond the isolated scattering events we have considered here (such as the dissociation of the cluster) are important for the formation of WD-WD binaries in high inclination triple systems. While we have treated scattering events in open clusters in isolation, this is likely not a good approximation, particularly at the large semi-major axes considered here (Geller & Leigh 2015). It is also possible that triple-binary scattering produces quadruples, which undergo strong KL oscillations over a wider range of parameter space (Pejcha et al. 2013). Scattering in globular clusters is more promising, as the rate of new triple formation can match the Galactic SN Ia rate (after correcting for the fraction of mass in globular clusters). Finally, we note that our calculations of the normalized cross sections were done assuming relatively compact triples (semi-major axis ratios of 10) and binaries of equal size. The distribution of semi-major axis ratios of triples in the Galaxy will certainly be broader than we assumed, so the true normalized cross sections will be different from the cross sections we calculated. Indeed, they will likely be lower due to the fact that scattering is generally more efficient in compact triples—see Section 4.5.3. These issues should be investigated more fully in a future work.

We conclude that scattering in the field cannot produce WD-WD binaries in high-inclination triples at a rate consistent with the SN Ia rate, but we cannot rule out the possibility that scattering in open clusters and globular clusters leads to high inclination triples at rates consistent with the SN Ia rate.

4.6.2. Application to free-floating planets

The dynamical effect of interloping stars or binaries on planetary systems has been studied extensively (see Adams 2010 for a review). Malmberg et al. (2007) measured the rate of close encounters between stars in an open cluster environment and found

132 that dynamical processes will alter the population of planetary systems similar to the Solar System. Malmberg & Davies (2009) found that these close encounters can produce eccentric orbits with an eccentricity distribution similar to the observed distribution of eccentricities of very eccentric exoplanets. Malmberg et al. (2011) calculated the long-term effect of distant flybys on planetary systems and showed that 10 per cent of systems are pushed from a configuration which is stable on ∼ timescales of 108 yr to a configuration which is unstable on these timescales. Most ∼ recently, Li & Adams (2015) calculated the cross sections for various outcomes and changes to the orbital parameters from scattering events between a planetary system and a binary star. In this subsection we apply our results to the free-floating planet population.

Microlensing surveys have claimed the existence of a large population of free-floating planets (Zapatero Osorio et al. 2000; Sumi et al. 2011). It is unknown whether these sub-brown dwarf objects form in isolation or if they were ejected from the planetary systems in which they were born. Numerical simulations indicate that it may be difficult for planet-planet scattering to fully account for the observed numbers of free-floating planets (Veras & Raymond 2012). While there is some observational evidence for the isolated formation of sub-brown dwarf objects in the Rosette Nebula (Gahm et al. 2013), it is nevertheless unclear if isolated formation is the dominant channel for the production of free-floating planets.

We here examine the efficiency of planet ionization due to scattering events. We perform three classes of scattering events: (1) a multi-planet system scattering off of a single or binary star system, (2) a circumbinary planetary system (i.e., a P-type orbit) scattering off of a single or binary star system, and (3) a planetary system in a binary (i.e., an S-type orbit) scattering off of a single or binary star system. We furthermore examine planet scattering in both open clusters and in the field. In these experiments the masses of the stars are set to 1 M and the planets are set to ⊙ 0.01 M . In the triple systems the inner object is given a semi-major axis of 1 AU ⊙ and the outer object is given a semi-major axis of 20 AU and the orbits are set to be coplanar. (In the case of a multi-planet system we instead set the semi-major axis of the inner planet to 4 AU to encourage planet-planet scattering.) In the case of

133 scattering off of a binary, the incoming binary is given a semi-major axis of 100 AU. 1 For the field the incoming velocity is set to 40 km s− and for open clusters it is set 1 to 3 km s− . Our assumed incoming velocity for open clusters is larger in this section than in Sections 4.6.1 and 4.6.3 due to the fact that the scattering experiments we perform in this subsection are prohibitively expensive at an incoming velocity of 0.3 1 km s− . Consequently we calculate the cross sections at this larger incoming velocity and later scale the results to a lower incoming velocity.

With these initial conditions we then calculate the cross section for planet ionization and present them in Table 4.8. In all systems in the field the normalized cross section is within an order of magnitude of unity. In open clusters the cross section is enhanced by one to two orders of magnitude because the incoming velocity is smaller by a factor of 10. Multi-planet–single scattering has the largest ∼ normalized cross section due to induced planet-planet scattering. Although this effect is also present in multi-planet–binary scattering, its normalized cross section is much lower simply due to the much larger area of the interloping binary. The physical cross section for planet ionization is slightly larger in multi-planet–binary scattering than multi-planet–single scattering.

The timescale for planet ionization can be written 1 tscatter = 2 πnσaˆ outv 12 1 2 1 1.1 10 n − aout − v − = × 3 1 yr. (4.53) σˆ 10 pc− 20 AU 3 km s− µ ¶ ³ ´ ³ ´ Even for the outcomes with the largest cross sections (ˆσ 70) the ionization ∼ timescale for a given system in a cluster is 20 Gyr. We noted above thatσ ˆ was ∼ 1 calculated for an incoming velocity of 3 km s− . For a more realistic incoming velocity 1 of 0.3 km s− ,σ ˆ would increase by two orders of magnitude due to gravitational 1 focusing. However, the scattering timescale is also proportional to v− , so the overall effect of a smaller incoming velocity would be to decrease the ionization timescale by one order of magnitude to 2 Gyr. This implies that planet ionization from ∼ scattering with interloping stellar systems is rare in stellar clusters; only 10 per ∼ cent of systems with planets on wide orbits would have lost them in a cluster with an age of 200 Myr. In the field, planet ionization occurs even less frequently due

134 Field environment Systemσ ˆ σ (AU2) Multi-planet–single 6.52 1.14 2608 457 ± ± Multi-planet–binary 0.32 0.05 3325 508 ± ± Circumbinary–single 3.21 0.08 1285 31 ± ± Circumbinary–binary 4.00 0.09 41648 950 ± ± S-type planet–single 0.68 0.02 272 7 ± ± S-type planet–binary 1.26 0.03 13147 309 ± ±

Open cluster environment Systemσ ˆ σ (AU2) Multi-planet–single 68.73 14.57 27493 5828 ± ± Multi-planet–binary 15.57 4.44 161879 46219 ± ± Circumbinary–single 53.59 6.69 21436 2675 ± ± Circumbinary–binary 15.20 3.51 158082 36466 ± ± S-type planet–single 64.43 2.03 25771 811 ± ± S-type planet–binary 7.57 0.77 78703 8046 ± ± Table 4.8: Cross sections for planet ionization. We include both the normalized cross section,σ ˆ, and the physical cross section, σ. Multi-planet systems consist of two planets, circumbinary systems consist of a planet orbiting a binary, and S-type planets consist of a planet orbiting a star in a binary system.

135 3 to the lower densities and cross sections. Assuming a stellar density of 0.1 pc− , a 1 typical velocity of 40 km s− , and a normalized cross section of 6, the scattering ∼ timescale is 4 1012 yr. Thus, fewer than 1 per cent of systems with planets on ∼ × wide orbits would have lost them in the lifetime of the Galaxy. Scattering from interloping systems can therefore produce some free-floating planets, but not at the level claimed by Sumi et al. (2011).

4.6.3. Stellar collisions during scattering events

Although the distances between stars are typically much larger than their radii, close encounters between stars can result in tidal interactions, mergers, and collisions. Candidates for observed outbursts from collisions or mergers include V838 Mon and V1309 Sco (Bond et al. 2003; Tylenda & Soker 2006; Tylenda et al. 2011; Kochanek et al. 2014). It is still unclear whether they should preferentially occur between stars born in the same system or between stars in different systems (Leigh et al. 2011; Perets & Kratter 2012). If collisions are typically between stars in the same system, then these collisions would be driven either by dynamical instabilities in the primordial system (in which case collisions will occur not long after the protostar reaches the main sequence, if it ever does) or due to KL oscillations. If, however, a substantial fraction of collisions are between stars in different systems, then this necessarily implies that scattering would be an important channel for stellar collisions.

Numerical calculations

Collisions are always a possibility in resonant scattering events because such events are chaotic. We here rerun our model system (ain = 1 AU, aout = 10 AU—see Section 4.3.1) but we now give the stars radii of 1 R . We do not include the effects ⊙ of tides. We halt the calculation after a single collision, so multiple collisions are not considered.

To calculate these cross sections we must account for the fact that some fraction of the observed collisions in triple-single and triple-binary scattering are due to KL oscillations and are not induced by scattering. To correct for this effect we run an

136 identical set of triples in isolation for the same length of time and calculate the fraction of triples that collide. This fraction ( few per cent) is then subtracted ∼ from the observed collision fraction from the scattering experiments before the cross section is calculated. We find that the normalized cross section for collision at the critical velocity isσ ˆ = 0.109 0.004 for binary-binary scattering,σ ˆ = 0.134 0.004 0 ± 0 ± for triple-single scattering, andσ ˆ = 0.235 0.007 for triple-binary scattering. 0 ± Because these calculations are all done in the equal mass case, they are biased towards scattering-induced collisions and away from KL-induced collisions because EKM oscillations will be stronger, leading to a larger collision fraction for triples in isolation. It is also plausible that at more extreme mass ratios ionizations will be a more common outcome (see Section 4.3.2), leading to fewer scattering-induced collisions. This implies that the collision cross sections derived above are likely upper limits. We do not calculate the velocity dependence of the collision cross sections, however, so we cannot rigorously assert that these cross sections are upper limits.

Rate calculation

We show in Section 4.4.1 that the velocity dependence of the collision cross section 2 6 for three-body interactions scales asv ˆ− forv ˆ 1 andv ˆ− forv ˆ 1 so long ≪ ≫ asv ˆ is much less than the escape speed of the stars. Collisions are therefore an exchange-like process. Because of this, the rate of stellar collisions from scattering may be estimated in the same way as the new triple formation rate was estimated in Section 4.6.1. We here follow the same analysis to estimate the rate of collisions due to scattering with one modification. Because we hold the stellar radii fixed, the ratio (R/a) varies as we vary a. Fregeau et al. (2004) found that the collision cross section scales linearly with the ratio (R/a) for binary-binary scattering, so we incorporate this factor into the rate calculation: v2 a πa σˆ η2 σ = πa2σˆ crit 0 = 0 0 , vˆ < 1, (4.54) 0 v2 a v2 and ³ ´ v6 a πa σˆ η6 σ = πa2σˆ crit 0 = 0 0 , vˆ > 1, (4.55) 0 v6 a a2v6 ³ ´ where a0 is the outer semi-major axis at whichσ ˆ0 was calculated of 10 AU and η is defined as in Section 4.6.1. Note that we assume here that the collision cross sections

137 for triple-single and triple-binary scattering carry the same (R/a) dependence as binary-binary scattering. Proceeding to integrate equation (4.37) as we did before, we find

πN nσˆ η2a a 1 η4 1 Γ = 0 0 0 ln crit + , (4.56) collision v a 2 v4 a2 · µ min ¶ µ ¶ crit ¸ where, as in equation (4.51), we have neglected amax.

We now evaluate Γcollision for the field, open cluster environments, and globular cluster environments using the same assumptions as we did in Section 4.6.1. We additionally assume that the mass of the Galactic disk is 5 1010 M (Dehnen & × ⊙ Binney 1998). Given a mean stellar mass of 0.36 M (Maschberger 2013), this implies ⊙ that the total number of systems in the Galactic disk is 1.4 1011. Making use of × our assumption that 10 per cent of systems are triples (Raghavan et al. 2010), this implies that the total number of triples is 1.4 1010, and the normalization constant × 9 of equation (4.39) is N0 = 10 for the field. We assume the same normalization constant for open clusters as we did in Section 4.6.1 of N = 2 105. Furthermore, 0 × 2 2 2 assuming a typical stellar mass of 0.36 M we find η 850 AU km s− . With ⊙ ∼ 6 1 6 1 these numbers we estimate Γ 2 10− yr− in the field, 9 10− yr− in collision ∼ × ∼ × open cluster environments. For globular clusters we assume that the binary fraction is 3 per cent (Milone et al. 2012) and the ratio of the triple fraction to the binary fraction is 0.2 as it is in the field. This implies a normalization constant for globular 6 4 1 clusters of N = 3 10 and a collision rate of 8 10− yr− in globular cluster 0 × ∼ × environments. The ratio between the estimated rate in the open clusters and the

field is much smaller here than in the case of our SN Ia rate estimate because acrit is much larger in open clusters, leading to smaller values of (R/a), and hence smaller collision cross sections.

These rates are much smaller than the observed rate of stellar mergers in 1 the Galaxy of 0.5 yr− (Kochanek et al. 2014). This implies that scattering is ∼ a relatively minor contributor to the stellar merger rate. Of course, since 10 per ∼ cent of all triples are formed at inclinations cos i 0.1, many of these systems | | ≤ will undergo merger events, and will therefore likely be a much more important contributor to the stellar merger rate. We also note that as in Section 4.6.1, our

138 assumption of dynamical isolation is likely invalid in an open cluster environment. We save a treatment of the full dynamics of open clusters for a future work.

4.6.4. How long do high inclination triples survive?

Although KL oscillations can drive interesting behavior in a diverse set of systems in isolation, real systems are not isolated. Perturbations to the gravitational potential of a hierarchical triple generally suppress the KL resonance. While several of these effects have already been studied in detail (e.g., relativistic precession), scattering has not.

Scattering will suppress KL oscillations on the timescale of an interaction in which the triple is scattered from a high-inclination state to a low-inclination state or some other configuration in which KL oscillations are not present (e.g., two binaries). Moreover, if this timescale is much shorter than the timescale of KL oscillations themselves, KL oscillations will not occur at all.

To calculate the lifetime of high inclination triples we performed a suite of scattering experiments over a range of outer semi-major axes with a semi-major axis ratio of 10. We performed experiments in two conditions: the field, for which 1 we take the incoming velocity to be 40 km s− ; and globular clusters, for which we 1 take the incoming velocity to be 6 km s− . The scattering timescale is calculated from the cross section for the triple to either disrupt or change its inclination from the range 80◦ 141◦). We take the number density of stars to be 0.25 pc− in the field and 10 3 pc− in globular clusters.

The scattering timescales for triple-single and triple-binary scattering, along 1 with tKL is shown in Fig. 4.19. The scattering timescale, tscat, is proportional to a− because, for fixed incoming velocity, σ/a2 v2 (Hut & Bahcall 1983) and v2 a. ∝ crit crit ∝ In the field the density of stars is so low that high inclination triples persist 5 for a Hubble time except for the very widest triples (aout > 10 AU). Globular ∼ clusters are dense enough, however, that high-inclination triples wider than few ∼ AU will be disrupted in a Hubble time. But even in globular clusters, triples as

139 Fig. 4.19.— The lifetime of high-inclination triple systems in globular clusters and in the field. Triple systems in the field persist for a Hubble time out to widths of 105 AU and triple systems in globular clusters persist for a Hubble time (black dotted line) out to widths of a few AU. We also include the KL timescale (black dashed line). For all but the widest triples in globular clusters the scattering time is much longer than tKL. All but the very widest triples in globular clusters therefore undergo many KL oscillations before disruption by scattering. The scattering timescale is proportional to the inverse of the outer semi-major axis (blue dotted line).

140 wide as 1000 AU will undergo many KL oscillations before disruption by scattering. This analysis ignores the cumulative effect of many perturbations to the angular momentum of the outer binary, however. At velocities much larger than the critical velocity, perturbations to the angular momentum of a binary become much larger than perturbations to its energy. Very distant passages cause the angular momentum of wide binaries to undergo a random walk and can generally lead to extremely large eccentricities before disruption occurs (Kaib & Raymond 2014). In the case of a triple with a wide outer binary, these perturbations can drive the triple to instability in less time than it would take the triple to disrupt due to scattering alone. Moreover, if the cumulative effect of many distant passages can be modeled as a global tidal field, the outer binary of a wide triple will undergo eccentricity oscillations similar to KL oscillations (Katz & Dong 2011). Since the cumulative effect of multiple distant perturbations can have a substantial impact on the orbital parameters of a triple system on long timescales, particularly in dense environments like globular clusters, these long-term effects should be more fully investigated.

4.7. Conclusions

We have explored the properties of binary-binary, triple-single, and triple-binary scattering events in a wide variety of contexts using over 400 million numerical scattering experiments. We have calculated the cross sections for the outcomes of these scattering events in several model systems and grouped these outcomes into a small number of outcome classes (Tables 4.3.1, 4.3.1, 4.3.1, 4.3.1, 4.3.1, and 4.3.1). These outcome classes can, in turn, be broadly grouped into “exchange-like” and “ionization-like” outcomes based on their velocity dependence in analogy to 2 binary-single scattering. Ionization-like outcomes exhibit av ˆ− dependence at large vˆ and exchange-like outcomes exhibit a steeper velocity dependence at largev ˆ (often 6 4 vˆ− , but sometimesv ˆ− —see Fig. 4.6). The statistical uncertainties on these cross sections are on the order of a few per cent or less. The systematic uncertainty is comparable, but only serves to increase the cross sections of ionization-like outcomes.

We find that the cross section for new triple formation is “exchange-like,” and at the critical velocity the normalized cross section for new triple formation isσ ˆ0 = 0.1

141 for both triple-single and triple-binary scattering, andσ ˆ0 = 0.02 for binary-binary 6 scattering. At high velocities the new triple cross section has av ˆ− dependence for binary-binary scattering. For triple-single scattering, the new triple cross section 6 also has av ˆ− dependence for velocities slightly in excess of the critical velocity, 2 but at high velocities, scrambles, which have av ˆ− dependence, are the dominant channel to form new triples. Similarly, the cross section for quadruple formation 6 from triple-binary scattering is an exchange-like process with av ˆ− dependence at high velocity. These results imply that quadruple formation and triple formation in environments dominated by binary-binary scattering are efficient only if the velocity is below the critical velocity (i.e., in cluster environments). In high velocity environments, triple-single scattering is the dominant channel to produce new triples.

We provide analytic fits for the velocity dependence of the ionization cross sections in binary-single, binary-binary, and triple-single scattering (equations 4.20 and 4.22). We also provide analytic fits for the velocity dependence of the new triple cross sections in binary-binary and triple-single scattering (equations 4.29 and 4.30).

We measure the dependences of the cross sections on various orbital parameters. We find that the dependence of the cross sections on the semi-major axis ratio can be attributed entirely to changes in the critical velocities of the various components, at least as long as the semi-major axis ratio is greater than 10 (Fig. 4.3). ∼ We find that there is no eccentricity dependence on the cross sections, except insofar as triples with more eccentric tertiaries come closer to the boundary of stability (Fig. 4.4). Weaker perturbations to the eccentricity of the tertiary can therefore cause the triple to destabilize, leading to a modest increase in the ionization cross section for large outer eccentricities. This increase is well modeled using our fit to the cross section for changes to the outer eccentricity in flyby outcomes. We also studied the mass dependence of the cross sections and find that in general cross sections increase for ionization-like outcomes in binary-binary and triple-binary scattering as the mass of a single star tends toward infinity (Fig. 4.5). In triple-single scattering the cross sections for all outcomes decrease in the high mass limit because the interloping system cannot be disrupted.

142 Almost all scattering events are simple flybys which leave the hierarchical structure intact. We calculate the cross section for changes to the orbital parameters after flybys in triple-single and triple-binary scattering events (Fig. 4.11). We find that changes to the eccentricity of the tertiary are well fit by a Gompertz function (equation 4.31). The normalized cross section for a large change in the inclination of the triple (∆ cos i 0.5) isσ ˆ 0.1 after correcting for changes to the inclination ∼ ∼ due to Kozai-Lidov oscillations (panel c of Fig. 4.11).

We study the properties of triples formed from scattering events and find that dynamically formed triples are extremely compact (Figs. 4.15 and 4.17). Indeed, most dynamically formed triples are very close to the stability boundary although longer-term simulations indicate that they are stable for at least hundreds of orbits. Because these triples are so compact, the timescale for Kozai-Lidov oscillations is very short, sometimes just a few times the outer orbital period (Figs. 4.16 and 4.18). These triples should therefore exhibit strong non-secular dynamics. We furthermore find that the inclination distribution of dynamically formed triples is approximately uniform in cos i and the distribution of outer eccentricities is approximately thermal. These results hold both for a model system and for a population study where we model scattering events in the field.

Because many-body gravitational dynamics are ubiquitous in astrophysics, our results have implications in a variety of different contexts. We find that scattering in the field cannot produce high inclination triple systems at a rate consistent with the SN Ia rate, particularly in ellipticals. This result poses problems for the triple scenario for SNe Ia because high inclination triples need to be produced after the stars that eventually comprise the inner binary of the triple evolve into WDs. If this does not occur KL oscillations will drive the inner binary to tidal circularization and will suppress future KL oscillations. We show in Section 4.6.1 that scattering in open clusters is more efficient and can produce high inclination triples at a rate a factor of 20 below the SN Ia rate. However, it is unlikely that our assumption ∼ of dynamical isolation in open clusters is valid, so the formation of high inclination triples may be more efficient than we have estimated when the full dynamics of the cluster are considered. A triple scenario where intermediate mass stars in binaries or

143 triple systems evolve to WDs and then scattering leads to high inclination tertiaries may therefore be an important contributor to the prompt component of the SN Ia delay time distribution. If the time for KL oscillations to drive the WD-WD binary to merger is much longer than the lifetime of the cluster, scattering in open clusters could also contribute to the delayed component of the delay time distribution. We save a complete exploration of scattering in cluster environments and its implications for SNe Ia for a future work.

Scattering can also lead to collisions between stars. We show that the velocity dependence for collisions in binary-single scattering due to three-body interactions 6 is exchange-like with a high-velocity dependence ofv ˆ− . We estimate the rate of 6 1 stellar collisions in the Galaxy due to triple scattering to be 2 10− yr− . In open ∼ × 6 1 cluster environments the estimated collision rate is comparable, 9 10− yr− . In ∼ × 4 1 globular cluster environments the estimated collision rate is larger, 8 10− yr− , ∼ × due principally to the higher number densities and higher velocity dispersion relative to open clusters. However, our assumptions about dynamical isolation are likely not valid in an cluster environments (Geller & Leigh 2015). The collision rate in clusters may therefore be much higher than our na¨ıve analysis suggests.

We additionally apply our results to several different types of planetary systems and find that scattering from external encounters is a negligible contributor to the free-floating planet population compared to planet-planet scattering. However, in dense open clusters nearly 10 per cent of planets on wide ( 100 AU) orbits will be ∼ ionized due to scattering with interloping systems.

We finally examine the stability of KL oscillations to scattering events (Fig. 4.19). We find that the cross section for the triple to move from a high- inclination to a low-inclination state is small enough that the timescale to scatter out of a regime in which KL oscillations take place is much longer than the Hubble time for systems in the field and all systems with outer semi-major axes

144 Chapter 5: Future directions

5.1. Modelling non-secular effects with single-orbit averaging

5.1.1. Motivation

In Chapter 2 I analyzed the orbital evolution of a hierarchical triple using the double-orbit averaged approximation. In Chapter 3 I showed numerically that for certain hierarchical triple systems, the evolution predicted by double-orbit averaging fails dramatically. However, failure of the double-orbit averaged approximation does not necessarily imply that analytic methods must be abandoned entirely. So long as the changes to the orbital parameters still take place on a timescale long relative to the inner orbital period, it is possible to average only over the inner orbit and obtain a set of differential equations describing the evolution of the orbital elements.

The advantage of single-orbit averaging is twofold. First, the timesteps can be long relative to the inner orbit, so in principle the integration can be faster than a general N-body calculation. Second, having analytic equations can provide some insight about the dynamics.

In this section we derive the single-orbit averaged equations of motion of a hierarchical triple for which one component of the inner binary is a test particle. The general case of three massive bodies is somewhat complicated by the fact that the outer orbit evolves as well, so it is omitted here.

5.1.2. Vectorial equations of motion

In Chapter 2 the approach used to derive the timescales of KL oscillations was Hamiltonian in nature. A set of canonical variables (the Delaunay orbital elements) was used to write down the Hamiltonian for the system, and the Hamiltonian was

145 then differentiated with respect to the conjugate momenta to obtain the equations of motion. An alternative, but equivalent, approach, known as the vectorial description, also exists. The orbit-averaging procedure is simpler using this technique so we employ it in this chapter. In the this derivation we follow Tremaine & Yavetz (2014), modifying the derivation where appropriate to produce single-orbit averaged equations of motion rather than double-orbit averaged equations of motion.

In the vectorial approach, the inner orbit is described by two vectors, j and e, rather than the more conventional Delaunay orbital elements. The vector j is the specific angular momentum vector normalized to the specific angular momentum of a circular orbit of the same energy. The vector e is the Runge-Lenz vector. Its magnitude is the eccentricity of the inner orbit and it points from the center of mass of the orbit to pericenter. More precisely, if we construct a coordinate system such that nˆ points perpendicular to the plane of the orbit and uˆ points toward pericenter, then

j = √1 e2 nˆ, (5.1) − and

e = e uˆ. (5.2)

Note that we have the following identities:

j e = 0, (5.3) · and

j2 + e2 = 1. (5.4)

Now consider the gravitational potential from a third body on the test particle.

Writing the position of the test particle as r and the position of the tertiary as r3, the gravitational potential may be written

Gm Φ= 3 . (5.5) − r r | − 3| 146 Under the assumption that the triple is hierarchical we may Taylor expand the potential in the small quantity r / r : | | | 3| Gm r r r2 3(r r )2 Φ 3 1+ · 3 + · 3 + . (5.6) ≃ − r r2 − 2r2 2r4 ··· 3 µ 3 3 3 ¶ The first term of the expansion is independent of r and is therefore dynamically unimportant. The second term of the expansion corresponds to the motion of the center of mass of the inner binary about the center of mass of the triple and must therefore be dropped to obtain the equations of motion of the orbital elements of the inner binary in its own frame of reference.

We may now describe the positions of the test particle more simply using the true anomaly, φ, and the semi-major axis, a1:

a (1 e2) r = 1 − . (5.7) 1+ e cos φ

The average of some function f(r) over a single orbit is given by

1 P f(r) = f(r(t)) dt, (5.8) h i P Z0 where P is the period given by Kepler’s third law:

4π2a3 P 2 = 1 . (5.9) Gm1 We may then write the integral in terms of the true anomaly to find

1 Gm 2π dt f(r) = 1 f(r, φ) dφ. (5.10) h i 2π a3 dφ s 1 Z0 Noting that the specific angular momentum is related to the areal velocity by

dφ J = r2 = Gm a (1 e2), (5.11) dt 1 1 − p we have

(1 e2)3/2 2π f(r, φ) f(r) = − dφ. (5.12) h i 2π (1 + e cos φ)2 Z0 147 To obtain the single-orbit averaged potential, we substitute equation 5.6 into equation 5.12. Since the integral in equation 5.12 will distribute over addition, we have Gm r2 3 (r r )2 Φ = 3 h i + h · 3 i . (5.13) h i − r − 2r2 2r4 3 µ 3 3 ¶ Now writing the radius vector as

r = r (cos φuˆ + sin φvˆ) , (5.14) the single-orbit averaged potential becomes

Gm 3(uˆ r )2 6(uˆ r )(vˆ r ) Φ = 3 r2 · 3 r2 cos2 φ · 3 · 3 r2 cos φ sin φ h i 2r3 − r2 − r2 − 3 µ 3 3 ­ ® ­ ® 3(vˆ r )2 ­ ® · 3 r2 sin2 φ . (5.15) r2 3 ¶ ­ ® Only the quantities in angled brackets need to be averaged. Substituting these quantities into equation 5.12, we have (1 e2)3/2 2π a2(1 e2)2 1 r2 = − 1 1 − 1 dφ = a2 2 + 3e2 , (5.16) 2π (1 + e cos φ)4 2 1 1 Z0 ­ ® ¡ ¢ (1 e2)3/2 2π a2(1 e2)2 cos φ sin φ r2 cos φ sin φ = − 1 1 − 1 dφ = 0, (5.17) 2π (1 + e cos φ)4 Z0 ­ ® (1 e2)3/2 2π a2(1 e2)2 cos2 φ 1 r2 cos2 φ = − 1 1 − 1 dφ = a2 1 + 4e2 , (5.18) 2π (1 + e cos φ)4 2 1 1 Z0 and ­ ® ¡ ¢ (1 e2)3/2 2π a2(1 e2)2 sin2 φ 1 r2 sin2 φ = − 1 1 − 1 dφ = a2 1 e2 . (5.19) 2π (1 + e cos φ)4 2 1 − 1 Z0 ­ ® ¡ ¢ The single-orbit averaged potential then reduces to Gm a2 3(uˆ r ) 3(vˆ r ) Φ = 3 1 2 + 3e2 · 3 1 + 4e2 · 3 1 e2 . (5.20) h i 4r3 1 − r2 1 − r2 − 1 3 µ 3 3 ¶ ¡ ¢ ¡ ¢ Recall that in this coordinate system the basis vectors are uˆ, which points towards pericenter; nˆ, which points perpendicular to the orbit; and vˆ, which is perpendicular to both. We therefore have

r2 =(uˆ r )2 +(vˆ r )2 +(nˆ r )2, (5.21) 3 · 3 · 3 · 3 148 so that (vˆ r )2 may be eliminated from equation 5.20. Furthermore, using · 3 equations 5.1 and 5.2 we obtain the single-orbit averaged potential:

Gm a2 3(j r )2 15(e r )2 Φ = 3 1 1 6e2 · 3 + · 3 . (5.22) h i − 4r3 − 1 − r2 r2 3 µ 3 3 ¶

In the vectorial formalism the equations of motion from some potential, Φ are h i given by (Musen 1954; Tremaine et al. 2009)

dj 1 = (j j Φ + e e Φ ) , (5.23) dt −√Gm3a1 × ∇ h i × ∇ h i and de 1 = (j e Φ + e j Φ ) , (5.24) dt −√Gm3a1 × ∇ h i × ∇ h i where the notation j is shorthand for the vector ∇ ∂ ∂ ∂ xˆ + yˆ + zˆ, (5.25) ∂jx ∂jy ∂jz and similarly for e. The gradients of the single-orbit averaged potential ∇ (equation 5.22) are therefore

3 Gm3a1 j Φ = 5 (j r3) r3, (5.26) ∇ h i 2 r3 · and

3 Gm3a1 5(e r3) e Φ = 2e · r . (5.27) ∇ h i 2 r3 − r2 3 3 µ 3 ¶

Substituting equation 5.22 into equations 5.23 and 5.24 we finally obtain the single-orbit averaged equations of motion:

dj 3 √Gm3a1 = 5 [(j r3)(j r3) 5(e r3)(e r3)] , (5.28) dt −2 r3 · × − · × and de √Gm a 15 e r 3 (j r ) = 3 1 3e 1 e2vˆ · 3 (j r )+ · 3 (e r ) . (5.29) dt − r3 − 1 − 1 − 2 r2 × 3 2 r2 × 3 3 · q 3 3 ¸ 149 To test the validity of equations 5.28 and 5.29 we show in Figure 5.1 the evolution of a hierarchical triple through a single KL oscillation that exhibits strong non-secular effects. We compare the evolution calculated by equations 5.28 and 5.29 to the evolution calculated using the N-body code rebound (Rein & Spiegel 2015), along with the evolution calculated using the double-orbit averaged equations of motion. The single-orbit averaged equations of motion capture non-secular effects that are missed by the double-orbit averaged equations of motion. While the qualitative behavior of the single-orbit averaged equations of motion is similar to the N-body evolution, it is not identical. This may be because when the inner orbit is very eccentric, most of the torque from the tertiary is imparted when the inner components are at apocenter. If the torque received is relatively large, it could invalidate the secular approximation for the inner orbit. These issues deserve further scrutiny.

5.2. High mass triples as LIGO sources

5.2.1. Motivation

The detection of gravitational waves will inaugurate a new domain of observational astronomy. As detector sensitivity has continued to improve, aLIGO and VIRGO are expected to detect gravitational waves within the next decade, with the dominant source of events predicted to be NS-NS mergers (Kalogera et al. 2004). But because gravitational waves are so weak, searches for gravitational waves must rely on template matching to pick the gravitational wave signal out of noise. The template waveforms are generated by calculating the gravitational waves generated by compact objects of various masses orbiting at various semi-major axes and orientations. However, it is assumed that by the time the binary is emitting gravitational waves at a frequency observable by the detector, the orbit of the binary would have lost nearly all of its initial eccentricity (typical eccentricities are thought to be less than 3 10− ).

However, studies of NS-NS mergers in triples have found that, due to the eccentricity oscillations induced by the tertiary, when the binary begins emitting gravitational waves observable to the detector, the orbit will retain a large

150 Fig. 5.1.— A comparison of a calculation of a KL oscillation performed by code using double orbit averaging (dotted line), a calculation using the N-body code rebound (solid line), and the single-orbit averaged equations of motion (dashed line). The single-orbit averaged equations of motion reproduce the oscillations in the eccentricity of the inner orbit on the timescale of the outer orbit that are missed by double-orbit averaging. Note that the single-orbit averaged code and the N-body code predict a substantially higher eccentricity than the double-orbit averaged code. The lines have been slightly offset in time for clarity.

151 eccentricity of e > 0.1 (Wen 2003; Antonini et al. 2014; Antognini et al. 2014). Since ∼ the gravitational waveform templates employed by aLIGO do not currently allow for non-circular orbits, these NS-NS mergers will be undetectable by aLIGO (Brown & Zimmerman 2010). It is currently unknown whether these eccentric mergers dominate the NS-NS merger rate. This question is also of direct relevance for short gamma-ray bursts since eccentric mergers can unbind more material than circular inspirals (Lee et al. 2010; Stephens et al. 2011).

If even a small fraction of NS-NS binaries have tertiaries, EKM-induced mergers could dominate the merger rate since the effect of the tertiary is to accelerate the merger by many orders of magnitude (Thompson 2011). However, the fraction of NS-NS binaries with tertiaries still needs to be studied. It is known that the high mass stellar progenitors of the NSs have very high multiplicities, and so a substantial fraction of them exist in triples. But to produce a NS-NS binary in a triple, the original triple must survive the mass loss and kicks from two SNe. While most SNe appear to eject one or members from the triple it is possible for a NS-NS binary to keep its tertiary if the orbital configuration and kick direction are favorable (Pijloo et al. 2012). In this section I estimate the fraction of surviving triples by combining binary population synthesis with a dynamical model of NS kicks in triple systems.

5.2.2. Methods

To determine the fraction of high mass triples that survive two SNe and the orbital properties of the resulting systems we proceed in two steps. First we perform binary population synthesis calculations on a large population of high mass binary stars. In the second step we then take the physical and orbital parameters of the surviving binaries and add a tertiary to the system. Given identical instantaneous mass loss events and identical kicks during the SNe (at identical times and orbital positions), we then determine whether the triple is disrupted. If the triple survives we record the orbital parameters of the resulting system.

The binary population synthesis calculations are performed using the BSE code (Hurley et al. 2002). The initial mass of the primary is drawn from a Salpeter IMF with a lower bound of 8 M and the mass of the secondary is drawn from a uniform ⊙

152 distribution bounded between 8 M and the mass of the primary. The initial ⊙ eccentricity is drawn from a uniform distribution and the semi-major axis is drawn from a logarithmically uniform distribution. BSE then evolves the binary through various stages of stellar evolution, accounting for changes to the orbital parameters when mass loss occurs. If the two stars come into close contact with each other BSE uses a numerical prescription to evolve them through common envelope evolution (see Hurley et al. 2002, for details). At the end of a star’s lifetime, BSE triggers a SN and applies a kick to the newly formed NS. BSE then calculates the effect of the kick and mass loss on the orbit of the binary to determine if it remains bound. If the binary becomes unbound the calculation halts, but if the binary remains bound BSE continues to evolve the other star until it goes SN. BSE then applies a new kick and instantaneous mass loss to the binary to determine the final orbital parameters of the binary (if it remains bound).

We then take all binaries that remain bound after two SN and add “dynamically inert” tertiaries. As with the inner binary, we draw the tertiary mass from a uniform distribution between 8 M and the mass of the primary. Similarly, we draw the ⊙ semi-major axis from a logarithmically uniform distribution and the eccentricity from a uniform distribution. We choose the inclination from a uniform distribution in its cosine. To ensure that the tertiary is dynamically inert, we check to see that if the two stars in the inner binary do not interact on the main sequence the tertiary does not drive them to interact due to KL oscillations. For reasons of computational efficiency we perform this check using equation 1.1, so this check is only valid at quadrupole order.

Once a dynamically inert tertiary has been added to the binary, we evolve the orbit of the tertiary through the same episodes of mass loss that the inner binary experiences and apply the two supernovae (both kicks and mass loss) to the triple. We then determine if the triple survives and, if so, we record its orbital parameters.

5.2.3. Results

We ran 109 binaries with BSE and found that 500,000 survived, implying a survival ∼ 4 rate of 5 10− . While binary population synthesis is not well understood (e.g., ×

153 Toonen et al. 2014), we are not concerned with the absolute rate of binary survival predicted by BSE. Rather, we are only concerned with the relative rate of survival of tertiaries. When we add tertiaries to the 500,000 systems we find that 2,500 ∼ ∼ survive, implying a relative survival rate of 0.5%.

We show in Figure 5.2 the distribution of the inner semi-major axis of surviving triples, in Figure 5.3 the distribution of the mutual inclination, and in

Figure 5.4 the distribution of the parameter ǫoct (see equation 2.50. The surviving triples are approximately uniform in cos i, implying that a large fraction will be at 2 high inclination. Moreover the distribution of ǫoct is peaked strongly near 10− , implying that the surviving triples are very compact and may undergo strong EKM oscillations. Indeed, as in the case of dynamically formed triples, these triples are close to the boundary of stability. Lastly, the inner binary of many of the surviving triples is very tight, with a characteristic separation being a few R . At these ⊙ distances an isolated binary will merge due to GW radiation within a Hubble time. Due to KL oscillations the merger time in many of these triples will therefore be much shorter than a Hubble time.

It is important to note that these results have several limitations. Toonen et al. (2014) found that there are substantial differences between binary population synthesis codes and that these codes predict very different evolution of populations. While our results concern only the relative difference that a tertiary adds, these results will be more robust if they can be reproduced with different binary population synthesis codes. Furthermore, while we currently ensure that the tertiary is dynamically inert and the beginning of our calculations, it is possible that after a mass loss episode a formerly inert tertiary could become dynamically active and lead to common envelope evolution between two otherwise well separated stars in the inner binary. Moreover, it is unclear how the evolution of such systems should be handled. While they are more complicated to study analytically or numerically, it is clear that KL-driven mergers do, in fact, occur in the Galaxy. While at present we ignore such issues simply to estimate a lower bound on the rate of compact object mergers in triples, these “messy” systems are of great interest as well.

154 Fig. 5.2.— The distribution of semi-major axes of the inner binaries of triples that survive two SNe. The distribution is bimodal with a peak at 1 AU, and a secondary ∼ peak at R . ∼ ⊙

Fig. 5.3.— The distribution of the mutual inclination of the surviving triples. The distribution is roughly uniform in the cosine of the inclination.

155 5.3. Conclusions

Triples are common throughout the universe, exist in many different contexts, and exhibit unique dynamics. In this dissertation we have explored the dynamics of triple and few-body systems over a broad range of timescales. Beginning with the longest timescales, we have extended the mathematical formalism of double-orbit averaged dynamics and applied these results to the timescales of KL and EKM oscillations in hierarchical triples. We then demonstrated using N-body calculations that oscillations in the eccentricity of the inner orbit on the timescale of the outer orbital period can in certain situations become dynamically important and lead to qualitatively different evolution. We then studied dynamical effects on the shortest timescales by performed a comprehensive study of gravitational scattering of hierarchical triples.

Because this work has been primarily dynamical in nature, we have been able to apply these results to a wide variety of contexts. We have shown that non-secular effects can lead to enhanced merger rates of SMBHs and other compact objects. We have shown that gravitational scattering cannot produce high inclination triples at a rate consistent with the SN Ia rate in the field, but that the rate may be comparable in open and globular clusters. We have shown that scattering in almost all environments takes place on timescales long enough to let at least one KL oscillation proceed uninterrupted. We have shown that gravitational scattering can produce head-on stellar collisions, though likely not at the rate observed in the Galaxy.

This work has many possible avenues for future research. As described in this chapter, a comprehensive analytic study of single-orbit averaging remains to be done so that non-secular effects in hierarchical triples may be understood in detail. Furthermore, there is a distinct possibility that compact object mergers in triples may be the dominant source of gravitational waves. Both of these questions will be of direct relevance to aLIGO and VIRGO as improvements in detector sensitivity bring the detection of gravitational waves within reach in the next few years. Though not described in detail in this chapter, other mechanisms to produce high inclination triple SN Ia progenitors must be explored. In particular, the effect of natal kicks on

156 WDs in triples and the effect of the Galactic tide on wide tertiaries must be studied. Order of magnitude estimates can show that these may be plausible resolutions to the SN Ia progenitor problem. Lastly, the KL effect is, fundamentally, a response of the inner binary of a triple to the tidal field of the outer binary. Any similar tidal field will produce a similar effect on a binary whether the field is produced by a tertiary or by some other mechanism like a global tidal field in a galaxy. The Galactic tide may therefore act on very wide binaries to produce eccentricity oscillations similar to the KL mechanism. However, the influence of the Galactic tide has only been studied at quadrupole order. Since the EKM demonstrates that adding the octupole order term of a tidal field qualitatively changes the dynamics, the effect of this term should be studied in detail in a future work. In particular, the octupole order term of a galactic tide could drive SMBHs to rapid merger.

157 Fig. 5.4.— The distribution of the parameter ǫoct (see equation 2.50). The vast majority of surviving triples are very compact and are near the boundary of stability.

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