secular evolution of compact triple systems: the … · orbit implies that the secondary star is a...

SECULAR EVOLUTION OF COMPACT TRIPLE SYSTEMS: THEINTERPLAY OF KOZAI RESONANCES, TIDAL FRICTION,

GRAVITATIONAL WAVE RADIATION AND MASS TRANSFER

by

SNEŽANA PRODAN

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

Copyright c© 2013 by SNEŽANA PRODAN

Abstract

SECULAR EVOLUTION OF COMPACT TRIPLE SYSTEMS: THE INTERPLAY OF

KOZAI RESONANCES, TIDAL FRICTION, GRAVITATIONAL WAVE RADIATION

AND MASS TRANSFER

SNEŽANA PRODAN

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2013

Ultra Compact X-ray Binaries (UCXBs) consist of a neutron star accreting mass

from a white dwarf and are found primarily in globular clusters. Here we focus on

UCXBs with orbital periods P . 30 min and luminosity variations with periods on the

order of hundreds of days. In order to understand, both analytically and via numerical

simulations, the dynamics of such triple systems we develop a model that considers

three body dynamics in the presence of additional effects such as tidal friction, mass

transfer and gravitational wave radiation. First we consider the dynamical state of

4U 1820-30, which is an UCXB located in the globular cluster NGC 6624. Its tight

orbit implies that the secondary star is a helium white dwarf. A feature of 4U 1820-30

is its large luminosity variation over a period of ' 170 days. We demonstrated that

the variations in the eccentricity of the inner binary arise from libration around a

stable fixed point deep in the Kozai resonance. Taking into account both tidal effects

and effects of mass transfer we set a lower limit for the tidal dissipation factor Q for

helium white dwarfs. Next, we explore the dynamics of additional three akin globular

cluster UCXBs 4U 1850-087, 4U 0513-40 and M15 X-2.

Finally, we use the dynamical model developed for UCXBs to examine white

dwarf–white dwarf mergers may lead to type Ia supernovae events. But since these

ii

mergers are driven by gravitational wave radiation the major issue is how to produce

enough binaries that are sufficiently tight to merge in a Hubble time and reproduce

the observed rates of these events. We investigate the role of tidal effects and GW

radiation in such systems. Our results indicate that tidal effects are important in the

regime of moderately high inclinations (85o ≤ i0 ≤ 89o and 97o ≤ i0 ≤ 105o) where,

combined with GW radiation they contribute to a dramatic decrease in GW merger

timescale. In the regime of high inclinations (90o ≤ i0 ≤ 96o), the inner binary suffers

direct collision and tidal effects do not alter the outcome of the evolution.

iii

“To my parents, Milka and Josip Prodan, and their love for science

and common sense”

iv

Acknowledgements

The work presented in this thesis would have never been completed without the

guidance, support and understanding of my supervisor, Prof. Norman Murray, to

whom I will always be grateful for sharing knowledge and ideas, as well as thoughtful

and constructive feedback over the years. I am thankful for helpful comments on

parts of my thesis to Todd Thompson, Andrew Cumming, Doug Hamilton, Phil Arras,

Natasha Ivanova, Fabio Antonini and Enrico Ramirez-Ruiz.

I am grateful to my Ph.D. committee members, Profs. Yanqin Wu and Jerry Mitro-

vica, for their valuable suggestions and discussions during the committee meetings

and doctoral defence. I am especially grateful to my external examiner, Prof. Cole

Miller, for excellent feedback on my thesis, constructive comments, and making my

external defence a pleasant experience.

During all these years I was lucky to get to know a number of amazing people

who have been a tremendous support and fantastic distraction, including Girija

Darmaraj, Erik Chan, Daniela Gonçalves, Sherry Yeh, Tony Chu, Kelsey Hoffman,

Vuk Radmilovic, Ilana MacDonald, Mubdi Rahman, Dariya Bezugla and Emiliano

Luigi Maiello. I am thankful to Nemanja Štulovic for sharing a part of this journey

with me and being a good friend afterwards. I am forever grateful to Fabio Antonini,

Marcelo Alvarez and Yulia Vasilyeva for being there for me, for their support and for

constantly not failing to make me laugh when things were not funny at all.

Above all I am grateful to my parents, Milka and Josip Prodan, who promoted

my interest in science and believed in me always, encouraged me to make my own

choices, no matter how bold and adventurous they were, and who taught me the

importance of common sense and reason.

v

Contents

1 Introduction 1

1.1 X-ray Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Ultra Compact X-ray Binaries . . . . . . . . . . . . . . . . . . . . 6

1.1.2 Ultra Compact White Dwarf–White Dwarf Binaries . . . . . . . . 8

1.1.3 Astrophysical relevance . . . . . . . . . . . . . . . . . . . . . . . . 10

2 ON THE DYNAMICS OF 4U 1820 -30 12

2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 UNDERSTANDING THE DYNAMICS OF THE

4U 1820-30 SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 The Kozai mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Kozai cycles in the presence of additional forces . . . . . . . . . . 23

The tidal bulge and the tidal Love number k2 . . . . . . . . . . . 24

2.2.3 Libration around the fixed point and the frequency of small

oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Why libration? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

The frequency of small oscillations . . . . . . . . . . . . . . . . . . 26

vi

2.3 MASS TRANSFER, TIDAL DISSIPATION,

AND CAPTURE INTO LIBRATION . . . . . . . . . . . . . . . . . . . . . 30

2.4 NUMERICAL RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1 Numerical model using the quadrupole approximation . . . . . 32

2.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.3 Resonant trapping and detrapping of 4U 1820-30 . . . . . . . . . 33

2.4.4 Numerical model using octupole approximation . . . . . . . . . 41

2.5 ON THE VALUE OF Q AND THE ORIGIN OF THE SMALL (OR

NEGATIVE) P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5.1 The nature of the third body . . . . . . . . . . . . . . . . . . . . . 47

2.6 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3 ON THE DYNAMICS OF UCXBs 61

3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1.1 4U 1850-087 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.1.2 4U 0513-40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.1.3 M15 X-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.1.4 Plan of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.2 OVERVIEW OF OUR DYNAMICAL MODEL . . . . . . . . . . . . . . . 66

3.2.1 Estimating the mass, the radius and the mass transfer rate of the

white dwarf secondary . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.2 The eccentricity and the period of small oscillations of the inner

binary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71


vii

3.3.1 Resonant trapping and detrapping . . . . . . . . . . . . . . . . . . 81

3.4 CONSTRAINING THE TIDAL DISSIPATION

FACTOR Q FOR THE WHITE DWARF

COMPANIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.5 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4 ON WD-WD MERGERS IN TRIPLE SYSTEMS 95

4.1 UNDERSTANDING THE DYNAMICS . . . . . . . . . . . . . . . . . . . 97

4.1.1 The Kozai–Lidov mechanism . . . . . . . . . . . . . . . . . . . . . 97

4.1.2 Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101


4.2.1 Numerical model using the octupole approximation . . . . . . . 104

4.2.2 High mutual inclination 91o ≤ i0 ≤ 96o . . . . . . . . . . . . . . . 106

4.2.3 Moderately high inclinations 85o ≤ i0 ≤ 90o and

97o ≤ i0 ≤ 102o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.3 DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Conclusions & Future Work 117

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.1.1 UCXBs in globular clusters . . . . . . . . . . . . . . . . . . . . . . 118

5.1.2 WD–WD mergers in triple systems . . . . . . . . . . . . . . . . . 119

5.2 Future Work and Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.2.1 Secular evolution of binary stars near massive black holes . . . 120

5.2.2 Secular evolution of dynamically formed triples in globular

clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

viii

5.2.3 Binaries Hosting Highly Eccentric Exoplanets: . . . . . . . . . . . 124

Bibliography 125

ix

List of Tables

2.1 4U 1820-30: Values of the System Parameters . . . . . . . . . . . . . . . . 21

3.1 Values of the Constrained Binary Parameters . . . . . . . . . . . . . . . . 70

3.2 4U 1850-087: Values of the System Parameters . . . . . . . . . . . . . . . 75

3.3 4U 0513-40: Values of the System Parameters . . . . . . . . . . . . . . . . 76

3.4 M15 X-2: Values of the System Parameters . . . . . . . . . . . . . . . . . 77

3.5 Values of the Tidal Dissipation Factor Q . . . . . . . . . . . . . . . . . . . 88

4.1 Values of the System Parameters . . . . . . . . . . . . . . . . . . . . . . . 97

x

List of Figures

1.1 Distribution of Low-Mass X-ray Binaries (open symbols) and High-Mass

X-ray Binaries (filled symbols) in galactic coordinates. Image credit:

Grimm et al. (2002). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Representation of the hierarchical triple system. . . . . . . . . . . . . . . 18

2.2 The period of small oscillations vs. the initial inclination for a system

similar to 4U 1820-30, with k2 = 0.01. . . . . . . . . . . . . . . . . . . . . 28

2.3 The period of small oscillations vs. aout. . . . . . . . . . . . . . . . . . . . 29

2.4 The eccentricity as a function of time (upper panel) and the phase space

(e versus ω) for our fiducial model. . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Phase portrait for four different initial eccentricities at initial inclination

i = 44.715o and initial ω = 90. . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 a) ω vs t. P/P vs t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7 e vs ω, phase space evolution plot. . . . . . . . . . . . . . . . . . . . . . . 38

2.8 ω vs t, resonant trapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.9 e vs ω, resonant trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.10 The eccentricity as a function of time (upper panel) and the phase space

(e versus ω) in the octupole approximation. . . . . . . . . . . . . . . . . . 42

xi

2.11 The eccentricity as a function of time (upper panel) and the argument

of periastron as a function of time ( ω versus t, lower panel) in the

quadrupole approximation using (e/0.009)2Q/k2 = 4.5× 109. . . . . . . 45

2.12 The mass transfer rate as a function of time (upper panel) and P/P

(lower panel, solid line) as a function of time in the quadrupole approx-

imation using (e/0.009)2Q/k2 = 4.5× 109. . . . . . . . . . . . . . . . . . 46

2.13 e as a function of time for the case where the semimajor axis is expand-

ing, Q = 8× 107. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.14 The eccentricity of the fixed point of the inner binary as a function of

time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.1 The X-ray luminosity, LX, versus the orbital period of the binary. . . . . 72

3.2 4U1850-087: The eccentricity as a function of time (upper panel) and

the phase space (e versus ω). . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3 4U 0513-40: The eccentricity as a function of time (upper panel) and the

phase space (e versus ω). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.4 M15 X-2: The eccentricity as a function of time (upper panel) and the

phase space (e versus ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.5 4U 1850-087: a) ω vs t. b)The eccentricity as a function of time. . . . . . 82

3.6 4U 1850-087: a) ω vs t. b) The eccentricity as a function of time. . . . . . 83

3.7 4U 0513-40: a) ω vs t. b)The eccentricity as a function of time. . . . . . . 84

3.8 4U 0513-40: a) ω vs t. b) The eccentricity as a function of time. . . . . . 85

3.9 M15 X-2: a) ω vs t. b)The eccentricity as a function of time. . . . . . . . 86

3.10 M15 X-2: a) ω vs t. b) The eccentricity as a function of time. . . . . . . . 87

3.11 4U 1850-087: a) ω vs t. b)The eccentricity as a function of time. . . . . . 89

xii

3.12 4U 0513-40: a) ω vs t. b)The eccentricity as a function of time. . . . . . . 90

3.13 M15 X-2: a) ω vs t. b)The eccentricity as a function of time. . . . . . . . 91

4.1 The timescale for semimajor axis to decay τa = a/a as a function of the

eccentricity of the inner binary . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2 The eccentricity as a function of time (upper panel) and the semimajor

axis and the periapse of the inner binary as a function of time (lower

panel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.3 The eccentricity as a function of time: upper panel shows the case

where we take into account tidal effects and the lower panel shows

the case where only GR precession and GW radiation are taken into

account with i0 = 89o. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4 The semimajor axis and the periapse of the inner binary as a function

of time: upper panel shows the case where we take into account GR,

GW and tidal effects, while the lower panel shows the case where only

GR precession and GW radiation are taken into account. . . . . . . . . . 111

4.5 The merger time as a function of initial inclination. . . . . . . . . . . . . 112

xiii

Chapter 1

Introduction

“Begin at the beginning," the King said gravely,

“and go on till you come to the end: then stop."

— Lewis Carroll, Alice in Wonderland

1.1 X-ray Binaries

The first extra-solar X-ray detection was the discovery of the extremely bright X-ray

source Scorpius X-1 in 1962 by Riccardo Giacconi and team (Giacconi et al., 1962). Sco

X-1 was detected during a rocket flight launched to look for X-rays from the Moon.

An optical star of 13th magnitude was found at the source location. This historic

discovery triggered X-ray astronomy.

The standard model for galactic X-ray sources, that of accreting neutron stars

or black holes, was suggested by Zel’dovich (1964); Salpeter (1964); Zeldovich &

Guseynov (1966). It was confirmed in 1971, when the first X-ray satellite, UHURU,

discovered the X-ray source Centaurus X-3, which is found to have X-ray pulsation

every 5 seconds (Giacconi et al., 1971). Such pulsation could only be produced by

a neutron star. The material from the Roche lobe overflowing blue giant star forms

1

Chapter 1. Introduction 2

an accretion disk. Ultimately it spirals inward and falls onto a neutron star. The

strong magnetic field of the neutron star channels the infalling material onto localized

hot spots on its surface where the X-ray emission occurs. Due to the spinning of

the neutron star the observer can see different features of these localized hot spots.

Therefore, the received X-ray flux is modulated at the stellar spin period giving these

characteristic X-ray pulsations. Continuous monitoring of this source showed slight

variations in the pulse period which were recognized as due to a Doppler shift, leading

to the conclusion that the source is a star moving in an orbit. X-rays from Cen X-3

were found to disappear for 11 hours roughly every 2 days, indicating that the system

is an eclipsing binary system.

Another new physical phenomena discovered and understood during the first

three decades of exploration are thermonuclear flashes on the neutron star surfaces

that are detected as powerful X-ray bursts (Grindlay et al., 1976) which are rapid

increases in luminosity typically a factor of 10 or greater. After enough of the material

from the donor star has been accreted on the surface of the neutron star, thermal

instabilities trigger nuclear fusion reactions causing an increase in temperature greater

than 109 Kelvins. This eventually gives rise to a runaway thermonuclear explosion of

hydrogen and/or helium on the surface of the neutron stars. Rising times are ≤ 1 s to

10 s, while decaying times ∼ 10 s to minutes. Intervals between them are from hours

to days. The amount of released energy is order of ∼ 1036 − 1038 erg s−1. These are

known as type I X-ray bursts and are characteristic only for neutron stars because they

have surface. On the other hand there are type II X-ray bursts that are seen in both

neutron star and black hole binaries and hence are most likely due to instabilities in

accretion disks. The two type of bursts differ by the burst profile. Type I bursts have

a rapid rise followed by a slow decline, while type II bursts start and stop abruptly

with no gradual decay from peak. The recurrence interval for type II bursts can be

separated by a few minutes while in the case of type I, the recurrence timescale is


Figure 1.1: Distribution of Low-Mass X-ray Binaries (open symbols) and High-Mass X-ray

Binaries (filled symbols) in galactic coordinates. Image credit: Grimm et al. (2002).

several hours or days.

There are currently more than 300 X-ray binaries (XRBs) known in the Milky Way,

with x-ray luminosities LX ∼ 1034 − 1038 erg s−1 . These x-ray binary systems are

concentrated toward the galactic center and the galactic plane. Some of them are

located in globular clusters (see Figure 1.1). There is as well a number of extragalactic

X-ray systems discovered in Large Magellanic Cloud, Small Magellanic Cloud and

other galaxies.

XRBs are divided into two classes based on the mass of the donor star: high-mass

X-ray binaries (HMXBs) and low-mass X-ray binaries (LMXBs). Donors in HMXBs are

massive early type stars with masses larger than 10M⊙. They have relatively hard

X-ray spectra with kT . 15 keV, where the X-ray emitting component is generally the

compact object: neutron star, black hole, or possibly a white dwarf . A fraction of mass

lost via the stellar wind of the massive donor star is captured by the compact object,

and produces X-rays as it falls onto the compact object. These systems often exhibit

regular pulsations and X-ray eclipses but not X-ray bursts. Normally they don’t form


an accretion disks but if they do the disks are very small. Their orbital periods range

from 1 day to 1 year. HMXBs belong to a young stellar populations with age . 107

years and they are concentrated toward the Galactic plane as Figure 1.1 shows.

HMXBs form from binary systems where the two stars have different mass. The

more massive star evolves faster and reaches the end of its life becoming a giant after

approximately a few million years. The outer layers of the evolved star are lost to its

companion. Eventually, it explodes in a supernova leaving behind either a neutron

star or a black hole. This explosion can disrupt the binary system. In cases where the

system remains intact, the orbits may be eccentric. After surviving the supernovae

event, the companion star comes to the end of its life and swells to form a giant. The

evolved companion loses its outer layers onto the neutron star or black hole. This

phase is referred to as the HMXB phase.

Donor stars in LMXBs are less massive than the compact object. They are low mass

late type or degenerate stars such as main sequence, white dwarfs or red giants with

masses smaller that 1M⊙. A typical low-mass X-ray binary has soft X-ray spectra

with kT ∼ 5− 10 keV. Almost all of the radiation is emitted in X-rays, and typically

less than one percent in visible light, making these objects among the brightest objects

in the X-ray sky, but relatively faint in visible light. The brightest part of the system is

the accretion disk around the compact object where the X-rays are emitted. The mass

transfer between the donor star and the compact object is via Roche lobe overflow.

The material is initially pulled into an accretion disk around the compact object. As

the material in the accretion disk slowly spirals into the enormous gravitational well

of the compact object, it gets heated to temperatures of order of millions of Kelvins,

causing the system to emit X-rays. Often these systems exhibit X-ray bursts. The

orbital periods of LMXBs range from ten minutes to hundreds of days. They belong

to the old stellar population, with age of order of (5− 15)× 109 years. LMXBs show a

fairly wide spread around the Galactic plane but are more concentrated toward the


Galactic center (see Figure 1.1).

The origin of the LMXBs is less clear. The standard formation theory argues that

they arise from primordial binaries consisting of a massive primary (the NS/BH

progenitor) and a low-mass secondary in a wide orbit. As a consequence of the stellar

evolution, the primary star evolves to a supergiant engulfing the low-mass secondary.

The low-mass star spirals into the envelope of the high-mass star. The frictional drag

that causes its orbit to shrink will, at the same time, eject the envelope of the massive

star. If the spiral-in ceases before the low-mass companion coalesces with the compact

helium core of the primary, a close binary is formed. The remaining helium core

continues to evolve and eventually explodes as s supernovae, resulting in a neutron

star or a black hole.

LMXBs in globular clusters form via different formation mechanisms. GC LMXBs

are 100 times more abundant than in the galactic field measured on a per star basis.

The reason for such overabundance lies in the existence of dynamical formation

channels in the dense cores of globular clusters (Katz, 1975; Clark, 1975); e. g. tidal

captures and three body interactions. In the tidal capture model, a neutron star

directly captures another star when the two stars experience a close encounter. During

the close encounter the neutron star raises a tidal bulge on the normal star. As the tidal

bulge changes orientation, orbital energy is dissipated as heat, which binds the two

stars together, forming the binary (Fabian et al., 1975). In the three body interaction

model, the neutron star interacts with a previously existing binary. As a consequence

of the interaction an unstable triple system is formed where the least massive star is

likely to be ejected leaving the neutron star in the final binary (Hills, 1976).


1.1.1 Ultra Compact X-ray Binaries

Ultra Compact X-ray Binaries (UCXBs) are a subclass of LMXBs with orbital periods

Porb . 1 hr. Short-period binaries like these require both components to be so close

that a single main sequence, hydrogen rich star would not fit, thus both components

must be compact stars. So tight orbits definitely point to white dwarf or helium

burning star donors that are transferring mass onto a neutron star or a black hole.

These stars are the only known stellar types that have the same size as the donor

Roche lobe corresponding to these orbital periods (Nelson et al., 1986). Apart from

such short orbital periods they distinguish themselves from other binaries by a very

low optical to X-ray ratio.

UCXBs are in the regime where the loss of angular momentum is dominated by

the emission of gravitation wave radiation and hence during their evolution the mass

transfer occurs via Roche-lobe overflow. The mass ratio of the donor and the accretor

decreases, which in turn leads to orbit expansion. Over a Hubble time, ∼ 14 Gyr

these systems can expand to orbital periods of about 80 min. The mass transfer rates

decrease at longer periods due to the weaker gravitational wave emission, slowing

down the expansion of the orbit. For this reason one would expect that UCXBs would

spend most of their life at long periods which is in contradiction with observation.

The lack of detection of longer period UCXBs still remains a puzzle; it could be due to

their lower mass transfer rates and hence their lower luminosity, or to some unknown

disrupting process.

The formation of these systems in not well understood either. One possibility

starts with two main sequence stars in a binary. The more massive star in the binary

evolves into a red giant and expands immensely. If the separation between the two

stars is small enough, the less massive companion can interact with the outer layers of

the evolved star which in turn can alter the further evolution of both the giant and


the companion. As a result, the giant may completely lose its outer layers leaving the

bare core. This bare core may become either a white dwarf or if it is massive enough,

a helium core in which nuclear burning continues to take place. In the latter case,

the remaining core of the giant may develop into a core collapse supernova. If only

this star becomes a supernova and the system is not unbound during the event, the

system can form a neutron star/ black hole – white dwarf/ helium star binary.

The outer layers of the initially more massive star may be accreted onto the

secondary or they may be lost from the system for good. All these cases have in

common the fact that the mass redistribution and the possible angular momentum

loss result in changing the separation between the stars.

If there is a phase where the companion finds itself in the outer layers of the giant,

which is known as the common envelope phase, the friction reduces the velocity of

the companion which implies a decrease in orbital separation and a transfer of energy

and angular momentum from the orbit to the envelope of the giant. The orbital energy

deposited in the envelope may cause the envelope to unbind from the giant’s core.

The description of the common envelope phase may sound simple but in reality

modelling it is extremely difficult due to the uncertainty of the efficiency with which

the orbital energy removes the stellar envelope of the primary.

In contrast to isolated binaries in the galactic disk, in globular clusters, dynamical

interactions are the dominant mechanism of formation of UCXBs. A physical collision

between a red giant and a compact object was first proposed by Verbunt (1987). In

this scenario, a collision leads to the formation of a bound system that later might

experience a common envelope phase and form a tight binary. The formation rate of

UCXBs by these encounters is sufficient to explain the observed number of UCXBs in

Galactic globular clusters or LMXBs in globular clusters near other galaxies (Ivanova

et al., 2005).


The effect of the formation of triples on the evolution of the binary population

in globular clusters has just started to be recognized as another possible scenario

for formation of compact systems. A triple consists of a binary star system orbited

by a third star. If the mutual inclination between the inner and the outer orbit of a

triple is large enough, the inner orbit will experience large periodic variations in its

eccentricity and inclination known as Kozai cycles (Kozai, 1962). These variations in

eccentricity and inclination could drive the inner binary to Roche lobe overflow. As

a result of this mass transfer, the inner binary either merges before the next Kozai

cycle, or starts stable mass transfer. During the periastron passage, at high eccentricity,

tidal interactions can be important. The combination of Kozai cycles and tidal friction

can shrink the orbit of the inner binary (Mazeh & Shaham, 1979; Kiseleva et al., 1998;

Eggleton & Kiseleva-Eggleton, 2001; Fabrycky & Tremaine, 2007) and be responsible

for production of short-period binaries (Makarov et al., 2009). This joint tidal- Kozai

mechanism can operate in a cluster if the Kozai cycle timescale (Innanen et al., 1997)

of a newly formed triple is shorter than the evaporation time of the triple due to the

interaction with the dense stellar field of the cluster (Ivanova, 2008). The dynamics

of compact triple systems is discussed in great details in Chapters 2 and 3. Work

described in Chapter 2 is published as Prodan & Murray (2012).

1.1.2 Ultra Compact White Dwarf–White Dwarf Binaries

Another class of binaries that deserve special attention are white dwarf – white dwarf

(WD–WD) binaries. These binaries are the most common compact binaries in the

Universe due to the fact that the vast majority of stars evolve into white dwarfs. In

addition most stars are formed in binary systems and about half of these binaries

have orbital periods short enough so that when they become giants their evolution

is strongly influenced by the presence of a companion (Duquennoy & Mayor, 1991;


Fischer & Marcy, 1992; Kroupa, 1995; Kouwenhoven et al., 2005, 2007; Zinnecker &

Yorke, 2007).

Observations of close binaries clearly indicate that the systems that interacted

in the past must have lost considerable amounts of angular momentum in order to

form such compact binaries. The currently favoured model for the loss of angular

momentum is common envelope evolution; as described above the details are rather

uncertain.

In globular clusters the situation is different. There are two important channels

for the formation of double white dwarfs in a globular cluster; exchange encounter

and physical collision. The exchange encounter, where the low mass companions

in preexisting binary tend to be replaced with the more massive star during the

close binary-single encounter, leads to the formation of a main sequence–white dwarf

binary. If the main sequence star has sufficient mass to evolve into a red giant on a

timescale shorter than the cluster lifetime, then such a binary can form, via common

envelope evolution, a close WD–WD binary. The other important channel is a physical

collision between a single WD and a red giant, which again involves common envelope

evolution. For details on formation channels in question we refer the reader to Ivanova

et al. (2006).

Once formed, the binary evolution of double white dwarfs is mainly driven by

gravitational wave radiation. In compact systems gravitational wave radiation can

drain the energy and angular momentum from the orbit very rapidly. Such a rapid

shrinking of the orbit can have dramatic effects on the binary. The binary can merge

on a timescale shorter than the age of the Universe. For example, a WD–WD binary

with orbital period of approximately 30 min is expected to merge within roughly

several millions of years (Benacquista et al., 2001). The outcome of such a merger

depends primarily on the masses of the binary components and could in general lead

to type Ia supernova event.


Short period WD–WD binaries could be dynamically produced in globular clusters

and thus globular clusters could be efficient type Ia supernovae factories. Thompson

(2011) explored the possibility that SNe Ia is produced triples, where the tertiary

induces Kozai cycles in the eccentricity and inclination of the inner binary. The

combination of Kozai cycles with gravitational wave radiation and tidal dissipation

expands the parameter space for merger events on timescales shorter than the age

of the Universe (Katz & Dong, 2012; Prodan et al., 2013). In the dense environment

of globular clusters, close WD–WD binaries may pick up a tertiary via either binary–

single or binary–binary scattering. We discuss Kozai-induced mergers in Chapter 4,

which is currently under review in the Astrophysical Journal.

1.1.3 Astrophysical relevance

In the 1980’s it was realized that compact binaries in our Galaxy are sources of low-

frequency gravitational waves even though a very small number of such sources

were known (e.g. Evans et al., 1987; Hils et al., 1990). For future planned space

based gravitational wave detectors such as eLISA, the most numerous expected

individually identifiable sources are galactic binaries with orbital periods under

∼ 15 min (Nelemans, 2013). Therefore, ultra compact binaries are interesting objects

from astrophysical point of view in a number of areas especially when observed with

eLISA.

Compact binaries are very important for understanding the processes governing

binary star evolution. They are the most evolved stages of the binary evolution, and to

reach such short periods they must have suffered extreme loss of angular momentum.

It is believed that this angular momentum loss occurs via one or even two common

envelope events (e.g. Nather et al., 1981; Webbink, 1984). This process is very poorly

understood (e.g. Taam & Sandquist, 2000; Nelemans & Tout, 2005) so further studies


of ultra compact binaries may provide insight into binary evolution in general.

Ultra compact binaries accrete hydrogen poor material which is a unique feature

to this population. That being the case, a combination of the accretion of hydrogen

poor material with a short timescale for accretion processes makes them excellent

laboratories for studying the accretion physics and testing physical theories under the

extreme conditions.

Using type Ia supernova as standard candles led to the discovery of the accelerated

expansion of the Universe, but even so it still remains uncertain which systems exactly

lead to type Ia explosions and how. The merger of two white dwarfs is one of the

proposed scenarios that could lead to these events. WD–WD compact binary as

progenitors of type Ia supernovae are observable with eLISA.

During the evolution of such compact systems, there are processes beside gravita-

tional wave radiation and mass transfer that affect the orbital evolution. Particularly

interesting are tidal effects in white dwarfs that may provide information about the

internal structure of the white dwarfs (e.g. Racine et al., 2007; Willems et al., 2008;

Valsecchi et al., 2012).

Discovering many thousands of ultra compact binaries with eLISA will open up

the possibility of mapping their distribution throughout the Galaxy especially the

inner regions of the Galaxy. This will inevitably lead to a new and different picture of

our Galaxy as well as contribute to our understanding of the structure of the Galaxy.

Chapter 2

ON THE DYNAMICS AND TIDAL

DISSIPATION RATE OF

THE WHITE DWARF IN 4U 1820 -30

“Simplicity is the keynote of all true elegance."

— Coco Chanel

2.1 INTRODUCTION

4U 1820-30 is a low mass X-ray binary (LMXB) located near the center of the globular

cluster NGC 6624. The binary orbital period is P1 ' 685 s, revealed in X-ray obser-

vations as a modulation with ∼ 2− 3% peak to peak amplitude (Stella et al., 1987).

Subsequently, Anderson et al. (1997) discovered a ∼ 16% peak to peak modulation

(period 687.6± 2.4 s) in the UV band from HST.

This short period, low amplitude variation is very stable, with P/P = (−3.47±

1.48)× 10−8yr−1 (Chou & Grindlay, 2001), which is consistent with the earlier mea-

12

Chapter 2. ON THE DYNAMICS OF 4U 1820 -30 13

surement of P/P = (−5.3± 1.1) × 10−8yr−1 from van der Klis et al. (1993a); this

stability led Chou & Grindlay (2001) to suggestion that this modulation reflects the

orbital period of the binary.

Both the short binary period and the type I X-ray bursts observed in this system

imply that the secondary star is a helium white dwarf, of mass m2 = (0.05− 0.08)M⊙,

accreting mass onto a primary neutron star (Rappaport et al., 1987). The distance to

the source is estimated to be 7.6± 0.4 kpc (Kuulkers et al., 2003).

It is striking that neither the magnitude nor the sign of the period derivative is

consistent with the prediction P/P > +8.8 × 10−8yr−1 of the standard evolution

scenario for compact binaries overflowing their Roche lobe (Rappaport et al., 1987).

It has been suggested that the negative period derivative is only apparent, i.e., that

it is not intrinsic to the binary, but instead reflects the acceleration of the binary in

the gravitational potential of the globular cluster which houses the binary (van der

Klis et al., 1993b). However, quantitative estimates show that the acceleration, while

of roughly the right magnitude, is unlikely to be large enough, by itself, to explain the

large discrepancy between the evolution scenario and the observations (van der Klis

et al., 1993b; King et al., 1993; Chou & Grindlay, 2001).

A second striking property of 4U 1820-303 is the much larger luminosity variation,

by factor of & 2, seen at a period of P3 ' 171 days (Priedhorsky & Terrell, 1984a; Chou

& Grindlay, 2001; Zdziarski et al., 2007). Analysis of the RXTE ASM (Rossi X-ray

Timing Explorer All Sky Monitor) data shows that this long period modulation does

not exhibit a significant period derivative, P3/P3 < 2.2× 10−4yr−1(Chou & Grindlay,

2001). The ratio between this long period and the binary orbital period is ' 2× 104,

which appears to be too high to be due to disk precession at the mass ratio of the

system (Larwood, 1998; Wijers & Pringle, 1999).

In this work we adopt the assumption of Grindlay (1988), that the 171 day period


is due to the presence of a third body in the system. The third (outer) star modulates

the eccentricity of the binary at long term period P3 ' P22 /(eP1), where P2 is the

orbital period of the third star and e is the eccentricity of the inner binary. Taking into

account only perturbations from the third star, the binary orbital period of 685 s and

∼ 171 day long-term modulation imply that the orbital period of the third star must

be ∼ 1 day. The presence of additional sources of precession, such as that due to tidal

distortion of the white dwarf secondary, requires a stronger perturbation from the

third body and hence a smaller orbit in order to modulate eccentricity of the inner

binary at the 171 day period. We show that the luminosity modulation arises from

variations in the eccentricity of the inner binary associated with libration around a

stable fixed point in the Kozai resonance.

Tidal dissipation in the white dwarf, driven by the eccentricity of the binary orbit,

tends to decrease both the eccentricity and the semimajor axis (hence period) of

the binary, which we suggest is responsible, in part, for the anomalous observed

period derivative—note that Rappaport et al. (1987) did not treat the effects of tidal

dissipation. The combination of tidal dissipation and mass transfer will result in a

lower value of P/P than that produced by conservative mass transfer alone.

For rapid enough dissipation, or, expressed another way, for low enough values

of the tidal dissipation parameter Q, P < 0 could result. We do not favor this as the

explanation for the observed negative period derivative; we show that such rapid

dissipation damps eccentricity within 10−3 of the system’s lifetime. Subsequently

the mass transfer takes over the evolution of the semimajor axis. In other words, we

would be incredibly lucky to observe the system in the short time that e is significant,

in the absence of another perturbing influence. We also show that, given the most

recent estimates for the acceleration of millisecond pulsars in the gravitational field of

the globular cluster, the cluster gravity does not appear to contribute significantly to

the observed period derivative of 1820-30.


Thus it appears that, while both tidal dissipation and acceleration in the gravita-

tional field of the cluster contribute negatively to the period derivative, they can not

fully explain it. Since we favor the hierarchical triple model as an explanation for

the origin of 171day period of luminosity variations, we suggest that the apparent

negative period derivative, which is a 2− σ result, may either be an observational

artifact or due to the some yet not understood physical processes.

The relation between the luminosity variations and the period derivative is deeper;

we argue that the (intrinsic) increase in the semimajor axis of the binary (driven by

Roche lobe overflow) leads to trapping of the system deep in the Kozai resonance. The

resonance transfers angular momentum from the inner binary to the third star, and

back, periodically, without affecting the semimajor axis of either orbit. However, the

dissipation associated with the strong tides when the forced eccentricity is largest does

remove energy from the orbit of the inner binary. This energy loss peaks when the

mutual inclination is small. It is well known that this coupled Kozai-tidal evolution

tends to leave the system with a mutual inclination between the two orbits near the

Kozai critical value (∼ 40); see, for example, Figure 4 in Wu et al. (2007) or Figure

7 in Fabrycky & Tremaine (2007). We show that the period of small oscillations is

naturally ∼ 170 days when the mutual inclination is close to the Kozai critical value.

Whether the evolution of the inclination in systems like 1820-30, which, unlike

the planetary systems, is know to undergo Roche lobe overflow, is a question we are

currently investigating.

This chapter is organized as follows. In section 4.1 we develop an analytic under-

standing of the system, describing the resonance dynamics, calculating the location

of the fixed point as a function of the system parameters (stellar masses, orbital

radii, and the mutual inclination of the two orbits), and the frequency (or period) of

small oscillations. In section 2.3 we describe a possible dynamical path by which the

system arrived at its present configuration. The dynamical history relies crucially


on both the Roche lobe overflow (which drives the system into resonance) and the

tidal dissipation, which tends to drive the mutual inclination toward the Kozai critical

value. In section 4.2 we describe the results of numerical integrations of the equations

of motion, presenting a fiducial model that reproduces the observed properties of 4U

1820-30. We also demonstrate trapping in the case of an expanding inner binary orbit,

and detrapping in the case of a shrinking binary orbit. In section 2.5 we use the model

to put constraints on the ratio of the tidal dissipation parameter Q and the tidal Love

number (k2) of the Helium white dwarf for our fiducial eccentricity. We discuss our

results, and those of previous workers, in section 4.3. We present our conclusion in

the final section. We give the details of the numerical model in the appendix A. In

appendix B we discuss in details adiabatic invariance of the action and how it governs

the evolution of the system by comparing analytic and numerical analysis.

2.2 UNDERSTANDING THE DYNAMICS OF THE

4U 1820-30 SYSTEM

The presence of a third body orbiting the center of mass of a tight binary will induce

changes in the orbital elements of the binary, changes that take place over a variety

of time scales. The changes are particularly dramatic if the mutual inclination of the

two orbits is large. Kozai (1962) showed that when the initial inclination between

inner and outer orbits has values between some critical inclination icrit and 180o − icrit,

both the eccentricity of the inner binary and the mutual inclination undergo periodic

oscillations known as Kozai cycles.

The period of the Kozai cycles is much longer than either the binary’s orbital period,

or the period of the outer orbit. This justifies the use of the secular approximation,

which involves averaging the equations of motion over the orbital periods of inner and


outer binaries; as a result, the averaged equations of motion predict that the semimajor

axes of both binaries are unchanged.

If the luminosity variations in 4U 1820-30 are due to Kozai cycles, the semimajor-

axis ratio aout/a ≈ 8, so in our analytic work we use the quadrupole approximation for

the potential experienced by the inner binary due to the third body. In our numerical

work we keep terms to octupole order, but we show that the higher order terms

change the quantitative results only slightly.

The angular momentum of the outer binary is much greater than that of the inner,

so that the orientation of the outer binary is, to a good approximation, also a constant

of the motion. In that case, after the averaging procedure, the final Hamiltonian has

one degree of freedom.

Kozai cycles are the consequence of a 1 : 1 resonance between the precession rates

of the longitude of the ascending node Ω and the longitude of the periastron v of

the inner binary. The condition for Kozai resonance, v− Ω = 0, is satisfied only for

high inclination orbits; for low inclinations, the line of nodes precesses in a retrograde

sense (Ω < 0), while the apsidal line precesses in a prograde sense.

We employ Delaunay variables to describe the motion of the inner binary. The

angular variables are the mean anomaly l, the argument of periastron ω, and the lon-

gitude of the ascending node Ω; of these, only ω appears in the averaged Hamiltonian

(see Figure 2.1. Their respective conjugate momenta are:

L = m1m2

√Ga

m1 + m2(2.1)

G = L√

1− e2 (2.2)

H = G cos i. (2.3)

The longitude of periastron is v ≡ Ω + ω. Recall that we are assuming that the

semimajor axis of the outer binary is large enough that the total angular momentum


H

Li

Ω

m2

m3

Figure 2.1: Representation of the hierarchical triple system (not to scale). Ω is the longitude

or the ascending node, omega is the argument of periastron. The longitude of periastron is

v ≡ Ω + ω, a dogleg angle since Ω and ω are not in the same plane. We are assuming that

the total angular momentum is dominated by that of the outer binary, so that i is effectively

the mutual inclination between the two binary orbits.


is dominated by that of the outer binary, so that i is effectively the mutual inclination

between the two binary orbits. We occasionally refer to the elements of the third star,

using a subscript ’out’ to distinguish them from those of the inner binary.

After averaging over l and lout, the Hamiltonian describing the motion of a tight

binary orbited by a third body, allowing for the effects of both tidal and rotational

bulges on the secondary, and for the apsidal precession induced by general relativistic

effects, is (Innanen et al., 1997; Ford et al., 2000; Fabrycky & Tremaine, 2007)

H =−3A

2

[− 5

3− 3H2

L2 +G2

L2 + 5H2

G2 + 5 cos 2ω

(1− G

2

L2 −H2

G2 +H2

L2

)]

−BLG − k2C

(35L9

G9 − 30L7

G7 + 3L5

G5

)− k2D

L3

G3 , (2.4)

where the term proportional to A is the Kozai term, the term proportional to B enforces

the average apsidal precession due to general relativity, and the terms proportional to

C and D represent the tidal and rotational bulges, respectively; the explicit appearance

of the tidal Love number k2 in the latter two terms highlights the fact that these terms

represent the effects of the white dwarf’s tidal and rotational bulges. The expressions

for the constants are

A =18

Φm2m3

(m1 + m2)2

(a

aout

)3 1(1− e2

out)3/2

(2.5)

B =32

Φm2

m1

rs

a(2.6)

C =116

Φm1

m1 + m2

(R2

a

)5

(2.7)

D =112

Φ(

R2

a

)5

f (Ωspin), (2.8)

where

Φ ≡ G(m1 + m2)m1

a. (2.9)

Recall that the semimajor axis and eccentricity of the outer body’s orbit are denoted

by aout and eout. The quantity rs ≡ 2Gm1/c2 in equation (3.15) is the Schwarzschild

radius of the neutron star.


As just noted, the term proportional to D accounts for the rotational bulge pro-

duced by the spin of the white dwarf. The spin is projected onto the triad defined by

the Laplace-Runge-Lenz vector, pointing along the apsidal line from the white dwarf

at apoapse toward the neutron star, and denoted by a subscript e, the total angular

momentum vector, subscript h, and their cross product, denoted by q. We have scaled

the spin to the orbital frequency (or mean motion) n, so that, e.g., Ωe ≡ Ωe/n. We

do so because we anticipate that for small eccentricity the white dwarf will be tidally

locked. Then f (Ωspin) ≡ 2Ω2h − Ω2

e − Ω2q is a dimensionless quantity of order unity.

For the fiducial values of the system parameters listed in table 1, A ≈ 1.73× 1044,

the ratios B/A ≈ 0.53, C/A ≈ 1.82, and D/A ≈ 2.54.

2.2.1 The Kozai mechanism

We start our discussion of the dynamics of the system by focusing on understanding

the Kozai mechanism, neglecting forces due to the tidal and rotational bulges of the

Helium white dwarf in the inner binary, and the effects of general relativity.

We locate the resonance by looking for a fixed point of the Hamiltonian; since we

are neglecting the tidal and rotational bulges, and the general relativistic precession,

we set B = C = D = 0 and differentiate the Hamiltonian with respect to ω, to

find ω f = 0, 90, 180, 270. The fixed points at ω f = 90 and ω f = 270 are stable.

Differentiating the Hamiltonian with respect to G, substituting ω = 90 (or 270) and

setting the result equal to zero, we find G4f = (5/3)H2 L2. In terms of the eccentricity,

e f =

√1− 5

3cos2 i f , (2.10)

where the subscript f indicates that this is the eccentricity of the stable fixed point.

The frequency of small oscillations around the fixed point (small librations) is

ω0 ≡[(

∂2H∂ω2

)ω f ,G f

(∂2H∂ G2

)ω f ,G f

]1/2

. (2.11)


TABLE 1. System parameters

Symbol Definition Value Reference

m1 Neutron star (primary) mass 1.4M⊙m2 White dwarf (secondary) mass 0.067M⊙ Rappaport et al. (1987)

m3 Third companion mass 0.55M⊙a1 Inner binary semimajor axis 1.32× 1010 cm Stella et al. (1987)

aout Outer binary semimajor axis 8.0a1

ein,0 Inner binary initial eccentricity 0.009

eout,0 Outer binary eccentricity 10−4

iinit Initial mutual inclination 44.715o

ωin,0 Initial argumet of periastron 90o

Ωin Longitude of ascending node 0

R2 White dwarf radius 2.2× 109 cm

k2 Tidal Love number 0.01 Arras (private communication)

Q Tidal dissipation factor 5× 107

Table 2.1: 4U 1820-30: Values of the System Parameters


Performing the derivatives,

ω0 = ωA

(18 + 90

H2 L2

G4f

)1/2(1−G2

f

L2 −H2

G2f+H2

L2

)1/2

, (2.12)

where we have defined

ωA ≡√

30A2

L2 =

√1532

nm3

m1 + m2

(a

aout

)3 1(1− e2

out)3/2

. (2.13)

The last factor in equation (2.12) is e f sin i f .

In terms of the eccentricity,

ω0 =32

√15 n

m3

m1 + m2

(a

aout

)3 e f sin i f

(1− e2out)

3/2. (2.14)

From equation (2.10) we see that the critical inclination for a Kozai resonance to

occur, in the absence of other dynamical effects, is icrit = cos−1√

3/5 ≈ 39.2. If

i > icrit, orbits started at ω = 90 with e < e f will librate around the fixed point,

so that ω remains between 0 and 180 (or an even more restricted range). From

equation (2.12) or (2.14), the period of small oscillations P0 ∼ 1/e f , a point that will

be important later.

In contrast, orbits started at ω = 0 and e > 0 will circulate (ω will range from

0 to 360). Librating and circulating orbits are separated by the separatrix, an orbit

that neither librates nor circulates. The width of the separatrix (as measured by the

excursion in e) depends only on the initial inclination: esep = [1− (5/3) cos2 i]1/2.

Examples of librating and circulating orbits (for a system including the effects of

GR and tidal bulges) are shown in section 4.2.

Note that, even for systems with i < icrit, where no stable Kozai fixed point exists,

both the mutual inclination and the eccentricity of the inner binary can undergo

oscillations with significant amplitude (although reduced compared to the case with

i > icrit).


Kozai cycles will be substantial only as long as the perturbation from the outer

body dominates over the other sources of apsidal precession in the inner binary orbit,

a point we now address.

2.2.2 Kozai cycles in the presence of additional forces

The physical effects represented by the terms proportional to B, C, and D are capable

of suppressing Kozai oscillations. We investigate their effects in this section.

As an aside, there is a small apsidal precession introduced by dissipative effects in

the He white dwarf, but this precession rate is negligible compared to the other three.

We mention it here because tidal dissipation has a major role to play in the capture

(or otherwise) of the system into the Kozai resonance.

The equations for the precession rates due to the Kozai mechanism, that due to

general relativity, and the tidal and rotational bulges of the white dwarf, are:

ωKozai =34

n(

m3

m1 + m2

)(a

aout

)3 1(1− e2

out)3/2× 1√

1− e2[2(1− e2) + 5 sin2 ω(e2 − sin2 i)

](2.15)

ωGR =32

n(

m1 + m2

m1

)(rs

a

) 1(1− e2)

(2.16)

ωTB =1516

n k2m1

m2

(R2

a

)5 8 + 12e2 + e4

(1− e2)5 (2.17)

ωRB =n k2

4m1 + m2

m2

(R2

a

)5 1(1− e2)2

×[ (

2Ω2h − Ω2

e − Ω2q

)+ 2Ωh cot i

(Ωe sin ω + Ωq cos ω

) ]. (2.18)

The Kozai term (equation 3.1) can be either positive or negative, depending on

the value of sin i. Both the white dwarf tidal bulge and the GR terms are positive,

so both tend to suppress Kozai oscillations. The term induced by the white dwarf

rotational bulge, on the other hand, can be of either sign, depending on the orientation


of the white dwarf spin. If the white dwarf is tidally locked and if its spin is aligned

(which we assume in our analytic model, but not in our numerical models), this

term contributes positive ω. In case of non-aligned spins the precession rate may be

negative (as we will see).

The tidal bulge and the tidal Love number k2

The tidal bulge of the white dwarf in 4U 1820-30 dominates the non-Kozai apsidal

precession rate, for physically plausible values of k2. We remind the reader that in

Newtonian gravitational theory the tidal Love number k2 is a dimensionless constant

that relates the mass multipole moment created by tidal forces on a spherical celestial

body to the gravitational tidal field in which it is immersed; in other words, k2 encodes

information about body’s internal structure.1

We use k2 = 0.01, which is computed by Arras (private communication) as the

ratio of the potential due to the perturbed mass distribution, to the external potential

causing the perturbed mass, under the assumption that our He white dwarf is a fluid

object.

Soft X-ray observations of the source indicate a rather small absorption, consistent

with that expected to be produced by the interstellar medium of the Galaxy; this rules

out any significant outflows from the accretion disk or the surface of the white dwarf.

This implies an absence of mass loss through the L2 Lagrangian point of the white

dwarf, which puts an upper limit on the eccentricity of the inner binary; according to

Regös et al. (2005), for our system parameters, the upper limit on the eccentricity of

inner binary is emax ' 0.07.

If 4U 1820-30 has a non-zero but small eccentricity, as indicated by the observed

1In a confusing usage, the apsidal precession constant, which is a factor of two smaller than thetidal Love number, but which we do not employ, is also denoted by k2.


luminosity variations, then in the absence of a third body, the precession rate of the

binary orbit is dominated by the tidal bulge induced in the white dwarf by the gravity

of the neutron star; from equations (3.2) and (3.3), the tidal bulge induces a precession

rate at least a few times that induced by GR:

ωTB

ωGR≈ 4

(k2

0.01

)(a

1.32× 1010 cm

)−4

. (2.19)

In order for the Kozai mechanism to produce significant variations in e, the Kozai-

induced precession rate must be comparable to or larger than the sum of the precession

rates produced by the other terms. For physically realistic values of k2, as we have

just seen, the precession rate induced by the tidal bulge of the white dwarf is by far

the largest, so if the Kozai effect is to be important, it must produce a precession rate

larger than ωTB.

2.2.3 Libration around the fixed point and the frequency of small

oscillations

Why libration?

For the values of the tidal Love number k2 and eccentricity listed in table 1, the period

of the precession rate induced by the tidal bulge, PTB = 2π/ωTB, is a factor of ten

shorter that the period of the observed luminosity variations. If this term set the

rate of precession, and the eccentricity varied as a result of this precession, then the

variations in X-ray luminosity would occur with a period substantially shorter than

the observed 170 days.

In order to produce a much longer period, some other term must tend to produce a

negative precession rate. When this negative precession rate is added to that produced

by the tidal bulge, the resulting period can be much longer than that produced by the

tidal bulge alone.


Under the assumption that the white dwarf is tidally locked (we show later it is

not), the only term capable of producing a negative precession rate is the Kozai term.

Hence we are led to look for a cancellation between the Kozai precession rate and the

precession rate induced by the tidal bulge.

However, it is not enough to ask for a rough cancellation. To get the observed

precession rate, the sum of all the terms must cancel to better than 10%. This requires

some fine tuning of the mutual inclination, a rather unsatisfactory situation.

On the other hand, if the system is captured into libration, then the sum of all

the precession terms is exactly zero. If the system is deep in the resonance, then the

period of libration is simply the period associated with small oscillations around the

fixed point. We show here that the period of small oscillations is naturally around 170

days, if the mutual inclination is near the critical value for Kozai oscillations.

The frequency of small oscillations

Setting the first derivative of the Hamiltonian (2.4) with respect to ω and G to zero,

we find the following expression for the location of the stable fixed points in the limit

of small eccentricity:

ω f = 90 , 270 (2.20)

e f =

√√√√18− 30 H2

L2 − BA − 120k2

CA − 3k2

DA

60 H2

L2 + 32

BA + 840k2

CA + 15

2 k2DA

. (2.21)

We can write the second of these as

e f =

√30 [cos2 icrit − cos2 i]

60 H2

L2 + 32

BA + 840k2

CA + 15

2 k2DA

, (2.22)

where

cos2 icrit ≡35− 1

30BA− 4k2

CA− 1

10k2

DA

. (2.23)


Evaluating the second derivative of the Hamiltonian at the fixed point we obtain

the expression for the frequency of small oscillation around the fixed point:

ω0 = ωA

[(18 + 90

H2 L2

G4f

)+ 2

BAL3

G3f+ k2

CA

(3150

L11

G11f− 1680

L9

G9f+ 90

L7

G7f

)

+ 12k2DAL5

G5f

]1/2

× e f sin i f , (2.24)

which should be compared to equation (2.12). As in the pure Kozai case, the period of

small oscillations P0 ∼ 1/e f .

Figure 2.2 shows P0 as a function of the initial inclination. As the initial inclination

increases above the critical value, the period of small oscillations decreases rapidly.

Increasing the initial inclination increases the magnitude of the Kozai torque; in the

absence of other torques, and for inclinations above the critical inclination, increasing

the magnitude of the Kozai torque is analogous to increasing the restoring force in a

harmonic oscillator, thereby increasing the frequency of oscillation. When there are

other torques in the problem, the critical inclination will change; for example, the

presence of a tidal bulge on the secondary increases the critical inclination.

Very near the critical inclination, the effective restoring force is small, ∼ e f sin i f ,

so the frequency of small oscillations is small, and the period of oscillations is large—

hence the rapid increase in P0 as the inclination decreases toward the critical inclination

(icrit ≈ 44 in Figure 2.2).

Figure 2.3 shows P0 as a function of aout. As expected from the nout ∼ a3out

dependance of ωA, the period of eccentricity oscillations increases rather rapidly with

aout.


Figure 2.2: The period of small oscillations vs. the initial inclination for a system similar to

4U 1820-30, with k2 = 0.01. The critical inclination is icrit ≈ 44.7. At large inclinations, well

above icrit, the period of libration is of order days. Only if i ≈ icrit is the period of order 170

days. The solid line is the prediction of equation (2.24); the solid circles come from numerical

integration of the equations of motion to quadrupole order, while the open squares come from

integration accurate to octupole order.


Figure 2.3: The period of small oscillations vs. aout. As the semimajor axis of the outer binary,

aout increases, the period of small oscillations is increasing too, which is expected from the

nout ∼ a3out dependance of ωKozai. The solid line is the prediction of equation (2.24), while the

solid circles and open squares are from numerical integrations accurate to quadrupole and

octupole order respectively.


2.3 MASS TRANSFER, TIDAL DISSIPATION,

AND CAPTURE INTO LIBRATION

We have shown that physically plausible values of k2 lead to a precession frequency

ωTB that is much larger than the observed frequency of luminosity variations in

4U 1820-30. We then appealed to an equally large precession, of the opposite sign,

produced by the Kozai interaction, to cancel the prograde precession caused by the

tidal bulge. In order to avoid fine tuning, we argued that the system has to be in

libration, so that the observed low frequency actually arises from libration, rather than

precession of the apsidal line of the binary orbit.

Whether the tidal bulge or GR effects produce a larger precession rate, we argue

that it is no coincidence that the magnitude of the Kozai precession rate is equal to

the sum of the other precession rates: the system will evolve so as to capture the orbit

into resonance, in which the sum of all the precession rates is zero.

Capture into libration in the Kozai resonance is a natural consequence of semimajor

axis expansion, the latter driven by mass loss from the white dwarf as a result of its

overflowing its Roche lobe. The action∫Gdω is an adiabatic invariant (for detailed

discussion see appendix B), since the semimajor axis of the binary orbit is expanding

on the accretion time scale m2/m2 ≈ 107 yr, much greater than either the orbital or

precession time scale. In contrast to mass transfer, tidal dissipation tends to shrink

the semi-major axis; if this effect dominates, trapping into the Kozai resonance is not

possible.

How does expansion of the inner orbit lead to capture into libration? As a

increases, the mutual torque between the two orbits will increase as well—the inner

orbit is expanding, effectively moving closer to the outer orbit. This increasing torque

corresponds to a deepening of the Kozai potential, and an expansion in the size of the


separatrix of the Kozai resonance. Orbits other than the separatrix have a fixed action,

while the action of the separatrix is increasing. If the increase in the action of the

separatrix grows to exceed the action of an initially circulating orbit, that circulating

orbit will be captured into resonance, and begin to librate. As a continues to expand,

the captured orbit will move closer and closer to the fixed point of the resonance,

librating with the frequency of small oscillations.

More quantitatively, mass transfer tends to increase a (Rappaport et al., 1982):

aMT =1

2/3− 1/q3× 223/6

5c5

[ K0.4242

]3/2 m1(G(m1 + m2))3/2

a9/2 , (2.25)

where q = m1/m2 and K = kθγ′/(µmp); k is the Boltzmann constant, µ is the mean

molecular weight, mp is the mass of the proton and θγ′ is the polytropic temperature.

The parameter K is given by the following mass-radius relation:

K = NnGm1−(1/n)2 R(3/n)−1, (2.26)

where Nn is a tabulated numerical coefficient (for n = 1.5 it is 0.4242; Chandrasekhar

(1939)).

Tidal dissipation in the white dwarf will tend to reduce the semimajor axis of the

binary. In the limit of small eccentricity,

(dadt

)TD≈ −38

3na

k2

Qm1

m2

(R2

a

)5

e2. (2.27)

We argue that the orbit must be expanding. (e/e)TD is 100 times shorter than

(a/a)TD, so unless something excites e (such as third body or thermal tides) we are

unlikely to catch the system in a phase where periastron, rp = a(1− e), is increasing

while a is decreasing.


2.4 NUMERICAL RESULTS

2.4.1 Numerical model using the quadrupole approximation

Our numerical model treats the gravitational effects of the third body in the quadrupole

approximation. We average over the orbital periods of both the inner binary and the

outer companion. We demonstrate in section 2.4.4 and in figures 2.2 and 2.3 that

treating the effects of the third body in octupole approximation does not qualitatively

change our findings. We include the following dynamical effects:

• periastron advance due to general relativity;

• periastron advance arising from quadrupole distortions of the helium white

dwarf due to both tides and rotation;

• orbital decay due to tidal dissipation in the white dwarf;

• loss of binary orbital angular momentum due to gravitational radiation;

• conservative mass transfer from the helium white dwarf to the neutron star

primary driven by the emission of gravitational radiation.

Note that the Kozai mechanism described in the previous section is included in the

three body gravitational dynamics. The equations used in our model are listed in the

appendix.

2.4.2 Results

We use as fiducial parameters m1 = 1.4M, m2 = 0.067M, and m3 = 0.55M. The

semimajor axis of the inner binary is a = 1.32× 1010 cm, chosen to match the observed

orbital period of 685 s. The radius of the Helium white dwarf is R2 = 2.2× 109 cm,

while the fiducial Love number is k2 = 0.01.


To reproduce the 171 day eccentricity oscillations (Figure 4.2), we use the following

initial parameters: aout = 8.0a (yielding Pout = 0.15 day). We start with e0 = 0.009,

ω0 = 90, iinit = 44.715, and eout,0 = 10−4.

Figure 4.2 shows the eccentricity oscillations of the inner binary, with a period of

171 days, over a decade. The amplitude of the eccentricity oscillations is of order of

7× 10−3, which is sufficient to enhance mass transfer enough to produce the observed

luminosity oscillations of a factor of & 2 (Zdziarski et al., 2007); see their Figure 3.

The amplitude of the eccentricity oscillations depends on the initial eccentricity, as

illustrated in Figure 2.5; a lower initial eccentricity produces eccentricity oscillations

with higher amplitude. If the system circulates, the amplitude of the eccentricity

oscillations is larger still.

Having the system trapped in libration about the fixed point explains both the

origin of the 171 day period luminosity variations, as well as the small amplitude of

the eccentricity oscillations; the observations require that magnitude of the eccentricity

oscillations be small so as to avoid overly large luminosity variations—a point we

return to below.

2.4.3 Resonant trapping and detrapping of 4U 1820-30

The mass transfer rate is determined by the inner binary mass and semimajor

axis. These parameters are reasonably well constrained from observations (Stella et al.,

1987; Anderson et al., 1997; Rappaport et al., 1987). The amount of tidal dissipation is

parameterized by the tidal dissipation factor Q, which for white dwarfs is not well

constrained at all. If we know the value of period derivative, P, we can constrain Q (or

more precisely, (e/0.009)2Q/k2, see equation 2.27) for the white dwarf in the system.

We argued at the end of §2.3 that the intrinsic P must be positive, since a shrinking

binary orbit and a decaying eccentricity quickly lead to mass transfer driving expan-


Figure 2.4: The eccentricity as a function of time (upper panel) and the phase space (e versus

ω) for our fiducial model. The period of the eccentricity oscillations is 171 days, and the

amplitude of the eccentricity oscillation is sufficient to produce the observed factor of 2− 3

variation in luminosity.


Figure 2.5: Phase portrait for four different initial eccentricities at initial inclination i = 44.715o

and initial ω = 90. Our fiducial orbit, with a libration period of 171 days and the amplitude

of the eccentricity oscillations sufficient to produce observed variations in the luminosity,

is labeled as the “current orbit in libration”; it is the same orbit presented on Figure 4.2.

The unlabelled librating orbit is very near the fixed point; it has a period of the eccentricity

oscillations that is close to but shorter than the observed period, while the variations induced

in the luminosity are too small compared to the observations. The circulating orbit produces

luminosity variations that are too large, as well as having an oscillation period that is too long.


sion of the binary orbit. There is a second argument against an intrinsic negative P:

if the orbit of the inner binary is shrinking, an initially librating orbit will quickly

become circulating, and the period of luminosity variations will change dramatically.

If we tune Q to the value that reproduces the observed negative period derivative

(Q = 2.5× 107, assuming k2 = 0.01) and let the system evolve, the system is driven

out of libration after about 1500 yr, as shown in Figure 2.7. As the figure shows, the

eccentricity of the inner binary decreases significantly due to tidal dissipation, which

in turn reduces the strength of tidal dissipation. With tidal dissipation weakened,

mass transfer will dominate the evolution of the semimajor axis and, as expected from

the standard evolutionary scenario, the semimajor axis starts to expand (not shown

in the figure). As long as there is some small eccentricity in the inner binary there

is some tidal dissipation present that tends to slow down the expansion rate of the

semimajor axis.

The reason for the detrapping is rather subtle. First, we note that the decrease in e

is not due to direct tidal damping; Equation (2.37) predicts (e/e)TD ∼ 105 yr, while

e changes by factor of 2 in 2000 yr. To verify this, we have set eTD = 0, and verified

that integration yields the same result. The reason for such a short time scale for

decrease in e lies in the fact that the spins do not remain tidally locked throughout

the evolution of the system and the evolution of the eccentricity is rather strongly

influenced by the their lack of pseudo-synchronism. Detailed discussion and figures

are given in appendix B.

On the other hand, if the observed negative period is not an intrinsic property of

the system, in other words, if the effect of mass transfer wins over the effect of tidal

dissipation, the action of the separatrix increases with time, and trapping will occur.

Figures 2.8 2.9 show a system initially put on a circulating orbit. As the integration

proceeds, the separatrix expands, eventually capturing the orbit, which then librates

for the duration of the integration.


Figure 2.6: a) ω vs t. We start the evolution of the system by placing the system in libration.

Here we use Q = 2.5× 107, which is the value required to reproduce the observed negative

period derivative. The action of the separatrix is decreasing, and the system is ejected from the

resonance after about 1500 yr. b) P/P vs t. The period derivative remains negative only for

about 1700 yr, which is 10−3 of the lifetime of the system; the eccentricity damps sufficiently

that the m term takes over.


88 90 920

0.01

0.02

ωin

[deg]

Time = 10 Years

ein

88 90 920

0.01

0.02

ωin

[deg]

Time = 1500 Years

ein

88 90 920

0.01

0.02

ωin

[deg]

Time = 1700 Years

ein

88 90 920

0.01

0.02

ωin

[deg]

Time = 2300 Years

ein

Figure 2.7: e vs ω, phase space evolution plot, showing that the orbit evolves from libration to

rotation, with the transition occurring between the 1500 and 1700 yr snapshots.


Figure 2.8: ω vs t. The system is placed in circulation; after about 27000 yr it gets trapped in

libration. Here we have used Q = 5× 107, while the initial eccentricity is e0 = 0.032; all other

parameters are the same as used in Figure 4.2.


Figure 2.9: e vs ω for the same integration as in Figure 2.8.


2.4.4 Numerical model using octupole approximation

In this subsection we treat gravitational effects of the third body in the octupole

approximation. As in the case of the quadrupole approximation, we derive our

equations of motion from the double averaged Hamiltonian (Ford et al., 2000; Blaes

et al., 2002; Thompson, 2011; Naoz et al., 2011) and we include all of the previously

listed dynamical effects. As Figure 2.10 demonstrates, the octupole approximation

does not change qualitatively our previous findings. All parameters, except the initial

inclination, used in the octupole approximation are listed in table 1. In order to

produce the 171 days period of the eccentricity oscillations, and the amplitude of

the eccentricity oscillation that produces the observed factor of 2− 3 variation in

luminosity, a slightly higher inclination is required (i = 45.1o).

2.5 ON THE VALUE OF Q AND THE ORIGIN OF THE

SMALL (OR NEGATIVE) P

The standard theory of Roche lobe overflow predicts P/P ≥ +8.8× 10−8 yr−1. The

measured P/P = (−3.47± 1.48)× 10−8 yr−1 is eight standard deviations away from

this value. We have argued in previous section that P/P should be positive, but even

if it is two or three standard deviation from the measured value, it is still five below

the predicted value. The origin of this discrepancy has been a puzzle since it was

discovered.

The suggestion that the binary has a finite eccentricity immediately suggests a

reason for the low value of P: tidal dissipation in the white dwarf will tend to reduce

the semimajor axis of the orbit, contributing a substantial negative term to P.

The tidal dissipation could in fact dominate the orbital evolution, overcoming the


Figure 2.10: The eccentricity as a function of time (upper panel) and the phase space (e

versus ω) in the octupole approximation. To produce the 171 days period of the eccentricity

oscillations, and the amplitude of the eccentricity oscillation that is sufficient to produce

the observed factor of 2− 3 variation in luminosity, slightly higher inclination is required

(i = 45.1o).


effects of mass transfer as seen in Figure 2.6. We do not argue for this point of view,

however, because it would be unlikely that the system could be observed in a stage of

the evolution that last only 10−3 of its lifetime. In addition, we believe that the system

is trapped in libration.

The observed P/P consists of at least three parts:(PP

)obs

=

(PP

)Roche

+

(PP

)accel

+

(PP

)TD

. (2.28)

The values of the observed and Roche terms were given above, and, as noted there,

they are not consistent with each other. The second term on the right hand side of

equation (2.28) represents the acceleration experienced by 1820-30 in the gravitational

field of its host globular cluster, while the third term on the right represents the effects

of tidal dissipation in the white dwarf secondary.

A natural explanation for the observed negative P might be provided by a combi-

nation of the last two effects, but still allow for the system to be trapped in resonance.

First, tidal dissipation reduces the intrinsic P/P substantially from that expected due

to Roche lobe overflow alone, but leaves P/P > 0. We then appeal to the argument of

van der Klis et al. (1993b), that the (P/P)accel term produces an apparent negative total

P. Indeed, given the most recent published estimate of amax/c = 7.9× 10−8yr−1 from

van der Klis et al. (1993b), it is plausible that we would observe a negative period

derivative, while the intrinsic (or physical) period derivative is in fact positive.

However, recent estimates for the cluster acceleration from millisecond pulsar

timing suggest a maximum of amax/c = 1.3× 10−9yr−1 (Lynch and Ransom, private

communication), an order of magnitude smaller than the estimate from van der Klis

et al. (1993b); if the smaller value holds up, the observed negative period derivative is

difficult to understand in the context of current models.

Given that the measured negative period derivative is significant only at the

two-sigma level, and that there is no clear physical explanation for such an orbital


decay, it is worth considering the possibility that the observed value is in error. If we

ignore the observed negative period derivative, and simply assume that the intrinsic

P is positive, we find a lower limit on Q given by (e/0.009)2Q/k2 > 3.15× 109. We

can get a firmer lower limit on Q by requiring the system to remain trapped in a

resonance for a considerable fraction of its lifetime. Given that m2 = 0.067M and

m2 ≈ 10−8 M yr−1, the lifetime during which this system can sustain its high X-ray

luminosity is estimated to be 7 million years, so a reasonable fraction of its lifetime

to remain trapped in a resonance is at least 105 yr. According to our model for

(e/0.009)2Q/k2 > 4.0× 109 the system remains trapped in the resonance for more

than 105 yr (see Figure 2.11), and as Figure 2.12 demonstrates, the mass transfer rate

remains within roughly 10% of its nominal value m2 ≈ 10−8 M yr−1. In this case,

the eccentricity of the inner binary will never damp down to a fixed point because it

is indirectly driven up by semimajor axis expansion due to mass loss on a timescale

of order 104 yr, which is at least an order of magnitude shorter then the timescale

for eccentricity damping due to tidal dissipation. As expected from the evolutionary

scenario the intrinsic period derivative is positive, but because of the effect of tidal

dissipation it is smaller than that due to Roche lobe overflow alone (see Figure 2.12);

the nominal value for the period derivative due to Roche lobe overflow alone for our

system parameters is P/P ≈ 1.3× 10−7 yr−1 (Rappaport et al., 1987).

Finally, we note that if the inner binary is in fact expanding, the eccentricity will

tend to increase as well. If the eccentricity is large enough, then Roche lobe overflow

will occur through both the inner and outer Lagrange points, in contradiction with

the low observed x-ray absorption. Figure 2.11 shows that the eccentricity, while

increasing with time, remains smaller than 0.07, consistent with the lack of mass loss

through the outer (L2) Lagrange point (Regös et al., 2005).


Figure 2.11: The eccentricity as a function of time (upper panel) and the argument of periastron

as a function of time ( ω versus t, lower panel) in the quadrupole approximation using

(e/0.009)2Q/k2 = 4.5× 109. The system remains trapped in the resonance for more than

105 yr which is a considerable fraction of the system lifetime. The eccentricity stays under the

limit of 0.07, a constraint imposed by the absence of L2 mass loss.


Figure 2.12: The mass transfer rate as a function of time (upper panel) and P/P (lower panel,

solid line) as a function of time in the quadrupole approximation using (e/0.009)2Q/k2 =

4.5× 109. The system remains trapped in the resonance for more than 105 yr, which is a

reasonable fraction of the system lifetime. The mass transfer rate is within 10% of its nominal

value m2 ≈ 10−8 M yr−1. P/P is lower than that due to Roche lobe overflow alone (dashed

line), but still > 0.


2.5.1 The nature of the third body

If the outer star is a white dwarf or a main sequence star, its mass is constrained to

be . 0.5M⊙ by the lack of an optical detection (Chou & Grindlay, 2001). The lack

of absorbing material along the line of sight to the X-ray source indicates that the

third star is not overflowing its Roche lobe. From the Roche lobe fitting formula of

(Eggleton, 1983),

R3 < RL ≈0.49q2/3

0.6q2/3 + ln(1 + q1/3)aout, (2.29)

where q is the mass ratio of the third star to the total mass of the inner binary.

This translates to R3 . 0.36R. From table 9 in Beatty et al. (2007), this implies

m3 . 0.39M. We conclude that the only stars with mass & 0.4M⊙ that will fit into

the outer orbit is a white dwarf or neutron star (or black hole). This leaves open the

possibility that the third star is a main sequence star with m . 0.4M.

According to Ivanova (2008), her Table 1, the fraction of hierarchical triples consist-

ing of a neutron star primary, a white dwarf secondary, and a white dwarf tertiary

formed via binary—binary encounters is about 1.4× 10−3. The fraction of triples

consisting of a neutron star-white dwarf binary orbited by a 0.4M (or lower) main

sequence star is similar. The fraction of neutron star-white dwarf-neutron star systems

is 2.1× 10−5. If this is the primary channel for formation of triple star systems, the

third star is likely to be either a white dwarf or a low mass (m < 0.4M) main

sequence star.

Far ultraviolet (FUV) observations made with Hubble Space Telescope revealed the

FUV period PFUV ' 693 s that is 1% longer than the X-ray period P1 ' 685 s (Wang &

Chakrabarty, 2010). The discovered FUV period may be consistend with a hierarchical

triple system or it may indicate a superhump system. If PFUV is considered to be a

beat period between the orbital period and the period of the third body, the required

semimajor axis of the third body would be aout ∼ 21× a. For such a distant third


body we do not find a librating solution. Wang & Chakrabarty (2010) suggested the

alternative explanation where PFUV is the positive superhump period. To confirm that

the system is a superhumper, the detection of the negative superhump is required.

2.6 DISCUSSION

The origin of the 170 day luminosity variations in 4U 1820-30 was first attributed

to the presence of a third body in the system by Grindlay (1988); this possibility

was expanded upon by Chou & Grindlay (2001) and more recently by Zdziarski

et al. (2007). Zdziarski et al. (2007) used a numerical model that calculates the time

evolution of an isolated hierarchical triple of point masses, using secular perturbation

theory up to octupole terms. Their model neglects the effects of tidal and rotational

distortion of the white dwarf, tidal friction, mass transfer and gravitational radiation

from the inner binary. Their calculations do include the GR periastron precession of

the inner binary.

Zdziarski et al. (2007) find a configuration that reproduces the 171 day oscillations

(assuming they are due to variations in e). They note that the GR precession rate is

near 170 days, and then choose a rather low neutron star plus white dwarf mass of

1.29 + 0.07M. With this choice, the period of the GR precession is ∼ 168 days. This

period is very near, but slightly shorter than, the observed 171 day period. To arrive

at the longer period, they choose the location and inclination of the third body so that

the Kozai torque results in a retrograde precession. When added to the GR precession,

this retrograde Kozai precession ensures the period of eccentricity oscillations will

be longer than 168 days. They are driven to a much lower magnitude for the Kozai

torque than employed in this chapter; they use aout = 8.66a and i0 = 40.96. They start

with ω = 0 and e = 10−4, ensuring that their solution circulates rather than librating.

They note that the apparent near equality between the Kozai and GR precession


rates is “a very remarkable coincidence”, but go on to say that they do not have any

explanation for this coincidence.

We have argued that the origin of the 171 day period of the luminosity variation of

LMBX 4U 1820-30 arises from libration in the Kozai resonance. This trapping explains

why the Kozai precession rate is comparable to the sum of the other precession rates

in the problem. If k2 is small enough, then the largest precession frequency in the

absence of a third body is that given by general relativity. In that case, the Kozai and

GR precession rates will sum to zero, i.e., the magnitude of the two precession rates

will be equal. Hence if k2 is small, then the expansion of the orbit of the inner binary

naturally explains the “remarkable coincidence” noted by Zdziarski et al. (2007). We

stress that, independent of the value of k2, the natural state of the system is likely to

be libration rather than circulation.

Trapping into libration is a consequence of mass-transfer driven orbital expansion

in the inner binary. We have pointed out that the apparent negative period derivative,

if it were intrinsic to the system, would not last for reasonable fraction of the system’s

lifetime. We find this to be an untenable situation.

The observed negative period derivative of the inner binary allows us to constrain

the tidal dissipation factor Q yielding a very firm lower limit of (e/0.009)2Q/k2 >

3.15× 109. We argue, however, that (e/0.009)2Q/k2 has to be yet higher, to trap and

maintain the system in libration around the stable Kozai fixed point. Our finding

indicates that if 4U 1820-30 is indeed a triple system, the negative period derivative is

not an intrinsic property of the system. However, as we showed in section 2.5 it does

not arise from the acceleration of the gravitation field of the globular cluster in which

4U 1820-30 resides, as suggested by van der Klis et al. (1993b).

In general, the eccentric orbit of a close binary system similar to 4U 1820-30 could

lead to a time-dependent irradiation of the secondary which could, in turn, give


rise to a thermal tide (Arras & Socrates, 2010). A thermal tidal torque opposes the

gravitational tidal torque, tending to force the secondary away from synchronous

rotation and to enhance the orbital eccentricity. An asynchronous spin may cause

large tidal heating rates, depositing heat in the interior of the secondary. In addition,

the irradiation of the stellar surface by the neutron star (or by the accretion disk) will

reduce the heat flux from the center of the white dwarf outward, so these irradiated

white dwarfs will be hotter than passively cooling white dwarfs. Since they are

hotter, they will have larger radii. The interplay between the two tidal torques would

eventually set the equilibrium spin state. As long as this equilibrium state is not

reached, the resulting bulge may oscillate, causing a periodic exchange of angular

momentum between the orbit and the spin of the white dwarf. This might provide an

alternate mechanism for producing the luminosity variations in 4U 1820-30. Since this

period is very stable, P3/P3 < 2.2× 10−4 according to Chou & Grindlay (2001), we are

currently looking into the possibility of such an interplay between gravitational and

thermal tidal torque as an explanation for 171 day period in 4U 1820-30.

We anticipate that the resonance trapping mechanism we have described in this

chapter is generic in Roche lobe overflow binaries in triple systems. The exact nature of

the librating orbit will vary with the properties of the particular system. For example,

for a binary with a larger semimajor axis, such that ωTB > ωGR, resonant trapping

will lead to |ωkozai| = ωGR.


APPENDIX A

EQUATIONS OF MOTION

The equations of motion we employ model the Kozai interaction, the dynamical effects

of the tidal bulge of the He white dwarf, GR periastron precession, the rotational

bulge of the He white dwarf, conservative mass transfer driven by the emission of

gravitational radiation, and tidal dissipation. We do not consider tides raised on

the neutron star primary. Detailed derivation of the equations representing Kozai

cycles with tidal friction and GR periastron precession can be found in Eggleton et al.

(1998) and Eggleton & Kiseleva-Eggleton (2001). Stellar masses are denoted by m1

(the mass of the neutron star primary), m2 (the mass of the white dwarf secondary)

M ≡ m1 + m2 (the inner binary mass), m3 (the mass of the outer companion), and

the reduced mass of the inner binary µ = m1m2/(m1 + m2). The mean motion of

the inner binary is n = 2π/P = [GM/a3]1/2. The inner binary orbital elements are:

semimajor axis a, eccentricity e, mutual inclination between the inner binary and the

outer binary orbit i, the argument of periastron ω, the longitude of ascending node Ω.

k2 is the tidal Love number, Q is the tidal dissipation factor, and R2 is the radius of

the white dwarf. The orbital parameters of the outer binary are denoted aout and eout.

G is Newtons constant and c is the speed of light.

Changes in the semimajor axis of the inner binary a are caused by tidal dissipation

and mass transfer:

a = aTD + aMT, (2.30)

where

aTD = −2a1tF

[1 + 15

2 e2 + 458 e4 + 5

16 e6

(1− e2)132

− Ωhn

1 + 3e2 + 38 e4

(1− e2)5

]

− 2ae2

1− e29tF

[1 + 15

4 e2 + 158 e4 + 5

64 e6

(1− e2)132

− 11Ωh18n

1 + 32 e2 + 1

8 e4

(1− e2)5

](2.31)


aMT = −23

am2

m2(2.32)

with m2 given by:

m2 = −6.21× 10−4(m1

M)

23

(Pperiastron

minutes

)−143 M

yr. (2.33)

For the zero eccentricity case, equation 2.33 is derived in detail in Rappaport et al.

(1987), where instead of the dependancy on periastron period Pperiastron they consider

dependancy on binary period.

The tidal friction time scale is

tF =16

(a

R2

)5 1n

m2

m1

Qk2

. (2.34)

The eccentricity of the inner binary e is affected by the Kozai torque and by tidal

dissipation:

ein = eKozai + eTD, (2.35)

where

eKozai =158

Gm3

a3out(1− e2

out)32 n

e√

1− e2 sin 2ω sin2 ß (2.36)

eTD = −9etF

[1 + 15

4 e2 + 158 e4 + 5

64 e6

(1− e2)132

− 11Ωh18n

1 + 32 e2 + 1

8 e4

(1− e2)5

]. (2.37)

The mutual inclination between the inner and the outer binary orbit, i is affected

by Kozai torques, by the rotational bulge, and by tidal dissipation:

i = iKozai + iRB + iTD, (2.38)

where

iKozai = −158

Gm3

a3out(1− e2

out)32 n

e2√

1− e2sin 2ω sin i cos i (2.39)

iRB =m1k2R5

22µna5

Ωh(Ωq sin ω−Ωe cos ω)

(1− e2)5 (2.40)

iTD = −Ωe sin ω

2ntF

1 + 32 e2 + 1

8 e4

(1− e2)5 −Ωq cos ω

2ntF

1 + 92 e2 + 5

8 e4

(1− e2)5 . (2.41)


Besides the negative precession rate of the argument of periastron due to Kozai

cycles, the total precession rate of the argument of periastron has additional positive

contributions from the tidal bulge, GR, the rotational bulge, and the tidal dissipation:

ωin = ωKozai + ωTB + ωGR + ωRB + ωTD, (2.42)

where

ωKozai =34

Gm3

a3out(1− e2

out)32 n

1√1− e2

[2(1− e2) + 5 sin2 ω(e2 − sin2 i)

](2.43)

ωTB =15(GM)

12

16a132

8 + 12e2 + e4

(1− e2)5m1

m2k2R5

2 (2.44)

ωGR =3(GM)

32

a52 c2(1− e2)

(2.45)

ωRB =M

12

4G12 a

72 (1− e2)2

k2R52

m2

×[ (

2Ω2h −Ω2

e −Ω2q

)+ 2Ωh cot i


) ](2.46)

ωTD = −Ωe cos(ω) cot i2ntF

1 + 32 e2 + 1

8 e4

(1− e2)5 +Ωq sin(ω) cot i

2ntF

1 + 92 e2 + 5

8 e4

(1− e2)5 . (2.47)

The precession of the longitude of ascending node is caused by Kozai cycles, rotational

bulge and tidal dissipation:

Ωin = ΩKozai + ΩRB + ΩTD, (2.48)

where

ΩKozai = − Gm3

a3out(1− e2

out)32 n

cos ι

4√

1− e2

(3 + 12e2 − 15e2 cos2 ω

)(2.49)

ΩRB =m1k2R5

22µna5 sin i

Ωh(1− e2)2

(−Ωq cos ω−Ωe sin ω

)(2.50)

ΩTD =Ωe cos ω

2n sin itF

1 + 32 e2 + 1

8 e4

(1− e2)5 −Ωq sin ω

2n sin itF

1 + 92 e2 + 5

8 e4

(1− e2)5 . (2.51)


APPENDIX B

ADIABATIC INVARIANCE OF THE ACTION

Time-dependent Hamiltonians, even those with just one degree of freedom, can be

difficult to solve. However, for Hamiltonians where the time dependance is sufficiently

slow, the problem is easier to tackle due to the existence of variables that are almost

constant. The approximate constants are the action variables of the Hamiltonian,

when the slow time dependance is neglected. Suppose that the time dependance

enters through a time dependent parameter κ(t). If the parameter κ varies very

slowly with time, treating κ as a time-independent parameter allows us to find action-

angle variables following the standard prescription. These action-angle variables are

functions of time through κ(t), which leads to the action no longer being a constant of

motion. However, when κ varies slowly with time, the action is nearly constant. Such

an action is known as an adiabatic invariant.

As described in section 2.3, capture in the resonance is a natural consequence

of semimajor axis expansion driven by mass transfer from the white dwarf. The

Hamiltonian of our system (see equation 2.4) is a function of the semimajor axis,

which is a parameter of H, playing the role of κ(t). In our case the semimajor axis is

not the only parameter varying with time; the masses of the inner binary vary with

time as well. Here we show, both analytically and via numerical integration, that the

change in the eccentricity is coupled to the change in the semimajor axis. When the

semimajor axis expands (respectively, contracts) the eccentricity of the stable fixed

point increases (decreases). We also demonstrate that the timescale for the change in

the eccentricity is a factor of & 150 shorter than the timescale for the semimajor axis.

To find the action we expand our Hamiltonian (equation 2.4) around the fixed


point:

G = G f + ∆ G (2.52)

ω = ω f + ∆ω (2.53)

Since we are expanding around the resonance, all terms ∝ ∆ G vanish. After some

algebra we find:

H = −A[− 10 + 12

G2f

L2 cos2 i f + 9G2

f

L2 + 15 cos2 i f +BALG f

+ k2CA

(35L9

G9f− 30

L7

G7f+ 3L5

G5f

)+ k2

DAL3

G3f

]− A

2L2

[18− 24 cos2 i f + 2

BAL3

G3f+ 30k2

CA

(105L11G1

f 1− 56

L9

G9f+ 3L7

G7f

)+ 12k2

DAL5

G5f

]∆ G2

− 15A

(1−G2

f

L2

)sin2 i f ∆ω2 (2.54)

which is similar to the Hamiltonian of the harmonic oscillator. Written more compactly

(and implicitly defining α(t), β(t) and C(t)):

H = C(t) + α(t)2

∆ G2 +β(t)

2∆ω2 = H0. (2.55)

We solve for ∆ G and evaluate the integral

J =2π

∫ ∆ωmax

0

(2(H0 − C(t))

α− β

α

) 12

d∆ω (2.56)

where ∆ωmax = (2(H0 − C(t))/β)12 . We find:

J =H0 − C(t)(αβ)

12

. (2.57)

Plugging in the corresponding terms from equation (2.54) yields:

J =L(a, m1, m2)

e f sin i f

P1(e f , a, m1, m2, i f )

P12

2 (e f , a, m1, m2, i f ), (2.58)


where

P1 =H0

A− 10− 12(1− e2

f ) cos2 i f + 9(1− e2f ) + 15 cos2 i f +

BA(1 +

12

e2f )

+4k2CA(2 + 15e2

f ) + k2DA(1 +

32

e2f ) (2.59)

and

P2 = 30[18− 24 cos2 i f + 2

BA(1 +

32

e2f ) + 120k2

CA(13 + 84e2

f ) + 12k2DA(1 +

52

e2f ).

(2.60)

Since the action J is an adiabatic invariant, we have:

dJdt

=∂J∂e

e +∂J∂a

a +(

∂J∂m2− ∂J

∂m1

)m2 = 0. (2.61)

The partial derivatives are:

∂J∂e

= − Je f

(1−

e f

P1

∂P1

∂e+

12

e f

P2

∂P2

∂e

)= Ce

Je f

(2.62)

∂J∂a

=Ja

(1 +

aP1

∂P1

∂a− 1

2aP2

∂P2

∂a

)= Ca

Ja

(2.63)

∂J∂m2

=J

m2

(1 +

m2

P1

∂P1

∂m2− 1

2m2

P2

∂P2

∂m2

)= Cm2

Jm2

(2.64)

∂J∂m1

=J

m1

(1 +

m1

P1

∂P1

∂m1− 1

2m1

P2

∂P2

∂m1

)= Cm1

Jm1

. (2.65)

Plugging these partial derivatives back into equation (2.61) yields:

0 = Ceee f

+ Caaa+

(Cm2 −

m2

m1Cm1

)m2

m2. (2.66)

The inner binary orbit is eccentric, which makes the mass transfer rate proportional

to periastron distance rp = a(1− e). Hence the m2 term can be decoupled into two

terms, one proportional to e and the other proportional to a:

m2

m2=

rp

rp= −3

2aa+

32

e1− e

. (2.67)

Combining equations (2.66) and (2.67) and solving for e leads to:

ee f

=

32

(Cm2 −

m2m1Cm1

)− Ca

Ce +32

e f1−e f

(Cm2 −

m2m1Cm1

) aa

. (2.68)


Plugging in the numerical values:

ee f≈ 150

aa

. (2.69)

Defining the time scales for the eccentricity and the semimajor axis to decay or increase

(depending on the value of Q) as τe = e f /e and τa = a/a, the timescales in equation

(2.69) are related by:

τe ∼ 6.7× 10−3τa. (2.70)

To demonstrate that the eccentricity evolution is indeed a consequence of the

action being an adiabatic invariant, we follow the evolution of the orbit around the

fixed point e f = 0.0155 and ω f = 90. Figure 2.13 shows e as a function of time

in a case where the semimajor axis is increasing, meaning that tidal dissipation is

sufficiently weak so that the evolution of the semimajor axis is dominated by mass

transfer (Q = 8× 107). The solid line presents e predicted by equation (2.69). For

t & 105 yr the numerical integration gives e ≈ 10−7 yr−1 corresponding to a timescale

150 times shorter than the timescale for the semimajor axis.

Despite the fact that the semimajor axis is expanding, the numerical integration

shows a transient phase (roughly the first 2000 years) where de/dt < 0, and a longer

phase (∼ 105 yr) where e is larger than predicted by equation (2.69). There are

contributions to the eccentricity evolution which we have ignored in our analytic

treatment; for example the spin of the white dwarf is not locked during the evolution

of the system. These un-modelled contributions are the source of the transient

behaviour.

To support this statement, we illustrate the eccentricity evolution in various cases

where we turn off different dynamical effects in Figure 2.14. The solid line presents a

result from the numerical integration that includes all dynamical effects in our model,

while the dotted line is the same integration with the eTD term set to 0; the result

shows that direct tidal dissipation on the eccentricity (equation 2.37) is not dynamically


Figure 2.13: e as a function of time for the case where the semimajor axis is expanding,

Q = 8× 107. We start integration exactly at the fixed point, where initial eccentricity is

e f ,0 = 0.01555. All other parameters are as listed in table 1. The solid line comes from the

analytic estimate where the action J is considered to be an adiabatic invariant. The dashed line

is a result of numerical integration. As expected from the action being adiabatic invariant, e is

positive. The difference in the magnitude of e within first 2× 105 yr is a result of our simplified

analytic calculation that does not include spin dynamics. Since our analytic estimate is valid

for small eccentricities, here we stop the integration when e > 0.1


significant. The dashed line presents the case where aTD is set to 0 (see equation 2.31);

The result shows that aTD has a significant influence on the eccentricity evolution.

The dash-dotted line shows the eccentricity when tidal dissipation factor Q is set to

infinity, but only in the differential equations that govern spin evolution. The long

dash-dotted line shows the eccentricity evolution in the case where Q is set to infinity

in the equations that govern the evolution of the spins together with aTD being set to 0.

The latter three cases demonstrate that the eccentricity starts increasing immediately

with the semimajor axis expansion, which is exactly the behaviour predicted by the

analytic analysis. After 5.5× 105 yr the eccentricity becomes & 0.1 and since our

analytic estimate is valid only for small eccentricities we stop the integration here.

The cause of the transient behaviour is the fact that the spin of the white dwarf is

not, contrary to our choice of initial conditions, tidally locked during the evolution

of the system. Whether the spin settles down in some Cassini state or other stable

configuration later during the evolution of the system is a possibility open to further

investigation.


Figure 2.14: The eccentricity of the fixed point of the inner binary as a function of time. Initial

conditions are the same as in Figure 2.13. The solid line is a result of the numerical integration

including all dynamical effects, while the dotted line is a result of the same integration but

with eTD = 0; these two results demonstrate that direct tidal dissipation on the eccentricity

does not significantly affect the evolution of eccentricity. The dashed line is the result of

integration where aTD = 0; this term has a more significant effect on the evolution of the

eccentricity. The dash-dotted line presents the case where we set Q = ∞, but only in the

equations that govern the spin evolution. The long dash-dotted line shows the eccentricity

evolution when Q = ∞ for the spins and aTD = 0; in this case we have the fastest increase in

the eccentricity.

Chapter 3

ON THE DYNAMICS OF UCXBs:

4U 1850-087, 4U 0513-40 AND M15 X-2

“The true mystery of the world is the visible, not

the invisible."— Oscar Wilde

3.1 INTRODUCTION

The possible existence of long period (∼ 100 d) variations in the luminosity of UCXBs

with orbital periods . 30 min raises the possibility that the binary is orbited by a

third body. The ratio between the orbital period and the period of the luminosity

variations is too large to be due to any kind of accretion disc precession or change

in the viewing angle (Larwood, 1998; Wijers & Pringle, 1999). These long period

variations in luminosity may be due to the actual change in the mass transfer rate

of the binary. The presence of the third body may induce periodic oscillations in the

eccentricity of the inner binary, which in turn will cause variations of the mass transfer

rate, with the same period. Using the idea described in Chapter 2, we show that this

61

Chapter 3. ON THE DYNAMICS OF UCXBs 62

long period can be explained as the period of libration around the stable fixed point

deep in a Kozai resonance. Since the semimajor axis of the inner binary expands on

roughly the accretion time scale (∼ 107 yr), the action is indeed an adiabatic invariant,

and as we demonstrated in detail in Chapter 2 the resonant trapping in libration

around the fixed point in a Kozai resonance is a natural consequence. In contrast,

tidal dissipation shrinks the semimajor axis. If tidal dissipation becomes a dominant

effect in the evolution of the semimajor axis, resonant trapping is no longer possible.

3.1.1 4U 1850-087

4U 1850-087 is a UCXB located in the galactic globular cluster NGC 6712. The distance

to the cluster is 6.8 kpc (Peterson & King, 1975; Harris, 1996). It was first detected as

an X-ray burster by Swank et al. (1976), which immediately indicates that the primary

is a neutron star. The source is located 6′′ or 0.1± 0.1 core radii from the cluster center

(Hertz & Grindlay, 1983). This system has been observed since the very beginnings

of observational X-ray astronomy, with detections by all the major satellites since

Uhuru. It shows variations of a factor of 10 in flux (Forman et al., 1978; Hoffman

et al., 1980; Warwick et al., 1981; Priedhorsky & Terrell, 1984b; Wood et al., 1984;

Warwick et al., 1988; Kitamoto et al., 1992; Christian & Swank, 1997; Juett et al., 2001).

Homer et al. (1996) reported a low amplitude periodicity in the Hubble Space Telescope

(HST) data of the ultraviolet counterpart of this source, where their periodogram can

be equally well fitted with a sinusoidal modulation at either of the two suspected

orbital periods of 20.6 min or 13.2 min. However, as Homer et al. (1996) point out, a

period as short as 13.2 min is very close to the 11 min period observed in 4U 1820-30

which is a much more luminous source. Since the larger luminosity corresponds to

higher mass transfer rate, 4U 1850-087 would be underlumiouns by a factor of ∼ 100

for a 13.2 min binary (Homer et al., 1996; Rappaport et al., 1987). Hence, 20.6 min is


interpreted as an orbital period even though it has yet to be confirmed. Such a short

period implies that a mass losing companion has to be a degenerate and low mass

star. Following the consideration of 4U 1820-30 from Rappaport et al. (1987), Homer

et al. (1996) derive the mass and the radius of the secondary to be 0.04M and 0.04R

respectively under the assumption that the secondary is a fully degenerate helium

white dwarf. Additionally, assuming a low-mass white dwarf donor and gravitation

radiation driven mass transfer, Homer et al. (1996) showed that the X-ray luminosity

of this system is consistent with the expected mass-transfer rate for a binary with

orbital period of 20.6 min.

An interesting feature of 4U 1850-087 is a possible long period of 0.72 yr reported

by Priedhorsky & Terrell (1984b) at which luminosity varies by a factor of 2− 3. Even

though, as we show further in the chapter, this long period fits well in the dynamical

picture given by our model, it needs to be further verified observationally.

3.1.2 4U 0513-40

4U 0513-40 is a low mass X-ray binary located in the globular cluster NGC 1851. The

distance to the source is 12 kpc (Harris, 1996). Far ultraviolet photometry obtained

by HST revealed a 17 min orbital modulation (Zurek et al., 2009). Observations

with BeppoSAX, Chandra, XMM-Newton, INTEGRAL confirmed the 17 min periodic

sinusoidal signal in soft X-ray (Fiocchi et al., 2011). The system is known to be an

X-ray burster (Galloway et al., 2008; Homer et al., 2001; Fiocchi et al., 2011) which

indicates that the primary is a neutron star. The short orbital period suggests a low

mass white dwarf secondary of ∼ 0.05M, as implied from mass- radius relation for

17 min period binaries from Deloye & Bildsten (2003).

4U 0513-40 shows very interesting variability in the X-ray flux on two different

time scales (Maccarone et al., 2010); a factor of ∼ 10 variation on timescales of weeks


and the variation of a factor of ∼ 2 in the luminosity when averaged over 1 yr

(Maccarone et al., 2010). This long time scale variation points toward a modulation in

the mass transfer rate, even though the mean luminosity agrees with that predicted

by gravitation radiation driven evolutionary scenario.

3.1.3 M15 X-2

M15, at a distance of 10.4 kpc (Harris, 1996), is the only globular cluster associated

with our galaxy known to house two bright LMXBs. In the early X-ray studies, a

single source 4U 2127+119 was first identified with the optical counterpart AC 211 by

Auriere et al. (1984) and further confirmed by a spectroscopic study by Charles et al.

(1986) showing signatures of an LMXB. A modulation in the optical and the X-ray

flux revealed the orbital period of 17.1 hr (Ilovaisky et al., 1993, and references within).

AC 211 is one of the brightest LMBXs in the optical and at the same time it has a low

X-ray luminosity ∼ 1036 erg s−1; the high optical-to -X-ray luminosity ratio led to a

conclusion that a very luminous central X-ray source is hidden behind the accretion

disk (Auriere et al., 1984). However, when X-ray bursts were detected by Ginga satellite

(Dotani et al., 1990) and later on with Rossi X-ray Timing Explorer (Smale, 2001), this

conclusion was highly dubious. This puzzling behaviour was finally understood when

Chandra observations resolved 4U 2127-119 into two X-ray sources (White & Angelini,

2001). One source is of course already known LMXB AC 211, while the second

one, named M15 X-2 which is actually the one producing X-ray bursts, is 2.5 times

brighter in X-rays than AC 211. The source is located 3.′′4 from the center of M15. The

optical and the FUV counterparts of M15 X-2 were identified in HST data by White &

Angelini (2001) and Dieball et al. (2005) respectively resulting in a determination of

the orbital period of 22.6 min by Dieball et al. (2005). The donor corresponding to such

a short period is a white dwarf of mass 0.02M ≤ M2,min ≤ 0.03M and of radius


0.02R ≤ R2,min ≤ 0.03R, while the presence of the X-ray bursts is in agreement

with a primary neutron star (Dieball et al., 2005).

So far no long period luminosity variations have been reported in M15 X-2. In

section 4.2 we discuss a constraint on the shortest expected long period assuming the

conservative mass transfer given by our model.

3.1.4 Plan of the chapter

In this chapter we apply the dynamical model described in Chapter 2 on the three

known UCXBs in the globular clusters just described: 4U 1850-087, 4U 0513-40 and

M15 X-2. We demonstrate that the suspected long period of 4U 1850-087 and 4U

0513-40 can be explained as libration in Kozai resonance with the period of small

oscillations around the fixed point. Our model gives a prediction for a yet undetected

long period of M15 X-2. As shown in Chapter 2 the crux of the dynamical history of

these systems is the interplay of mass transfer via Roche lobe overflow, which drives

the systems into a resonance) and the tidal dissipation, which tends to dump the

mutual inclination close to the Kozai critical inclination. This interplay allows us to

infer a constraint on the tidal dissipation factor Q for white dwarf donors in these

systems. In section 3.2 we give a review of the dynamical model developed in Chapter

2 and estimates for the systems’ parameters. The numerical results are presented in

section 4.2. In section 3.4 we use the model to provide a constraint on the ratio of the

tidal dissipation factor Q and the tidal Love number k2 of the white dwarf donors for

our fiducial eccentricities. We discuss our results and the results of other authors in

section 4.3.


3.2 OVERVIEW OF OUR DYNAMICAL MODEL

In Chapter 2 on the dynamics of 4U 1820-30, as well as in Prodan & Murray (2012),

we argue that the observed long period modulation of the luminosity (∼ 170 day)

is caused by the presence of a third body orbiting the center of mass of the binary.

Taking into account the presence of additional precessions, such as those due to tidal

and rotation distortion of the secondary, tidal dissipation and apsidal precession due

to general relativistic effects (GR), we show that the luminosity modulation arises from

the variations in the eccentricity of the inner binary associated with libration around

the stable fixed point deep in the Kozai resonance. Kozai resonance is 1 : 1 resonance

between the precession rate of the longitude of periastron v and the precession rate

of the longitude of the ascending node Ω of the inner binary. The condition for Kozai

resonance is satisfied only in cases where the mutual inclination is above its critical

value of 39.2. Here we list the equations for all four precession rates:

ωKozai =34

n(

m3

m1 + m2

)(a

aout

)3 1(1− e2

out)3/2× 1√

1− e2[2(1− e2) + 5 sin2 ω(e2 − sin2 i)

](3.1)

ωGR =32

n(

m1 + m2

m1

)(rs

a

) 1(1− e2)

(3.2)

ωTB =1516

n k2m1

m2

(R2

a

)5 8 + 12e2 + e4

(1− e2)5 (3.3)

ωRB =n k2

4m1 + m2

m2

(R2

a

)5 1(1− e2)2

×[ (

2Ω2h − Ω2

e − Ω2q

)+ 2Ωh cot i


) ]. (3.4)

The sign of the Kozai term depends on sin i, while the contribution from the tidal

bulge of the secondary and the GR term are always positive. The contribution from

the rotational bulge of the secondary is positive if the secondary is tidally locked

and if the spins are aligned; in the opposite case it is negative. We adopt k2 = 0.01,


which is computed for the helium white dwarf in 4U 1820-30 by Arras (private

communication) as the ratio of the potential due to the perturbed mass distribution, to

the external potential causing the perturbed mass, under the assumption that our He

white dwarf is a fluid object. Assuming that the system is tidally locked, to produce

these observed long period modulations, we are looking toward the cancellation better

than 10% between the Kozai term and all other terms. It turns out that the dominant

contribution, other than the Kozai precession, comes from the tidal bulge for our

fiducial value of k2. Such situation requires some fine tuning of inclination. However,

this fine tuning can be avoided if the system is trapped in the libration around the

stable fixed point, because then the sum of all these precession rates on average adds

up to zero. Therefore, these long periods of libration are associated with the period of

small oscillations around the fixed point.

Assuming the conservative mass transfer and absence of mass loss through the

L2 Lagrangian point at the mass ratios of these three systems, the maximum allowed

eccentricities are order of ∼ 0.05, as listed in Table 1 (Regös et al., 2005). As demon-

strated in Chapter 2, the periods of small oscillation of order of several hundred days

require such small eccentricities and arise naturally when the mutual inclination is

close to its critical value for producing Kozai cycles.

As shown in Chapter 2, via both analytical and numerical calculations, in order to

trap the system in the resonance and then to maintain it trapped for at least 105 yr

(which is about ∼ 1% of the lifetime for such systems), the semimajor axis has to

expand. In another words the evolution of the semimajor axis has to be dominated

by the mass transfer. Since the semimajor axis expands on a timescale much larger

than any orbital or precession timescale in the system, the action of the Hamiltonian

that describes the dynamics of such systems is an adiabatic invariant (for detailed

discussion see Chapter 2). The semimajor axis expansion of the inner binary lead to

the increase of the torque between the two orbits which is equivalent to deepening


of the Kozai potential. The fact that the action is an adiabatic invariant means that

the action of all orbits other than the separatrix remain constant, while the action of

the separatrix increases. Once the action of the separatrix exceeds the action of the

initially circulating orbit, that orbit is captured into the resonance and starts to librate.

The semimajor axis expansion eventually leads the orbit close to the fixed point where

is librates with the period of small oscillations.

On the other hand, if tidal dissipation, which tends to shrink the semimajor

axis, dominates the evolution of the semimajor axis, the system does not remain

trapped in the resonance. Rapid tidal dissipation dumps the eccentricity of the inner

binary within the 10−3 of the system’s lifetime leading toward the loss of long period

modulation of luminosity. Therefore, we would have to be incredibly lucky to observe

the system during such a short phase when the eccentricity modulation induced by

the tertiary is important. Hence we argue that the expansion of the semimajor axis

driven by Roche lobe overflow leads to trapping of the system deep in the Kozai

resonance. In the resonance, the angular momentum is transferred back and forth

periodically between the inner binary and the third star without affecting by any

means the semimajor axis of both binaries. However, when the forced eccentricity is

at its maximum and the mutual inclination between the two orbits at its minimum,

strong tidal dissipation reaches its maximum in removing the energy from the inner

orbit. Such coupled Kozai-tidal evolution tends to bring the mutual inclination toward

its critical value (∼ 40 Wu et al., 2007; Fabrycky & Tremaine, 2007).

3.2.1 Estimating the mass, the radius and the mass transfer rate of

the white dwarf secondary

To model the evolution of the system due to the mass transfer we follow the prescrip-

tion of Rappaport et al. (1987). Assuming that the secondary is a polytrope with index


n = 3/2, completely degenerate and hydrogen depleted, its mass-radius relation is

given by (Rappaport et al., 1987):

R2

R= 0.0128

(m2

M

)− 13

(3.5)

Since the secondary fills its Roche lobe, we have:

P =9π√

2(Gm2)

− 12 R

322 (3.6)

Combining equation 3.5 and equation 3.6 we obtain the mass-period relation (Rappa-

port et al., 1987):

m2 = 0.769(

Pmin

)−1

M (3.7)

Equation 3.7 puts a constraint on the mass of the donor star and knowing its mass we

constrain the radius of the donor using equation 3.5. For each binary we list the mass

and the radius of the donor star, as well as X-ray luminosity, maximum eccentricity

and the long period, in Table 1. The estimated masses and radii of the secondary for

each system listed in Table 1 are in agreement with the approximate values quoted in

the introduction for the masses of the secondary given by the mass-radius relation for

appropriate orbital period by Deloye & Bildsten (2003). For the mass of the neutron

star primary we adopt a canonical mass of 1.4M.

The mass transfer rate is given by (for detailed derivation see Rappaport et al.,

1987):

m2 = 6.21× 10−4(

m1

M

) 23(

Pmin

)− 143 M

yr(3.8)

Equation 3.8 gives the following expression for X-ray luminosity (Rappaport et al.,

1987; Homer et al., 2001):

LX = 1.06× 1038(

m1

1.4M

) 53(

R1

10 km

)−1( P11.4 min

)− 143

erg s−1 (3.9)

Both equation 3.8 and equation 3.9 are scaled with respect to the parameters of 4U

1820-30. We plot equation 3.9 in Figure 3.1 and we demonstrate that this equation,


TABLE 1. Constrained binary parameters

4U 1850-087

Symbol Definition Value Citation

m2 White dwarf (secondary) mass 0.04M Homer et al. (2001)

R2 White dwarf (secondary) radius 2.78× 109 cm

LX X-ray luminosity 1× 1036 erg s−1 Kitamoto et al. (1992)

emax Maximum inner binary 0.05 Regös et al. (2005)

eccentricity

P0 Long period 0.72 yr Priedhorsky & Terrell (1984b)

4U 0513-40


m2 White dwarf (secondary) mass 0.045M


LX X-ray luminosity 3× 1036 erg s−1 Callanan et al. (1995)


eccentricity

P0 Long period 1 yr Maccarone et al. (2010)

M15 X-2


m2 White dwarf (secondary) mass 0.034M


LX X-ray luminosity 0.74× 1036 erg s−1 Homer et al. (1996)


eccentricity

P0,min Minimum long period 1 yr

Table 3.1: Values of the Constrained Binary Parameters


originally derived to describe the evolution of 4U 1820-30, accounts reasonably well

for the X-ray luminosity of all three UCXBs: 4U 1850-087, 4U 0513-40 and M15 X-2.

3.2.2 The eccentricity and the period of small oscillations of the

inner binary

To describe the motion of the inner binary we employ Delaunay variables: the mean

anomaly l, the argument of periastron ω, and the longitude of the ascending node

Ω. In the Hamiltonian averaged over l and lout only ω appears . Their respective

conjugate momenta are:

L = m1m2

√Ga

m1 + m2(3.10)

G = L√

1− e2 (3.11)

H = G cos i. (3.12)

The averaged Hamiltonian is given by (Innanen et al., 1997; Ford et al., 2000; Fabrycky

& Tremaine, 2007; Prodan & Murray, 2012):

H =−3A

2

[− 5

3− 3H2

L2 +G2

L2 + 5H2

G2 + 5 cos 2ω

(1− G

2

L2 −H2

G2 +H2

L2

)]

−BLG − k2C

(35L9

G9 − 30L7

G7 + 3L5

G5

)− k2D

L3

G3 , (3.13)

where the term proportional to A is the Kozai term, the term proportional to B

enforces GR apsidal precession, and the terms proportional to C and D represent the

tidal and rotational bulges, respectively. The expressions for the constants are:

A =18

Φm2m3

(m1 + m2)2

(a

aout

)3 1(1− e2

out)3/2

(3.14)

B =32

Φm2

m1

rs

a(3.15)

C =116

Φm1

m1 + m2

(R2

a

)5

(3.16)


Figure 3.1: The X-ray luminosity, LX, versus the orbital period of the binary. The solid line

represents the equation 3.9 while dots are averaged observed X-ray luminosities. As shown,

equation 3.9 accounts reasonably well for the X-ray luminosity of 4U 1850-087, 4U 0513-40 and

M15 X-2.


D =112

Φ(

R2

a

)5

f (Ωspin), (3.17)

where

Φ ≡ G(m1 + m2)m1

a. (3.18)

We denote the semimajor axis and eccentricity of the outer body’s orbit by aout and

eout; rs ≡ 2Gm1/c2 is the Schwarzschild radius of the neutron star.

To obtain the the frequency (or the period) of the small oscillation, we evaluate the

second derivative of the Hamiltonian given by equation 3.13 at the fixed point:

ω0 = ωA

[(18 + 90

H2 L2

G4f

)+ 2

BAL3

G3f+ k2

CA

(3150

L11

G11f− 1680

L9

G9f+ 90

L7

G7f

)

+12k2DAL5

G5f

]1/2

× e f sin i f , (3.19)

where e f and icrit are the eccentricity of the fixed point and the critical mutual

inclination given by (Prodan & Murray, 2012):

e f =

√30 [cos2 icrit − cos2 i]

60 H2

L2 + 32

BA + 840k2

CA + 15

2 k2DA

(3.20)

cos2 icrit =35− 1

30BA− 4k2

CA− 1

10k2

DA

. (3.21)

The argument of periastron of the fixed point is ω f = 90, 270.

Under the assumption that there is no mass loss through the Lagrangian point L2,

we list in Table 1 the maximum possible eccentricities in the inner binaries given by

Regös et al. (2005) for the mass ratio of the binary. 4U 1850-087 and 4U 0513-40 have

suspected but yet not confirmed long periods of 0.72 yr (Priedhorsky, 1986) and 1 yr

(Maccarone et al., 2010) respectively. Their eccentricities are well below the maximum

possible eccentricities given in Table 1. Adopting the listed value for the maximum

eccentricity, we estimate using equations 3.20 and 3.19 the value for the minimum

possible period of small oscillations of M15 X-2 to be of order of 1 yr. This estimate

provides values on the verge of overflowing L2 point, hence in reality we would expect

this period to be longer and the eccentricity to be smaller.



In our numerical calculation we treat the gravitational effects of the third star in

the quadrupole approximation that includes the Kozai resonance described before.

We demonstrated in Chapter 2 that introducing the octupole approximation does

not qualitatively change our results. Our equations of motion are obtained from

the Hamiltonian averaged over the orbital periods of the inner and the outer binary.

The equations of motion, given in appendix A of Chapter 2, include the following

dynamical effects:


• periastron advance arising from quadrupole distortions of the white dwarf

secondary due to both tides and rotation;

• orbital decay due to tidal dissipation in the white dwarf secondary;

• loss of binary orbital angular momentum due to gravitational radiation;

• conservative mass transfer from the white dwarf secondary to the neutron star

primary driven by the emission of gravitational radiation.

The initial conditions that give us appropriately long periods for each binary

are listed in tables 2-4. These parameters are used throughout this chapter unless

otherwise stated.

Figures 3.2-3.4 show the eccentricity oscillations with corresponding long periods

for UCXB 4U 1850-087, 4U 0513-40 and M15 X-2. The amplitude of the eccentricity

oscillations is ∼ 5× 10−3 in the case of 4U 1850-087 and 4U 0513-40, while in the case

of M15 X-2 the amplitude of eccentricity oscillations is ∼ 3× 10−3. These eccentricity

oscillations are sufficient to produce the observed luminosity variations of a factor

of ∼ 2− 3 (Zdziarski et al., 2007). Trapping the system in libration around the fixed


TABLE 2. 4U 1850-087: System parameters


m1 Neutron star (primary) mass 1.4M

m2 White dwarf (secondary) mass 0.04M Homer et al. (2001)

m3 Third companion mass 0.55M

a1 Inner binary semimajor axis 1.95× 1010 cm Homer et al. (1996)







R2 White dwarf radius 2.78× 109 cm see section 3.2.1





TABLE 3. 4U 0513-40: System parameters



m2 White dwarf (secondary) mass 0.045M see section 3.2.1


a1 Inner binary semimajor axis 1.65× 1010 cm Zurek et al. (2009)











point provides a natural explanation for the origin of the long periods and the small

eccentricity oscillations required to produce observed luminosity variations.

In order to keep the fiducial eccentricity of M15 X-2 below the maximum eccentric-

ity given in Table 1 at all times and to have sufficiently large eccentricity oscillation

to produce luminosity variations of a factor of 2− 3, we require in our numerical

calculations a long period of 3.4 yr, which is a factor of ∼ 3 larger than the estimated

minimum long period listed in Table 1. Note that any long period variations in

luminosity for M15 X-2 remain undetected until this date.


TABLE 4. M15 X-2 : System parameters



m2 White dwarf (secondary) mass 0.034M see section 3.2.1


a1 Inner binary semimajor axis 2.1× 1010 cm Dieball et al. (2005)










Table 3.4: M15 X-2: Values of the System Parameters


Figure 3.2: 4U1850-087: The eccentricity as a function of time (upper panel) and the phase space

(e versus ω) for our fiducial model. The long period (period of the eccentricity oscillations) is

0.72 yr, and the amplitude of the eccentricity oscillation is sufficient to produce the observed

factor of 2− 3 variation in luminosity.


Figure 3.3: 4U 0513-40: The eccentricity as a function of time (upper panel) and the phase space

(e versus ω) for our fiducial model. The long period (period of the eccentricity oscillations)

is 1 yr, and the amplitude of the eccentricity oscillation is sufficient to produce the observed

factor of ∼ 2 variation in luminosity.


Figure 3.4: M15 X-2: The eccentricity as a function of time (upper panel) and the phase space

(e versus ω) for our fiducial model. The long period (period of the eccentricity oscillations) is

3.4 yr, and the amplitude of the eccentricity oscillation is sufficient to produce the observed

factor of ∼ 2 variation in luminosity.


3.3.1 Resonant trapping and detrapping

As argued in Prodan & Murray (2012) and in Chapter 2 the semimajor axis of the

inner binary has to expand. This expansion is expected in the standard evolutionary

scenario (see Introduction). As the orbit expands due to the mass transfer, the action

of the separatrix increases adiabatically on the accretion timescale, leading to trapping

in a resonance around the fixed point. One might expect that tidal effects could be

dominant at such a small separation. There are two arguments against a shrinking

semimajor axis. The first argument is that the phase where the semimajor axis shrinks

and the eccentricity decays due to tidal dissipation is short-lived; only a few thousand

years until the mass transfer driven expansion dominates the evolution. The second

argument is that the shrinking orbit drives the initially librating orbit out of resonance

into circulation which would change dramatically the period of luminosity variations.

Thus, as shown previously for the case of 4U 1820-30, the evolution of the system is

dominated by mass transfer but the rate of expansion of the orbit is decreased due to

tidal dissipation.

In this section we demonstrate that just this physical picture applies in the case of

these three binaries as well and it can indeed explain the origin of their long periods.

Figures 3.5, 3.7 and 3.9 show the systems initially put on circular orbit with a choice

of Q such that the orbit expands. As integration proceeds, the separatrix expands and

eventually captures the initially circulating orbit in libration around the fixed point.

Once captured, the system librates for at least 105 yr which is a reasonable fraction

of the system’s lifetime during which it can be observed in such a state (see section

3.4). Figures 3.6, 3.8 and 3.10, on the other hand, show the case where the systems is

initially on librating orbit with Q such that semimajor axis shrinks. In all three cases

detrapping from the resonance occurs fairly quickly making observation of the system

in such a state highly unlikely.


Figure 3.5: 4U 1850-087: a) ω vs t. Initially we place the system in circulation; after about

40000 yr it gets trapped in libration. Here we have used Q = 1 × 108, while the initial

eccentricity is e0 = 0.044; all other parameters are the same as listed in Table 2. For this choice

of parameters, the system remains in libration for about 105 yr. b)The eccentricity as a function

of time. The eccentricity does not exceed significantly the estimated maximum value of 0.05.


Figure 3.6: 4U 1850-087: a) ω vs t. We start the evolution of the system by placing the system

in libration. Here we use Q = 2× 107, which is the value required to shrink the semimajor

axis. The action of the separatrix is decreasing, and the system is ejected from the resonance

after about 35000 yr. b) The eccentricity as a function of time.


Figure 3.7: 4U 0513-40: a) ω vs t. Initially we place the system in circulation; after about

20000 yr it gets trapped in libration. Here we have used Q = 5 × 107, while the initial

eccentricity is e0 = 0.02; all other parameters are the same as listed in Table 2. For this choice

of parameters, the system remains in libration for about 3× 105 yr. b)The eccentricity as a

function of time. The eccentricity does not exceed significantly the estimated maximum value

of 0.05 for at least 3× 105 yr .


Figure 3.8: 4U 0513-40: a) ω vs t. We start the evolution of the system by placing the system

in libration. Here we use Q = 2× 107, which is the value required to shrink the semimajor

axis. The action of the separatrix is decreasing, and the system is ejected from the resonance

after about 1800 yr. b) The eccentricity as a function of time.


Figure 3.9: M15 X-2: a) ω vs t. Initially we place the system in circulation; after about 20000 yr

it gets trapped in libration. Here we have used Q = 6× 107, while the initial eccentricity is

e0 = 0.015; all other parameters are the same as listed in Table 2. For this choice of parameters,

the system remains in libration. b)The eccentricity as a function of time. The eccentricity does

not exceed significantly the estimated maximum value of 0.04 .


Figure 3.10: M15 X-2: a) ω vs t. We start the evolution of the system by placing the system in

libration. Here we use Q = 2× 107, which is the value required to shrink the semimajor axis.

The action of the separatrix is decreasing, and the system is ejected from the resonance after

about 140 yr. b) The eccentricity as a function of time.


TABLE 5. Tidal Dissipation Factor Q

4U 1850-087

Symbol Value

( e0.018)

2 Qk2

6× 109

4U 0513-40

Symbol Value

( e0.007)

2 Qk2

5× 109

M15 X-2

Symbol Value

( e0.005)

2 Qk2

6× 109

Table 3.5: Values of the Tidal Dissipation Factor Q

3.4 CONSTRAINING THE TIDAL DISSIPATION

FACTOR Q FOR THE WHITE DWARF

COMPANIONS

In this section we constrain the tidal dissipation factor Q for the white dwarf compan-

ions simply by asking that the system remains trapped in the resonance for at least

105 yr. Estimated mass transfer rates for these systems are (5× 10−10-10−9)M/ yr

(see equation 3.8) giving the lifetime of ∼ 7× 107 yr during which these systems can

sustain the observed luminosity. Therefore a reasonable fraction of time to remain

trapped in the resonance is at least 105 yr. Figures 3.11 to 3.13 demonstrate that for the

fiducial values of Q or more precisely e2Q/k2 listed in table 5 these systems remain

trapped in the resonance for reasonable fraction of their lifetime during which the

maximum eccentricity does not exceed values given in Table 1.


Figure 3.11: 4U 1850-087: a) ω vs t.Here we have used Q = 6× 107; all other parameters are

the same as listed in Table 2. For this choice of parameters, the system remains in libration

for about 105 yr. b)The eccentricity as a function of time. The eccentricity does not exceed the

estimated maximum value of 0.05 during the integration.


Figure 3.12: 4U 0513-40: a) ω vs t. Here we have used Q = 5× 107; all other parameters are

the same as listed in Table 2. The system remains in libration during the integration. b)The

eccentricity as a function of time. The eccentricity does not exceed the estimated maximum

value during integration time.


Figure 3.13: M15 X-2: a) ω vs t. Here we have used Q = 6× 107; all other parameters are

the same as listed in Table 2. The system remains in libration during the integration. b)The

eccentricity as a function of time. The eccentricity does not exceed the estimated maximum

value of 0.04 during the integration.


3.5 DISCUSSION

A long term luminosity periodicity in UCXBs has been suspected for a while now but

the observations are not sufficiently good to confirm it with reasonable certainty. The

only system for which the long periodicity is certain is the 11 min binary 4U 1820-30.

The detection of X-ray bursts in this object (Grindlay et al., 1976) led to extensive

observations and hence a very well sampled light curve. Several authors suggested

that this long periodicity may be due to the presence of a third body (Grindlay, 1988;

Chou & Grindlay, 2001; Zdziarski et al., 2007). In Chapter 2 we attribute the origin of

the long period variations in the luminosity to the libration in Kozai resonance with

frequency of small oscillations around the fixed point. In addition to the perturbations

from a third body, we consider tidal effects, GR precession and mass transfer driven

by gravitational wave radiation. Also we show that trapping in a resonance is a

consequence of the expansion of the orbit of the inner binary driven by mass-transfer.

The model developed for 4U 1820-30 predicts long periodicities in the light curve of

4U 1850-087 and 4U 0513-40 as well as of M15 X-2. Requiring the systems to remain

trapped in a resonance for a reasonable fraction of their lifetime allows us to put the

upper limit on the tidal dissipation factor for the white dwarf donors. The actual

values are listed in Table 5. Obtaining the lower limit for tidal dissipation factor Q of

order of f ew× 107 is in agreement with the results of Piro (2011), Fuller & Lai (2011),

Prodan & Murray (2012) and Chapter 2.

The three binaries examined in this chapter have similar properties to 4U 1820-30.

For two of them, 4U 0513-40 and 4U 1850-087, a long period variations have been

suggested (Priedhorsky & Terrell, 1984b; Maccarone et al., 2010). Therefore we suggest

that these three systems may be triples as well and further more we would anticipate

that majority of UCXBs in the globular clusters are actually triples. Short-period

binaries like these are quite likely to acquire a third body in the dense environment


of globular clusters. Comparing the confirmed orbital periods in the field to those in

globular clusters, the trend seems to indicate that field UCXBs have orbital periods of

order of 40 min while those in globular clusters have periods . 20 min. Such a trend

hints at different formation scenarios operating in these two environments. Very long

period variations which cannot be due to accretion disk precession or a change in the

viewing angle, seem to be the characteristic of UCXBs in globular clusters. To check

this speculation it is necessary to determine orbital periods for more of these systems

and obtain a better statistical sample.

Several mechanisms have been proposed to explain why the accretion rates differ

from those expected based on the binary parameters. Other than the presence of the

third body, which we discussed in great detail, two viable candidates two are: tidal disk

instabilities (Osaki, 1995) and irradiation of the donor responsible for the modulation

of the mass transfer rate (Hameury et al., 1986). Their common characteristic is that

both cause more stochastic variations than those expected from the presence of a

third body. The light curves of these binaries are not sampled as well as the light

curve of 4U 1820-30. Even though the data may indicate the potential presence of a

regular modulation, there are lots of irregularities that may be the consequence of a

combination of regular modulation due to the third body and these other mechanisms

causing aperiodic variations.

The basic idea of the tidal disk instability model is that the mass transfer rate is

assumed to be constant and all outbursts of accretion onto the primary are caused by

intrinsic instabilities in the accretion disk. During the minimum luminosity phase of

the long period cycle the disk is compact. The thermal instability produces only quasi-

periodic episodes of accretion that are observed as normal outbursts. In each normal

outburst the accreted mass is less than the mass transferred during the quiescent

phase because tidal removal of angular momentum from the disk is inefficient. As

both the mass and the angular momentum of the disk are gradually built up, the


radius of the disk expands further with each successive outburst until it eventually

exceeds the critical radius for tidal instability. At this point, the final normal outburst

triggers the tidal instability producing a precessing eccentric disk, which is observed

as a superhump. The eccentric disk enhances greatly the tidal torque, resulting

in the superoutburst that significantly clears out the disk mass. At the end of the

superoutburst, the disk returns to the starting compact state.

The second model considers a mass loss instability in the donor star as a conse-

quence of illumination of its atmosphere by the X-ray flux emitted by the compact

object. During the quiescent phase the donor does not fill in completely its Roche lobe

causing the accretion rate to be lower but still sufficient to heat up the external layers

of the donor’s atmosphere. As these layers are heated up slowly, by an X-ray flux that

is comparable to the stellar flux at the vicinity of the L1 point, they expand. Ultimately,

the heating brings the atmosphere in the unstable regime where matter flows through

the L1 point at a high rate. Eventually the shielding by the accretion disk may prevent

X-ray flux from reaching the L1 region, which will cause the outburst to cease. By this

time the heated layers have been transported to the disk. The outburst stops when the

entire disk is accreted onto the compact object.

Tidal instabilities in the accretion disk may explain the variations in the accretion

rate on weeks timescale, such as those seen in 4U 0513-40, but most likely not the

long period variations (Maccarone et al., 2010). There are no studies of the irradiation

induced mass transfer in the context of white dwarfs and hence it is very difficult to

make any conclusive statements. The observations clearly show that irradiation of

white dwarf donor in 4U 0513-40 is significant (Maccarone et al., 2010), but the same

is not observed in 4U 1820-30 which is a brighter system. Provided that one finds

a reasonable explanation for this discrepancy, the irradiation induced mass transfer

model could be feasible. Unquestionably, to understand the details of the dynamics

and evolution of UCXBs more observations are required.

Chapter 4

ON WD-WD MERGERS IN TRIPLE

SYSTEMS: THE ROLE OF KOZAI

RESONANCE WITH TIDAL

FRICTION

“I can believe anything provided it is incredible."

— Oscar Wilde

The merger of two white dwarfs (WDs) driven by gravitational wave (GW) radi-

ation has been suggested as a possible mechanism leading to production of type Ia

supernovae (Iben & Tutukov, 1984; Webbink, 1984; Howell, 2011). For this mechanism

to be observationally relevant, the merger rate has to be comparable to the rate of

Ia supernovae events. Even though a large fraction of stars are born as binaries,

only a small fraction are tight enough to merge via GW emission within a Hubble

time. But if these binaries are in hierarchical triples, the presence of a tertiary on a

highly inclined orbit can induce eccentricity oscillations in the inner binary via secular

95

Chapter 4. ON WD-WD MERGERS IN TRIPLE SYSTEMS 96

resonance (Kozai, 1962; Lidov, 1962). The GW radiation timescale (TGW) and the time

scale associated with dissipative tides are both a strong function of eccentricity; hence

the presence of a third body on a highly inclined orbit pumping the eccentricity of

the inner binary to high values can dramatically decrease the merger time scale of

the binary (Blaes et al., 2002; Miller & Hamilton, 2002; Wen, 2003; Antonini & Perets,

2012).

This mechanism was invoked by Thompson (2011) to enhance the rate of WD–WD,

NS–WD and NS–NS mergers due to GW radiation. Such mergers are responsible for

production of exotica such as Ia supernovae, γ-ray bursts (GRBs) and other transients.

Thompson (2011) showed that the GW merger timescale for compact object binaries

in triple systems was significantly decreased from that of the binary alone, which

allowed for a larger range of the semimajor axes that could lead to a merger in a

Hubble time, as well as an increased rate of prompt mergers (< 108 yr). For detailed

discussion on how common these systems are, formation scenarios and rates we refer

the reader to Thompson (2011).

In this chapter we explore the role of tides in WD–WD merger events. We demon-

strate that in the range of high inclinations (91o ≤ i0 ≤ 96o), the outcome of the

evolution is a direct collision of the two WDs, in which tidal effects do not play a

significant role. Tidal effects do play a significant role in the range of moderately high

inclinations (85o ≤ i0 ≤ 90o and 97o ≤ i0 ≤ 102o) where they dramatically decrease

TGW .

In Section 4.1 we describing the Kozai–Lidov dynamics in the presence of additional

forces and dissipation due to tides and gravitational wave radiation. We describe

relevant timescales for our dynamical problem. In Section 4.2 we describe the results

of numerical integrations of the equations of motion. We discuss our findings in

Section 4.3.


TABLE 1. System parameters

Symbol Definition Value

m1 White dwarf (primary) mass 0.8M⊙m2 White dwarf (secondary) mass 0.6M⊙m3 Third companion mass 1.0M⊙a Inner binary semimajor axis 0.05AU

aout Outer binary semimajor axis 1AU


eout,0 Outer binary eccentricity 0.5

iinit Initial mutual inclination 85o − 102o

ωin,0 Initial argument of periastron 0


R White dwarf radius 5× 108 cm

k2 Tidal Love number 0.1

Q Tidal dissipation factor 107

Table 4.1: Values of the System Parameters

4.1 UNDERSTANDING THE DYNAMICS

4.1.1 The Kozai–Lidov mechanism

The presence of a third body on a hierarchical orbit around the centre of mass of a

binary will affect the orbital elements of the binary on a variety of timescales. The

induced changes in the orbital elements of the binary will be particularly striking if the

mutual inclination between the inner and the outer orbit is high. The two orbits will

exchange angular momentum, causing both the eccentricity of the inner binary and

the mutual inclination to undergo periodic oscillations known as Kozai cycles (Kozai,


1962). Kozai (1962) showed that the mutual inclination required for having Kozai

cycles is icrit ≤ i ≤ 180o − icrit, where icrit ≈ 39.2o is a critical inclination. Kozai cycles

result from a 1 : 1 resonance between the longitude of the periapse v and the longitude

of the ascending node Ω and therefore the condition for Kozai resonance, v− Ω = 0,

is fulfilled only for inclinations in the Kozai regime (icrit ≤ i ≤ 180o − icrit). For

inclinations outside of the Kozai regime, the apsidal line precesses in a prograde sense

(v > 0), while the line of nodes precesses in a retrograde sense (Ω < 0). For prograde

orbits (i ≤ 90o) these cycles are out of phase, meaning that when the eccentricity

reaches its maximum, the mutual inclination reaches its minimum and vice versa.

On the other hand, for retrograde orbits (i > 90o) these cycles are in phase: both the

eccentricity and the mutual inclination reach maximum values simultaneously. The

period of a Kozai cycle is significantly longer than either the orbital period of the

inner or the outer binary, suggesting the use of the secular approximation. The secular

approximation consists of averaging the equations of motion over the orbital periods

of the inner and the outer binary. Such averaged equations allow for exchange of

angular momentum between the two orbits but not variations in the energy, so that

the semimajor axes of both orbits remain unchanged. The relevance of the secular

approximation is discussed further in Section 4.3. The maximum eccentricity attained

in Kozai cycles in the absence of additional forces is given by:

emax =

(1− 5

3cos2i0

) 12

(4.1)

where i0 is the initial mutual inclination between the inner and the outer orbit. Note

that equation 4.1 is given in the test particle limit where the secondary is treated as a

massless particle.

Kozai cycles can be suppressed by other dynamical effects that induce periapse

precession in the inner binary. We take into account the following additional sources

of periapse precession: apsidal precession due to tidal and rotational bulges, apsidal


precession due to general relativity (GR) and the apsidal precession due to tidal

dissipation, which is negligible in comparison to the other two. Tidal dissipation as

well as gravitational wave radiation play a major role in driving the merger of the

inner binary and is discussed in more details later in the chapter.

We consider an inner WD–WD binary with semimajor axis ain, eccentricity ein,

argument of periastron ωin , masses m1 and m2 and two equal radii R. The third

body with mass m3, semimajor axis aout, eccentricity eout and argument of periastron

ωout is on a larger orbit around the center of mass of the inner binary. The mutual

inclination between the two orbits is i. The mean motion of the inner binary is

n = [G(m1 + m2)/a3in]

1/2 Fiducial values of the system parameters used throughout

this chapter are listed in Table 1. Subscript “0” denotes initial values (e.g. ein,0).

The equations for the precession rates due to the Kozai–Lidov mechanism, GR,

tidal and rotational bulges raised on both WDs in the quadrupole approximation are

(Eggleton & Kiseleva-Eggleton, 2001):

ωKozai =34

Gm3

a3out(1− e2

out)32 n

1√1− e2

[2(1− e2) + 5 sin2 ω(e2 − sin2 i)

](4.2)

ωGR =3(G(m1 + m2))

32

a52 c2(1− e2)

(4.3)

ωTB =15(G(m1 + m2))

12

16a132

8 + 12e2 + e4

(1− e2)5

(m1

m2+

m2

m1

)k2R5 (4.4)

ωRB =(m1 + m2)

12

4G12 a

72 (1− e2)2

k2R5 ∑i=1,2

1mi

[ (2Ω2

ih −Ω2ie −Ω2

iq

)+2Ωih cot i

(Ωie sin ω + Ωiq cos ω

) ](4.5)

ωTD =cot i2ntFi

∑i=1,2

(−Ωie cos ω

1 + 32 e2 + 1

8 e4

(1− e2)5

+Ωiq sin ω1 + 9

2 e2 + 58 e4

(1− e2)5

)(4.6)


where tF1 = 16

( aR)5 1

n

(m2m1

)Qk2

is the tidal friction time scale for the star with mass

m1. A similar expression with two indices 1 and 2 swapped holds for the tidal friction

time scale tF2 for the star with mass m2. The three components of the spin, Ωi, are the

projection along the Laplace-Runge-Lenz vector, pointing along the apsidal line from

the WD secondary at the apoapse toward the WD primary, denoted by Ωe, along the

total angular momentum vector, Ωh, and their cross product, denoted by Ωq.

As equation 4.2 shows, the term driving Kozai cycles can be either positive or

negative depending on the value of sin i. On the other hand, the terms driving the

precession due to GR and the tidal bulge are always positive and therefore tend to

promote periapse precession. The effect of these two terms is to lower the maximum

eccentricity attainable by the system, while the critical inclination increases (Eggleton

& Kiseleva-Eggleton (2001); Miller & Hamilton (2002); Wu & Murray (2003); Fabrycky

& Tremaine (2007), see their Figure 3). The term induced by the rotational bulge

may have either positive or negative value. We assume that initially the system is

tidally locked and that the spins are aligned, so this term tends to increase the rate of

precession and hence suppress Kozai cycles. Precession due to both the rotational and

tidal bulges are parametrized by the tidal Love number k2 which is a dimensionless

constant that relates the mass of the multipole moment created by tidal forces on the

spherical body to the gravitational tidal field in which that same body is immersed.

Furthermore, k2 encodes information on the internal structure of the body in question1.

Tidal dissipation in the stars in the inner binary, due to either an eccentric orbit

or to asynchronous rotations, becomes important when the separation between the

stars is of order of a few stellar radii (Mazeh & Shaham, 1979; Eggleton & Kiseleva-

Eggleton, 2001). During the phases of high eccentricity the periapse distance may

become sufficiently small as to lead to strong tidal dissipation. During these phases of

1The apsidal precession constant, which is a factor of two smaller than the tidal Love number, butwhich we do not utilize, is often denoted by k2 as well.


high eccentricity the tidal dissipation will drain energy from the orbit, but not angular

momentum. The energy loss results in a reduction of the semimajor axis and therefore

enhances the rate of dissipation. Since the angular momentum remains conserved

during this process, the eccentricity is damped as well until the orbit eventually

circularizes and the system settles at a separation of only a few stellar radii. Tidal

dissipation is parametrized by the tidal dissipation factor Q, defined as the ratio of

the energy stored in the tidal bulge to the energy dissipated per orbit.

Another source of dissipation in WD–WD binaries is gravitational wave radia-

tion which drains both energy and angular momentum from the orbit. Like tidal

dissipation, it tends to shrink and circularize the orbit. During the Kozai cycles, if

the amplitude of the eccentricity oscillations is sufficiently large, the GW radiation

becomes much stronger than in the circular case, leading to mergers on timescales

much shorter than a single Hubble time, THubble (Blaes et al., 2002; Miller & Hamilton,

2002; Thompson, 2011; Antonini & Perets, 2012). As the inner eccentricity reaches

values close to 1, dissipation due to tides and GW radiation become comparable and

neither can be neglected, as we discuss in more detail in next section ( see figure 4.1).

4.1.2 Timescales

The GW merger timescale in the limit of high eccentricity is (Peters, 1964):

TGW =385

ac

(a3c6

G3m1m2M

)(1− e2)

72

' 5.4× 1012 yr

(0.672M3⊙m1m2M

)( a0.05AU

)4(1− e2)

72 , (4.7)

where M = m1 + m2. As seen from eqn 4.7, in the absence of a third body TGW is

greater than the Hubble time, THubble ' 14 Gyr, for a > 0.01AU.

As discussed in the previous section, GR precession tends to promote periapse

precession and therefore suppress Kozai cycles. In general, GR precession decreases


the maximum possible eccentricity attainable by the binary at a fixed initial inclination

and increases the critical inclination required for undergoing Kozai cycles (Blaes et al.,

2002; Fabrycky & Tremaine, 2007; Prodan & Murray, 2012). The timescale for GR

precession is:

TGR =13

ac

(ac2

GM

) 32

(1− e2)

' 1.8× 103 yr(

1.4M⊙M

) 32 ( a

0.05AU

) 52(1− e2). (4.8)

We consider the effects of tidal forces, where both rotational and tidal bulges tend

to suppress Kozai cycles in a similar manner as GR precession (Equation 4.8). The

timescale for precession induced by the tidal bulge is given by (Prodan & Murray,

2012; Fabrycky & Tremaine, 2007):

TTB =16

15k2

( aR

)5(

a3

GM

) 12(

m2

m1+

m1

m2

)−1 (1− e2)5

8 + 12e2 + e4

' 5.9× 1013 yr( a

0.05AU

) 132(

5× 108 cmR

)5(1.4M⊙M

) 12 (1− e2)5

8 + 12e2 + e4 ,(4.9)

where R is the radius of the white dwarf and k2 = 0.1. The Kozai timescale is (Innanen

et al., 1997; Holman et al., 1997):

TKozai =43

(a3MGm2

3

) 12 ( aout

a

)3(1− e2

out)12

' 22 yr( a

0.05AU

) 32(

M1.4M⊙

) 12(

M⊙m3

)(aout/a

20

)3

(1− e2out)

12 . (4.10)

The condition for the inner binary to undergo Kozai oscillation is that the Kozai

timescale is shorter than the timescale of any of the suppressing effects. Furthermore,

the Kozai timescale is strongly dependent on the ratio of the inner and the outer

binary semimajor axis, which sets a limit on the maximum allowed aout, beyond which

the Kozai oscillations are ineffective. During the evolution, depending on the type of

stars in the inner binary (i.e. compact objects or main sequence stars) and tightness


of its orbit, dissipative effects due to gravitational wave radiation or tides may be

significant. Either of the two may lead to shrinkage and/or circularization of the inner

orbit, which increases the ratio of the inner and outer semimajor axis and thus reduces

the effectiveness of Kozai oscillations. Consequently, the Kozai timescale may become

larger than either TGR or TTB and at this point the evolution of the system toward

merger would be completely dominated by non-Kozai effects.

To emphasize the importance of Kozai oscillations for rapid mergers of compact

objects and considering only the presence of a third body, Thompson (2011) gives the

following order-of-magnitude estimate of the merger time:

Tmerge ∼ 25153

ac

(a3c6

G3m1m2M

)cos6i

∼ 4.5× 104 yr

(0.672M3⊙m1m2M

)( a0.05AU

)4(

cosicos(88o)

)6

, (4.11)

which shows a strong dependance on mutual inclination. Equation 4.11 is only valid

in the Kozai regime, 39o ≤ i ≤ 141o and fails to account for additional sources of

apsidal precession such as GR precession and precession due to the tidal and/or

rotational bulge. As pointed out by Thompson (2011), this expression significantly

underestimates the merger time in certain regions of parameter space and hence

should be used only to obtain a rough estimate. For detailed discussion we refer the

reader to the Section 4 of Thompson (2011).

The timescale for the semimajor axis to decay due to either tidal dissipation or GW

radiation is strongly dependent on the eccentricity:

τa|TD =( a

a

)TD

= −92

tF1− e2

e2

[1 + 15

4 e2 + 158 e4 + 5

64 e6

(1− e2)132

− 1118

1 + 32 e2 + 1

8 e4

(1− e2)5

]−1

(4.12)

τa|GW =( a

a

)GW

= − 5c5a4(1− e2)72

64G3m1m2(m1 + m2)

[1 +

7324

e2 +3796

e4]−1

(4.13)

As figure 4.1 shows, during the phases of the Kozai cycle where e ∼ 1, τa|TD ≈


τa|GW and hence tidal dissipation can not be ignored. We demonstrate in Section 4.2.3

that the merger timescale of the WD–WD binary in the regime of moderately high

inclinations (85o ≤ i0 ≤ 90o and 97o ≤ i0 ≤ 102o) is shorter by at least an order of

magnitude when tidal dissipation is taken into consideration than when gravitational

wave radiation alone is accounted for.

On the other hand, in the regime of highly inclined orbits (91o ≤ i0 ≤ 96o), neither

of the two sources of dissipation affects the merger time of the inner binary since it

occurs at the first Kozai maximum. The timescale to reach Kozai maximum is too

short for tides or gravitational wave radiation to kick in (for details see Section 4.2.2).

Such collisions at the first Kozai maximum were noted in the work Thompson (2011)

as well.


4.2.1 Numerical model using the octupole approximation

In our numerical model we treat the gravitational effects of the third body in the

octupole approximation, meaning we average over the orbital periods of both the inner

binary and the outer companion and retain terms up to (ain/aout) to 3rd order. Beside

the perturbations due to the presence of the third body via Kozai–Lidov mechanism,

we include the following dynamical effects:


• periastron advance arising from quadrupole distortions of the white dwarf due

to both tides and rotation;

• orbital decay due to tidal dissipation in the white dwarf;


Figure 4.1: The timescale for semimajor axis to decay τa = a/a as a function of the eccentricity

of the inner binary for our fiducial model. The solid line represents τa due to tidal dissipation

(TD) while the dashed line shows τa due to gravitational wave radiation (GW). As the

eccentricity of the inner binary exceeds e ≈ 0.95 the decay rate of the semimajor axis due to

tidal dissipation becomes comparable to gravitational wave radiation, and therefore cannot be

neglected.


• loss of binary orbital angular momentum due to gravitational radiation.

The equations used in our model are those of Blaes et al. (2002) for the octupole terms,

which are based on those derived in Ford et al. (2000), combined with equations from

Prodan & Murray (2012) for tidal effects. For the discussion of the relevance of the

secular approximation we refer reader to Section 4.3.

Following Thompson (2011), as fiducial parameters we use: m1 = 0.8M⊙, m2 =

0.6M⊙, and m3 = 1M⊙. The semimajor axis of the inner binary is a = 0.05AU and

the semimajor axis of the outer binary is aout = 1AU. We use the crude approximation

that the radius of both white dwarfs in the inner binary is R = 5× 108 cm, while the

fiducial Love number is k2 = 0.1 and tidal dissipation factor is Q = 107. For the initial

eccentricities and arguments of a periapse we take: ein,0 = 0.1, eout,0 = 0.5, ωin,0 =

ωout,0 = 0 throughout this chapter. Initially we take the WDs to be tidally locked.

We only evolve systems within the following range of initial mutual inclinations:

85o ≤ i0 ≤ 102o. We consider system merged when rp ≤ 2Rsum = 4R. Next we discuss

the results of our model for: high mutual inclination 91o ≤ i0 ≤ 96o and moderately

high inclinations 85o ≤ i ≤ 90o and 97o ≤ i0 ≤ 102o.

4.2.2 High mutual inclination 91o ≤ i0 ≤ 96o

In this subsection we consider the parameter space where the initial mutual incli-

nation is in the range 91o ≤ i0 ≤ 96o. For such high initial mutual inclinations, as

the eccentricity of the inner binary reaches its first maximum, the periapse of the

inner binary, rp, becomes less than 2× (R1 + R2) = 4× R, and we consider that

a collision/merger has occurred. Figure 4.2, upper panel, shows the evolution of

the eccentricity of the inner binary. Within 20 yr, the first eccentricity maximum

occurs leading to the collision of the two white dwarfs. The lower panel of figure 4.2

shows the evolution of the semimajor axis and the periapse of the inner binary. The


semimajor axis remains constant while the periapse rapidly becomes of order of a few

R. In this high inclination regime the sources of additional apsidal precession such as

tidal and rotational bulges as well as tidal dissipation do not influence the outcome of

the evolution. The time scale for the collision induced by the presence of a third body

via Kozai oscillation is too short for any kind of tidal interaction to matter, which we

confirm by repeating the calculation where we neglect tidal effects. The possibility

of Kozai cycles leading straight to a collision was noted in Thompson (2011) even

though effects of rotational bulges and tidal dissipation were not taken into account.

The range of initial mutual inclinations leading to direct collision becomes larger for

wide range of the inner semimajor axes and moderately hierarchical triples when

non secular approach is used, as in work of Katz & Dong (2012) (see Section 4.3 for

detailed discussion).

4.2.3 Moderately high inclinations 85o ≤ i0 ≤ 90o and

97o ≤ i0 ≤ 102o

In this subsection we explore the parameter space of moderately high initial mutual

inclinations: 85o ≤ i0 ≤ 90o and 97o ≤ i0 ≤ 102o. In the first case, calculations are done

including all the dynamical effects captured in our model: the secular perturbations

due to the presence of the third body, tidal effects such as tidal dissipation and tidal

and rotational bulges, GR precession, and GW radiation. In the second case, we

neglect tidal effects and we take into account only GR precession as a source of

additional apsidal precession and GW radiation as the only source of dissipation

in the system. By design, the maximum eccentricity in the case of moderately high

initial mutual inclinations is not high enough to lead to a collision. Instead, during

the phases of high eccentricity the tidal dissipation and GW radiation are strong. As a

consequence both the eccentricity and the semimajor axis of the inner binary decrease


Figure 4.2: The eccentricity as a function of time (upper panel) and the semimajor axis and the

periapse of the inner binary as a function of time (lower panel) for our fiducial model. The

solid line represents the semimajor axis while the dashed line shows the periapse evolution

for i0 = 95o. The horizontal dotted line represents the double sum of the radii of the two

white dwarfs. At the first eccentricity maximum the periapse is less than 4R = 2× 109 cm. We

consider rp ≤ 4× R as the condition for collision/merger in the inner binary.


and eventually the periapse becomes comparable to the sum of the radii of the two

white dwarfs. Figures 4.3 and 4.4 show the time evolution of the eccentricity, the

semimajor axis and the periapse of the inner binary with and without tidal effects

taken into account for i0 = 89o. The high eccentricity induced by the third body

results in strong dissipation even with just GW radiation taken into account, which by

itself shortens dramatically the merger timescale. However, comparing the merger

timescale in the two cases clearly demonstrates that the merger timescale is at least an

order of magnitude shorter when tidal dissipation is taken into account.

Figure 4.5 shows the merger timescale dependence on the initial mutual inclination

for the cases with and without the tidal effects included. In the high inclination regime

(91o ≤ i0 ≤ 96o), the two white dwarfs collide at the first eccentricity maximum on a

very short timescale (∼ 20 yr), so the suppressing effects due to additional sources

of apsidal precession and dissipation are insignificant. On the other hand, in the

moderately high inclination regime, tidal dissipation shortens the merger timescale for

an additional order of magnitude as seen in figure 4.5. This implies that a combination

of the perturbations due to the presence of a third body and tidal dissipation and GW

radiation indeed dramatically shorten the merger timescale of WD–WD binaries in

comparison to Hubble time, THubble when the system is in moderately high inclinations

regime (85o ≤ i0 ≤ 90o and 97o ≤ i0 ≤ 102o).

4.3 DISCUSSION

In Thompson (2011), the relevance of WD–WD, NS–WD, and NS–NS mergers driven by

GW radiation was explored for events like Ia supernovae, GRBs, and other transients.

It was explicitly demonstrated that the GW merger timescale for these compact

binaries becomes shorter by up to a few orders of magnitude due to the eccentricity

oscillations induced by the presence of a third body at high inclination with the respect


Figure 4.3: The eccentricity as a function of time: upper panel shows the case where we take

into account tidal effects and the lower panel shows the case where only GR precession and

GW radiation are taken into account for our fiducial model with i0 = 89o. The tidal dissipation

factor is Q = 107. We terminate the integrations when rp ≤ 4R. As the upper panel shows,

including tidal dissipation leads to a merger timescale about an order of magnitude shorter

than the merger time due to GW radiation alone.


Figure 4.4: The semimajor axis and the periapse of the inner binary as a function of time:

upper panel shows the case where we take into account GR, GW and tidal effects, while

the lower panel shows the case where only GR precession and GW radiation are taken into

account. The plot represents our fiducial model, with i0 = 89o. The solid line represents the

semimajor axis while the dashed line shows the periapse evolution. The dotted line represents

the double sum of the radii of the two white dwarfs. As the upper panel shows, including

tidal dissipation leads to a merger timescale about an order of magnitude shorter than the

merger time due to GW radiation alone.


Figure 4.5: The merger time as a function of initial inclination. The open squares come

from numerical integration of the equations of motion that include tidal dissipation and GW

radiation while solid circles come from integration that includes only GW radiation. In the

high inclination regime (91o ≤ i0 ≤ 96o), the collision of the two white dwarfs occurs at the first

eccentricity maximum on a very short timescale (∼ 20 yr) showing that the suppressing effects

due to additional sources of apsidal precession and dissipation are insignificant. Including tidal

dissipation in the moderately high inclinations regime (85o ≤ i0 ≤ 90o and 97o ≤ i0 ≤ 102o)

leads to shorter merger timescale up to an order of magnitude comparing to the merger time

for GW radiation alone.


to the inner orbit, compared to the case of an isolated binary. The triple scenario

allows for a wider range of binary semimajor axes leading to a merger in a Hubble

time, enhancing the population of compact object binaries capable of producing Ia

supernovae and GRBs.

In this chapter, we build on work of Thompson (2011) and explore the combined

effect of tides and GW radiation on the merger timescale of WD–WD binaries in

triple systems. We examine the evolution of the WD–WD binary in the presence

of a third body at aout/a ∼ 20. When the mutual inclination is sufficiently high,

the third body perturbs the inner binary orbit causing periodic oscillations in the

eccentricity and mutual inclination via the Kozai–Lidov mechanism. As discussed

in Section 4.1, tidal effects become important during the phases of high eccentricity

where the inner binary periapse is of order of few stellar radii. In the case of high

mutual inclination (91o ≤ i0 ≤ 96o), the outcome of the evolution is a direct collision

of the WDs at the first eccentricity maximum, where the Kozai torque is the dominant

torque and all other dynamical effects are insignificant. In other words, none of the

additional sources of precession such as GR and tidal and/or rotational bulges have

timescale short enough to affect the maximum possible eccentricity or suppress Kozai

cycles, contrary to what was expected in Thompson (2011). For the same reason the

strong tidal dissipation or GW radiation do not affect the evolution of the system

even though the eccentricity reaches values close to 1 as seen in Section 4.2.2. Such

collisions have been discussed by several authors as possible channels for production

of type Ia supernovae (Rosswog et al., 2009; Raskin et al., 2009, 2010; Katz & Dong,

2012; Hamers et al., 2013). As shown here and in Thompson (2011), this scenario

seems very promising for retrograde orbits with i0 ≥ 90o. Globular clusters, where

the third star can be captured in binary-binary interactions (Ivanova, 2008; Ivanova

et al., 2006) may produce such retrograde triples. Further study of such triples formed

via binary-binary interactions is a subject of our future work.


In Katz & Dong (2012), collisions of white dwarfs occur in moderately hierarchical

triple systems (3 . rp,out/a . 10) for wide range of a, where the inner binary reaches

high eccentricities via Kozai–Lidov mechanism. A similar scenario is studied in

Hamers et al. (2013). Katz & Dong (2012) use a symplectic three body integrator

while Hamers et al. (2013) combine the secular three body dynamics with a detailed

prescription for stellar, binary and triple evolution prior to the formation of the WD–

WD binary. Considering the evolution on the main sequence is important because most

of the systems that could be possible progenitors of WD–WD binaries (large a, small

aout/a) experience mergers while they are on main sequence (Hamers et al., 2013).

Therefore the rate of direct collisions described in Katz & Dong (2012) contributes only

a small fraction to the SNe Ia events in field. There is hope for getting this mechanism

to work in globular clusters, where binary-binary interactions produce triples with a

flat distribution in cos i, but in the field, for a primordial triple, it is not clear what

cos i0 should be once the stars have evolved to WDs.

Katz & Dong (2012) find that the secular treatment of the Kozai–Lidov mechanism

breaks down for most of their systems. The reason lies in the assumed ratio R/a,

which for their choice of parameters requires 1− e > 10−6 in order for tides to be

significant as for collisions to occur. This is not the case in our work since we are

focused on very compact WD–WD binaries. For our choice of R/a, the two WDs

collide when 1− e ∼ 10−3. The secular treatment of the Kozai–Lidov mechanism

gives an adequate description in this case (see equation 16 in Katz & Dong (2012)).

Our choice of parameter space is relevant for close compact object binaries in the

field, as well as for those in globular clusters. For example, Rosswog et al. (2009) and

Raskin et al. (2009) consider WDs that reside in the dense core of the globular clusters

and similar dense stellar environments where the fact that stars are sufficiently

close to each other makes collisions very likely. The authors propose WD–WD

collision scenario with low impact parameter (close to head on collision) as a possible


formation channel for type Ia supernovae in such environments. They carry out 3D

hydrodynamical calculations of thermonuclear explosion of colliding WDs. Rosswog

et al. (2009) investigate the outcome of direct, head on collisions of several mass pairs,

while Raskin et al. (2009) investigate the outcome of a collision of a single mass pair

(2× 0.6M⊙) with three different impact parameters. Both papers establish that a

collision of moderately massive WD pair naturally leads to shock-triggered ignition

and a synthesis of substantial amounts of Ni for low impact parameter even if the

total mass of the pair is below the Chandrasekhar limit. As emphasized by both

groups, one should keep in mind that the outcome of the WD–WD collision hinge

on several factors: masses and nuclear compositions, their relative speed and impact

parameters. Results from Raskin et al. (2009) imply that the WD–WD collision with

high impact parameters (i.e. grazing collision) lacks violent shocks seen in the cases

with low impact parameters. Instead, there is negligible nuclear burning as well as a

negligible amount of 56Ni produced by the initial interaction. In this scenario both

WDs become unbound and form a rotating disk of white dwarf debris that cools

down and eventually collapses into a single compact object. In both studies the rates

of collisions that may result in explosion are low and still subject to uncertainties (i.e.

core-collapse evolution of the globular cluster). But, such rates indicate that WD–WD

collisions are not unlikely and hence they can contribute a modest fraction of type

Ia events. The likelihood of low impact parameter collisions due to the eccentricity

oscillations induced by the presence of a third body remains unknown. One should be

cautious about claiming that these events can explain the SNe Ia rates, but as already

pointed out these events definitely do contribute to the overall rate.

In the regime of moderately high inclinations previously described additional

sources of precession affect the evolution of the system by suppressing the Kozai–

Lidov mechanism as demonstrated in Section 4.2.3. We showed that including tidal

dissipation together with GW radiation shortens the merger timescale by a factor of


a few to an order of magnitude compared to when only GW radiation is taken into

account (see Figure 4). The fact that tidal dissipation can speed up the mergers driven

by GW radiation implies that the very prompt merger rate (< 108 yr) may be higher

than found in Thompson (2011). The tidal dissipation rate is determined by the tidal

dissipation factor Q. In this work we take Q ' 107 as obtained in Prodan & Murray

(2012), a value in a reasonable agreement with the findings of Piro (2011) and Fuller &

Lai (2011).

Chapter 5

Conclusions & Future Work

5.1 Conclusions

“Life is an adventure of passion, risk, danger,

laughter, beauty; a burning curiosity to go with the

action to see what it is all about, to go search for a

pattern of meaning, to burn one’s bridges because

you’re never going to go back anyway, and to live to

the end."— Saul D. Alinsky

In this thesis, we have presented the results of several applications of the interplay

of Kozai resonances, tidal friction, gravitational wave radiation and mass transfer,

which are discussed individually in each chapter. We emphasize below some of the

important, often recurring, themes. We discuss possible future work and directions in

section 5.2.

117

Chapter 5. Conclusions & Future Work 118

5.1.1 UCXBs in globular clusters

The work in Chapter 2 provides an estimate for the lower limit of the tidal dissipation

parameter Q for a Helium white dwarf. It also elucidates the possible evolutionary

history of 4U 1820-30, i.e., how the system arrived at a state where the secular

dynamics are not dominated by the effects of the white dwarf’s tidal bulge, despite

the fact that the white dwarf is overflowing its Roche lobe in an orbit with a period of

685 s.

We suggest that the system is trapped in the Kozai resonance. This resonance

trapping is responsible for the observed 171 day period, which we interpret as the

period of small oscillations around a stable fixed point in the Kozai resonance. If the

system is not librating, one requires significant fine tuning to obtain the observed 171

day period.

We provide a lower limit on the tidal dissipation rate, as measured by the factor Q;

(e/0.009)2Q/k2 > 4× 109.

In Chapter 3 we extend our dynamical study to three more UCXBs in globular

clusters: 4U 1850-087, 4U 0513-40 and M15 X-2. These three UCXBs have orbital

periods . 20 min. Two of them, 4U 1850-087 and 4U 0513-40, have suspected long

period luminosity variations of order of ∼ 1 yr. There is insufficient observational

data to make any statements regarding the long periodicity in the light curve of M15

X-2 at this point. The properties of these three systems are quite similar to 4U 1820-30,

which prompt us to model their dynamics in the same manner. As in the case of 4U

1820-30, we interpret the suspected long periods as the period of small oscillations

around a stable fixed point in the Kozai resonance. We provide a lower limit on the

tidal dissipation factor Q which is in agreement with results obtained for the case of

4U 1820-30.

Further exploration of the long term (tidal and mass overflow-driven) evolution of


this and similar short period ultra compact X-ray binaries is clearly warranted. We

anticipate that the resonance trapping mechanism we have described in Chapters 2

and 3 is generic in Roche lobe overflow binaries in triple systems. Inclusion of thermal

tides into the dynamics of these systems may introduce an alternative explanation for

the origin of the long period modulation of the light curve. For the particular case of

4U 1820-30, better modelling of the gravitational potential in the host globular cluster,

NGC 6624, would allow for an upper limit on Q.

5.1.2 WD–WD mergers in triple systems

WD–WD mergers driven by gravitational wave radiation are believed to lead to

SNe type Ia explosions. Explaining the observed rates of these events requires a

sufficient number of WD–WD binary systems that can merge in a Hubble time. The

gravitational wave radiation driving these mergers is a slow process, so only tight

systems can merge reasonably fast. If there is a third body in the system, however, the

combination of the Kozai resonance, gravitational wave radiation and tidal dissipation

can significantly expand the parameter space for which these mergers become feasible;

an option we explore in great detail in Chapter 4. In the range of high inclinations

(91o ≤ i0 ≤ 96o), the outcome of the evolution is a direct collision of the two WDs, in

which tidal effects do not play a significant role. On the other hand, in the range of

moderately high inclinations (85o ≤ i0 ≤ 90o and 97o ≤ i0 ≤ 102o), tidal effects do

play a significant role in dramatically decreasing the merger timescale. The choice of

parameter space in this work is relevant for close compact object binaries in the field

as well as in globular clusters.

The enhanced rate of production of Type Ia SN in globular clusters has been

expected (e.g. Ivanova et al., 2006). Voss & Nelemans (2012) and Washabaugh &

Bregman (2013) looked into the archival Hubble Space Telescope images of nearby


galaxies that have hosted a SNe Ia explosion to examine the rate at which these events

occur in globular clusters. They did not find evidence for globular clusters coincident

with supernovae positions, which may be due to: poorly determined supernovae

positions, archival images that are used in their work target the inner part of the

galaxies, and the distance limit of 100 Mpc at which is possible to observe globular

clusters. Therefore, both groups of authors conclude that a more dedicated survey is

necessary for testing the theoretical estimates.

5.2 Future Work and Directions

My Future Work and Directions build on doctoral research and focuses on dynamics

in three different astrophysical systems:

• Binaries near massive black holes

• Ultra Compact X-ray Binaries (UCXBs) in globular clusters (GC)

• Planets in binary systems.

5.2.1 Secular evolution of binary stars near massive black holes

Most stars are believed to be in binaries or in even higher multiplicity systems (e.g.

Raghavan et al. 2010), both in the field and in the dense stellar environments of

globular clusters and galactic nuclei. In the vicinity (within ∼ 1 pc) of the Galactic

massive black hole (MBH), where the gravitational potential is dominated by its mass,

binaries are on orbits that are bound to the MBH and can form a hierarchical triple

system with the MBH in which the binary orbit around the Galactic center is the outer

orbit of the triple. If the orbit of a binary is highly inclined with respect to its orbit

around the MBH, strong oscillations of the inner orbit eccentricity (Kozai cycles) are


induced on a secular timescale. This timescale is often shorter than the timescale

over which gravitational interactions with background stars would significantly affect

either the internal or external orbit of the binary (Antonini & Perets, 2012). The

induced high eccentricities could lead to rapid tidal energy dissipation, which will

very rapidly “shrink” the inner binary orbit. A combination of perturbations due to

the MBH and tidal effects in the binary could therefore provide an effective channel

for formation of close/contact binaries and their products, including stellar mergers,

X-ray binaries, supernovae and γ- ray bursts.

The dynamical processes in the Galactic Center that can affect the stellar population,

and their related timescales, are:

• Binary evaporation due to the dynamical interactions with the field stars (TEV);

• Resonant relaxation – a process where coherent changes in angular momentum

are induced by the orbit-averaged mass distribution of the surrounding stars

(TRR);

• Vector resonant relaxation – a process that non-coherently changes the direction

but not the magnitude of the angular momentum of the outer orbit (TRR,v);

• Two-body relaxation (TNR).

The timescale TKozai, which is associated with perturbations from the MBH on the

binary, is typically the shortest, meaning that the binary will undergo several Kozai

cycles before it evaporates due to interactions with the stellar background (TEV) or

leaves the main sequence (Antonini & Perets, 2012). A combination of Kozai cycles

with tidal effects may significantly shrink the orbit of the binary which in turn can

lead to the formation of close/contact binaries and their products. Both TRR and TRR,v

are too long to significantly alter the orbit of either the inner binary or the external


orbit around the MBH. Before these two processes become important, binaries are

more likely to evaporate.

An analysis performed by Prodan, Antonini & Perets (in prep.) demonstrates

that the combination of perturbations due to the MBH and tidal effects in the binary

could lead to the formation of close/contact binaries and their products, including

stellar mergers, young massive stars, X-ray binaries, blue stragglers, supernovae and

γ- ray bursts. It is anticipated that the described combination of dynamical effects

will alter the expected initial period distribution of binaries. Comparing the outcome

of this model with upcoming observations that will resolve binaries in the Galactic

Center will provide deeper understanding of the dynamical processes that govern the

evolution of these binaries. Stellar mergers are thought to be a potential explanation

for the origin of some of the young massive stars, blue stragglers and red novae (Soker

& Tylenda, 2003; Perets & Fabrycky, 2009; Antonini et al., 2010, 2011). Such stellar

mergers could be effectively produced via the Kozai mechanism. Constraining the

rates for such events theoretically is crucial for planning and interpretation of both

already existing and upcoming data in our Galactic Center as well as in other galaxies.

5.2.2 Secular evolution of dynamically formed triples in globular

clusters

It has been found that 30− 40 per cent of all formed triples are affected by interactions

between the inner and outer binary (Ivanova, 2008). A combination of triple dynamics

with tidal effects may lead to significantly alter the formation of interacting binaries

and their products. This combination of effects is not taken into account in the

standard binary population synthesis approach. However this approach can provide

the initial conditions for my model. I will focus on Ultra Compact X-ray Binaries

(UCXBs), consisting of a neutron star accreting mass via Roche lobe overflow from a


companion star, and on white dwarf–white dwarf binaries (WD–WD).

Three of four UCXBs that have orbital periods of P . 30 min suggest that there is

a long periodicity (∼ 1yr) in their light curve: 4U 0513- 40, 4U 1850-087 and M15 X-2.

Applying the model, as described in the case of 4U 1820-30 (Prodan & Murray, 2012),

to these three additional sources we obtained qualitatively similar results where the

long periods can be related with periods of small oscillations around the fixed point

deep in the Kozai resonance (see Chapter 3). This result implies that the mechanism

described above could be the formation mechanism for UCXBs in globular clusters

and therefore requires further investigation.

It is believed that white dwarf–white dwarf mergers (WD–WD) may lead to type

Ia supernovae events. In order for these mergers to be driven by gravitational wave

radiation within a Hubble time, Thompson (2011) suggested that the needed tight

binaries are produced in triple systems. In such systems, the third body, orbiting the

center of mass of the WD–WD binary, induces eccentricity oscillations in the inner

binary via the Kozai mechanism, driving the binary to very high eccentricities. This

in turn reduces the gravitational wave merger timescale (TGW). In Prodan et al. (2013),

we investigated the role of tidal effects in these events and demonstrated that tidal

effects are important in the regime of moderately high inclinations (85o ≤ i0 ≤ 90o and

97o ≤ i0 ≤ 102o) where, combined with GW radiation, they contribute to a dramatic

decrease in TGW . In the regime of high inclinations (91o ≤ i0 ≤ 96o), the inner binary

suffers a direct collision, and tidal effects do not alter the outcome of the evolution.

This mechanism is relevant for close compact object binaries in the field and also

for those in globular clusters, where the described mechanism could be one of the

channels for production of SNe Ia.


5.2.3 Binaries Hosting Highly Eccentric Exoplanets:

A significant fraction of planets have highly eccentric orbits. There are several expla-

nations for the origin of these highly eccentric orbits. These include mean-motion

resonances, planet–planet scattering, planet–disk interactions, Kozai mechanism, stel-

lar encounters, and jet-induced excitation (Namouni, 2007). Of all the proposed

explanations, only the Kozai mechanism requires the presence of a third body on an

inclined orbit (& 40). As previously described, the two orbits exert a torque on one

other and exchange angular momentum, causing the orbit of the planet to become

highly eccentric on secular timescales. During the phases of high eccentricity tidal

dissipation in the planet becomes important. It tends to significantly shrink and

circularize the orbit of the planet. This mechanism is known as Kozai migration –

even a distant companion (∼ 1000 AU) can still be responsible for migration (Wu &

Murray, 2003). Kozai migration is responsible for the formation of some hot Jupiters

(Fabrycky & Tremaine, 2007).

The current sample of exoplanets shows two striking features that are known to

be the signature of high-eccentricity migration: high-eccentricity (i.e. e & 0.5, see ??)

and the tendency for quite a number of massive planets (& 1MJ) to have spin-orbit

misalignments of order |40| (Wu et al., 2007; Fabrycky & Tremaine, 2007). In fact,

the majority of highly eccentric planets are massive (& 1MJ), which enhances the

effectiveness of Kozai migration. Although Kozai migration is unlikely to explain the

formation of all of the eccentric planets, it nevertheless can plausibly account for a

significant fraction of them.

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