Secular Stellar Dynamics near Massive Black Holes Ann-Marie Madigan

Secular Stellar Dynamics near Massive Black Holes Ann-Marie Madigan
Secular Stellar Dynamics
near Massive Black Holes
Ann-Marie Madigan
Secular Stellar Dynamics
near Massive Black Holes
PROEFSCHRIFT
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van de Rector Magnificus prof. mr. P. F. van der Heijden,
volgens besluit van het College voor Promoties
te verdedigen op donderdag 16 februari 2012
klokke 11.15 uur
door
Ann-Marie Madigan
geboren te Dublin
in 1982
Promotiecommissie
Promotor:
Prof. dr. K. H. Kuijken
Co-Promotors:
Dr. Y. Levin (Monash University)
Dr. C. Hopman
Overige leden:
Prof. dr. A. Ghez (University of California, Los Angeles)
Prof. dr. R. Genzel (Max Planck Institute for Extraterrestrial Physics)
Prof. dr. S. F. Portegies Zwart
Prof. dr. S. Tremaine (Institute for Advanced Studies, Princeton)
This research has been supported by a TopTalent fellowship from the Netherlands Organisation for Scientific Research (NWO).
for Ev and Dave
ISBN 978-94-6191-145-2
Cover design by Dave Madigan
vii
Contents
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1
2
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4
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Chapter 2. The Eccentric Disk Instability
2.1 Introduction . . . . . . . . . . . . . . . . . . .
2.2 The eccentricity instability . . . . . . . . . . .
2.3 N -body simulations . . . . . . . . . . . . . . .
2.3.1 Initial conditions . . . . . . . . . . . . .
2.4 Results . . . . . . . . . . . . . . . . . . . . . .
2.5 Application to S-stars . . . . . . . . . . . . . .
2.5.1 Bimodal eccentricity distribution in disk
2.6 Critical discussion . . . . . . . . . . . . . . . .
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Chapter 1. Introduction
1.1 Massive black holes . . . . . . . . . . . .
1.2 Mysteries at the Galactic center . . . . . .
1.3 Stellar dynamics near massive black holes
1.4 Thesis summary . . . . . . . . . . . . . .
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Chapter 3. Secular Stellar Dynamics
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 N -body simulations . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Model of Galactic nucleus . . . . . . . . . . . . . . . . .
3.2.2 Illustrative simulations . . . . . . . . . . . . . . . . . . .
3.3 Statistical description of resonant relaxation . . . . . . . . . .
3.3.1 The autoregressive moving average model ARMA(1, 1) .
3.4 Interpretation of the ARMA parameters and extension
of parameter space . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Non-resonant relaxation (NR): the parameter σ . . . . .
3.4.2 Persistence of coherent torques: the parameter φ . . . .
3.4.3 Magnitude of resonant relaxation steps: the parameter θ
3.5 Results: ARMA analysis of the N -body simulations . . . . . . .
3.6 Monte Carlo simulations and applications . . . . . . . . . . . .
3.6.1 Angular momentum . . . . . . . . . . . . . . . . . . . .
3.6.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.3 Initial conditions and boundary conditions . . . . . . . .
3.6.4 Time steps . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.5 Treatment of the loss cone . . . . . . . . . . . . . . . . .
viii
3.7 Results . . . . . . . . . . . . . . . . . . .
3.7.1 Evolution to steady-state . . . . . .
3.7.2 A depression in the Galactic center
3.7.3 Dynamical evolution of the S-stars
3.8 Summary . . . . . . . . . . . . . . . . . .
3.A Description of N -body code . . . . . . . .
3.B Energy evolution and cusp formation . .
3.C Precession due to power-law stellar cusp
3.D Equations of ARMA(1,1) model . . . . .
3.D.1 Variance . . . . . . . . . . . . . . .
3.D.2 Autocorrelation function . . . . . .
3.D.3 Variance at coherence time tφ . . .
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Chapter 4. Build-up of Stars in the Galactic Center
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Nuclear star cluster formation . . . . . . . . . . . . .
4.1.2 Observational constraints from extra-galactic nuclei .
4.1.3 Evidence from the Galactic center . . . . . . . . . . .
4.2 Numerical methods and scenarios . . . . . . . . . . . . . .
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Evolution of orbital eccentricities . . . . . . . . . . .
4.3.2 The high-eccentricity statistic . . . . . . . . . . . . .
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.A Statistical constraints on eccentricity from the h-statistic . .
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Chapter 5. Secular Dynamical Anti-Friction
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Secular dynamical anti-friction . . . . . . . . . . . . . . . . . .
5.2.1 Increase in orbital eccentricity . . . . . . . . . . . . . . .
5.2.2 Comparison with theories in the literature . . . . . . . .
5.3 Numerical method and simulations . . . . . . . . . . . . . . .
5.3.1 Secular horseshoe orbits . . . . . . . . . . . . . . . . . .
5.3.2 Separation of potential into pro- and retro- grade orbits
5.3.3 Increase in orbital eccentricity of IMBH . . . . . . . . .
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Nederlandse samenvatting (Dutch Summary)
119
Curriculum Vitae
123
Acknowledgments
125
1
Introduction
Stars near massive black holes move on elliptical orbits which precess
slowly, exerting persistent gravitational torques on each other. In this thesis we present four important consequences of these gravitational torques:
1) A new instability which exposes the inherently unstable nature of eccentric stellar disks in galactic nuclei, 2) A density depression of stars
near massive black holes due to enhanced angular momentum relaxation
and tidal disruptions, 3) A signature of high-eccentricity orbits for stars
formed by Hills’ mechanism at large radii from the massive black hole, and
a directly-observable statistic that can highlight these populations, and 4)
A new dynamical process which we call "secular dynamical anti-friction"
which boosts the orbital eccentricity of hypothesized intermediate-mass
black holes as they spiral into massive black holes.
Ann-Marie Madigan 2011
2
Introduction
Figure 1.1 – Composite image of the Galactic center (2280′′ × 840′′ ) with the Hubble Space Telescope (near-infrared, in yellow), Spitzer Space Telescope (infrared, in red) and Chandra X-ray Observatory (X-ray, in blue/violet). The bright white cluster at lower middle-right is the location of the
MBH SgrA*. Credit: X-ray: NASA/CXC/UMass/D. Wang et al.; Optical: NASA/ESA/STScI/D.Wang
et al.; IR: NASA/JPL-Caltech/SSC/S.Stolovy.
1.1 Massive black holes
M
ASSIVE black holes (MBHs) are found in the nuclei of most, if not all,
galaxies (Ferrarese & Ford 2005). Embedded in dense star clusters, they
dominate the gravitational potential out to their dynamical radius where the
mass in stars becomes comparable to that of the MBH. If one understands the
dynamics of stars within this radius, one can provide a theoretical framework
for some of the most energetic phenomena in the universe such as the tidal
disruption of stars and launching of relativistic jets. Compact stellar remnants
on high-eccentricity orbits within these clusters, such as neutron stars, white
dwarfs and stellar-mass black holes, interact with the MBH producing gravitational waves potentially detectable by future space observatories.
1.2 Mysteries at the Galactic center
In the center of our Galaxy, at a distance of ∼ 8 kpc (Eisenhauer et al. 2003;
Ghez et al. 2008; Gillessen et al. 2009), there quietly lives a MBH1 . Though
highly under-luminous with respect to its Eddington luminosity (Melia & Falcke
2001), its presence is inferred by the effect it has on the orbits of stars that live
in the surrounding dense nuclear star cluster. Just as (exo-)planets revolve
around their sun, so too do the stars in the Galactic center orbit the MBH on
near-Kepler ellipses.
1 At
this distance 1′′ ∼ 0.038 pc.
Secular Stellar Dynamics
3
Figure 1.2 – Left - The central ∼ 20′′ of our Galaxy imaged in near-infrared. Right - The S-star
orbits: the result of 16 years of tracking the motions of stars around the MBH in the Galactic center.
Credit: ESO/Gillessen et al. (2009).
Obscuration from interstellar dust in the plane of the Galaxy results in a
high optical extinction of ∼ 30 magnitudes towards the Galactic center. For
this reason, observations of the stars near the MBH are taken at longer, nearinfrared wavelengths. The development of near-infrared instruments, adaptive optics and integral field spectroscopy has lead to dramatic progress in our
knowledge of the Galactic center in the past two decades. Angular resolution
has improved by more than an order of magnitude to ∼ 5 × 10−4 arcseconds,
and sensitivity by three to five magnitudes to Ks ∼ 16−18, for spectroscopy and
imaging respectively (Genzel et al. 2010). Using astrometric tracking of the orbits of luminous B-stars, astronomers have revealed that the MBH (known as
SgrA*) has a mass of M• ≃ 4 × 106 M⊙ . The most intensely-studied of these
stars is called S0/S0-2; with an orbital period of just 15.2 years (Schödel et al.
2003) its orbit has been fully traced and shown to be a bound, high-eccentricity
Kepler ellipse.
The proximity of the Galactic center makes it unique among galactic nuclei
in that one can resolve individual stars of different ages and potentially test
decades-old predictions of the dynamics of stars near MBHs. One such prediction is a lack of young stars near the MBH; its immense tidal force prohibits
star formation from the collapse and fragmentation of molecular gas clouds
(Sanders 1992; Morris 1993). We do however observe many (∼ 200) early type
stars within the central parsec. They can be separated roughly into two groups.
The first is a population of O and Wolf-Rayet (WR) stars, ∼ 6 ± 2 Myr old, with
masses & 20M⊙ (Paumard et al. 2006). The majority form a well-defined,
warped disk with projected radii 0.8′′ − 12′′ , which rotates clockwise on the sky
(Levin & Beloborodov 2003; Genzel et al. 2003; Lu et al. 2009; Bartko et al.
2009). The second population is an apparently isotropic distribution of B-stars
4
Introduction
(< 10 ∼ 100 Myr). Those that lie within the central 0.8′′ are often collectively
referred to as the “S-stars”; see Figure 1.2. Their kinematics reveal randomlyinclined and highly-eccentric orbits (Ghez et al. 2005; Gillessen et al. 2009).
Their proximity to the MBH, combined with their young ages, are indicative of
a “paradox of youth” (Ghez et al. 2003; Eisenhauer et al. 2005; Martins et al.
2008). They could not have formed in-situ, but due to their young ages they
have not had much time to dynamically migrate to their current locations.
Another prediction is that we would expect, due to gravitational interactions, that the population of late type stars in the Galactic nuclear star cluster
is dynamically relaxed and forms a steep density power-law cusp around the
MBH (e.g., Bahcall & Wolf 1976). However, the late type k-giants, 1−10 Gyr old,
do not show a central concentration as is theoretically expected for a steadystate stellar cusp (Do et al. 2009; Buchholz et al. 2009; Bartko et al. 2010), a
fact reflected in the flattening of the surface brightness distribution of diffuse
stellar light (Yusef-Zadeh et al. 2011). Instead they suggest the presence of a
∼ 0.3 pc scale core, a puzzle referred to as the “conundrum of old age” (Merritt
2011).
This thesis aims to address these types of puzzles at the Galactic center. To
begin, we must first ask, how do stars move and gravitationally interact with each
other around MBHs? This simple question is key to understanding the dynamics
of stars near MBHs – the dynamics responsible for illuminating the MBH both
through highly-energetic phenomena and gravitational-wave astrophysics.
1.3 Stellar dynamics near massive black holes
One can picture stars moving on orbits determined by the combined smooth
gravitational potential of the MBH and surrounding cluster stars. The orbit of
a star can be described by its six phase space parameters (r, v) or alternatively
by its energy and angular momentum (E, J). In a Kepler potential, the stellar
orbit is an ellipse with specific energy and angular momentum
E=−
GM•
,
2a
(1.1)
J = Jc (1 − e2 )1/2 ,
(1.2)
√
where e is the eccentricity and Jc = GM• a is the circular (maximum) angular
momentum of a star with semi-major axis a. Due to the additional potential
of the surrounding stars in the cluster (and general relativistic effects), the
stellar orbit will experience apisidal precession (see Figure 1.3) which can be
described as a rotation in the eccentricity vector in the retrograde (prograde)
direction.
In reality, the stars are not moving in a smooth potential – they feel gravitational forces from each other and are perturbed from their original orbits. To
calculate the timescale over which a star will be significantly perturbed away
Secular Stellar Dynamics
5
Figure 1.3 – Left - Illustration of the apsidal precession of a stellar orbit (retrograde with respect
to the star’s motion) due to the additional Newtonian potential from surrounding stars near the
MBH. Right - Schematic diagram illustrating coherent torques between two stellar orbits.
from its original orbit (known as the relaxation time), we follow the outline of
the derivation2 by Rauch & Tremaine (1996).
We consider a spherical volume of radius R centred on a point mass – an
MBH of mass M• – populated uniformly
with N ≫ 1 stars of mass m which
p
move with average velocity V ∼ GM• /R. As N m ≪ M• , the stellar orbits
are Kepler ellipses which precess on a timescale
tprec ∼
M•
P,
Nm
(1.3)
where P is the orbital period of order ∼ R/V . The relaxation time for a single
test star may be derived considering a spherical shell of outer radius r ≪ R
and inner radius 1/2 r, centred on the test star. The number of stars within this
shell is n ∼ N (r/R)3 ; the instantaneous number fluctuates by n1/2 . Hence the
test star experiences a fluctuating gravitational acceleration of magnitude
a∼
Gmn1/2
,
r2
(1.4)
on a timescale tf ∼ r/V . The typical impulse is therefore
δv ∼ atf ∼
Gmn1/2
.
rV
(1.5)
As successive impulses are uncorrelated, the total impulse in time t is
(∆v)2 ∼ (δv)2
2 We
2
G2 m2 N
t
2m N t
∼
t
∼
V
.
tf
R3 V
M•2 P
will ignore factors of order unity throughout the derivation.
(1.6)
6
Introduction
∆v is independent of r. Hence the shells spanning the radii (1/2 r, r), (r, 2r) . . .
all contribute equally to the relaxation rate of the test star. We account for this
by multiplying the total rate by the Coulomb logarithm ln Λ, which represents
the number of octaves that lead to to the relaxation of the test star. Defining
the non-resonant relaxation time as (∆v/V )2 = t/tnr
rel , we find
M•2
P.
(1.7)
m2 N ln Λ
The fluctuating force changes both energy and angular momentum of the stellar orbit, such that
1/2
1/2
∆E
m(N ln Λ)1/2 t
∆J
t
∼
.
(1.8)
∼ nr
∼
E
Jc
trel
M•
P
tnr
rel ∼
The characteristic timescale of this non-resonant relaxation in the Galactic center is greater than 5 Gyr at all radii (Merritt 2010). This means that, under
this mechanism, stars will retain memory of their initial orbital parameters for
very long times. It is this form of stellar relaxation that underlies many longstanding predictions about the distribution of stars around MBHs.
However, as demonstrated by Rauch & Tremaine (1996), rapidly fluctuating forces are not the dominant mechanism for stellar relaxation near MBHs.
Stars orbiting in the gravitational potential of a MBH move along nearly-closed
Kepler ellipses, which remain fixed in space over long timescales. It is therefore
instructive to imagine the mass of each star spread out along its orbit, with the
density at each segment inversely proportional to the local speed in the orbit.
Instead of a system of point particles orbiting the MBH, one can imagine a collection of massive stellar ‘wires’. Over a timescale P ≪ t ≪ tprec , the wires
remain fixed in space and exert persistent mutual torques on each other (see
Figure 1.3). The specific torques are of the order
GmN 1/2
.
(1.9)
R
As the gravitational potential remains stationary over this timescale, there is
no additional energy relaxation, but the torques change the angular momenta
of the stellar wires, ∆J ∼ τ t. The apisidal precession of the stellar wires limits
the timescale over which the torques act on each other. For a time t ≪ tprec ,
∆J
N 1/2 m t
.
(1.10)
∼
Jc
M•
P
τ∼
For t ≫ tprec , the change in angular momentum is a random walk with increments of the order
1/2
∆J
t
N 1/2 m tprec
∼
Jc
M•
P
tprec
(1.11)
1/2 1/2
t
m
.
∼
M•
P
Secular Stellar Dynamics
7
Defining the resonant relaxation3 time as (∆J/Jc )2 = t/trr
rel ,
trr
rel ∼
M•
P.
m
(1.12)
The resonant relaxation time is shorter than the non-resonant timescale by a
factor (mN/M• ) ln Λ, depending on the eccentricity of the stellar orbit (Gürkan
& Hopman 2007). In the Galactic center this difference in timescale can be
greater than two orders of magnitude, making secular torques between stellar
orbits the driving mechanism for angular momentum relaxation. This thesis
explores the consequences of these secular torques, and applies the results to
the observations of stars in the Galactic center.
1.4 Thesis summary
In Chapter 2, we present a new secular gravitational instability present in coherently eccentric disks of stars around MBHs in galactic nuclei. The instability
separates, on less than a precession timescale, both the scalar values and the
directions of the eccentricity vectors of the stellar orbits within the disk. We
apply our results to the Galactic center and propose that the O/WR stars fragmented from a gaseous, coherently eccentric disk and were promptly subject
to this instability. The observed bi-modality in the eccentricity distribution of
the disk may be explained in this way. Binary stars propelled to high eccentricity orbits may have played a role in the formation of the S-stars, and of the
hyper-velocity stars in the Galactic halo through Hills (1988) mechanism.
We investigate the long-term angular momentum evolution of stars around
MBHs in Chapter 3. We perform Monte Carlo simulations which use an autoregressive moving average (ARMA) equation for angular momentum evolution
to capture the statistics of both two-body and resonant relaxation. We show
that for a single-mass system in steady state, a depression is carved out near
an MBH as a result of tidal disruptions. In context of the Galactic center, the
extent of the depression is about 0.1 pc, of similar order to but less than the
size of the observed “hole” in the distribution of bright late-type stars. We furthermore predict that the velocity vectors of stars around an MBH are locally
not isotropic. As a final application, we use the ARMA code to evolve higheccentricity S-star orbits that result from the tidal disruption of binary stars,
and find that RR predicts more highly eccentric (e > 0.9) orbits than have been
observed to date. In this chapter we also describe in detail our special purpose
N -body code, which we use in extensive N -body simulations to calibrate the
ARMA equation.
In Chapter 4 we investigate the build-up of stars in the Galactic center. We
contrast two formation scenarios for the intermediate-aged B-stars which are
3 The term ‘resonant’ refers to the 1 : 1 resonance in the azimuthal and radial frequencies in a
Kepler ellipse. Resonant relaxation is a secular relaxation process in that the position of the star
along its orbit is dynamically unimportant.
8
Introduction
on the order of ∼ 100 Myr old – episodic star formation in nuclear disks, and
tidal disruptions of binaries by the MBH. We perform N -body simulations to
show that after 6 Myr of gravitational interactions with the O/WR disk, the
two scenarios lead to very different dynamical outcomes. In particular, we find
a high-eccentricity signature at large radii for the stars in the tidal disruption
scenario. We present a new statistic, h, that uses directly-observable kinematical stellar data to highlight high-eccentricity populations, and show how this
statistic may distinguish between the two scenarios.
In Chapter 5 we present a new gravitational dynamical process in galactic
nuclei containing MBHs, which we call secular dynamical anti-friction (SDAF).
We show that the gravitational torque exerted on the orbit of a massive body
by less massive bodies comes from a force acting parallel to its direction of
precession - a dynamical anti-friction which affects the angular momentum of
the orbit, not the energy. Together with Chandrasekhar’s dynamical friction
the effective result of this anti-friction is to increase the orbital eccentricity of
the more massive object. N -body simulations of the inspiral and coalescence
of intermediate-mass black holes into MBHs show this increase in eccentricity,
but as of yet the underlying mechanism has eluded a complete description. In
this chapter we show that the theory of SDAF can account for such an increase.
Secular Stellar Dynamics
9
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2009, ApJ, 692, 1075
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2
A New Secular Instability of Eccentric Stellar Disks
Around Supermassive Black Holes, with Application to the
Galactic Center
We identify a new secular instability of eccentric stellar disks around supermassive black holes. We show that retrograde precession of the stellar
orbits, due to the presence of a stellar cusp, induces coherent torques that
amplify deviations of individual orbital eccentricities from the average, and
thus drive all eccentricities away from their initial value. We investigate the
instability using N -body simulations, and show that it can propel individual orbital eccentricities to significantly higher or lower values on the order
of a precession time-scale.
This physics is relevant for the Galactic center, where massive stars are
likely to form in eccentric disks around the SgrA* black hole. We show that
the observed bimodal eccentricity distribution of disk stars in the Galactic
center is in good agreement with the distribution resulting from the eccentricity instability and demonstrate how the dynamical evolution of such
a disk results in several of its stars acquiring high (1 − e ≪ 0.1) orbital
eccentricity. Binary stars on such highly eccentric orbits would get tidally
disrupted by the SgrA* black hole, possibly producing both S-stars near the
black hole and high-velocity stars in the Galactic halo.
Ann-Marie Madigan, Yuri Levin and Clovis Hopman
The Astrophysical Journal Letters, 697, 44M (2009)
12
The Eccentric Disk Instability
2.1 Introduction
M
OST well-observed galaxies with stellar bulges show evidence for supermassive black holes (SMBHs) in their centers (Kormendy 2004). An important feature of stellar dynamics in galactic nuclei is that for orbits contained
well within the SMBH’s radius of influence, the precession time tprec is many
orders of magnitude larger than their period P .1 Therefore, for a study of
long-term (secular) gravitational dynamics, it is useful to think of the stars
not as particles but as massive elliptical wires stretched along the stars’ eccentric orbits (Rauch & Tremaine 1996). Gravitational torques between the
wires can become the dominant mode of orbital eccentricity evolution, as is
seen in the case of resonant relaxation (Rauch & Tremaine 1996; Gürkan &
Hopman 2007; Touma et al. 2009; Eilon et al. 2008). Secular effects have a
significant impact on the stability of stellar cusps (Tremaine 2005; Polyachenko
& Shukhman 1981; Polyachenko et al. 2008), on gravitational wave emission
from in-spiraling compact objects (Hopman & Alexander 2006a,b), on tidal
disruption of stars (Rauch & Ingalls 1998) and, as we show here, on the origin
of the S-stars in the Galactic center.
In this chapter we describe a mechanism for generating high-eccentricity
stars via a fast secular instability of an eccentric stellar disk (Section 2.2). We
use N -body simulations to investigate the non-linear development of the instability (Section 2.3 and Section 2.4) and to explore its dependence on the initial
eccentricity and mass of the disk. We then show that the instability may be
relevant for young stars in the Galactic center, and may provide an interesting
channel for formation of the S-stars and of high-velocity stars in the Galactic
halo (Section 2.5). We critically discuss our results in Section 2.6.
2.2 The eccentricity instability
Consider a thin stellar disk of mass Mdisk around a SMBH of mass M• which
is embedded in a power-law stellar cusp of mass Mcusp. Our two main assumptions are that (1) Mdisk ≪ Mcusp , so that precession of individual orbits
is driven by the cusp, and (2) initially the stellar eccentricity vectors
e=
1
v × (r × v) − r̂
GM•
(2.1)
are aligned and similar in magnitude so that the disk precesses coherently as
a whole (for a typical power-law cusp, the orbital precession rate depends
only weakly on the semi-major axis; see below). Here |e| corresponds to the
magnitude of the eccentricity of the orbit and r and v are the position and
velocity vectors respectively. Both assumptions are realistic for a disk of young
stars formed in a galactic center via the gravitational capture of part of an infalling molecular cloud: observations reveal a massive spherical stellar cusp
1 This is in contrast to the rapidly precessing stellar orbits typical for globular clusters or galaxies.
Secular Stellar Dynamics
13
(Genzel et al. 2003; Schödel et al. 2007), while hydrodynamical simulations
tentatively show similar, aligned initial orbital eccentricities of young stars2
(Sanders 1998; Alexander et al. 2008; Bonnell & Rice 2008; Hobbs & Nayakshin 2009). The orbits precess in the direction opposite to the orbital rotation
of the stars. The precession time-scale scales as
tprec ∼
M•
P (a)f (e)
N (< a)m
∝a
γ−3/2
(2.2)
f (e)
p
where P (a) = 2π a3 /GM• is the period of a star with semi-major axis a,
N (< a) is the number of stars in the cusp within a, m is the individual mass of
the stars, and γ is the power-law index for the space-density ρ(r) ∝ r−γ . For
a power-law cusp with γ = 3/2, formed in the special case when the SMBH
grows adiabatically (Young 1980), tprec is constant for all a.
Observations of the Galactic center indicate that there is a stellar cusp in the
inner parsec, with published measurements of the slope not significantly different from γ = 3/2, e.g. γ = 1.4 ± 0.1 (Genzel et al. 2003) and γ = 1.19 ± 0.05
(Schödel et al. 2007). It is important to stress that these observations only
show a small and biased subset of the stellar content in that region. They
are restricted to massive young stars, that have not yet relaxed, and to giants,
which have relaxed, but are affected by hydrodynamical collisions with other
stars (e.g. Alexander 1999). These two stellar types are not expected to dominate the density. Instead, theoretical models accounting for mass-segregation
show that the density at the regions of interest is mostly determined by stellar black holes and main sequence stars, with cusp values between γMS = 1.4
and γBH = 2.0 (Alexander et al. 2008). We conclude that both theory and
observations yield values of γ close to 3/2 and the dependence of tprec on a is
weak.
A key element of the eccentric instability is that f (e) in Equation (2.2)
is an increasing function of eccentricity (e.g. Binney & Tremaine 2008), so
that an orbit that is slightly more eccentric than average will lag behind in
precession. Such a star will feel a strong, coherent torque from the other stars
in the disk, in the opposite direction of its angular momentum vector. As a
result, its angular momentum decreases in magnitude, causing its eccentricity
to increase even further. In this way, very high eccentricities can potentially be
achieved. Conversely, if a stellar orbit is slightly less eccentric than the average,
it will have a higher precession rate, thus experiencing a torque which acts to
decrease its eccentricity further.
2 Bonnell & Rice (2008) make two large simulations, investigating star-formation in the Galactic
center. In their Run 1 they find 0.6 < e < 0.7, while in Run 2 the inner edge of the disk is
circularized, and hence the stellar eccentricities range between 0 in the inner disk and ∼ 0.6 in the
outer. While the precise nature of the circularization, or its absence, has not been investigated, it
is clear that it depends on the initial conditions.
14
The Eccentric Disk Instability
We thus find that the initial eccentricity vector distribution is unstable, due
to a combination of retrograde precession and df /de > 0, and that all eccentricities are driven away from the initial eccentricity of the disk. It can be shown
analytically that the initial growth timescale of the eccentric instability scales
as
−1/2
1/2 p
d
1
Mcusp
e 1 − e2
.
(2.3)
τgrowth ∼ tprec
Mdisk
de f (e)
For the Galactic center this amounts to several precession time-scales. We now
demonstrate the eccentricity instability using N -body simulations and look at
its impact on the young stars in the Galactic center.
2.3 N-body simulations
We summarize a newly-developed code which we have used in these simulations. Our integrator is based on a 4th-order Wisdom-Holman algorithm (Wisdom & Holman 1991; Yoshida 1990), where the forces on a particle are split
into a dominant Keplerian force and perturbation forces. We use Kepler’s equation to integrate the orbit (Danby 1992), keeping the position of the central
object fixed. We use adaptive time-stepping to deal with close encounters and
the algorithm is not strictly energy conserving. We implement time-symmetry
to reduce the energy error and the K2 kernel from Dehnen (2001) for gravitational softening. The overall fractional energy error for these simulations is
typically ∼ 10−8 or less. We describe the code in detail in Chapter 3.
2.3.1
Initial conditions
We have three main components in our simulations. (1) An eccentric disk with
surface density profile Σ ∝ r−2 (cf. Paumard et al. 2006), consisting of 100
equal-mass stars, with semi-major axes 0.05 pc≤ a ≤ 0.5 pc. We vary the masses
of the stars for different simulations. We initialize the eccentricity vectors of all
stars pointing in roughly the same direction, though they are scattered in their
orbital phase. The initial rms inclination is ∼ 0.01. (2) A SMBH of 4 × 106 M⊙
(Ghez et al. 2008), and (3) a smooth stellar cusp with density profile index
γ = 1.5 and a mass of 0.5 × 106 M⊙ within 1 pc (Schödel et al. 2007; Genzel
et al. 2003). The stars within the disk precess due to the cusp and experience
a prograde general relativistic precession which amounts to apsidal precession
of the orbit by an angle
6πGM•
(2.4)
δφprec =
a(1 − e2 )c2
per orbit. Precession due to self-gravity of the disk is minimal. To verify our
results, we also run simulations with twice the number of half-as-massive stars
and various softening parameters. We discuss the use of different values of γ
(e.g. Bahcall & Wolf 1976) in Section 2.6.
Secular Stellar Dynamics
0
1e+06
tyears
2e+06
15
3e+06
1
1 - ecc
0.1
0.01
0.001
0.0001
0
1000
2000
3000
4000
∆t/T0
5000
6000
7000
8000
Figure 2.1 – Eccentricity evolution of stars in an eccentric disk (Mdisk = 8 × 103 M⊙ ) as a
function of time. ∆t is elapsed time expressed in units of the initial period of the innermost orbit
T0 (tprec ∼ 1200 T0 ). The smoothly varying behavior is characteristic of the secular nature of the
instability.
2.4 Results
With a mass of 8 × 103 M⊙ and an initial eccentricity of 0.6, we let the disk
of stars evolve for several precession times (tprec ∼ 0.6 - 0.7 Myrs or ∼ 1200
orbits at a = 0.05 pc). In Figure (2.1) we show a selection of stars from this
run which best demonstrates the dispersion of the disk’s initial eccentricity,
while Figure (2.2) follows the evolution of the eccentricity vectors (this time
for Mdisk = 4 × 103 M⊙ ). The circles in the latter plot indicate the stars experiencing the highest torque (τ ) and hence angular momentum changes. Note
that the highest eccentricity orbits are precessing behind the bulk of the stars
as expected. We find a monotonically decreasing relationship between the rate
of change of angular momentum, i.e. torque on each orbit, and semi-major
axis a.
The effects of the instability on the stellar eccentricity distribution are evident after ∼ 0.5−1 tprec and the strongest torques are experienced over a few
precession times before the disk loses coherence.
We explore the dependence of the final range of eccentricities on the mass
and initial eccentricity of the disk. To this end, we run a set of simulations with
a disk of initial eccentricity 0.6 for various disk masses (4, 6, 8, 10 ×103 M⊙ ),
and a set with Mdisk = 104 M⊙ for various initial eccentricities (0.1, 0.4, 0.5,
16
The Eccentric Disk Instability
1
ey
0.5
0
-0.5
0 orbits
800
1600
2400
3200
high τ
-1
-1
-0.5
0
0.5
1
ex
Figure 2.2 – Evolution of the eccentricity vectors of all the stars in the disk (Mdisk = 4 × 103 M⊙ )
for different times. The outermost circle indicates where |e| = 1; the inner circle indicates the
initial value of the eccentricities |e| = 0.6. An arrow points to the initial positions. The eccentricity
vectors spread out as the stars complete more and more orbits, and precess with retrograde motion.
Those experiencing the highest torques (τ ) are over-plotted with small circles. As the eccentric disk
instability predicts, the slowest precessing orbits have the highest eccentricities. In addition, orbits
with low eccentricity clump together at late times. We believe non-linear effects are responsible
for this feature.
0.6, 0.7, 0.9). The results are presented in Table (2.1). Higher initial eccentricities and masses of the disk are correlated with higher resulting eccentricities.
We also see a correlation between the highest eccentricities and inclinations
reached, with the largest values occurring at the smallest radii.
In contrast to the secular instability described in Tremaine (2005), relativistic effects in this case do not promote stability once the relativistic prograde
precession rate exceeds that of the retrograde precession due to the cusp. The
highest eccentricity orbit precesses with prograde motion at a higher rate than
the average and again experiences a torque in the opposite direction of its angular momentum.
2.5 Application to S-stars
The region surrounding the SMBH in the Galactic center (Gillessen et al. 2009)
is host to several interesting populations of young stars, which include one
Secular Stellar Dynamics
17
Table 2.1 – Simulation Parameters and Statistics of S-stars
Mdisk
(M⊙ )
4 × 103
...
...
...
6 × 103
...
8 × 103
...
...
...
1 × 104
...
...
...
...
...
...
...
...
...
...
...
edisk
ehigh a
ihigh b
(rads)
abin
(AU)
rt
(pc)
a•
(pc)
rp < rt c
(%)
0. 4
...
0.6
...
0.6
...
0. 4
...
0.6
...
0.1
...
0.4
...
0.5
...
0.6
...
0.7
...
0.9
...
0.8755
...
0.9691
...
0.9981
...
0.8796
...
0.9993
...
0.5115
...
0.9580
...
0.9773
...
0.99991
...
0.99998
...
1.0
...
0.4662
...
0.7960
...
1.824
...
0.6643
...
2.170
...
0.2935
...
1.440
...
1.348
...
2.328
...
3.0312
...
3.125
...
0.1
1
0.1
1
0.1
1
0.1
1
0.1
1
0.1
1
0.1
1
0.1
1
0.1
1
0.1
1
0.1
1
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
3.57 × 10−5
3.57 × 10−4
0.0026
0.026
0.0026
0.026
0.0026
0.026
0.0026
0.026
0.0026
0.026
0.0026
0.026
0.0026
0.026
0.0026
0.026
0.0026
0.026
0.0026
0.026
0.0026
0.026
0
0
0
0
4
8
0
0
3
7
0
0
1
2
0
3
14
14
32
32
58
59
Notes
a
b
c
Disks with Mdisk & 6 × 103 M⊙ and initial eccentricity edisk & 0.6 produce
very high eccentricities, possibly leading to tidal disruptions of binaries.
Mean of highest five eccentricities reached during simulation.
Mean of highest five inclinations (in radians).
Percentage of stars that enter the tidal radius.
well-defined clockwise disk of O and Wolf-Rayet (WR) stars, and a group of
counter-clockwise moving O and WR stars, possibly a dissolving second disk
(Levin & Beloborodov 2003; Nayakshin & Cuadra 2005; Genzel et al. 2003;
Paumard et al. 2006; Lu et al. 2008; Bartko et al. 2008). These O and WR stars
are located at projected distances of 0.05 pc to 0.5 pc, and were most likely
formed as a result of gravitational fragmentation of a gaseous disk (Levin &
Beloborodov 2003). More puzzling in terms of origin is the population of Bstars known as the ‘S-stars’ (Schödel et al. 2002; Ghez et al. 2005; Eisenhauer
et al. 2005; Gillessen et al. 2009). At distances of 0.003−0.03pc from the SMBH,
in-situ formation seems highly improbable due to the tidal field of SgrA* (Levin
2007). In addition, their young age (∼ 20−100 Myr) imposes a tight constraint
on formation scenarios in that they cannot have travelled far from their place of
birth. One possibility is the formation of the S-stars from the disruption of stellar binaries (Hills 1988; Gould & Quillen 2003). Perets et al. (2007) propose
the surrounding stellar bulge as the source of the binaries, while Löckmann,
Baumgardt, & Kroupa (2008) (LBK) invoke two stellar disks in the Galactic
center to explain the origin. In their simulations the stars experienced Kozai-
18
The Eccentric Disk Instability
type torques which induced strong changes in their inclination and eccentricity.
The eccentricity of many of the stars became greater than 0.9, and LBK argued
that if these stars were binaries they would be tidally disrupted by the SMBH,
thus producing the S-stars and high-velocity stars via Hills’ (1988) mechanism.
However, the gravitational influence of a stellar cusp, which was neglected
by LBK, strongly suppresses Kozai-type dynamics. This was recently discussed
in Chang (2009). Chang’s treatment was highly simplified; in particular, his
expressions for Kozai evolution were valid only if the stellar orbits were inside
the inner edges of both disks. We have performed LBK-type simulations and
have confirmed Chang’s general conclusion that a realistically massive stellar
cusp entirely suppresses the production of high-eccentricity stars via a Kozaitype mechanism.
In contrast, the eccentricity instability is capable of driving several stars to
near-radial orbits, as is evident in Figure (2.1). For capture, the pericenter rp
of the binary’s orbit must come within the tidal radius rt = (2M• /mbin )1/3 abin .
This results in a captured star with semi-major axis a• that scales as a• ∼
(M• /mbin)2/3 abin , where abin is the semi-major axis of the binary itself, mbin
is its combined mass and the eccentricity of the captured star can be approximated as 1−e ∼ (mbin /M• )1/3 .
We summarize our results for varying masses and initial eccentricities of
the disk in Table (2.1). All simulations have evolved for ∼ 7 Myr. For a given
semi-major axis of a binary system, we calculate the tidal radius at which the
binary will be disrupted and finally the percentage of stars that come within
this radius. It is clear that for many parameters of our disk, stars have rp < rt .
For example, a disk with an initial eccentricity of 0.6 and mass of 104 M⊙ , 14
% of stars in the disk will enter the tidal radius. Thus, if some of these stars are
binaries, Hills’ mechanism for producing the S- and high-velocity stars (Brown
et al. 2006) could be at work. An important element in this scenario is the
binary survival rate at the radii of the disk; however, Perets (2009) show that
for a = 0.1 AU the evaporation rate is > 107 yr.
The scalar resonant relaxation time in the Galactic center is everywhere
larger than the age of the disks. It follows that since the post-capture eccentricity of the stars is of order 1 − e ≈ 10−2 , most of the S-stars cannot have
formed in the current disk. The lifetime of many S-stars could be of the order
of 100 Myr, and thus earlier (∼ 10 − 100 Myr) starburst episodes could have
contributed to their present population. In that case, the scalar relaxation time
is short enough to randomize the eccentricities (Hopman & Alexander 2006a;
Perets et al. 2009). We note that the scenario presented by LBK has the same
limitation.
2.5.1
Bimodal eccentricity distribution in disk
In Figure (2.3) we compare the observed stellar eccentricity distribution of
the clock-wise disk (Bartko et al. 2008) with the simulated post-instability ec-
19
Secular Stellar Dynamics
24
7
6
(a)
(b)
20
16
4
3
12
% Stars
# Stars
5
2
8
1
5.7 Myr
0
4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 2.3 – (a) Observed eccentricity distribution of clock-wise stellar disk in the Galactic center,
reproduced with permission from Bartko et al. (2008). (b) Simulated distribution for a disk with
an initial eccentricity of 0.6 and Mdisk = 104 after 5.7 Myr. Both the observed and simulated
eccentricity distributions exhibit bimodality.
centricity distribution for all stars with an initial disk eccentricity of 0.6 (c.f.
Bonnell & Rice 2008). We draw attention to the qualitative similarities in the
two plots, in particular the double-peaked profile. This bimodality is a natural consequence of the eccentric disk instability as all eccentricities are driven
away from the mean. Without the instability it is unclear how to generate such
a distribution.
2.6 Critical discussion
So far, we have described the eccentricity instability for an idealized initial
disk configuration. We now demonstrate the effectiveness of the instability
for a broader scope of parameters and extend our disk model to include (1)
a significant range in both magnitude and direction of the stellar eccentricity
vectors, (2) heavily inclined stellar orbits, and (3) different values of γ, the
power-law index of the cusp, which determines the precession rate within the
disk. We discuss each of these in turn.
(1) We simulate disks with initial eccentricity vectors scattered between a
range of opening angles θ ∈ [0, 2π]. As anticipated, significant spreading of
the vector decreases the effectiveness of the instability. However, we find sufficiently high eccentricity orbits to generate S-stars up to θ ∼ π radians, which
suggests that the instability remains important for substantially less favorable
circumstances. Next we vary the magnitudes of the eccentricities throughout
the disk, basing our initial conditions on Run 1 and 2 of Bonnell & Rice (2008).
Run 1 (0.6 < e < 0.7) is in excellent agreement with our previous results. The
instability in Run 2 (0 < e < 0.6) however, is not efficient at generating the
20
The Eccentric Disk Instability
highest e orbits and hence the S-stars. These conclusions are transparent; the
stars which experience the greatest torque are those at small a (see Section 2.3)
and if initially circular, need a greater torque to produce the highest e orbits.
We find that the most important condition, for Mdisk = 104 , is for innermost
orbits to have e > 0.4. Future work on gas dynamics of eccentric disks will tell
whether this presents a serious problem for our scenario.
(2) We study disks with stellar rms inclinations in the range [0, 0.3] and
find that the instability is robust in all cases. However, above the limit 0.2, the
instability appears ineffective in generating the highest e (> 1 − 10−3 ) orbits.
(3) We vary the power-law slope of the cusp. For γ < 3/2, due to the
shallowness of the cusp, the innermost orbits lag behind in precession and are
more susceptible to being pushed to high e orbits. While this improves the
ability of the instability to produce S-stars, the instability is not as effective
at generating low e orbits. Conversely, an increase in γ (> 3/2) results in
the innermost orbits precessing faster than the bulk of the stars and hence
suppresses the generation of the highest e orbits and lengthens the time-scale
on which it occurs. We conclude that the production of S-stars is unlikely
for a cusp with γ > 7/4. In all cases examined (1 < γ < 2), the instability
proves robust and, as emphasized in Section 2.2, there is both observational
and theoretical evidence that the slope of the cusp in the Galactic center is
sufficiently close to 3/2 for the instability to be highly effective.
The eccentricity instability described in this work is relevant for an eccentric
disk embedded in a stellar cusp. We note that the instability is not directly
applicable for the eccentric disk in M31 (Tremaine 1995) as the mass of the
disk is much greater than the mass of the stellar bulge enclosed within the
same radius. Indeed we find that such a disk remains coherent over several
precession times, although the innermost stars can undergo significant angular
momentum changes as described by Cuadra et al. (2008). The situation is
different in the Galactic center, where massive stars were likely born from an
eccentric gaseous disk, and where there is strong observational evidence for a
heavy spherical stellar cusp (Genzel et al. 2003; Schödel et al. 2007). We have
shown that the instability may have strongly influenced the development of the
eccentricities of the massive stars observed, and in particular, can account for
the bimodality in their eccentricity distribution. Furthermore, combined with
Hills’ (1988) mechanism, the eccentricity instability may have played a role in
the formation of the S-stars and of the high-velocity stars in the Galactic halo.
Acknowledgments We thank Atakan Gürkan, Richard Alexander, Simon
Portegies Zwart, Mher Kazandjian, Jihad Touma and Sergei Nayakshin for useful discussions, and Mark Hayden for comments on the text. AM is supported
by a TopTalent fellowship from the Netherlands Organization for Scientific Research (NWO), YL by a Vidi fellowship from NWO and CH by a Veni fellowship
from NWO.
Secular Stellar Dynamics
21
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The Eccentric Disk Instability
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3
Secular Stellar Dynamics near Massive Black Holes
The angular momentum evolution of stars close to massive black holes
(MBHs) is driven by secular torques. In contrast to two-body relaxation,
where interactions between stars are incoherent, the resulting resonant relaxation (RR) process is characterized by coherence times of hundreds of
orbital periods. In this chapter, we show that all the statistical properties
of RR can be reproduced in an autoregressive moving average (ARMA)
model. We use the ARMA model, calibrated with extensive N -body simulations, to analyze the long-term evolution of stellar systems around MBHs
with Monte Carlo simulations.
We show that for a single-mass system in steady-state, a depression is
carved out near an MBH as a result of tidal disruptions. Using Galactic
center parameters, the extent of the depression is about 0.1 pc, of similar order to but less than the size of the observed “hole” in the distribution of bright late-type stars. We also find that the velocity vectors of
stars around an MBH are locally not isotropic. In a second application,
we evolve the highly eccentric orbits that result from the tidal disruption
of binary stars, which are considered to be plausible precursors of the “Sstars” in the Galactic center. We find that RR predicts more highly eccentric
(e > 0.9) S-star orbits than have been observed to date.
Ann-Marie Madigan, Clovis Hopman and Yuri Levin
The Astrophysical Journal 738, 99 (2011)
24
Secular Stellar Dynamics
3.1 Introduction
T
HE gravitational potential near a massive black hole (MBH) is approximately equal to that of a Newtonian point particle. As a consequence,
the orbits of stars are nearly Keplerian, and it is useful, both as a mental picture and as a computational device, to average the mass of the stars over their
orbits and consider secular interactions between these ellipses, rather than interactions between point particles. These stellar ellipses precess on timescales
of many orbits, due to deviations from the Newtonian point particle approximation: there is typically an extended cluster of stars around the MBH, and
there is precession due to general-relativistic (GR) effects. Nevertheless, for
timescales less than a precession time, torques between the orbital ellipses are
coherent.
It was first shown by Rauch & Tremaine (1996) that such sustained coherent torques lead to much more rapid stochastic evolution of the angular momenta of the stars than normal relaxation dynamics. They called this process
resonant relaxation (RR). RR is potentially important for a number of astrophysical phenomena. Rauch & Ingalls (1998) showed that it increases the tidal
disruption rate, although in their calculations the effect was not very large since
most tidally disrupted stars originated at distances that were too large for RR
to be effective (in Section 3.7.2 we will revisit this argument). By contrast with
tidally disrupted main-sequence stars, inspiraling compact objects originate at
distances much closer to the MBH (Hopman & Alexander 2005). The effect of
RR on the rate of compact objects spiralling into MBHs to become gravitational
wave sources is therefore much larger (Hopman & Alexander 2006). RR also
plays a role in several proposed mechanisms for the origin of the “S-stars”, a
cluster of B-stars with randomized orbits in the Galactic center (GC; e.g. Hopman & Alexander 2006; Levin 2007; Perets et al. 2007, see Section 3.7.3 of this
chapter).
There have been several numerical studies of the RR process itself, which
have verified the overall analytic predictions. However, since stellar orbits
in N -body simulations (e.g. Rauch & Tremaine 1996; Rauch & Ingalls 1998;
Aarseth 2007; Harfst et al. 2008; Eilon et al. 2009; Perets et al. 2009; Perets &
Gualandris 2010) need to be integrated for many precession times, the simulations are computationally demanding. All inquiries have thus far have
been limited in integration time and/or particle number. In order to speed
up the computation, several papers have made use of the picture described
above, where the gravitational interaction between massive wires are considered (Gauss’s method, see e.g. Rauch & Tremaine 1996; Gürkan & Hopman
2007; Touma et al. 2009). It is common, when possible, to use the the FokkerPlanck formalism to carry out long-term simulations of the stellar distribution
in galactic nuclei (Bahcall & Wolf 1976, 1977; Lightman & Shapiro 1977; Murphy et al. 1991; Hopman & Alexander 2006,b). The current formalism however
is not directly applicable to the case when RR plays an important role. At the
Secular Stellar Dynamics
25
heart of all current Fokker-Plank approaches is the assumption of a randomwalk diffusion of angular momentum, whereas RR is a more complex relaxation mechanism based on persistent autocorrelations.
In this work, we will show that the auto-regressive moving average (ARMA)
model provides a faithful representation of all statistical properties of RR. This
model, calibrated with special-purpose N -body simulations, allows us to carry
out a study of the long-term effects of RR on the stellar cusp, thus far out of
reach.
The plan of the chapter is as follows. In Section 3.2, we present an extensive
suite of special purpose N -body simulations, which exploit the near-Keplerian
nature of stellar orbits and concentrate on the stochastic orbital evolution of
several test stars. We use these simulations to statistically examine the properties of RR for many secular timescales. In Section 3.3, we introduce the
ARMA model for the data analysis of the N -body simulations. This description
accommodates the random, non-secular (non-resonant) effects on very short
timescales, the persistent autocorrelations for intermediate timescales less than
a precession time, and the random walk behavior for very long times (the RR
regime). In Section 3.4 we extend the ARMA model, using physical arguments,
to a parameter space that is larger than that of the simulations. In Section 3.5
we calibrate the parameters using the results of the N -body simulations. Once
the free parameters of the ARMA model are determined, we then use them in
Section 3.6 as an input to Monte Carlo (MC) simulations which study the distribution of stars near MBHs. In Section 3.7 we show that RR plays a major role
in the global structure of the stellar distribution near MBHs, and in particular
for the young, massive B-type stars near the MBH (the “S-stars”) in our GC. We
summarize in Section 3.8.
3.2 N-body simulations
We have developed a special-purpose N -body code, designed to accurately integrate stellar orbits in near-Keplerian potentials; see Appendix 3.A for a detailed description. We wrote this code specifically for the detailed study of
RR. Such a study requires an integration scheme with an absence of spurious
apsidal precession and one which is efficient enough to do many steps per stellar orbit while integrating many orbits to resolve a secular process. The main
features of this code are the following:
1. A separation between test particles and field particles. Field particles
move on Kepler orbits, which precess due to their averaged potential and
general relativity, and act as the mass that is responsible for dynamical
evolution. Test particles are full N -body particles and serve as probes of
this potential (Rauch & Tremaine 1996; Rauch & Ingalls 1998).
2. The use of a fourth-order mixed-variable symplectic (MVS) integrator
(Yoshida 1990; Wisdom & Holman 1991; Kinoshita et al. 1991; Saha &
Tremaine 1992). The MVS integrator switches between Cartesian coordi-
26
Secular Stellar Dynamics
nates (in which the perturbations to the orbit are calculated) to one based
on Kepler elements (to calculate the Keplerian motion of the particle under the influence of central object). We make use of Kepler’s equation
(see, e.g., Danby 1992) for the latter step.
3. Adaptive time stepping and gravitational softening. To resolve the periapsis of eccentric orbits (Rauch & Holman 1999) and close encounters
between particles, we adapt the time steps of the particles. We use the
compact K2 kernel (Dehnen 2001) for gravitational softening.
4. Time symmetric algorithm. As MVS integrators generally lose their symplectic properties if used with adaptive time stepping, we enhance the
energy conservation by time-symmetrizing the algorithm.
Our code is efficient enough to study the evolution of energy and angular momentum of stars around MBHs for many precession times, for a range of initial
eccentricities.
3.2.1
Model of Galactic nucleus
We base our Galactic nucleus model on a simplified GC template1 . It has three
main components. (1) An MBH with mass M• = 4 × 106 M⊙ which remains at
rest in the center of the coordinate system. (2) An embedded cluster of equalmass field stars m = 10 M⊙ , distributed isotropically from 0.003 pc to 0.03 pc,
which follow a power-law density profile n(r) ∝ r−α . The outer radius is
chosen with reference to Gürkan & Hopman (2007), who show that stars with
semi-major axes larger than the test star’s apoapsis distance rapo = a(1 + e)
contribute a negligible amount of the net torque on the test star. The field
stars move on precessing Kepler orbits, where the precession rate is determined
by the smooth potential of the field stars themselves (see Appendix 3.C) and
general relativity. The precession of the field stars is important to account for
because for some eccentricities the precession rate of the test star is much lower
than that of the “typical” field star, such that it is the precession of the latter
that leads to decoherence of the system. The field stars do not interact with
each other but they do interact with the test stars if the latter are assumed to
be massive. In this way the field stars provide the potential of the cluster but
are not used as dynamic tracers. (3) A number of test stars, used as probes
of the background potential, that are either massless or have the same mass
as the field stars, m = 10M⊙. We consider both massless and massive stars in
order to study the effects of resonant friction (Rauch & Tremaine 1996). The
test stars have semi-major axes of a = 0.01 pc and a specified initial eccentricity
e.
Following Equation (17) from Hopman & Alexander (2006), who use the
M• −σ relation (Ferrarese & Merritt 2000; Gebhardt et al. 2000) which correlates the mass of a central black hole with the velocity dispersion of the host
1 Due to the effect of GR precession, the system is not scale-free and the masses need to be
specified.
Secular Stellar Dynamics
27
Table 3.1 – Galactic Nucleus
Model
Parameter
Numerical Value
M•
m
α
rh
rmin
rmax
ateststar
N (< rmax )
4 × 106 M⊙
10 M⊙
1.75
2.31 pc
0.003 pc
0.03 pc
0.01 pc
1754
galaxy’s bulge, we define the radius of influence as
GM•
= 2.31 pc
rh =
σ2
M•
4 × 106 M⊙
1/2
.
(3.1)
The number of field stars within radius r is
N (< r) = Nh
r
rh
3−α
,
(3.2)
where we assume that the mass in stars within rh equals that of the MBH, Nh =
M• /m = 4 × 105 . We take α = 7/4 (Bahcall & Wolf 1976), the classic result
for the distribution of a single-mass population of stars around an MBH, which
relies on the assumption that the mechanism through which stars exchange
energy and angular momentum is dominated by two-body interactions. Hence
the number of field stars within our model’s outer radius is N (< 0.03 pc) =
1754. We summarize the potential for our Galactic nucleus model in Table 3.1.
We evolve this model of a galactic nucleus for a wide range in eccentricity of the test stars, e = 0.01, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 0.9 and 0.99. For each
initial eccentricity, we follow the evolution of a total of 80 test stars, both
massless and massive, in a galactic nucleus. Typically we use four realizations of the surrounding stellar cluster for each eccentricity, i.e., 20 test stars
in each simulation. They have randomly oriented orbits with respect to one
another, so that they experience different torques within the same cluster. The
simulations
p are terminated after 6000 orbital periods (henceforth denoted by
P = 2π a3 /GM• ) at a = 0.01 pc, roughly equivalent to three precession times
for a star of eccentricity e = 0.6, deep into the RR regime for all eccentricities,
meaning that several coherence timescales have passed; see Section 3.5 for
verification. Using our method of analysis, it would not be useful for our simulations to run for longer times as the stars would move significantly away from
their initial eccentricities. In addition, the autocorrelation functions (ACFs) of
their angular momentum changes drop to zero before this time; see Figure 3.4.
28
3.2.2
Secular Stellar Dynamics
Illustrative simulations
In Section 3.3, we present a description that captures all the relevant statistical
properties of RR, and in particular has the correct autocorrelations. Here we
consider several individual simulations for illustration purposes, in order to
highlight some interesting points and motivate our approach. We define energy
E of the test star as
GM•
,
(3.3)
E=
2a
and dimensionless angular momentum2 J as
p
J = 1 − e2 .
(3.4)
Secular torques should affect the angular momentum evolution, but not the
energy evolution of the stars (in the picture of interactions between massive
ellipses of the introduction, the ellipses are fixed in space for times less than a
precession time, t < tprec , such that the potential and therefore the energy is
time-independent). It is therefore of interest to compare the evolution of these
two quantities. An example is shown in Figure 3.1. As expected, the angular
momentum evolution is much faster than energy evolution; furthermore, the
evolution of angular momentum is much less erratic, which visualizes the long
coherences between the torques.
In Figure 3.2 we show the eccentricity evolution for a sample population of
stars with various initial orbital eccentricities. There is significant eccentricity
evolution in most cases, but almost none for e = 0.99, the reasons for which
we elucidate in Section 3.6.
To quantify the rate at which the energy E and angular momentum J
change as a function of eccentricity e, we calculate the E and J relaxation
timescales. We define these as the timescale for order of unity (circular angular momentum) changes in energy E (angular momentum J). We compute the
following quantities:
E2
∆t,
(3.5)
tE =
h(∆E)2 i
tJ =
Jc2
∆t,
h(∆J)2 i
(3.6)
where ∆E and ∆J are the steps taken in a time ∆t, and we take the mean h i
over 80 test stars in each eccentricity bin.
We plot the resulting timescales as a function of eccentricity in Figure 3.3,
taking several ∆t values to get order-of-magnitude estimates for these timescales.
We find no significant difference between the results for the massless and massive test stars. We note that tE is only weakly dependent on e, whereas tJ is
2 Throughout this chapter, we use units in which angular momentum J and torque τ are normal√
ized by the circular angular momentum for a given semi-major axis, Jc = GM• a. All quantities
are expressed per unit mass.
Secular Stellar Dynamics
29
t [Myr]
0
0.1
0.2
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
∆E/E0
∆J/Jc
-0.16
-0.18
0
1000
2000
3000
t [P]
4000
5000
6000
Figure 3.1 – Evolution of angular momentum J and energy E, normalized to the initial values, of
a massless test star with semi-major axis a = 0.01 pc and eccentricity e = 0.6. Time is in units
of the initial orbital period P (top axis shows time in Myr). The coherence of RR can be seen in
the J evolution (precession time tprec is ∼ 2000 orbits), whereas the E evolution displays a much
slower, more erratic, random walk.
t [Myr]
0
0.1
0.2
1
0.9
0.8
0.7
e
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1000
2000
3000
t [P]
4000
5000
6000
Figure 3.2 – Eccentricity evolution of a sample of massless test stars in a fiducial galactic nucleus
model. We show four realizations for five different initial eccentricities. Evolution is rapid for
most eccentricities; however, the stars with the highest value e = 0.99 have sluggish eccentricity
evolution.
30
Secular Stellar Dynamics
10000
x = E, ∆t = 4000
∆t = 2000
∆t = 1000
∆t = 500
x = J, ∆t = 4000
∆t = 2000
∆t = 1000
∆t = 500
tx [Myr]
1000
100
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
e
Figure 3.3 – Timescale for order of unity (circular angular momentum) changes in energy E
(angular momentum J) for massless test stars in our N -body simulations, presented on a log scale
of time in units of Myr; see Equations (3.5) and (3.6). We find a weak trend for lower E relaxation
times with increasing eccentricity (from ∼ 400 Myr to 200 Myr). J relaxation occurs on very short
timescales (∼ 20 Myr) at eccentricities of 0.8 < e < 0.9. Stellar orbits with low eccentricities have
very long J relaxational timescales (∼ 1 Gyr) as the torque on a circular orbit drops to zero. This
timescale increases again at very high eccentricities.
of the same order of magnitude as tE for e → 0 and e → 1, but much smaller
for intermediate eccentricities. It is in the latter regime that secular torques
are dominating the evolution. In the following sections, we will give a detailed
model for the evolution of angular momentum. Since the focus of this chapter is mainly on secular dynamics, we do not further consider energy diffusion
here, but in Appendix 3.B we discuss the timescale for cusp formation due to
energy evolution, and compare our results to other results in the literature.
3.3 Statistical description of resonant relaxation
As a result of the autocorrelations in the changes of the angular momentum of
a star, as exhibited in Figure 3.1, RR cannot be modeled as a random walk for
all times. This is also clear from physical arguments. Rauch & Tremaine (1996)
and several other papers take the approach of considering two regimes of evolution: for times t ≪ tprec , evolution is approximately linear as the torques
continue to point in the same direction. For times t ≫ tprec , the torque autocorrelations vanish and it becomes possible to model the system as a random
walk. Here we introduce a new approach, which unifies both regimes in a
single description. This description is also useful as a way of quantifying the
statistical properties of RR.
We study the evolution of the angular momenta of the test stars using a
time series of angular momenta at a regular spacing of one period. This choice
Secular Stellar Dynamics
31
t [Myr]
0.0001
0.001
0.01
0.12
0.1
ρ(data)
ρ(theory)
0.1
0.08
ρt
0.06
0.04
0.02
0
-0.02
-0.04
1
10
100
t [P]
1000
10000
Figure 3.4 – Autocorrelation function for the differences ∆J in one particular simulation (noisy
orange line) as a function of time in units of the initial orbital period P . The test star is massless
and has initial eccentricity e = 0.6. The autocorrelation function is significantly larger than zero
(if much less than one) for several hundred orbits. The smooth red line gives the theoretical ACF
for an ARMA(1, 1) model with φ1 = 0.995, θ1 = −0.976.
is arbitrary, and below we show how our results generalize to any choice of
time steps. The normalized ACF can be written as
ρt =
h(∆Js+t − h∆Ji)(∆Js − h∆Ji)i
,
h(∆Js − h∆Ji)2 i
(3.7)
where ∆Js is the change in angular momentum between subsequent measurements at time s and it is assumed that the time series is stationary. A typical
ACF from our simulations is shown in Figure 3.4. The fundamental feature of
this ACF is that it is significantly larger than zero for hundreds of orbits, but
with values much less than unity.
Our goal in this section is to define a process that generates a time series
that has the same ACF as that of the changes in angular momentum of a star.
Such a time series has all the statistical properties that are relevant for the
angular momentum evolution of the star, such as the same Fourier spectrum.
We have two motivations for doing this. First, defining such a process implies
that once the parameters of the model are calibrated by use of the N -body
simulations, RR is fully defined – as it is a random Gaussian process that is also
stationary, for sufficiently short times, and as such is entirely described by its
ACF. Second, a generated time series with the same statistical properties as the
time series generated in the physical process can be used in MC simulations to
solve for the long term evolution of RR.
32
3.3.1
Secular Stellar Dynamics
The autoregressive moving average model ARMA(1, 1)
We now introduce the ARMA model, which is often used in econometrics (see,
e.g., Heij et al. 2004). We show that this model can generate time series that
reproduce the ACF and variance of ∆Jt . The model is ad hoc in that it does not
have a direct physical foundation. However, we will find physical interpretations for the free parameters of the model in Section 3.4. We use a form of the
model, ARMA(1,1), with one autoregressive parameter, φ1 , and one moving
average parameter, θ1 . The model can then be written as
(1)
(1)
∆1 Jt = φ1 ∆1 Jt−1 + θ1 ǫt−1 + ǫt .
(3.8)
The label “1” in this equation refers to the fact that the data have a regular
spacing of one period. In this equation, φ1 and θ1 are free parameters, and the
random variable ǫ is drawn from a normal distribution with
(1)
hǫ(1) i = 0;
2
hǫt ǫ(1)
s i = σ1 δts
(3.9)
where σ1 is a third free parameter and δts is the Kronecker delta. For such a
model, the variance of the angular momentum step is
h∆Jt2 i =
1 + θ12 + 2θ1 φ1 2
σ1 ,
1 − φ21
(3.10)
and the ACF can be derived analytically (see Appendix 3.D for details):
θ1 /φ1
(t > 0).
(3.11)
ρt = φt1 1 +
1 + (φ1 + θ1 )2 /(1 − φ21 )
From this expression, we see that the decay time of the ACF is determined by
the parameter φ1 , and θ1 captures the magnitude of the ACF. As an example, we
plot the ACF in Figure 3.4 with parameters typical for RR. We emphasize the
need for both an autoregressive parameter and a moving average parameter
to reproduce the slow decay of the ACF. Neither parameters can replicate the
features on their own.
A useful reformulation of the ARMA(1, 1) model, which can be found using
recursion, is
∞
X
ψk ǫt−k ,
(3.12)
∆Jt =
k=0
where
ψk = (φ1 + θ1 )φ1k−1
(k > 0);
ψ0 = 1.
(3.13)
Once the model parameters (φ1 , θ1 , σ1 ) are found for a time step of one
period, δt = P , the variance after N steps can be computed. The displacement
after N steps is given by
N
X
n=1
∆Jn =
∞
N X
X
n=1 k=0
ψk ǫn−k ,
(3.14)
Secular Stellar Dynamics
33
where (n − k) ≥ 0. We denote the variance of the displacement after N steps
by
!2 +
* N
X
,
(3.15)
V (N ) ≡
∆Jn
n=1
such that
V (N ) =
N X
∞ X
N X
∞
X
n=1 k=0 m=1 l=0
=
σ12
ψk ψl hǫn−k ǫm−l i
N X
∞ X
N X
∞
X
ψk ψl δn−k,m−l ,
(3.16)
n=1 k=0 m=1 l=0
which can be decomposed as
V (N )σ1−2 = N +
N m−1
X
X
ψm−n +
m=2 n=1
+
∞ X
N
X
k=1 m=1
N
−1
X
k=1
ψk (N − k)
(3.17)
min(k+m−1,N )
X
ψk ψk+m−n .
n=1
This expression can be cast in a form without summations by repeatedly using the properties of the independent normal random variables ǫt in Equation
(3.9), resulting after some algebra in
2φN − φ1 − 1
φ1 + θ1
2N − 1 + 1
V (N )σ1−2 = N +
1 − φ1
1 − φ1
(3.18)
2 φ1 (1 − φN
φ1 + θ1
1 )
+
N −2
.
1 − φ1
1 − φ21
Note that in the special case that N = 1, Equation (3.10) is recovered. We plot
this expression in Figure 3.5.
Three characteristic timescales are clearly discernible in this figure. On a
short timescale, the angular momentum diffusion is dominated by the usual
two-body gravitational scattering (i.e., non-resonant relaxation,
√ hereafter NR).
In this regime the angular momentum deviation scales as t. On an intermediate timescale the orbital angular momentum evolves due to the nearly
constant orbit-averaged secular torques from the other stellar orbits, and thus
the angular momentum deviation scales linearly with time. On a yet longer
√
timescale the secular torques fluctuate due to orbital precession, and the t
scaling is reinstated, albeit with a much higher effective diffusion coefficient.
Thus the ARMA-driven evolution is in agreement with what is expected for a
combination of RR and NR acting together; see also Rauch & Tremaine (1996)
and Eilon et al. (2009) for discussion.
34
Secular Stellar Dynamics
0.0001
t [Myr]
0.01
0.001
0.1
1
1
<variance after time t>
0.1
0.01
0.001
0.0001
1e-05
1e-06
1e-07
1e-08
1
10
100
1000
10000
100000
t [P]
Figure 3.5 – Variance in angular momentum J after time t (V (t); from Equation (3.18)) as a
function of time in units of the initial orbital period P , with φ1 = 0.999, θ1 = −0.988 and
σ1 = 10−4 . Note that there is first a linear part (“NR”), then a quadratic part (“coherent torque”),
and then again a linear part (“RR”).
For the typical values of φ1 and θ1 that we find, these different regimes can
be identified in the expression for the variance V (N ); Equation (3.18). For very
short times, the expression is dominated by the first term, which can thus be
associated with NR. The second term and third term are both proportional to
N 2 for short times (this can be verified by Taylor expansions assuming N φ1 ≪
1; the linear terms cancel), and as such represent coherent torques; and they
become proportional to N for large time, representing RR. In Figure 3.5 we
plot Equation (3.18) for typical values of the parameters of the model.
Equation (3.18) is useful for finding the variance after N periods, but most
importantly to define a new process with arbitrary time steps. We can now
define a new ARMA(1, 1) process with larger time steps, but with the same
statistical properties. If the new process makes steps of N orbits each, we write
the new model as
(N )
(N )
∆N Jt = φN ∆N Jt−N + θN ǫt−N + ǫt
;
2
,
h[ǫ(N ) ]2 i = σN
(3.19)
where the parameters [φN , θN ] are related to (φ1 , θ1 ) by3
φN = φ(N P ) = φN
1 ;
3 For
θN = θ(N P ) = −[−θ1 ]N .
(3.20)
φ1 we use the fact that the theoretical ACF of an ARMA(1, 1) model decays on a timescale
P/ ln φ1 . We have no interpretation for the functional form of θ1 , which was found after some experimenting. We confirm numerically that this form leads to excellent agreement with simulations
with arbitrary time; see Figure 3.6.
Secular Stellar Dynamics
35
350
0.1
1
3
10
300
250
dN/dJ
200
150
100
50
0
0.495
0.5
J
0.505
Figure 3.6 – Monte Carlo simulation of RR with the model parameters φ1 = 0.999, θ1 = −0.988,
and σ1 = 10−6 (chosen for purely illustrative purposes), and the time steps in units of the orbital
period indicated in the legend. After 104 orbits, the distribution is very similar for all time steps
chosen, confirming that the diffusion in J is independent of the time step.
Since after N small steps of one period, the variance should be the same as
after 1 single step of time N orbits, we can find the new σ to be (see Equations
(3.10), (3.18))
1 − φ2N
2
(3.21)
σN
=
2 + 2φ θ V (N ).
1 + θN
N N
Once the ARMA parameters for steps of one orbital period are found, they
can be generalized to a new ARMA model via Equation (3.19), so that the time
step equals an arbitrary number of periods (not necessarily an integer number),
and the new parameters are found by the use of Equations (3.20) and (3.21).
We will exploit these relations in MC simulations (Section 3.6), in which we
use adaptive time steps. We have tested the equivalence of the models by considering an ARMA diffusion process over a fixed time interval but with varied
time steps. We show an example in Figure 3.6 where we have taken a delta
function for the initial J-distribution, and have plotted the resulting final distributions after 104 orbits for several different time steps which vary by two
orders of magnitude. We find excellent agreement between these distribution
functions which verify the generalization of (φ1 , θ1 , σ1 ) to (φN , θN , σN ).
3.4 Interpretation of the ARMA parameters and extension
of parameter space
The ARMA(1, 1) model presented here contains the three free parameters
(φ1 , θ1 , σ1 ). In this section, we interpret these parameters in the context of
a stylized model of a galactic nucleus. Our purpose in doing so is twofold.
36
Secular Stellar Dynamics
First, it relates the ARMA model to the relevant physical processes that play a
part in the angular momentum evolution of stars near MBHs, and gives insight
into how the parameters depend on the stellar orbits, in particular on their
eccentricity. This allows us to define, and discuss, RR in the language of the
ARMA model. Second, in Section 3.5 we calibrate the parameters as a function of eccentricity using N -body simulations for one specific configuration —
as specified in Section 3.2.1. In order for the parameters to be also valid for
other conditions than those studied, one needs to understand how they vary
as a function of the quantities that define the system. The model presented in
this section, which is based on physical arguments, allows us to generalize our
results to models we have not simulated (models with different stellar masses,
semi-major axes, etc.) The arguments presented here do not aspire to give a
full theoretical model of RR, but rather parameterize the model in a simplified
way that is useful for generalization to other systems. We will use this model
in our MC simulations (Section 3.6).
3.4.1
Non-resonant relaxation (NR): the parameter σ
For very short times NR dominates the angular momentum evolution of a star
and the variance of J is expected to be of order P/tNR after one period, where
tNR = ANR
M•
m
2
1 1
P
N< ln Λ
(3.22)
is the NR time (Rauch & Tremaine 1996), defined as the timescale over which
a star changes its angular momentum by an order of the circular angular momentum at that radius. In this expression, N< is the number of stars within the
radius equal to the semi-major axis of the star, Λ = M• /m, and ANR = 0.26,
chosen to match the value of tE from the N -body simulations; see Figure 3.3.
Since this variance should be approximately equal to σ12 , it follows that
σ1 = fσ
m
M•
r
N< ln Λ
.
ANR
(3.23)
Here fσ is a (new) free parameter which, calibrated against our N -body simulations, we expect to be of order unity (the deviation from unity will quantify
the difference between the energy relaxation time tE and the non-resonant
component of the angular momentum relaxation time tJ ). In our N -body simulations we find that σ1 is an increasing function of e; see Figures 3.12 and
3.14. This suggests that NR is also an increasing function of e which we attribute to the increase in stellar density at small radii which a star on a highly
eccentric orbit passes through at periapsis. To account for this in our theory we
fit fσ as a function of e.
Secular Stellar Dynamics
3.4.2
37
Persistence of coherent torques: the parameter φ
We now define a new timescale over which interactions between stars remain
coherent, the coherence time tφ , which corresponds to the time over which secular torques between orbits remain constant. This is the minimum between
the precession time of the test star and the median precession time of the field
stars. The latter timescale is of importance as there exists at every radius an
orbital eccentricity which has equal, but oppositely directed, Newtonian precession due to the potential of the stellar cluster and GR precession. A star with
this particular orbital eccentricity remains almost fixed in space; for our chosen
model of a galactic nucleus the eccentricity at which this occurs is e ∼ 0.92 at
0.01 pc. Its coherence time, however, does not tend to infinity, as the surrounding stellar orbits within the cluster continue to spatially randomize.
The timescale for the Newtonian precession (i.e., the time it takes for the
orbital periapsis to precess by 2π radians) is given by
M•
cl
P (a)f (e, α)−1
(3.24)
tprec = π(2 − α)
N< m
(see Appendix 3.C for derivation), where N< is the number of stars within the
semi-major axis a of the test star, P (a) its period, and
f (e, α = 7/4)−1
h
i
= 0.975(1 − e)−1/2 + 0.362(1 − e) + 0.689 .
(3.25)
The timescale for precession due to general relativity is given by the first order
(in 1/c2 ) relativistic correction to the Newtonian result for the angle swept out
in one orbit
2
1
2 ac
tGR
=
P (a)
(3.26)
(1
−
e
)
prec
3
GM•
(Einstein 1916). The combined precession time is
−1
1
1
.
− GR
tprec (a, e) = cl
tprec (a, e) tprec (a, e) (3.27)
As stated, the coherence time for a particular star is the minimum of its
precession time and that of the median precession time of the surrounding
stars,
tφ = fφ min [tprec (a, e), tprec (a, ẽ)] ,
(3.28)
p
where ẽ is the median value of the eccentricity of the field stars (so ẽ = 1/2
for a thermal distribution) and fφ is a second (new) free parameter.
We now have all the ingredients to define the RR time. This is the timescale
over which the angular momentum of a star changes by order of the circular
momentum. The angular momentum evolution is coherent over a time of order
38
Secular Stellar Dynamics
tφ , and the change in angular momentum during that time is τ tφ , where τ
is the torque exerted on the star’s orbit. Normalized to the circular angular
momentum, this torque is (Rauch & Tremaine 1996)
√
m N<
e,
(3.29)
τ = Aτ
M• P
where the linear dependence on eccentricity was determined by Gürkan & Hopman (2007). We adopt their value of Aτ = 1.57 which was determined for a different stellar mass profile m = 1M⊙ , α = 1.4 where M• = 3×106 M⊙ , rh = 2pc.
Assuming that the evolution is random for longer times, this leads to the definition
2
1
tRR (E, J) =
tφ .
(3.30)
τ tφ
The theoretical ACF of an ARMA(1, 1) model in Equation (3.11) shows that
it decays on a timescale P/ ln φ1 . Hence we find that φ scales as exp(−P/tφ ).
However, even if the coherence time is very long, the evolution of the orbit
may not be driven by secular dynamics. The secular torques on an orbit are
proportional to its eccentricity so if the eccentricity is very small secular effects
are weak. As a result, the evolution is dominated by two-body interactions,
which have a vanishingly short coherence time. Within our formalism, this
effect can be accounted for by multiplying tφ by a function S,
P
(3.31)
φ1 = exp −
Stφ
with
S
=
1
1 + exp(k [e − ecrit ])
(3.32)
which interpolates between the secular and two-body regime. We remind the
reader that the subscript “1” is used to emphasize that this is the value of φ for
a time step of one orbital period. The exponential transition between NR and
RR regimes is somewhat arbitrarily chosen but fits the data reasonably well.
k > 0 is a new parameter which determines the steepness of the transition (a
third free parameter), and ecrit is defined as the critical eccentricity at which
the NR and RR times are equal (Equations (3.22), (3.30)):
s
1/2
P
ln Λ
.
(3.33)
ecrit (a, e) =
2
ANR Aτ tφ
Inserting relevant values into the above equation and solving numerically
using a Newton-Raphson algorithm, we calculate ecrit ∼ 0.3 at a = 0.01 pc
which is in good agreement with the N -body simulations. We plot ecrit as a
39
Secular Stellar Dynamics
1
0.8
ecrit
0.6
0.4
0.2
0
0.001
0.01
0.05
a [pc]
Figure 3.7 – Critical eccentricity ecrit for which NR and RR mechanisms work on comparable
timescales, as a function of semi-major axis a from the MBH in our Galactic nucleus model. The
shaded region in this plot indicates the eccentricities for which RR is the dominant mechanism.
The lower values of ecrit decrease with distance to the MBH as RR becomes more effective (τ
and tφ increase with decreasing a). At a ∼ 0.004 pc this value rises again in our model as tNR
decreases rapidly as r −1/4 . The higher values of ecrit are due to GR effects which decrease tφ
toward the MBH and substantially reduce the effectiveness of RR. The lower values of ecrit are
due to the fact that τ ∝ e.
function of a in Figure 3.7. RR becomes more effective at lower a as the mass
precession time increases, and the value of ecrit decreases. At the lowest a
there are two roots to Equation (3.33); this is due to GR effects which compete
with mass precession and decrease the coherence time to such an extent as to
nullify the effects of RR at high eccentricities. These results are derived for
a simplified galactic nucleus model and will change for different assumptions
on the mass distribution; nevertheless we find the concept of ecrit useful for
understanding the relative importance of RR and NR at different radii.
3.4.3
Magnitude of resonant relaxation steps: the parameter θ
For times less than tφ , there are coherent torques between stellar orbits; see
Equation (4.6). The expected variance after a time tφ is then
h∆Jφ2 i = A2τ
m
M•
2
N<
tφ
P
2
e2 .
(3.34)
Alternatively, we can use the ACF (see Appendix 3.D for derivation) to find
h∆Jφ2 i = σ12
tφ
P
2
(θ1 + φ1 )(θ1 + 1/φ1 )
.
1 − φ21
(3.35)
40
Secular Stellar Dynamics
Equating Equations (3.34) and (3.35) gives two values for θ1
" #
s
1
4(1 − φ21 )τ 2 P 2
1
1
2
−
+ φ1 ±
+ φ1 − 2 +
θ1 =
2
φ1
φ21
σ12
(3.36)
Taking the positive root we find an excellent match to the data. For values of
e > ecrit the first term 12 ( φ11 + φ1 ) ≈ 1 and recognizing the Taylor expansion of
− exp(−x) ≈ x − 1 we simplify this expression to
s
!
fθ
1
4(1 − φ21 )τ 2 P 2
2
θ1 = − exp −
(3.37)
+ φ1 − 2 +
2 φ21
σ12
where fθ is a fourth free parameter.
We summarize this section by clarifying that we have four free parameters
in our ARMA model (fθ , fφ , fσ , k) fully determinable by our N -body simulations, with which we calibrate our model for use in MC simulations.
3.5 Results: ARMA analysis of the N-body simulations
We now return to the N -body simulations, and use the time series of angular
momenta that are generated to calibrate the ARMA model. These parameters
fully define both the RR and NR processes. Once they are known, they can be
used to generate new angular momentum series that have the correct statistical
properties, much like what is done in regular MC simulations. We will exploit
this method in Section 3.6.
When a test star has mass, there will be a back-reaction on the field stars
known as resonant friction, analogous to dynamical friction (Rauch & Tremaine
1996). In our simulations, we consider both the case that the test star is massless and that it has the same mass as the field stars to calibrate the model
parameters. Differences in the parameters then result from resonant friction.
We use the “R” language for statistical computing to calculate the ARMA
model parameters from our simulations. We input the individual angular momentum time series of a test star, and use a maximum likelihood method to
calculate φ1 , θ1 and σ1 . We find that the drift term h∆Ji for stars of all eccentricities is O(10−6 ).
In Figure 3.8 and 3.9 we show scatter plots of the model parameters 1 − φ1
and 1 + θ1 , for simulations in which the test stars are massless and massive
respectively. We present these quantities rather than φ1 and θ1 themselves
because they span several orders of magnitude and, physically speaking, the
relevant question is by how much the parameters differ from (minus) unity.
For intermediate eccentricities (0.4 . e . 0.9), 1 − φ1 is much smaller than
unity, indicating that the angular momentum autocorrelations are indeed persistent, and there are significant torques between stellar orbits. However, the
Secular Stellar Dynamics
1 + θ1
1
41
e=0.01
e=0.1
e=0.2
e=0.3
e=0.4
e=0.6
e=0.8
e=0.9
e=0.99
0.1
0.01
0.001
0.01
0.1
1
1 - φ1
Figure 3.8 – Scatter plot of (1 − φ1 ) vs (1 + θ1 ) as found from data-analysis on the N -body runs,
for the case of massless test stars. For most cases, the parameters cluster, leading to a concentration
of points around small values; reasons for exceptions are discussed in the text. From inspection of
several individual cases, we see that the test stars of intermediate eccentricities which have values
φ1 , |θ1 | . 1 (to the bottom left of plot) have non-zero autocorrelations for long times, similar to
Figure 3.4. This is as expected from Equation (3.11). Test stars with φ1 , θ1 ≈ 0 have ACFs that
are close to zero everywhere, even for short times.
1 + θ1
1
e=0.01
e=0.2
e=0.4
e=0.6
e=0.8
e=0.9
e=0.99
0.1
0.01
0.001
0.01
0.1
1
1 - φ1
Figure 3.9 – Scatter plot of (1 − φ1 ) vs (1 + θ1 ) as found from data analysis on the N -body runs,
for the case of massive (m = 10M⊙ ) test stars. We find that, for most eccentricities, the values for
1 − φ1 and 1 + θ1 cluster around those for the massless cases.
42
Secular Stellar Dynamics
10
massive
massless
1 - φ1
1
0.1
0.01
0.001
0
0.2
0.4
0.6
0.8
1
e
Figure 3.10 – Median value of 1 − φ1 , plotted on a log scale, for both massless and massive
runs, computed for 80 test stars in each eccentricity bin. Closed boxes denote the 45th and 55th
percentiles, while the lines indicate the one sigma values. Note that for very high eccentricities
the values increase, which in the interpretation of Section 3.4 means that the coherence time has
decreased. This is consistent with the fact that for e ≥ 0.92, GR precession becomes dominant
over mass precession. The deviation at low eccentricities between values for massless and massive
particles may due to resonant friction.
coherent torques are mixed with NR noise (the classical NR as treated in Binney & Tremaine (2008)), and as a result 1 + θ1 ≪ 1 as well. These values for
1 − φ1 and 1 + θ1 confirm that although the ACF for RR is finite for long times,
it is much smaller than unity — which would be the case if there was no NR
noise at all.
For very small and very large eccentricities, we find φ1 ≈ θ1 ≈ 0, so there
are no persistent torques of significance. The reason for the absence of coherent torques on the high and low eccentricity limits are different: for very
small eccentricities, the secular torque on a stellar orbit approaches zero as
τ ∝ e (Gürkan & Hopman 2007). At the high eccentricity end, there are significant torques, but the coherence time is so short due to general relativity, that
the persistence of these torques is negligible, and there is no coherent effect
(quenching of RR: see Merritt & Vasiliev 2010). Test stars with eccentricities
of 0.2 ≤ e ≤ 0.4 have a large scatter in their 1 − φ1 and 1 + θ1 values. This
is due to their proximity to ecrit , the eccentricity at which NR and RR compete
for dominance at this radius in our model.
In Figures 3.10 and 3.11 we compare the median values of 1 − φ1 and 1 + θ1
for both massive and massless test stars. There is significant scatter in the data
near 0.2 ≤ e ≤ 0.4, where NR relaxation competes with RR, as can be seen
from the 45th, 55th and one sigma (34.1 − 84.1) percentiles. The differences
between the median values for massless and massive test stars are very small,
such that we did not see a strong feedback effect due to resonant friction,
Secular Stellar Dynamics
10
43
massive
massless
1 + θ1
1
0.1
0.01
0.001
0
0.2
0.4
0.6
0.8
1
e
Figure 3.11 – Median value of 1 + θ1 , plotted on a log scale, for massless and massive runs, computed for 80 test stars in each eccentricity bin. Closed boxes denote the 45th and 55th percentiles,
while the lines indicate the one sigma values. The results for massless and massive particles follow
each other closely except at the low-eccentricity end where there is a large scatter in the data.
0.0003
massive
massless
0.00028
0.00026
σ1
0.00024
0.00022
0.0002
0.00018
0.00016
0
0.2
0.4
0.6
0.8
1
e
Figure 3.12 – Data for the model parameter σ1 as a function of eccentricity for massless and
massive test stars. Plotted are the 45th to 55th percentiles (open boxes), one sigma values (lines),
and 50th percentile for each eccentricity. The results for massive test stars are shifted along the
x-axis for clarity. The values follow each other closely with the exception of low eccentricities.
except for small stellar eccentricities. Resonant friction appears to drag stars
away from circular angular momentum. The same deviation is seen for the σ1
model parameter in Figure 3.12.
From here on, we use only the results of the massless simulations as we have
a larger range of test star eccentricities and have found the ARMA parameters
44
Secular Stellar Dynamics
Table 3.2 – Median values of 1−φ1 , 1+θ1
and σ1 for massless test stars as a function
of eccentricity.
e
0.01
0.1
0.2
0.3
0.4
0.6
0.8
0.9
0.99
1 − φ1
1 + θ1
σ1 (10−4 )
0.992
0.993
0.991
0.569
0.006
0.005
0.004
0.004
0.047
1.004
1.002
1.003
0.609
0.024
0.025
0.031
0.033
0.156
1.853
1.828
1.866
1.923
2.027
2.184
2.379
2.460
2.626
(φ1 , θ1 , σ1 ) to be similar in value. We refer the reader to Table 3.2 for exact
quantities.
The ARMA model parameters for both very low eccentricities (e ∼ 0.01) and
high eccentricities (e & 0.4) tend to cluster together. For these cases one form
of relaxation, either NR or RR, dominates the form of the ACFs, and taking
mean values of (φ1 , θ1 ) gives an accurate representation of the population.
However, near the ecrit boundary where the two relaxation effects compete for
dominance, (φ1 , θ1 ) do not consistently cluster but rather choose NR or RR
values which can vary significantly. Hence we choose the median value in each
case to compare with our theory.
In Figures 3.13 and 3.14 we compare the experimental median values of
(1 − φ1 ), (1 + θ1 ) and σ1 to our theoretical model. We find that good agreement
between the N -body simulations and the model is obtained by assuming values
for the free parameters (fφ , fθ , k) = (0.105, 1.2, 30). The value of fφ shows that
the coherence time tφ is much less than the precession time tprec ; see Equation
(3.28). Fitting fσ as a function of e we find
√
fσ = 0.52 + 0.62e − 0.36e2 + 0.21e3 − 0.29 e.
(3.38)
3.6 Monte Carlo simulations and applications
Our aims in this section are to (1) explore the long-term evolution of stars
around an MBH, (2) investigate the depletion of stars around an MBH due to
RR in context of the “hole” in late-type stars in the GC, and (3) study the evolution of the orbits of possible S-star progenitors from different initial setups.
Direct N -body simulations of secular dynamics are challenging due to the inherent large range in timescales. The precession time is orders of magnitude
larger than the orbital time, and in order to study relaxation to a steady-state,
the system needs to evolve for a large number of precession times. Currently
it is not feasible to do this using N -body techniques. Instead we draw on MC
Secular Stellar Dynamics
10
45
1 - φ1: data
fit
1 + θ1: data
fit
1 - φ1, 1 + θ1
1
0.1
0.01
0.001
0
0.1
0.2
0.3
0.4
0.5
e
0.6
0.7
0.8
0.9
1
Figure 3.13 – Median value of (1 − φ1 ) and (1 + θ1 ), compared with the theoretical values from
(3.31, 3.37). There is a sharp transition between eccentricities 0.3 . e . 0.4 where the two
effects of NR and RR compete. A second transition occurs at high eccentricities (e > 0.92) as the
coherence time, and hence φ1 , decreases due to general relativistic effects. The theory does not
exactly match the median value of the data at 0.2 < e < 0.3. However we can vary the value of
ecrit to reconcile this difference and we find that this is not important for our results in the next
section.
0.00027
data
fit
0.00026
0.00025
0.00024
σ1
0.00023
0.00022
0.00021
0.0002
0.00019
0.00018
0
0.1
0.2
0.3
0.4
0.5
e
0.6
0.7
0.8
0.9
1
Figure 3.14 – Median values of the model parameter σ1 (related to non-resonant relaxation) and
theoretical fit, as a function of eccentricity for massless test stars.
simulations to study long term secular evolution. For this purpose, we have
developed a two-dimensional (energy and angular momentum) MC code. The
main new aspect compared to earlier work is that it implicitly includes evolution due to RR. The angular momentum diffusion is based on the ARMA(1,
1) model presented in Section 3.3. For the energy evolution we use a similar
46
Secular Stellar Dynamics
method as in Hopman (2009), which was in turn based on Shapiro & Marchant
(1978). A key feature of this method is a cloning scheme, which allows one
to resolve the distribution function over many orders of magnitude in energy,
even though the number of particles per unit energy is typically a steep powerlaw, dN/dE ∝ E −9/4 . We will not describe the cloning scheme here; for details
we refer the reader to Shapiro & Marchant (1978) and Hopman (2009).
We base our model of a galactic nucleus on parameters relevant for the GC;
see Section 3.2. We set the MBH mass as M• = 4 × 106 M⊙ , which affects the
physics through the GR precession rate and the loss cone. We linearize the
system by considering the evolution of test stars on a fixed background stellar
potential. We assume a single-mass distribution of stars of either m = 1M⊙ or
m = 10M⊙, with the number of stars within a radius r, N (< r), following a
power-law profile as in Equation (3.2).
In Figure 3.15 we plot the NR and RR timescales for stars with different
orbital eccentricities as a function of semi-major axis for this model with m =
10M⊙. A cuspy minimum arises in the RR timescales where GR precession
cancels with Newtonian mass precession. In their GC model Kocsis & Tremaine
(2010) find, using Eilon et al. (2009) parameters, the cuspy minimum of RR
near r = 0.007 pc (see their Figure 1), which falls precisely within the range
of values in our model (0.004 − 0.02 pc). Figure 3.16 shows the RR time as
a function of eccentricity e for different stellar masses within 0.01 pc. Models
with different enclosed masses will be used in Section 3.7.3.
In reality, several stellar populations are present in the GC, and our choice
for a single mass model is therefore a simplification. The slope α = 1.75
is smaller than would be expected for strong mass-segregation (Alexander &
Hopman 2009; Keshet et al. 2009), but was chosen to make the model selfconsistent as a collection of interacting single mass particles will relax to have
this distribution (Bahcall & Wolf 1976).
3.6.1
Angular momentum
For the angular momentum evolution of a test star, we go through the following
steps: we first calculate the ARMA model parameters for one orbital period
(φ1 , θ1 , σ1 ) through Equations ((3.31), (3.37), (3.23)). We then find the model
parameters (φN , θN , σN ) for the time step δt, where N (= δt/P ) ∈ R > 0, using
Equations (3.20) and (3.21). Finally, the step in angular momentum is given
by Equation (3.19)
(N )
(N )
∆N Jt = φN ∆N Jt−N + θN ǫt−N + ǫt
h[ǫ
(N ) 2
] i=
2
σN
.
;
(3.39)
Secular Stellar Dynamics
47
1000
t [Myr]
100
tnr
trr e=0.1
e=0.2
e=0.4
e=0.6
e=0.8
e=0.9
e=0.99
10
1
0.0001
0.001
0.01
0.1
a [pc]
Figure 3.15 – RR and NR timescales in units of Myr as a function of semi-major axis a in our
Galactic nucleus model for different eccentricities.
1000
trr [Myr]
100
10
N = 800
444
200
100
80
50
no GR
1
0.2
0.3
0.4
0.5
0.6
e
0.7
0.8
0.9
1
Figure 3.16 – Resonant relaxation timescale trr for varying stellar mass (N m where m = 10M⊙ )
within 0.01 pc as a function of e. In this plot the density power-law index α = 1.75. The e at
which RR is most efficient decreases with decreasing mass. The solid black line indicates the value
of trr if the effects of GR precession are not taken into account (and is independent of number N ).
In our fiducial model the optimal efficiency (shortest trr ) is for N m = 1000M⊙ . At this radius,
GR effects are important for RR and trr is not independent of N .
48
3.6.2
Secular Stellar Dynamics
Energy
The energy of a star is given in units of the square of the velocity dispersion at
the radius of influence, vh2 , such that E = rh /2a. The step in energy during a
time δt is
1/2
δt
,
(3.40)
∆E = ξE
tNR
where ξ is a normal random variable with zero mean and unit variance, and
tNR is given by Equation (3.22) with ANR = 0.26.
3.6.3
Initial conditions and boundary conditions
For our steady-state simulations, we initialize stars with energies E = 1 and
angular momenta drawn from a thermal distribution, dN (J)/dJ ∝ J. We
follow the dynamics of stars between energy boundaries 0.5 = Emin < E <
Emax = 104 (for our model this corresponds to 2.31 pc > a > 1.155 × 10−4 pc).
Stars that diffuse to E < Emin are reinitialized, but their time is not set to
zero (zero-flow solution; Bahcall & Wolf 1976, 1977). Stars with E > Emax
are disrupted by the MBH and also reinitialized while keeping their time. Stars
have angular momenta in the range Jlc < J < 1, where Jlc is the angular
momentum of the loss cone. Stars with J < Jlc are disrupted and reinitialized,
while stars with J > 1 are reflected such that J → 2 − J.
3.6.4
Time steps
In the coherent regime, the change
√ in angular momentum during a time δt is
of order δJ ∼ τ δt ∼ Aτ e(m/M• ) N< (δt/P ). The changes in angular momentum should be small compared to either the size of the loss cone Jlc , or the
separation between J and the loss cone, J − Jlc ; and compared to the distance
between the circular angular momentum Jc and J, i.e., compared to 1 − J. We
therefore define the time step to be
δt = ft τ (a, e)−1 min [Jc − J, max (|J − Jlc |, Jlc )] ,
(3.41)
where ft ≪ 1. After some experimenting we found that the simulations converge for ft ≤ 10−3 , which is the value we then used in our simulations.
3.6.5
Treatment of the loss cone
Stars with mass m and radius R⋆ on orbits which pass through the tidal radius
1/3
2M•
rt =
R⋆
(3.42)
m
are disrupted by the MBH. The loss cone is the region in angular momentum
space which delineates these orbits and is given by
p
Jlc = 2GM• rt .
(3.43)
Secular Stellar Dynamics
49
Our prescription for the time step, which is similar to that used in Shapiro
& Marchant (1978), ensures that stars always diffuse into the loss cone and
cannot jump over it. Stars are disrupted by the MBH when their orbital angular
momentum is smaller than the loss cone, and they pass through periapsis. Only
if floor(t/P ) − floor [(t − δt) /P ] > 0, do we consider the star to have crossed
periapsis during the last time step and hence to be tidally disrupted (or directly
consumed by the MBH in case of a compact remnant). In that case we record
the energy at which the star was disrupted and initialize a new star, which
starts at the time t at which the star was disrupted.
3.7 Results
3.7.1
Evolution to steady-state
We first consider a theoretical benchmark problem, that of the steady-state
distribution function of a single-mass population of stars around an MBH.
With m = 1M⊙ this solution appears after ∼ 100 Gyr (10 energy diffusion
timescales; see Appendix 3.B). In order to highlight the differences caused by
RR relative to NR, we define the function
g(E, J 2 ) ≡
E 5/4 d2 N (E, J 2 )
.
J 2 d ln Ed ln J 2
(3.44)
For a Bahcall & Wolf (1976) distribution without RR, this function is constant.
In Toonen et al. (2009), simulations similar to those presented here were used,
except that angular momentum relaxation was assumed to be NR. It was shown
that indeed g(E, J 2 ) is approximately constant for all (E, J) for the NR case;
see Figure A1 in that paper.
In Figure 3.17 we show a density plot of the function g(E, J 2 ) from our
steady-state simulations, which illustrates several interesting effects of RR. At
low energies, g(E, J 2 ) is constant which is to be expected since the precession
time is of similar magnitude as the orbital period, such that there are no coherent torques. At higher energies (E & 10), there are far fewer stars than
for a classical Bahcall & Wolf (1976) cusp. This is a consequence of enhanced
angular momentum relaxation, where stars are driven to the loss cone at a
higher rate than can be replenished by energy diffusion, and was anticipated
by one-dimensional Fokker-Planck calculations (Hopman & Alexander 2006,
see their Figure 2). At the highest energies considered (E > 100) most orbits
accumulate close to the loss cone, and there are a few orbits populated at larger
angular momenta, with a “desert” in between.
To further illustrate the reaction of the stellar orbits to RR, in Figure 3.18
we show the distribution of eccentricities of stars in slices of energy space. As
in Figure 3.17 we see that for higher energies the distribution is double peaked,
with most stars accumulating at the highest eccentricities, and another peak at
eccentricities around e = 0.2. This distribution can be understood as follows:
50
Secular Stellar Dynamics
RR is very effective at intermediate eccentricities, where torques are strong and
the coherence time is very long. As a result, the evolution of such eccentricities
occurs on a short timescale. At high eccentricities, the coherence time is short
due to GR precession, while at low eccentricities the torques are very weak.
At such eccentricities, evolution is very slow. The eccentricity therefore tends
to “stick” at those values, leading to the two peaks in the distribution. It is
interesting to note that N -body simulations by Rauch & Ingalls (1998) also
show a rise of the angular momentum distribution near the loss cone. This
result however may be affected by the new dynamical mechanism recently
described by Merritt et al. (2011).
In Figure 3.19 we show the rate of direct captures of stars by the MBH per
log energy. Loss cone theory (Frank & Rees 1976; Lightman & Shapiro 1977;
Cohn & Kulsrud 1978; Syer & Ulmer 1999; Magorrian & Tremaine 1999) yields
that the disruption rate continues to rise with decreasing energy for energies
larger than the “critical energy4 ”, which in our case is at E < 1. This is in
accordance with Figure 3.19. As the figure shows, the rate drops quickly; this
is due to the depletion of stars, and was also found (though to a lesser extent,
compare their Figure 4) in Hopman & Alexander (2006).
A limiting factor of our approach in this section is that the potential of
our Galactic nucleus does not evolve; hence we cannot look at collective effects (e.g., instabilities). Ideally an iterative process should be used to selfconsistently find the steady-state solution; we have shown that RR significantly
depletes stars at inner radii. To what extent this will affect our steady-state
solution is beyond the scope of the chapter. We note the RR time does not
formally depend on the stellar number density until such energies and angular
momenta that GR precession becomes important.
3.7.2
A depression in the Galactic center
Early studies of integrated starlight at the GC detected a dip in the CO absorption strength within 15′′ of the MBH, Sgr A* (Sellgren et al. 1990; Haller et al.
1996), which indicated a decrease in the density of old stars (see also Genzel et al. 1996, 2003; Figer et al. 2000, 2003; Schödel et al. 2007; Zhu et al.
2008). Recent observations by Do et al. (2009), Buchholz et al. (2009), and
Bartko et al. (2010) have confirmed this suggestion and, in particular, revealed
that the distribution of the late-type stars in the central region of the Galaxy is
very different than expected for a relaxed density cusp around an MBH.
Do et al. (2009) use the number density profile of late-type giants to examine the structure of the nuclear star cluster in the innermost 0.16 pc of the
GC. They find the surface stellar number density profile, Σ(R) ∝ R−Γ , is flat
with Γ = 0.26 ± 0.24 and rule out all values of α > 1.0 at a confidence level of
99.7%. The slope measurement cannot constrain whether there is a hole in the
4 The critical energy is approximately the energy where the change in angular momentum per
orbit equals the size of the loss cone.
Secular Stellar Dynamics
1
51
0.2
0.18
0.16
0.14
0.1
J2
0.12
0.1
0.08
0.01
0.06
0.04
0.02
0.001
0
1
10
100
E
1000
10000
Figure 3.17 – Two-dimensional distribution of stars, normalized such that for an isothermal Bahcall & Wolf (1976) profile the distribution should be constant (see Equation (3.44)). Note that
angular momentum is in units of the circular angular momentum, so that the loss cone (indicated
by the red line) is not constant. At high energies it is clear that there is a depletion of stars, and
that the distribution of angular momenta is far from isotropic.
6
E=3
E=10
E=30
E=100
E=300
E=1000
5
dN/de
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
e
Figure 3.18 – Eccentricity distribution at fixed energies. For high energies the distribution of
eccentricities is bimodal due to RR, whereas for low energies the distribution is isotropic with a
cutoff at the loss cone.
52
Secular Stellar Dynamics
1
0.1
-1
-1
N0 dNdisr/dlogE dt [Gyr ]
0.1
r [pc]
0.01
0.001
Disruption rate per star
0.01
0.001
0.0001
1e-05
1e-06
1
10
100
E
1000
10000
Figure 3.19 – Disruption rate per star per log E, in units of Gyr−1 , as a function of energy. The
rate drops quickly as the stellar distribution is depleted due to RR.
stellar distribution or simply a very shallow power law volume density profile.
Buchholz et al. (2009) show that the late-type density function is flat out to
10′′ with Γ = −0.7 ± 0.09 and inverts in the inner 6′′ , with power-law slope
Γ = 0.17 ± 0.09. They conclude that the stellar population in the innermost
∼ 0.2 pc is depleted of both bright and faint giants. These results are confirmed
by Bartko et al. (2010) who find that late-type stars with mk ≤ 15.5 have a flat
distribution inside of 10′′ .
This result has changed our perception of the nuclear star cluster at the
center of our Galaxy. Stellar cusp formation theory predicts a volume density profile n(r) ∝ r−α of relaxed stars with α = 7/4 for a single mass population (Bahcall & Wolf 1976) and has been confirmed in many theoretical
papers with different methods, including Fokker-Planck (e.g., Cohn & Kulsrud
1978; Murphy et al. 1991), MC (e.g., Shapiro & Marchant 1978; Freitag & Benz
2002), and N -body methods (Preto et al. 2004; Baumgardt et al. 2004; Preto
& Amaro-Seoane 2010). Theory also predicts that a multi-mass stellar cluster
will mass segregate to a differential distribution with the more massive populations being more centrally concentrated. The indices of the power-law density
profiles for the different mass populations can vary between 3/2 < α < 11/4
(Bahcall & Wolf 1977; Freitag et al. 2006; Alexander & Hopman 2009).
There have been two types of solutions put forward in the literature to
explain the discrepancy between cusp formation theory and observations of
the old stellar population in the GC. The first maintains that the stellar cusp
in the GC is relaxed as predicted, but additional physical mechanisms have
depleted the old stars in the central region. Such mechanisms include strong
mass segregation (Alexander & Hopman 2009) and envelope destruction of
giants by stellar collisions (see Dale et al. 2009, and references within). The
Secular Stellar Dynamics
53
second solution proposes an unrelaxed cusp, either by the depletion of the
stars due to the infall of an intermediate-mass black hole (IMBH; Levin 2006;
Baumgardt et al. 2006), or from a binary black hole merger (Merritt & Szell
2006; Merritt 2010). Kinematical information for the old stellar population in
the GC however shows a lack of evidence for any large-scale disturbance of
the old stellar cluster (Trippe et al. 2008), and Shen et al. (2010) find that the
Galaxy shows no observational sign that it suffered a major merger in the past
9 − 10 Gyr. In addition to this, the dynamics of the S-stars and of the MBH
greatly restricts the parameter space available for the presence of an IMBH in
the GC (Gualandris & Merritt 2009). For a summary see the latest review by
Genzel et al. (2010).
Here we propose that these observations can be explained in part by a depression of stars carved out by RR acting together with tidal disruption. The
underlying reason is that stars move rapidly in angular momentum space and
enter the loss cone at a higher rate than are replaced by energy diffusion. We
show that when we initialize the system as a cusp that, even though steadystate is reached in longer than a Hubble time, the depression forms on a shorter
timescale.
We perform simulations of stars, both for m = 1M⊙ and m = 10M⊙ ,
initially distributed in a Bahcall-Wolf (BW) cusp around the MBH, such that
dN/dE ∝ E −9/4 . We present first the results for m = 10M⊙ , chosen to reflect
the dominant stellar population at small radii. We plot the stellar distribution
function f (E), which scales as E 1/4 for BW cusp, at various times during the
simulation as the stars move in angular momentum space due to RR; see Figure 3.20. In ∼ 1 Gyr a depression is carved out of the population of stars,
significantly depleting the stellar population around the MBH out to ∼ 0.2 pc.
To conclusively demonstrate that RR is the reason for this depression, we also
show the distribution of stars for simulations in which we have switched RR
off (by setting φ1 = θ1 = 0, σ1 6= 0) such that angular momentum relaxation
follows a random walk. In this case no depression forms5 . These simulations
show that, as the cusp is destroyed on timescales of a few Gyr due to RR, a BW
cusp is not a solution to the distribution of stars around an MBH. We see the
same appearance of a depression when we substitute our non-resonant parameters (both energy and angular momentum) for those of Eilon et al. (2009).
In Figure 3.21, we plot the number density of stars as a function of radius
from the MBH. We do this by sampling each stellar orbit randomly in mean
anomaly. Black lines indicate the initial BW number density slope of −1.75 and
a slope of −1. The slope begins to deviate, albeit gently, from the BW solution
at ∼ 0.5 pc and decreases to −1 at ∼ 0.1 pc. We present results from m = 1M⊙
simulations in Figures 3.22 and 3.23 even though, due to mass segregation,
we do not expect low-mass stars to dominate the potential so close to an MBH.
5 The fact that the distribution moves away from the BW solution is attributed to a factor of
∼two discrepancy in our energy relaxation time due to the outer cut-off in radius in our N -body
simulations; see Appendix 3.B
54
Secular Stellar Dynamics
1
10
r [pc]
0.01
0.1
0.001
t=0
250 Myr
1 Gyr
2.5 Gyr
f(E)
1
0.1
0.01
0.001
1
10
100
log(E)
1000
10000
Figure 3.20 – Carving out of a depression in the distribution of stars around an MBH for simulations in which m = 10M⊙ . The broken orange line shows the stellar distribution at the start of the
simulation (the solid black line above indicates a slope of 1/4). The thick colored lines show the
results for RR while the corresponding (in color and style) thin lines show results for simulation
in which RR is switched off. The energy, or equivalently radius, at which RR becomes efficient at
carving out the depression is clearly seen on the scale of ∼ 0.1 pc.
r [’’]
0.01
0.1
1
1e+07
10
t=0
250 Myr
1 Gyr
2.5 Gyr
1e+06
100000
n(r)
10000
1000
100
10
1
0.0001
0.001
0.01
r [pc]
0.1
Figure 3.21 – Number density of stars as a function of radius (in parsec and arcsecond) from the
MBH for m = 10M⊙ simulations. The stellar distribution begins to deviate from the BW solution
at ∼ 0.5 pc. The black lines indicate a slope of −1.75 (top) and −1.0 (bottom right).
Secular Stellar Dynamics
1
10
r [pc]
0.01
0.1
55
0.001
t=0
1 Gyr
5 Gyr
10 Gyr
f(E)
1
0.1
0.01
0.001
1
10
100
log(E)
1000
10000
Figure 3.22 – Carving out of a depression in the distribution of stars around an MBH for simulations in which m = 1M⊙ . The timescale on which the depression develops is longer than for
simulations in which the potential is built up of 10M⊙ stars. Stars which have an energy E & 1000
survive longer than those with 100 & E & 1000 as RR is ineffective so close to the MBH due to
GR.
r [’’]
0.01
0.1
1
1e+07
10
t=0
1 Gyr
5 Gyr
10 Gyr
1e+06
100000
n(r)
10000
1000
100
10
1
0.0001
0.001
0.01
0.1
r [pc]
Figure 3.23 – Number density of stars as a function of radius (in parsec and arcsecond) from the
MBH for m = 1M⊙ simulations. The depression in stars takes an order of magnitude longer in
time to develop than the m = 10M⊙ simulations.
The major difference between the two simulations is the timescale on which
the depression forms, 1 Gyr and 10 Gyr, respectively.
To reproduce the radial extent of the depletion of old stars in the observations of Do et al. (2009), Buchholz et al. (2009), and Bartko et al. (2010),
the size of the hole in energy space must be significantly larger, by a factor of
56
Secular Stellar Dynamics
5 − 10, than what we find in our simulations6 . In addition, a hole in energy
space alone cannot give rise to an inverted profile in surface number density.
We stress however that the potential in our model does not evolve in response
to RR. As the depression develops and the number of stars at small radii decreases, the energy relaxation timescale, tNR ∝ N −1 , will increase. As N is
not important for the RR timescale for all energies but the largest (GR precession at small radii affects the precession rate), this could further enlarge the
depression as the rate of stars flowing inward toward the MBH decreases.
In Figure 3.24, we demonstrate the increase in the NR timescale in response
to the hole in stars. We use a fit to the stellar density profile at 1 Gyr from
Figure 3.21 to recalculate tNR , normalizing the profile such that the mass in
stars at the radius of influence is equal to that of the MBH. We plot the RR
timescale without GR precession as relativistic effects are unimportant at radii
greater than ∼ 10−2 ; see Figure 3.15. As the outer extent of the hole at ∼
10−1 pc is at a much larger distance from the MBH, the RR timescale at this
scale will not be affected by changes in N . The increase in the radius at which
tNR = tRR strongly suggests that the outer radius of the hole should increase
in response to the change in stellar potential.
One implication of this depression is that we expect no gravitational wave
bursts to be observed from the GC (Hopman et al. 2007). For the implications of the existence of a depletion of stars in the vicinity of an MBH on predicted event rates of gravitational wave inspirals of compact objects (EMRIs)
and bursts, see Merritt (2010).
3.7.3
Dynamical evolution of the S-stars
At a distance of ∼ 0.01 pc from the MBH in the GC, there is a cluster of young
stars known as the S-stars. This cluster has been the subject of extensive observational and theoretical work, for which we refer the reader to Alexander
(2005) and Genzel et al. (2010). The stars are spectroscopically consistent with
being late O- to early B-type dwarfs, implying that they have lifetimes that are
limited to between . 10 and 400 Myr (Ghez et al. 2003; Eisenhauer et al. 2005;
Martins et al. 2008). The upper constraint comes from the limit on the age of
a B-type main-sequence star, while the lower constraint comes from the spectroscopic ID of a specific star (S2/S0-2) (Ghez et al. 2003). One particularly
interesting question concerns the origin of these stars. There is a consensus
in the literature in that they cannot have formed at their current location due
to the tidal field of the MBH. Several scenarios incorporating the formation of
these stars further out from the MBH, and transport inward through dynamical
processes, have been proposed (e.g. Gould & Quillen 2003; Alexander & Livio
2004; Levin 2007; Perets et al. 2007; Löckmann et al. 2008, 2009; Madigan
et al. 2009; Merritt et al. 2009; Griv 2010).
6 The absence of a BW cusp in the GC can also simply be explained by the long timescale necessary for steady-state formation; see Appendix 3.B
Secular Stellar Dynamics
57
100000
10000
t [Myr]
1000
100
10
1
tNR cusp
tNR hole
tRR
0.1
0.001
0.01
0.1
1
r [pc]
Figure 3.24 – NR (energy) timescale changing in response to the stellar potential (m = 10M⊙ )
hole
as it develops from a BW cusp to one with a hole in energy space (from tcusp
NR to tNR ). The RR
timescale is plotted here without GR precession as M• /mP and, at large radii, does not depend
on the number of stars. Hence this timescale should remain constant above ∼ 10−2 pc as the NR
timescale increases. The radius at which tRR is equal to tNR moves outwards in response to the
changes and hence the outer radius of the hole in stars will increase.
Many leading hypotheses involve binary disruptions (Hills 1988, 1991). In
this scenario a close encounter between a binary system and an MBH leads to
an exchange interaction in which one of the stars is captured by the MBH, and
the other becomes unbound and leaves the system with a high velocity. For
capture, the periapsis rp of the binary’s orbit (with combined mass mbin and
semi-major axis abin ) must come within the tidal radius
rt =
2M•
mbin
1/3
abin .
(3.45)
The captured star will remain with a semi-major axis acap that scales as acap ∼
(M• /mbin )2/3 abin , and its eccentricity can be approximated as 1−e ∼ (mbin /M• )1/3 .
A unifying aspect of all proposed formation mechanisms is that none of
them lead directly to orbital characteristics that are in agreement with those of
the S-stars; in particular, the binary disruption mechanism leads to highly eccentric orbits. Since the local RR time is shorter than the maximum age of the
S-stars, it has been suggested that following the arrival of the S-stars on tightly
bound orbits to the MBH, a subsequent phase of RR has driven them to their
current distribution (Hopman & Alexander 2006; Levin 2007). RR evolution
of S-stars has been studied by Perets et al. (2009) by means of direct N -body
simulations. They find that high-eccentricity orbits can evolve within a short
time (20 Myr), to an eccentricity distribution that is statistically indistinguishable to that of the S-stars. They conclude that the S-star orbital parameters are
58
Secular Stellar Dynamics
consistent with a binary disruption formation scenario. However, these simulations do not include GR precession, which we have found to be important for
the angular momentum evolution of stars at these radii.
3.7.3.1 Simulations
We use our ARMA code, which includes GR precession, to study the evolution of
the S-star orbital parameters within the binary disruption context. We initialize
our test stars on high eccentricity orbits, e = 0.97 (mbin ∼ 20M⊙), with initial
semi-major axes corresponding to those reported by Gillessen et al. (2009)
for the S-stars. We assume a distance to SgrA* of 8.0 kpc (Eisenhauer et al.
2003; Ghez et al. 2008; Gillessen et al. 2009). For this distance, one arcsecond
corresponds to a distance of 0.0388 pc. We do not include S-stars that are
associated with the disk of O/WR stars at larger radii for two reasons. The
first is that they most likely originated from the disk with lower initial orbital
eccentricities than we simulate here. Second, RR is not the most effective
mechanism for angular momentum evolution at their radii; torques from stars
at the inner edge of the disk will dominate. We run the simulation one hundred
times for each test star to look statistically at the resulting distribution.
We simulate two different initial setups. The first, which we call the continuous scenario, assumes that the tidal disruption of binaries is an ongoing
process, as is most likely in the case considered by Perets et al. (2007), where
individual binaries are scattered into the loss cone by massive perturbers. The
same initial conditions also hold if low angular momentum binaries are generated as a result of triaxiality of the surrounding potential. For this study we
start evolving the stellar orbits at times that are randomly distributed between
[0, tmax ], and continue the simulation until the time t = tmax . In our second
setup, which we call the burst scenario, we consider a situation in which Sstars are formed at once on eccentric orbits at t = 0, and follow their evolution
until t = tmax ; this enables us to compare directly to the results of Perets
et al. (2009) in whose simulations all stars commence their evolution simultaneously. A burst of S-star formation, over a short period of a few Myr, could for
example result from an instability in an eccentric stellar disk (Madigan et al.
2009); we note that the instability is even more effective for a depressed cusp
(Madigan 2010).
We use two different models for the galactic nucleus. For each we recompute the ARMA parameters (φ, θ, σ) and tNR for the new profile and re-run
the MC simulations using Equations (3.39) and (3.40) with the updated values.
Model A is similar to the galactic nucleus model discussed in Section 3.2. The
stellar cusp has a density profile n(r) ∝ r−α with α = 1.75, and the number
of stars is normalized with N (< 0.01 pc) = 222 and m = 10. We chose this
model such that it faithfully reflects the dominant stellar content at a distance
of 0.01 pc, where RR is most important and where the S-stars are located. We
use Model B to simulate the response of the S-stars to a depression in the GC,
Secular Stellar Dynamics
59
Table 3.3 – S-star Parameters
Model A
tmax (Myr)
a
6
10
20
100
Burst
Continuouse
Burst
Continuous
Burst
Continuous
0.012
2e-4
.40
.47
.23
.09
0.147
0.002
.29
.40
.32
.17
0.007
0.05
.24
.33
.46
.29
5e-4
0.02
.07
.16
.72
.55
Burst
Continuous
Burst
Continuous
Burst
Continuous
0.002
1.5e-5
.45
.50
.17
.08
0.04
0.0002
.35
.47
.29
.16
0.100
0.006
.22
.38
.45
.26
0.0005
0.06
.07
.19
.70
.55
d
p
b
f > 0.97
DRc
Model B
pa
b
f > 0.97
DRc
Notes
a
b
c
d
e
Two-dimensional (a, e) Kolmogorov-Smirnov p-values.
Fraction of S-stars with e > 0.97 at end of simulation.
Disruption rate (fraction) of S-stars.
Stars initialized at t = 0.
Stars randomly initialized between [0, tmax ].
as found in the previous section. This model has a smaller density power-law
index of α = 0.5 (which is within the 90% boundaries of the bootstrapped
values for α in the GC as found by Merritt (2010)) and N (< 0.01 pc) = 148.
Less mass at small radii leads to slower mass precession, pushing to a lower
eccentricity the value at which mass precession cancels with GR precession;
see Figure 3.16 for an illustration of this effect where α = 1.75. Interestingly
for all values of N (< 0.01 pc) . 120, this eccentricity lies at ∼ 0.7 a location in
eccentricity space where we do not observe S-stars, which causes a flattening
of the observed cumulative distribution.
3.7.3.2 Results
For both models we find the cumulative eccentricity distribution of the stars at
different times and compare them to the observed distribution from Gillessen
et al. (2009); see Figures 3.25 and 3.26. In Table 3.3, we present results from
nonparametric two-dimensional, two-sample, Kolmogorov-Smirnov (2DKS) testing7 between the observed cumulative distribution and the simulations. The
two-sample KS test checks whether two data samples come from different distributions (low p values).
We also give the fraction of stars that have an eccentricity e > 0.97 at the
end of the simulation. The total disruption rate (DR) gives an indication of
how many progenitors would be needed to make up the population of S-stars
observed today.
7 We
use a version due to Fasano & Franceschini (1987). See also Peacock (1983).
60
Secular Stellar Dynamics
Model A: The best fit to the S-star observations are for the burst scenario
with tmax = 6 − 10 Myr. Greater values of tmax produce too many low eccentricity stars in our simulations; as the RR time is longer at these eccentricities
stars will tend to remain at these low values. In the continuous scenario the
simulations conclude with many more high eccentricity stars as they do not
have as much time to move away from their original eccentricities. Hence the
best-fit result for this scenario is tmax = 10 − 20 Myr. In both cases the 2DKS
p-values ≪ 1, suggesting that no simulated distribution matches very well the
observed. Both scenarios predict a large number (7 − 45%) of high eccentricity
stars with e > 0.97.
Model B: The best match to the observations is the burst scenario with
tmax = 20 Myr, though the continuous scenario has a relatively high p-value
at a time of tmax = 100 Myr. In both cases a longer tmax is preferred as this
model has less mass, and hence GR precession will dominate the precession
rate of these stars down to lower eccentricities, rendering RR less effective at
high eccentricities. Again the p-values are low in most cases due to the excess
of high eccentricity orbits relative to the observations.
For both models, simulations with tmax > 100 Myr produce too many low
eccentricity stars with respect to the observations. Simulations with tmax =
200 − 400 Myr have increasingly smaller p-values for this reason. The small
number of observed low eccentricity S-stars could in principle constrain the
time S-stars have spent at these radii (their “dynamical lifetimes”). Although
the S-stars can in theory have ages up to ∼ 400 Myr, we find that, in the binary
disruption context, their dynamical lifetimes at these radii are < 100 Myr. Furthermore, as noted by Perets et al. (2009), the lack of observed low eccentricity
S-stars is difficult to reconcile with formation scenarios which bring the stars
to these radii on initially low eccentricity orbits. The RR timescale trr is long
and stars will retain their initial values for long times.
For all values of tmax there is a clear discrepancy between the lack of high
eccentricity S-stars orbits observed and the results from both formation scenarios and models. To further illustrate this problem we show a scatter plot from
our simulations in Figure 3.27, of eccentricity e as a function of semi-major
axis a after 6 Myr. We over-plot the observed S-star parameters, indicating
separately those associated with the disk. It is clear that while the simulations can reproduce the range of (a, e) in which the S-stars are observed, our
theory over-predicts the amount of high eccentricity orbits with respect to the
observations. Such orbits at these radii are expected; stars with high orbital
eccentricities, although they experience large torques (Equation (4.6)), have
short coherence times (Equation (3.28)) and hence long RR times (Equation
(3.30)). Consequently, their angular momentum evolution is sluggish. At any
given time, we would expect to see a population of stars in this region. Therefore, we find it difficult to explain why so few high eccentricity S-stars have
been observed; the maximum is e = 0.963 ± 0.006 (S14; Gillessen et al. 2009);
e = 0.974 ± 0.016 (S0-16; Ghez et al. 2005).
Secular Stellar Dynamics
1
61
Observations
Thermal
6 Myr
10 Myr
20 Myr
100 Myr
0.8
N(<e)
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
e
Figure 3.25 – Cumulative distribution of eccentricities for the burst scenario in which the stars’
initial time is zero. The red step function is the observed cumulative eccentricity distribution of Sstars in the GC as found by Gillessen et al. (2009). The dashed orange line is a thermal distribution
N (< e) = e2 . The other lines are results from MC simulations, where the evolution time tmax and
eccentricity e are indicated in the legend. Thick (thin) lines are the results for Model A (B).
1
Observations
Thermal
6 Myr
10 Myr
20 Myr
100 Myr
0.8
N(<e)
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
e
Figure 3.26 – Cumulative distribution of eccentricities for the continuous scenario in which the
stars’ initial time t is randomly distributed between [0, tmax ]. The red step function is the observed
cumulative eccentricity distribution of S-stars in the GC from Gillessen et al. (2009). The dashed
orange line is a thermal distribution N (< e) = e2 . The other lines are results from MC simulations,
where the maximum evolution time tmax and initial eccentricity e are indicated in the legend.
Thick (thin) lines are the results for Model A (B).
62
Secular Stellar Dynamics
1
Simulation
S Stars
Disk Stars
0.8
e
0.6
0.4
0.2
0
0.001
0.01
0.1
1
a [pc]
Figure 3.27 – Scatter plot of eccentricity, e, of the S-stars, as a function of semi-major axis, a,
on a log scale for the Model B burst scenario. The small orange dots represent the results of our
simulations after 6 Myr, the blue squares are data from Gillessen et al. (2009) and the black open
circles are those S-stars consistent with being members of the CW disk.
An interesting solution to this problem is that there are S-stars with higher
orbital eccentricities but that their orbital elements have not yet been determined due to the observational detection limits for stellar accelerations.
Schödel et al. (2003) show that there is a bias against detecting high eccentricity orbits at the average S-star radius of 0.01 pc (see their Figure 14). Future observations will be able to reveal if this can account for the discrepancy
(Weinberg et al. 2005). The solution on the other hand may involve a new
stellar dynamical mechanism which is not accounted for in our ARMA model.
Merritt et al. (2011) have recently discovered a new dynamical barrier in angular momentum space, termed the “Schwarzschild barrier”, which can reflect
stars orbiting close to the MBH from very high eccentricity orbits (e > 0.99) to
less eccentric ones. This could well explain why not so many S-stars have been
observed on highly eccentric orbits as RR alone predicts.
3.8 Summary
We have used several techniques to study the angular momentum evolution
of stars orbiting MBHs, with a focus on secular effects. First, we carried out
N -body simulations to generate time series of angular momentum changes
for stars with various initial eccentricities. We quantified the evolution in an
ARMA(1, 1) model. We then used this model to generate new, much-longer
time series in our MC code, which enabled us to study the steady-state distribution of stellar orbits around an MBH, and the evolution of the angular
momentum distribution of possible progenitors of the S-stars.
Secular Stellar Dynamics
63
We have shown that the steady-state angular momentum distribution of
stars around an MBH is not isotropic when RR is a dominant relaxation mechanism. For the GC this corresponds to the innermost couple of tenths of parsecs. We note that Schödel et al. (2009) find a slight degree of anisotropy in
the velocity distribution of late-type stars in the GC within the innermost 6"
(∼ 0.2 pc). It is of particular interest to note that close to the MBH, we find
that the angular momentum distribution rises more steeply than a linear distribution, which may have consequences for the stability of the system. Tremaine
(2005) and Polyachenko et al. (2007) showed that if (1) the distribution rises
faster than linearly, and (2) the distribution of angular momenta vanishes at
J = 0, the system is only neutrally stable (or even unstable if it is flattened).
The second condition is also satisfied, because of the presence of the loss cone:
N = 0 for J < Jlc . In a later paper, Polyachenko et al. (2008) found that a
non-monotonic distribution of angular momenta is required for the system to
develop a “gravitational loss cone instability”. This condition is also satisfied
for a range of energies. An assessment of the importance and consequences of
these instabilities is beyond the scope of this work.
We have furthermore demonstrated the presence of a depression in the distribution of stars near MBHs due to resonant tidal disruption. This depression
forms on a timescale shorter than the Hubble time, even though a steady-state
solution may not yet be reached. While it is tempting to invoke this mechanism to explain the “hole” in late-type stars in the GC, we stress that our
fiducial galactic nucleus model, a single-mass BW distribution of stars around
an MBH, is greatly simplified compared to reality and produces an outer hole
radius which is too small to explain the observations. We expect the GC to be
far more complex, with ongoing star formation, multi-mass stellar populations,
mass segregation, physical collisions between stars and additional sources of
relaxation (e.g., the circumnuclear ring, the CW disk). Nevertheless, there is
merit to our approach: even with our simplified picture, we illustrate the importance of RR on stellar dynamics close to an MBH.
Finally we have shown that the (a, e) distribution of the S-stars does not
match that expected by the binary disruption scenario in which the stars start
their lives on highly eccentric orbits and evolve under the RR mechanism. We
surmise that there are a number of high eccentricity S-stars whose orbital parameters have not yet been derived, though the “Schwarzschild barrier” recently discovered by Merritt et al. (2011) may provided a dynamical solution
to this problem. We have confirmed the result by Perets et al. (2009) in that
it is unlikely that the S-stars were formed on low eccentricity orbits. We place
a constraint on the dynamical age of the S-stars of 100 Myr from the small
number of low eccentricity orbits observed.
Acknowledgments A.M. is supported by a TopTalent fellowship from the
Netherlands Organisation for Scientific Research (NWO), C.H. is supported by
a Veni fellowship from NWO, and Y.L. by a VIDI fellowship from NWO. We are
64
Secular Stellar Dynamics
very grateful to Atakan Gürkan for his significant contribution to this project.
A.M. thanks Andrei Beloborodov, Pau Amaro-Seoane and Eugene Vasiliev for
stimulating discussions. We thank Tal Alexander and Ben Bar-Or for their comments on the first draft of this chapter, particularly with respect to energy diffusion. Finally, we thank the anonymous referee for a comprehensive reading
of this chapter and insightful comments.
Secular Stellar Dynamics
65
3.A Description of N-body code
We have developed a new integrator, specifically designed for stellar dynamical
simulations in near-Keplerian potentials. A distinctive feature of this code is
the separation between “test” particles and “field” particles (see, e.g., Rauch
& Tremaine 1996). The field particles are not true N -body particles; they
move on Keplerian orbits which precess according to an analytic prescription
(see Equation (3.62)), and do not directly react to the gravitational potential
from their neighbors. Conversely, the test particles are true N -body particles,
moving on Keplerian orbits and precessing due to the gravitational potential
of all the other particles in the system. This reduces the force calculation from
O(N 2 ) → O(nN ), where N is the number of field particles and n is the number
of test particles in the simulation. Both test and field particles’ orbits precess
due to GR effects. The stellar orbits rotate in their planes at each time-step dt
GR
by an angle |δφ| = 2π(dt/tGR
prec ), with tprec given by Equation (3.26).
The main difference between our code and many others, is that instead of
considering perturbations on a star moving on a straight line with constant
velocity, we consider perturbations on a star moving on a precessing Kepler
ellipse. Our algorithm is based on a mixed-variable symplectic method (Wisdom & Holman 1991; Kinoshita et al. 1991; Saha & Tremaine 1992), so called
due to the switching of the coordinate system from Cartesian (in which the
perturbations to the stellar orbit are calculated) to one based on Kepler elements (to determine the influence of central Keplerian force). These forms of
symplectic integrator split the Hamiltonian, H, of the system into individually
integrable parts, namely a dominant Keplerian part due to the central object,
i.e., MBH, and a smaller perturbation part from the surrounding particles, i.e.,
stellar cluster,
H = HKepler + Hinteraction.
(3.46)
The algorithm incorporates the well-known leapfrog scheme where a single
time step (τ ) is composed of three components, kick (τ /2)-drift (τ )-kick (τ /2).
The kick stage corresponds to Hinteraction and produces a change of momenta
in fixed Cartesian coordinates. The drift stage describes motion under HKepler
and produces movement along unperturbed Kepler ellipses. To integrate along
these ellipses, we exploit the heavily-studied solutions to Kepler’s equation.
This is a less computationally efficient method than direct integration, but accurate to machine precision. This is of utmost importance in simulations involving RR as spurious precession due to build-up in orbital integration errors
will lead to erroneous results. The computation is performed in Kepler elements, using Gauss’ f and g functions. We use universal variables which allow
the particle to move smoothly between elliptical, hyperbolic, and parabolic orbits, making use of Stumpff functions. For details we refer the reader to Danby
(1992). For the simulations presented in this chapter, we use a fourth-order
integrator (Yoshida 1990) which is achieved by concatenating second order
leapfrog steps in the ratio (2 − 21/3 )−1 : −21/3 (2 − 21/3 )−1 : (2 − 21/3 )−1 .
66
Secular Stellar Dynamics
0.0008
0.0007
0.0006
δφ/φ
0.0005
0.0004
0.0003
0.0002
0.0001
0
0
0.1
0.2
0.3
t [tprec]
0.4
0.5
0.6
Figure 3.28 – Fractional difference between the theoretical value for apsidal precession and that
achieved by our integrator for a test star of eccentricity e = 0.99 as a function of time in units of
the precession time of the test-star. The angle through which the test star has precessed is given
by φ. The analytical formula is adjusted for the test star’s wandering in semi-major axis.
We use adaptive time stepping to resolve high eccentricity orbits at periapsis, and to accurately treat close encounters between particles, hence our
integrator is not symplectic. To reduce the energy errors incurred with the loss
of symplecticity, we implement time symmetry in the algorithm. These steps
must be sufficient to avoid artificial apsidal precession of stellar orbits, particularly at high eccentricity where Wisdom-Holman integrators are known to
perform poorly (Rauch & Holman 1999). To demonstrate this we compare the
rate of change, as a function of time, of the direction of the eccentricity vector
of a test star using our N -body integrator with our analytical prediction. Figure 3.28 shows our results in terms of the angle through which the test star has
precessed, φ, for e = 0.99, a = 0.01 pc orbits in the gravitational potential as
described in Section 3.2.1. We take the average of nine simulations for half a
precession time of the test star. Extrapolating to a full precession time we find
a fractional difference of δφ/φ = 0.00148 between our analytical estimate and
that of the test star precession rate. This verifies that the long RR timescale
at high eccentricities, as shown in Figure 3.3, is not a result of artificial rapid
apsidal precession. To further demonstrate the accuracy of the integrator with
respect to the apsidal precession rate, we run additional calibration simulations for high-eccentricity orbits (e = 0.96, 0.98) in order to compare their angular momentum relaxation timescale (calculated using Equation (3.5)) with
the formula we derive for RR in Equation (3.30). Figure 3.29 demonstrates the
agreement between increase of the RR timescale with eccentricity as retrieved
from the simulations and that expected from theory.
An additional feature of the code is the option to implement “reflective”
Secular Stellar Dynamics
10000
theory
∆t = 2000
∆t = 1000
∆t = 500
∆t = 200
1000
tJ [Myr]
67
100
10
1
0.95
0.96
0.97
0.98
0.99
1
e
Figure 3.29 – Timescale for angular momentum J changes of the order of the circular angular
momentum for massless test stars in our N -body simulations, presented on a log scale of time in
units of Myr; see Equation (3.6). We take several ∆t values to get order-of-magnitude estimates
for this timescale. Over-plotted in red is the theoretical expectation for this timescale based on
Equation (3.30).
walls, which bounce field particles back into the sphere of integration. Without these walls, particles on eccentric orbits nearing apoapsis move outside
the sphere of interest and alter the density profile of the background cluster
of particles, and hence the (stellar) potential. With the walls “on”, a particle
reaching the reflective boundary is reflected directly back but with a different
random initial location on the sphere, to avoid generating fixed orbital arcs in
the system which would artificially enhance RR. The reflective walls are most
useful in situations where we want to examine how a test particle reacts in a
defined environment with a specific background (stellar) potential. The prescription we use for the time step (τ ) is based on orbital phase, the collisional
timescale between particles and free-fall timescale between particles. We treat
gravitational softening between particles with the compact K2 kernel (Dehnen
2001). We have parallelized our code for shared-memory architecture on single
multiple-core processors using OPENMP.
3.B Energy evolution and cusp formation
Many different definitions of relaxation times are used in the literature. In
Section 3.2.2 we consider the energy diffusion timescale tE , defined as tE ≡
E 2 /DEE where DEE ≡ h(∆E)2 i/∆t is the energy diffusion coefficient. This is
the timescale for order unity steps in energy. In order to be explicit, we now
compare our timescale (see Figure 3.3 and Equation (3.5)) with a commonly
68
Secular Stellar Dynamics
used reference time for relaxation processes defined in Spitzer (1987):
tr =
2
vM
1
3 h(∆v|| )2 iv=vM
3
vM
.
= 0.065
(Gm)2 n ln Λ
(3.47)
We note that this is equivalent to the timescale given
√ by Binney & Tremaine
(2008) for a Maxwellian velocity distribution, vM = 3σ, where vM (r) is the
mean velocity of a star and σ(r) is the one-dimensional velocity dispersion of
the stellar distribution at radius r,
tr =
0.34σ 3
.
(Gm)2 n ln Λ
(3.48)
The velocity dispersion within a stellar cusp with power-law density profile
n(r) ∝ r−α can be calculated from the Jeans equation for spherical isotropic
systems as
Z r ′
1
dr M (r′ )n(r′ )
G
=
v2
(3.49)
σ 2 (r) ≡ −
n(r) 0
r′2
1+α c
where M (r) is the mass within a radius r and vc is the circular velocity. For the
comparison of tE and tr we take Keplerian parameters for the circular velocity
at a given radius such that
σ2 =
1 GM•
1+α r
(3.50)
(Alexander 2003), and set the Coulomb logarithm to Λ = M• /m (Bahcall &
Wolf 1976). For the cusp model used in our N -body simulations we find from
Equations (3.22) and (3.48)
2
E2
1 1
M•
tE ≡
= 0.26
P = 2.18 tr ,
(3.51)
DEE
m
N< ln Λ
with tr = 156 Myr, tE = 340 Myr for r = 0.01 pc. Thus with this definition, the
energy of a star changes by order unity in roughly twice a relaxation time.
We compare our result for the energy relaxation time to that found by
Eilon et al. (2009) using an entirely different integrator. Their N -body simulations incorporate a fifth-order Runge-Kutta algorithm with optional pairwise
KS regularization (Kustaanheimo & Stiefel 1965), without the need for gravitational softening.
√ We use their numerically derived value (see their Table 1)
of hαi ≡ hαΛ / ln Λi, averaged over all stars in the cluster, and their expresE
sion for the energy relaxation time TNR
= (M• /m)2 P/(N α2Λ ), and compare
with our Equation (3.22) with its own numerically derived parameter. We find
an energy relaxation timescale which is a factor of ∼ four longer than in their
Secular Stellar Dynamics
f(E)
10
69
α = 7/4
t = 1 tE
t = 3 tE
t = 10 tE
1
0.1
1
10
100
E
1000
10000
Figure 3.30 – Stellar distribution function f (E) for single mass stars as a function of energy E for
different times t = (1 − 10) tE , where tE is the Keplerian energy diffusion timescale evaluated at
E = 1. steady-state is reached within 10 tE , the solution of which is a BW cusp with α = 7/4,
represented by the dotted green line.
paper (i.e., their energy diffusion timescale is about half of tr ). Our maximum
radius at 0.03 pc means we do not simulate several “octaves” in energy space
which could contribute to energy relaxation. In a Keplerian cusp, the variance
in the velocity changes of a star, (∆v)2 , is not independent of radius r but scales
with the power-law density index α. In our simulations α = 7/4, and hence
the number of octaves from the cutoff radius, r = 0.03 pc, out to the radius of
influence, rh = 2.31 pc, can account for at most a factor of ∼ two difference
in tE . The inner cutoff radius at r = 0.003 pc contributes a smaller factor of
difference to the energy diffusion timescale.
We now consider the timescale for building a cusp due to energy diffusion.
For this purpose, we use our MC code without angular momentum evolution.
In Figure 3.30, we show the distribution function for several times, where the
system was initialized such that all stars start at E = 1, corresponding to half
the radius of influence. From this figure, we find that it takes about 10 tE (E =
1, r = 0.5 rh ) or 8 tE (r = rh ) to form a cusp. The result that it takes much
longer than tE is also implicitly found in Bahcall & Wolf (1976). They give an
expression for DEE (called c2 (E, t) in their paper); evaluating the pre-factor
in their expression due to only the contribution of bound stars in steady-state,
BW
one finds that c2 (E = 1) = 16/tBW
cusp , where tcusp is the reference time in which
Bahcall & Wolf (1976) find a cusp is formed.
Baumgardt et al. (2004) and Preto et al. (2004) were the first to show the
formation of a cusp in direct N -body simulations. We use Equations (3.48) and
(3.51) to calculate the cusp formation times in Table 1 of Preto et al. (2004)
70
Secular Stellar Dynamics
in terms of our energy diffusion time. We find that in their simulations a cusp
forms on a timescale of the order of 10 tE . We conclude that energy diffusion
proceeds in their simulations at a rate similar to ours. However, Preto et al.
(2004) state that a cusp forms within one relaxation time. These two statements, apparently contradictory, are reconciled in two ways. First, Preto et al.
(2004) define the relaxation time at a radius such that the mass in stars is equal
to twice that of the MBH. Second, they include the non-Keplerian potential in
their calculation of the relaxation time. It is valid to assume Keplerian parameters for the stellar velocity dispersion at the distances we are interested in close
to the MBH. However, neglecting the contribution of the surrounding stellar
mass leads to a significant numerical difference in the calculation of the relaxation time near the radius of influence compared with those which incorporate
the non-Keplerian potential.
To verify this, and to reconcile our conclusions, we adapt our definition of
the energy diffusion time by incorporating the non-Keplerian potential,
vc3
=
G[M• + M (< r)]
r
3/2
=
GM•
r
3/2 t′E = 1 + AE α−3
α−3
Mh
r3−α
1+
M• rh3−α
3/2
tE ,
3/2
, (3.52)
(3.53)
where A = 2
Mh /M• , Mh is the mass inside the radius of influence rh , t′E is
the new energy diffusion timescale and we have used the relation E = rh /2a.
The new energy diffusion coefficient is
′
DEE
= 1 + AE α−3
−3/2
DEE .
(3.54)
In Figure 3.31 we show the results of MC simulations in which the new
energy diffusion coefficient is used. The system was again initialized such that
all stars start at E = 1. A BW cusp (α = 7/4) forms in about one relaxation
time evaluated at E = 0.287, the energy at which Mh = 2M• . This result
emphasizes the importance of taking the stellar potential into account for the
relaxation timescale.
For a simplistic single mass (m = 1 M⊙ ) GC model (see Section 3.2.1), we
find that at the radius of influence the Keplerian energy diffusion timescale
is tE = 12 Gyr and a cusp forms in ∼ 100 Gyr. Alternatively we find from
Equation (3.53) a cusp formation time of t′E ∼ 90 Gyr. We stress that the
time for cusp formation is much longer than the Hubble time, even if we halve
it to account for the outer cutoff in radius in our N -body simulations which
suppresses the energy diffusion timescale. This is in contradiction to what is
often claimed; one cause of the discrepancy is that in, for example, Bahcall
& Wolf (1976) and Lightman & Shapiro (1977), energy diffusion proceeds at
a rate exceeding the relaxation rate by a factor > 10, in contrast to what we
find in our N -body simulations. We note here that if energy diffusion did in
Secular Stellar Dynamics
f(E)
10
71
α = 7/4
t = 0.3 tE'
t = 0.6 tE'
t = 1 tE'
1
0.1
1
10
100
E
1000
10000
Figure 3.31 – Stellar distribution function f (E) for single mass stars as a function of energy E for
different times t = (0.3 − 1) t′E , where t′E is the energy diffusion timescale which incorporates
the non-Keplerian stellar potential and is evaluated at E = 0.287, r = 4.023 pc. The stars start in
a delta function at E = 1 (and hence generate an artificial peak at this position) and form a cusp
within one relaxation time 1 t′E .
fact proceed at this higher rate, then the energy diffusion timescale for the Sstars in the GC would be on the order of ∼ 20 Myr which would have profound
consequences for the (E, J) evolution of the cluster. It thus appears that the GC
cannot have reached a steady-state (see also Merritt 2010), even though masssegregation enhances the rate of cusp formation by a factor of a few (Preto &
Amaro-Seoane 2010; Hopman & Madigan 2010).
3.C Precession due to power-law stellar cusp
The presence of a nuclear stellar cluster in the vicinity of an MBH adds a small
correction δU (r) to the Keplerian gravitational potential energy U = −α/r
of the system. As a consequence, paths of finite motion (i.e., orbits) are no
longer closed, and with each revolution the periapsis of a star is displaced
through a small angle δφ. Given a power-law density profile of the stellar
cluster n(r) ∝ r−α , we can calculate this angle using the formula
Z π
2
∂
r2 δU dφ ,
(3.55)
δφ =
∂L L 0
p
where L = GM• a (1 − e2 ) is the unnormalized angular momentum of a Keplerian orbit (Landau & Lifshitz 1969). We continue to use specific parameters
in the following equations. The stellar mass within radius r is
3−α
r
,
(3.56)
M (r) = M0
r0
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Secular Stellar Dynamics
M0 , r0 being proportionality constants which we normalize to the radius of
influence, r0 = rh , M0 = M• . Thus the potential felt by a massless particle due
to this enclosed mass is
Z r
δU = −
F (r′ )∂r′
0
(3.57)
GM• rhα−3 2−α
r
(α 6= 2).
=
(2 − α)
Inserting δU into Equation (3.55) yields
Z π
GM• rhα−3 ∂
2
4−α
δφ =
r
dφ .
(2 − α) ∂L L 0
(3.58)
We apply a change
L with normalized angular
√ of coordinates here, replacing
√
momentum J = 1 − e2 , such that L = GM• aJ. Expressing radius r in
terms of Kepler elements, semi-major axis a, eccentricity e and angle from
position of periapsis φ,
r=a
yields
J2
1 − e2
√
,
=a
1 + e cos φ
1 + 1 − J 2 cos φ
3−α
2
a
f (e, α)
(2 − α) rh
N< m
2
,
f (e, α)
=
(2 − α)
M•
(3.59)
δφ =
(3.60)
where m is the mass of a single star, N< is the number enclosed within radius
r and
4−α !
Z ∂
1 π
J2
√
f (e, α) =
dφ .
(3.61)
∂J J 0 1 + 1 − J 2 cos φ
The function f (e, α) is calculated numerically, and fit, for a given α. This returns a cluster precession time of
M•
−1
P
(a)
(α =
6 2),
(3.62)
tcl
=
π(2
−
α)f
(e,
α)
prec
N< m
where P (a) is the period of an orbit with semi-major axis a.
3.D Equations of ARMA(1,1) model
We will use Equations (3.8) and (3.9) in our calculations; for clarity we repeat
them here, dropping the label “1”,
∆Jt = φ∆Jt−1 + θǫt−1 + ǫt ,
hǫi = 0;
hǫt ǫs i = σ 2 δt,s .
(3.63)
(3.64)
Secular Stellar Dynamics
3.D.1
73
Variance
2
h∆Jt2 i = φ2 h∆Jt−1
i + 2φθh∆Jt−1 ǫt−1 i + 2φh∆Jt−1 ǫt i
+ θ2 hǫ2t−1 i + 2θhǫt−1 ǫt i + hǫ2t i
(3.65)
2
Using h∆Jt2 i = h∆Jt−1
i, expanding ∆Jt−1 in the second and third terms, and
2
applying hǫt ǫs i = σ δt,s yields
h∆Jt2 i(1 − φ2 ) = σ 2 (2φθ + θ2 + 1),
(3.66)
which returns Equation (3.10):
h∆Jt2 i =
3.D.2
1 + θ2 + 2θφ 2
σ .
1 − φ2
(3.67)
Autocorrelation function
The ACF for the ARMA model as described by Equation (3.63), is defined as
ρt =
h∆Js+t ∆Js i
h∆Jt2 i
(3.68)
(t > 0).
Expanding the numerator gives
h∆Js+t ∆Js i = φ2 h∆Js+t−1 ∆Js−1 i + φθh∆Js+t−1 ǫs−1 i
+ φh∆Js+t−1 ǫs i + φθh∆Js−1 ǫs+t−1 i
+ φh∆Js−1 ǫs+t i + θ2 hǫs+t−1 ǫs−1 i
+ θhǫs+t−1 ǫs i + θhǫs+t ǫs−1 i + hǫs ǫs+t i,
(3.69)
where h∆Js+t ∆Js i = φ2 h∆Js+t−1 ∆Js−1 i. Recursively expanding the ∆J terms,
and again using hǫt ǫs i = σ12 δt,s , reduces the fourth and fifth terms to zero, and
greatly simplifies the expression to
h∆Js+t ∆Js i =
σ 2 φt (θ + φ)(θ + 1/φ)
.
1 − φ2
Normalizing by the variance returns Equation (3.11):
θ/φ
t
ρt = φ 1 +
1 + (φ + θ)2 /(1 − φ2 )
3.D.3
(t > 0).
(3.70)
(3.71)
Variance at coherence time tφ
To calculate the variance at the coherence time, h∆Jφ2 i, we begin by calculating
the variance after some time t,
!+ Z Z
* t
t
t
X
2
=
hτ (t1 )τ (t2 )idt1 dt2
∆Jn
0
0
n=0
(3.72)
Z Z
t
= h∆Jt2 i
t
ρ(t1 −t2 ) dt1 dt2 .
0
0
74
Secular Stellar Dynamics
Here ρ(t1 −t2 ) is the ACF of the torque at (t1 − t2 ) > 0, normalized with
h∆Jt2 i to be comparable with Equation (3.11),
θ/φ
(t1 −t2 )
.
(3.73)
ρ(t1 −t2 ) = φ
1+
1 + (φ + θ)2 /(1 − φ2 )
From Equation (3.31) we can write
(t1 − t2 )
.
= exp −
tφ
(t1 −t2 )
φ
Inserting this into Equation (3.72), writing
θ/φ
A = h∆Jt2 i 1 +
1 + (φ + θ)2 /(1 − φ2 )
(3.74)
(3.75)
and taking an upper limit of t = tφ yields
h∆Jφ2 i
=
A
Z
tφ
dt1
0
=
σ2
Z
0
tφ
P
2
tφ
(t1 − t2 )
exp −
dt2
tφ
(θ + φ)(θ + 1/φ)
.
1 − φ2
where we have approximated (e + 1/e − 2) ≈ 1.
(3.76)
Secular Stellar Dynamics
75
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4
On the build-up of stars in the Galactic center
The central parsec of our galaxy is dense with stars of various ages. These
populations may well derive from different formation processes and carry
signatures of their origins. In this chapter we present a directly-observable
statistic, h, that uses kinematical stellar data to highlight high-eccentricity
populations. We perform N -body simulations to show how this statistic
may be used on intermediate-aged stars, ∼ 100 Myr, in the Galactic center
to distinguish between two specific scenarios for their formation: episodic
star formation in nuclear disks and tidal disruptions of binaries by the massive black hole. The most distinguishing feature between the two scenarios
is a disparate |h|-signature at large radii, p & 7′′ , where h|h|i = 0.12 ∼ 0.48
in the nuclear disk scenario and h|h|i = 0.1 ∼ 0.14 in the binary disruption
scenario. This signature is already detectable with current observational
techniques, and new data should soon distinguish between the two scenarios.
Ann-Marie Madigan et al. (2012)
To be submitted to The Astrophysical Journal
80
Build-up of Stars in the Galactic Center
4.1 Introduction
T
HE nuclear star cluster (NSC) in the central parsec of our galaxy is, in addition to the massive black hole (MBH) SgrA*, composed of stellar populations of various ages. The question arises as to where these stars originate
— in situ, for example in a compact fragmenting gas disk, or outside this central region proceeding dynamical transportation inward? The answer to this
question is relevant not only for piecing together the formation history of our
Galactic NSC, but for understanding the mass accumulation in extragalactic
NSCs in general, which is likely connected to the formation of MBHs at their
centers (Hopkins & Quataert 2010a,b).
Due to the long stellar relaxation timescales in galactic nuclei containing
MBHs, we expect their formation histories to be recorded in the morphology
and kinematics of the cluster and its stellar populations. Extragalactic nuclei
have been explored in this context using integral field spectroscopy and photometry, the results of which we briefly review below. In this chapter we focus
on our own Galactic center (GC). Due to its proximity, it is possible to constrain
formation scenarios using data from individual stellar phase space parameters,
rather than collective position/velocity moments.
4.1.1
Nuclear star cluster formation
There are two basic scenarios for the build-up of stars in NSCs. The first scenario involves the merger of multiple star clusters following their inspiral towards the centers of galaxies via dynamical friction with background stars. Initial research focused on globular clusters (Tremaine et al. 1975), but recently
Agarwal & Milosavljević (2011) have considered young massive star clusters
formed in the inner part of the galaxy disk. The second scenario is in situ formation in nuclear stellar disks as a result of gas migration into the center of
galaxies (Milosavljević 2004). The timescales for the inspiral of star clusters
in the first scenario and gas migration in the second tell us that both scenarios
are theoretically possible and the reality may well be a combination of both
mechanisms, each varying in importance depending on factors such as galaxy
type and redshift. However the ubiquity of nuclear stellar disks in the local
neighborhood (Milky Way; M31; NGC 4244; M32) lends weight to the in situ
scenario. Assuming we do not live in a special epoch, it suggests that such
structures occur with regularity and are important contributors to the stellar
mass in NSCs (e.g., Lauer et al. 1993; Tremaine 1995; Levin & Beloborodov
2003; Bender et al. 2005; Paumard et al. 2006; Chang et al. 2007; Lu et al.
2009; Bartko et al. 2010).
4.1.2
Observational constraints from extra-galactic nuclei
Observations of extragalactic NSCs frequently reveal, in addition to old, red
spherical bulges, thin, blue disks of young stars . 200 Myr in age (Seth et al.
Secular Stellar Dynamics
81
2006, 2008; Seth 2010), which support an in situ formation scenario for NSCs.
Seth et al. (2006) propose a model in which stars form episodically in compact
nuclear disks. As observed NSC disks all have populations younger than 1 Gyr,
there must exist a mechanism that disrupts the disk on such a timescale. The
authors propose that the stellar disk loses angular momentum or heats vertically within this time to form an older spheroidal structure. As two-body relaxation is too slow a process, we require a more rapid dynamical mechanism.
One possibility is torquing on the stellar disk from successive gas accretion
(Seth et al. 2006) or due to an inner molecular ring (e.g., the circumnuclear
disk in the GC; see Haas et al. 2011, and references therein), or, when formed,
from the resulting new stellar disk (this chapter, Nayakshin et al. 2006; Löckmann & Baumgardt 2009; Löckmann et al. 2009). The duty cycle in this model,
that is the period between star formation episodes, can be roughly estimated
knowing the mass in the currently observed disk and the total mass in the NSC
needed to build up over a Hubble time. Using nearby galactic NSCs to estimate
this period yields a range between ∼ 0.1 − 1 Gyr (see discussion in Seth et al.
2006). Further motivation for this model is found in the results of Hartmann
et al. (2011) who compare observations and dynamical models of NGC 4244
and M33 to results from N -body simulations of the formation and growth of
NSCs through the merger of star clusters. The authors confirm that NSCs cannot form from purely stellar dynamical mergers and that gas dissipation is
necessary for at least half of the stellar mass in extragalactic nuclei.
4.1.3
Evidence from the Galactic center
The Galactic NSC is unique among nuclei in that one can resolve individual
stars of different ages. The spectroscopic detection is currently limited in magnitude to mk ∼ 15 − 16. Early type stars in this magnitude range correspond
to O-/WR-/B- dwarfs, giants and supergiants, and late type stars to K-giants
and cool red supergiants (Bartko et al. 2010). While the late type stars may be
too old to retain memory of their initial orbital conditions, the kinematics of
the early type stars should, in some ways, reflect their original distribution. We
briefly discuss their possible origins below; for a comprehensive review of the
star formation history at the Galactic center see Genzel et al. (2010).
The ∼ 200 early type stars within the central parsec can be categorized
into two main groups. The first is a population of O- and Wolf-Rayet (WR)
stars, ∼ 6 ± 2 Myr old, with masses & 20M⊙ (Paumard et al. 2006). The
majority form a well-defined, warped disk with projected radii 0.8′′ −12′′ , which
rotates clockwise on the sky (Levin & Beloborodov 2003; Genzel et al. 2003; Lu
et al. 2009; Bartko et al. 2009); we will refer to this population as the “O/WR
disk”. The second population is an apparently isotropic (though a number may
be part of the O/WR disk) distribution of B-stars, mk > 15. Those that lie
within the central 0.8′′ are often collectively referred to as the “S-stars”; their
kinematics reveal randomly-inclined and highly eccentric orbits (Ghez et al.
2005; Gillessen et al. 2009). Their proximity to the MBH, which prohibits star
82
Build-up of Stars in the Galactic Center
formation due to its immense tidal force (Morris 1993), combined with their
young ages (< 10 Myr ∼ 100 Myr) imply a “paradox of youth” (Ghez et al.
2003; Eisenhauer et al. 2005; Martins et al. 2008). Currently, it is not clear
from the data whether or not the S-stars and the other B-stars in the central
half-parsec form two distinct populations.
While the inspiral and subsequent tidal disruption of a young star cluster can produce a disk-like configuration similar to the O/WR disk (Gerhard
2001; Gürkan & Rasio 2005), the theory suffers from inconsistencies with respect to observations. A short dynamical friction timescale is imposed by the
young ages of the stars, which it turn places implausible constraints on the initial location and mass of the cluster (Kim & Morris 2003)1 . The necessity for
an extremely dense cluster core invokes the presence of an intermediate-mass
black hole (Hansen & Milosavljević 2003; Gürkan et al. 2004; Levin et al. 2005;
Berukoff & Hansen 2006), but the formation of such an object in star clusters
is still debated (e.g., Glebbeek et al. 2009) and there are stringent mass and
orbital separation constraints from the MBH (Portegies Zwart et al. 2002; Yu
& Tremaine 2003; Hansen & Milosavljević 2003; Kim et al. 2004; Gualandris
& Merritt 2009). Finally, there is disparity of a factor of ∼ 20 between the
expected total X-ray luminosity of young, low-mass stars in the cluster and that
from Chandra observations (Nayakshin & Sunyaev 2005). For these reasons,
this scenario is disfavored for the recent build-up of stars in the GC, though
it may well be a relevant or even dominant component in some extragalactic
nuclei.
The leading theory for the origin of the O/WR disk is the in situ formation from fragmentation of in-falling or colliding gas clumps at the GC (Morris
1993; Sanders 1998; Levin & Beloborodov 2003; Nayakshin & Cuadra 2005;
Levin 2007; Wardle & Yusef-Zadeh 2008, 2011). The density profile, radial
extent and sharp inner edge of the disk (Lu et al. 2009; Bartko et al. 2010) is
reflected in simulations (Bonnell & Rice 2008; Hobbs & Nayakshin 2009). The
low levels of X-ray luminosity in the SgrA* field (Nayakshin & Sunyaev 2005)
can be explained by a top-heavy initial-mass function, a condition that has so
far been fulfilled observationally (Maness et al. 2007; Bartko et al. 2010) and
in simulations (Alexander et al. 2008) (but see Löckmann et al. 2010). Additional support comes from recent analysis of Fermi-LAT data, which reveal
two large gamma-ray bubbles symmetric about the Galactic plane, extending
50◦ above and below the GC (Su et al. 2010). The authors propose that they
were born in a large episode of energy injection in the GC – an accretion event
on to SgrA*, or a nuclear starburst in the last ∼ 10 Myr, the latter being coincident with the formation of the O/WR disk, as explored by Zubovas et al.
(2011). Assuming, as evidence suggests, the O/WR disk formed in situ around
the MBH, it is not unreasonable to expect that previous epochs of gas migration
and nuclear stellar disk formation took place in the GC.
1 A problem exacerbated if the core in the GC exists for the bulk of the mass in stars (Antonini
& Merritt 2011).
Secular Stellar Dynamics
83
The origin of the B-stars however, is much less clear than that of the O/WR
disk. Their ages are ambiguous, and range from a spectroscopically confirmed
< 10 Myr for the S-star S2/S0-2 (Ghez et al. 2003; Eisenhauer et al. 2005;
Martins et al. 2008), to an upper-limit of ∼ 100 Myr on the main sequence of
the lower-mass B-stars. This upper limit does not preclude the B-stars having
formed with the O/WR stars, but they may well be from an older population or
even form a continuous distribution in age. To discuss various possible formation mechanisms, we must first distinguish between studies on the S-stars and
those involving the outermost B-stars.
One of the leading theories for the formation of the S-stars is the tidal capture by SgrA* of close B-star binaries by Hills mechanism (Hills 1988, 1991;
Gould & Quillen 2003). The most plausible theory for populating these infalling binaries is dynamical relaxation by massive perturbers such as giant
molecular clouds (Perets et al. 2007) within the central 10 −100 pc, although
viable alternatives certainly exist (e.g., Madigan et al. 2009; Löckmann et al.
2009). The challenge in the binary disruption scenario is that the S-stars are
captured on orbits of very high eccentricity and yet we observe them to have
a near-thermal distribution (Gillessen et al. 2009). Two studies have investigated this issue: direct N -body simulations but without general relativistic
precession by Perets et al. (2009), and our own study (Madigan et al. 2011)
utilizing a Monte Carlo “ARMA” code with general relativistic precession. Perets
et al. (2009) find that resonant relaxation is sufficiently rapid to randomize
the orbital eccentricities from high-eccentricity values to those observed within
∼ 20 Myr, while we find that GR precession undermines this process close to the
MBH and leaves the stars with too high eccentricities as compared with observations. Further study is necessary to see if the newly identified “Schwarzschild
barrier” will resolve this issue (Merritt et al. 2011).
It is attractive to assume that the B-stars outside the central arcsecond have
also formed via Hills mechanism, as proposed by Perets & Gualandris (2010).
However in this case, as the authors point out, the high eccentricity issue is
acute, since neither two-body nor resonant relaxation will be able to significantly change the orbital eccentricities of the B-stars at large radii within the
range of their lifetimes. This sets up a prediction which observational data
from this population will be able to verify or refute.
In this chapter we 1) point out that there is another mechanism for dynamically relaxing the eccentricity distribution – the formation and presence of the
O/WR stellar disk, and 2) devise a statistic which is particularly sensitive to
identifying, or ruling out, stellar distributions with high orbital eccentricities.
We investigate two scenarios – the massive perturber plus binary disruption
scenario (Perets et al. 2007), and one based on the model of Seth et al. (2006),
wherein we propose that the B-stars formed in a nuclear stellar disk ∼ 100 Myr
ago – and examine the resulting orbital parameters of the B-stars after 6 Myr
of interaction with the O/WR disk. The chapter is organized as follows. In
Section 4.2 we discuss the two formation scenarios in detail and describe our
84
Build-up of Stars in the Galactic Center
suite of N -body simulations. We show that the new statistic can robustly differentiate between the two scenarios for the origin of the B-stars in the GC in
Section 4.3. We summarize our study in Section 4.4.
4.2 Numerical methods and scenarios
Here we contrast two main formation scenarios for the intermediate-aged population of B-stars in the Galactic center. We perform N -body simulations to
investigate whether their original orbital eccentricities will be preserved over
6 Myr, having been subjected to torquing from the O/WR disk. We use our
special-purpose N -body integrator, which is described in detail in Madigan
et al. (2011). Our integrator is based on a mixed-variable symplectic algorithm (Wisdom & Holman 1991; Kinoshita et al. 1991; Saha & Tremaine 1992)
and designed to accurately integrate the equation of motion of a particle in
a near-Keplerian potential with minimum spurious precession. In the simulations for this study, we use direct N -body particles which move in Kepler
elements along ellipses under the influence of the central object, and calculate perturbations to their orbits in Cartesian co-ordinates from surrounding
N -body particles (Danby 1992). We define semi major axis, a, and eccentricity,
e, of stellar orbits with respect to a stationary MBH in our simulations,
a=−
GM•
,
2E
(4.1)
and
e= 1−
J2
GM• a
1/2
,
(4.2)
where M• is the mass of the MBH, and J and E are the specific orbital angular
momentum and energy of a star. The periapse of the orbits of these stellar
particles precess with retrograde motion due to Newtonian mass precession
from the additional smooth potential from surrounding cluster of stars. The
time to precess by 2π radians is
tcl
prec = π(2 − α)
M•
P (a)f (e, α),
N (< a)m
(4.3)
where N (< a) is the number of stars within a given a, m is the mass of a
single star, f (e, α) is a function which depends on the eccentricity of the orbit
and the power-law index α of the surrounding cluster of stars, and P (a) =
2π(a3 /GM• )1/2 is the orbital period of a star with semi-major axis a. We
include the first post-Newtonian general relativistic effect, that is prograde
apisidal precession with a timescale,
tGR
prec =
ac2
1
P (a).
(1 − e2 )
3
GM•
(4.4)
Secular Stellar Dynamics
85
Our simulations have four main components chosen to represent the Galactic center ∼ 6 Myr in the past. The first is a MBH of mass M• = 4 × 106 M⊙
(Ghez et al. 2008). Second is a smooth stellar cusp with power-law density
profile n(r) ∝ r−α , α ∈ [0.5, 1.75], normalized with a mass of 1.5 × 106 M⊙
within 1 pc (Schödel et al. 2007; Trippe et al. 2008; Schödel et al. 2009). We
choose to use a smooth potential for the cusp in our simulations as this greatly
decreases the required computation time. Traditional two-body gravitational
relaxation has no impact on the stars since its characteristic timescale is greater
than 5 Gyr at all radii (see figure 3 from Merritt 2010). Resonant relaxation
(Rauch & Tremaine 1996) occurs on a shorter timescale, and we model it explicitly to use in our initial conditions (see Figure 4.1) using the ARMA code
described in detail in Madigan et al. (2011). The numerical simulations of
Löckmann & Baumgardt (2009) show that even with resonant relaxation, the
cusp contributes relatively little to relaxation of stellar disks as compared with
disk stars themselves, and Madigan et al. (2009) find that in the case of coherently eccentric disks, significant angular momentum changes can occur on
timescales less than 1 Myr. Hence we are confident that choosing a smooth
stellar cusp over a grainy one will not greatly affect our results.
The third component in our simulations is the most dynamically important
structure: an eccentric stellar disk (e = 0.3, 0.6) with surface density profile
Σ(a) ∝ a−2 , made up of Ndisk = 100 equal-mass N -body particles with total
mass Mdisk = 104 M⊙ and semi-major axes 0.03 pc ≤ a ≤ 0.5 pc. This disk
represents the O/WR disk. In our basic model, the O/WR disk is formed instantaneously, i.e., fully formed at t = 0, with an inclination opening angle of
1◦ . However, we also run a number of simulations where we model its formation using a “switch-on” multiplicative function for the disk mass, such that the
mass of a single star is
t − t0
Mdisk
,
(4.5)
tanh
m(t) =
Ndisk
τ
where t0 is −1 years (so that the O/WR disk has mass at t = 0) and the growth
timescale τ is 1 × 105 years (Bonnell & Rice 2008). Secular gravitational interactions with the O/WR disk will affect the angular momenta of the B-stars. We
anticipate the greatest eccentricity change for stars at similar radii to the inner
edge of the disk – the torques are much greater at these radii, as
GMdisk edisk e
τ∼
δφ,
(4.6)
a
where τ is specific torque on a stellar orbit, edisk is a typical eccentricity of a
star in the disk and δφ is the angle between them.
The fourth component of our simulations, the B-stars, depends on the scenario we are simulating; we describe them both here.
Dissolved Disk (DD) Scenario: In this scenario, the B-stars form in-situ around
the MBH in a nuclear stellar disk during an earlier episode (∼ 100 Myr) of
86
Build-up of Stars in the Galactic Center
gas infall and fragmentation: we call this the dissolved disk scenario as the
disk structure should have ‘puffed up’ due, mainly, to vector resonant relaxation. The disk should not be entirely dissipated within 100 Myr however –
two-body relaxation acts on much longer timescales and vector resonant relaxation is suppressed in high-frequency, short-wavelength modes (Kocsis &
Tremaine 2010).
We simulate the B-stars in this scenario as a stellar disk with semi-major
axes between 0.03 pc ≤ a ≤ 0.7 pc, consisting of 100 N -body particles with
equal masses of 100M⊙. The density profile is Σ(a) ∝ a−2 while the initial
eccentricities and opening angle of the disk vary (the basic model draws from a
thermal eccentricity distribution), and the inclination with respect to the O/WR
is systematically explored. These initial conditions are crude out of necessity –
we have a poor understanding of the true initial parameters – but nevertheless
we are able to capture the essential dynamics of the interaction with the O/WR
disk.
Binary disruption (BD) Scenario: This formation scenario invokes enhanced
relaxation from massive perturbers (Perets et al. 2007) which propel binaries
from outside the central parsec onto near-radial orbits where they can be disrupted by the MBH near periapsis via Hills’ mechanism. Thus, in this scenario
the B-stars are members of binaries that have been left bound to the MBH as
their partners were ejected into the halo at high velocities. They have a spatially isotropic distribution, represented by 100 N -body particles with masses
100M⊙ distributed between 0.03 pc ≤ a ≤ 0.7 pc. They are initialized with a
density profile Σ(a) ∝ a−2 and begin their lives in the Galactic center on Kepler orbits with very high eccentricities as e = 1 − rt /a ∼ (mbin /M• )1/3 (see,
e.g., Pfahl 2005), where rt = abin (M• /mbin )1/3 is the tidal radius of the MBH,
and mbin and abin are the mass and semi-major axis of the binary.
We use our ARMA code (see Madigan et al. 2011, for details) to simulate
the evolution of orbital eccentricities for B-stars in the BD scenario without the
dynamical influence of the O/WR disk but including two-body and resonant relaxation. In doing so, we confirm the result by Perets & Gualandris (2010) that
stellar orbits remain at very high eccentricities outside ∼ 0.1 pc. We find higher
mean eccentricity values at all radii however, for both steep (α = 1.75) and
shallow (α = 0.5) cusp profiles; see Figure 4.1 in which we plot mean orbital
eccentricities as a function of semi-major axis at 60 and 100 Myr. We simulate
both a burst scenario in which all B-stars begin at t = 0, and a continuous scenario in which they are randomly initialized between t = 0 and t = tmax ; the
latter best reflects binary disruptions due to massive perturbers but the former
can be directly compared with the simulations of Perets & Gualandris (2010).
As the authors use full N -body simulations, they do not include the entire stellar cusp and hence precession rates for stars are lower than in our simulations,
contributing to a higher resonant relaxation rate and hence larger eccentricity
changes. We use a fit of the resulting eccentricity distribution after 100 Myr in
the continuous case as our initial conditions for our BD scenario.
Secular Stellar Dynamics
87
1
0.95
<e>
0.9
0.85
60 Myr burst
100 Myr burst
60 Myr continuous
100 Myr continuous
0.8
0.75
0.01
0.1
1
a (pc)
Figure 4.1 – Mean B-star orbital eccentricities as a function of semi-major axis in the BD scenario.
Stars begin with initial eccentricity e = 0.98 and evolve due to resonant and non-resonant relaxation over 60 and 100 Myr. In the burst scenario, stars are initialized at t = 0; in the continuous
scenario, they are initialized randomly between t = 0, tmax . For this plot we use our ARMA code
(Madigan et al. 2011) with M (< 1 pc) = 1.5 × 106 M⊙ and α = 1.75.
Our simulation runs are organized to explore a wide range of parameters
relevant for the GC; see Table 4.1 for specifics of each run. An important variation in the models is the stellar cusp index of which we choose two values
– a steep stellar cusp α = 1.75, and a shallow stellar cusp α = 0.5 – the latter motivated by recent observations of the old stellar population in the GC
(Buchholz et al. 2009; Do et al. 2009; Bartko et al. 2010; Yusef-Zadeh et al.
2011). This variation reveals itself in the precession rate of stars at different
radii, and hence the persistence of stellar torques. Another difference is the
mean eccentricity of the B-stars and the O/WR disk stars, which will affect the
results through the strength of the torques between the two groups.
In addition, we run simulations with both coherently eccentric (or lopsided)
O/WR disks where the eccentricity vectors of the stellar orbits overlap (Bonnell
& Rice 2008; Hopkins & Quataert 2010a,b), and non-coherent ones. The DD
simulations vary in the opening angle of the B-star disk and the inclination angle between the two disks. The relative orientation of O/WR disk is important
as B-star orbits like to orient themselves with this new disk.
A more realistic stellar mass function for the O/WR and B-star populations
should not have an impact on our results; due to the equivalence principle,
the stars will respond to the gravitational potential in the same way, regardless
of their mass. A reduction in the average mass in stars however, with a corresponding increase in their number, will reduce the coherence of the stellar
torques – in this case the B-star population will retain memory of their initial
conditions over an even longer timescale.
88
Build-up of Stars in the Galactic Center
Table 4.1 – Model parameters
Model1
hei2
α3
hei4O/WR
sw 5
θ6
hii7
h|h|i8
BD1
BD2
BD3
BD4
BD5
BD6
0.97
0.97
0.97
0.97
0.93
0.93
0.5
0.5
0.5
0.5
1.75
1.75
0.6 c
0.6 c
0.6
0.3 c
0.6 c
0.3 c
0
1
0
0
0
0
-
-
0.1
0.1
0.11
0.1
0.12
0.14
DD1
DD2
DD3
DD4
DD5
DD6
DD7
DD8
DD9
DD10
0.6
0.6
0.6
0.6
0.6
0.3
0.6
0.6
0.6
0.6
0.5
0.5
0.5
0.5
0.5
0.5
1.75
0.5
0.5
0.5
0.6 c
0.6 c
0.6 c
0.6 c
0.6
0.6 c
0.6 c
0.6 c
0.6 c
0.3 c
0
0
0
0
0
0
0
0
1
0
40
40
40
40
40
40
40
60
40
40
80
25
121
163
80
80
80
80
80
80
0.15
0.28
0.48
0.18
0.12
0.43
0.23
0.29
0.14
0.19
Notes
1
2
3
4
5
6
7
8
BD: binary disruption. DD: dissolved disk.
Mean initial eccentricity of B-stars across all radii.
Index of power-law density cusp profile.
Mean initial eccentricity of O/WR stars. c: coherent disk.
Switch-on function on/off (1/0).
Initial opening angle B-star disk in arcdeg.
Mean initial inclination of B-star disk with respect to
O/WR disk in arcdeg.
Mean absolute value of h for B-stars with p > 7′′ at end of
simulation.
4.3 Results
4.3.1
Evolution of orbital eccentricities
We run the simulations for 6 Myr to follow the orbital vector and scalar angular
momentum change in the B-stars in response to the stellar disk. The B-stars
with small semi major axes not only experience a greater torque, τ ∝ 1/a, but
have lower angular momentum, J, and hence the percentage change in angular
momentum is high. This brings about a rapid change in their eccentricities. In
contrast, B-stars with large semi-major axes experience smaller torques, less
percentage change in J and retain memory of their initial orbital eccentricities.
Hence these large a stars can constrain their formation scenario. This is as
Perets & Gualandris (2010) found for resonant relaxation, but the change in
eccentricities of low-radii B-stars over 6 Myr under the persistent torques of
the O/WR disk proves to be more rapid than resonant relaxation over 100 Myr.
We show this effect in Figure 4.2, plotting the eccentricity, e, of B-star orbits
as a function of semi major axis, a, for different times during the simulation.
The BD scenario produces the most dramatic signature as stars with large semi
Secular Stellar Dynamics
89
1
0.9
0.8
0.7
<e>
0.6
0.5
0.4
0.3
0.2
0.1
DD
0
0
0.05
0.1
0.15
0.2
a (pc)
0.25
0.3
0.35
0.4
1
0.8
<e>
0.6
0.4
0.2
BD
0
0
0.05
0.1
0.15
0.2
0.25 0.3
a (pc)
0.35
0.4
0.45
0.5
Figure 4.2 – Mean eccentricity and one standard deviation of B-stars after 6 Myr of evolution in
the DD6 scenario (top) and BD1 scenario (bottom) as a function of semi major axis a. Stars at
large a do not move far from their original eccentricity values. In particular, all stars in the BD
scenario with a > 0.26 remain almost fixed with e > 0.96.
major axes retain their high orbital eccentricities. In the DD scenario, the Bstars with large semi major axes have similar eccentricities as their initial values
and so are highly sensitive to their initial conditions.
For the B-stars outside of the central arcsecond, only a very small portion of
their orbits has been mapped out, sufficient to determine their “instantaneous”
proper motion but insufficient to determine their eccentricities and other orbital parameters. In the next subsection we develop a statistic which uses the
star’s sky position and proper motion velocities and is particularly sensitive in
identifying distributions with high orbital eccentricity.
90
4.3.2
Build-up of Stars in the Galactic Center
The high-eccentricity statistic
The basic idea for identifying stars with high eccentricities is straightforward:
the radial orbit in three spacial dimensions also appears as a radial orbit in
projection on the sky. This was noted by Genzel et al. (2003) and revisited by
Paumard et al. (2006) and Bartko et al. (2009). These authors have utilized
the j versus p diagram, where j is the normalized angular momentum along
the line-of-sight (z-axis),
jz
jz(max)
xvy − yvx
.
=
pvp
j=
(4.7)
Here vx , vy are the right ascension and declination velocities of a star at (x, y)
on the sky,
vp = (vx2 + vy2 )1/2
(4.8)
is the projected velocity (i.e., proper motion), and
p = (x2 + y 2 )1/2
(4.9)
is the projected radius from the MBH. The quantity j is ∼ 1, ∼ 0, ∼ −1 depending on whether the stellar orbit projected on the sky is mainly clockwise (CW)
tangential, radial, or counterclockwise (CCW) tangential with respect to the
projected radius. To compare different stellar populations, Genzel et al. (2003)
and Paumard et al. (2006) define three j ranges: CW tangential (j ≥ 0.6), CCW
tangential (j ≤ −0.6) and radial (−0.3 ≥ j ≤ 0.3).
Here we show that the quantity j is not the one that is most sensitive to
the high-e stellar distributions. This is because stars on radial orbits spend
the majority of their orbital period near apoapsis, and hence can have low vp
with respect to the circular velocity at their projected radii p; this increases
their value of j and imparts a more tangential orbit in projection. Instead, we
propose to use the high-eccentricity statistic, h: jz normalized to the maximum
angular momentum at that radius (i.e. taking circular velocity at p):
jz
Jp
xvy − yvx
= √
,
GM• p
h=
(4.10)
where we have used the Keplerian circular velocity such that vcirc (p) = (GM• /p)1/2
(i.e., we do not include the stellar mass within radius p).
We contrast the j and h versus p diagrams for both scenarios in Figure 4.3.
To directly compare with observations, we work in units of arcsecond using
Secular Stellar Dynamics
DD
j
BD
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.1
h
91
1
10
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
0.1
1
0.1
1
10
-1
t=0
6 Myr
-1.5
0.1
1
p (arcsec)
10
t=0
6 Myr
-1.5
10
p (arcsec)
Figure 4.3 – (Top) Distribution of projected, normalized orbital angular momentum on the sky,
j, for stars in the DD6 scenario (left) and BD1 scenario (right) at t = 0, 6 Myr. In plotting the
parameter j we lose evidence of high eccentricity orbits as stars on near-radial orbits spend most
of their orbital period near apoapsis and their vp value will be lower than the circular velocity at
their projected radius p. (Bottom) Distribution of projected orbital angular momentum on the sky
normalized to the maximum angular momentum at that projected radius p, for stars in the DD6
scenario (left) and BD1 scenario (right) at t = 0, 6 Myr. This new statistic, h, illuminates high
eccentricity orbits by focusing them at zero.
the conversion 1′′ = 0.0388 pc which corresponds to a distance to SgrA* of
8.0 kpc (Eisenhauer et al. 2003; Ghez et al. 2008; Gillessen et al. 2009). The
advantage of the h-parameter is not obvious in the DD scenario as there are
few very-high eccentricity orbits. However, in the BD scenario, the B-stars at
large radii retain their high eccentricities over the 6 Myr simulation, and their
h values are centered about zero in contrast with the j-parameter.
We plot the mean value of |h| as a function of projected radius p at the beginning of the simulation and after 6 Myr in Figure 4.4. The most distinguishing feature between the two scenarios is the disparate |h|-signature at large
radii, p > 7′′ , where h|h|i ∼ 0.12−0.48 in the DD scenario and h|h|i ∼ 0.1−0.14
in the BD scenario. These results are dependent on binning but the qualitative
trend is not, and they can be directly compared to observations using a χ2 analysis which can take observational errors into account. The mean values of |h|
for stars with large projected radii, p > 7′′ , are listed in Table 4.1.
Simulations BD1−4 (shallow cusp, α = 0.5) produce B-stars with higher
eccentricities, and hence lower mean |h| values, at large radii in comparison
92
Build-up of Stars in the Galactic Center
0.9
DD5
DD6
DD7
DD8
DD9
DD10
0.8
0.7
< |h| >
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
1
10
p (arcsec)
0.9
BD1
BD2
BD3
BD4
BD5
BD6
0.8
0.7
< |h| >
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
1
10
p (arcsec)
Figure 4.4 – Mean absolute h values as a function of projected distance, p, for several variations of
the DD scenario (top) and BD scenario (bottom). The DD results are dependent on disk orientation
angle with respect to the line of sight. All variants of the BD scenario produce low |h| values at
large projected radii, the most distinguishable feature between the two scenarios.
with simulations BD5−6 (steep cusp, α = 1.75); the stellar orbits of the latter, at large radii, precess relatively slowly and secular changes in J are more
efficient as torques can act on the stellar orbits over a longer time. The BD3
simulation with a non-coherent O/WR disk has lower |h| values at low radii
than the others. This is expected as the stellar torques at the inner edge of the
disk will not be aligned and hence be weaker. This is also seen in DD5 in which
the O/WR disk is non-coherent. The DD6 simulation, in which the B-stars are
initialized in an e = 0.3 disk, and DD8 with a larger opening angle in the B-star
disk, produce higher |h| values for a similar reason. The simulations employing
the “switch-on” function show no differences with respect to the basic models.
The inclination angle between the two disks plays an important role in deter-
Secular Stellar Dynamics
0.9
93
DD1
DD2
DD3
DD4
0.8
0.7
< |h| >
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
1
10
p (arcsec)
Figure 4.5 – Mean absolute h values as a function of projected distance, p, for four variations of
the DD scenario which vary only in inclination angle between the O/WR disk and B-star disk. The
DD results are dependent on disk orientation angle with respect to the line of sight.
1
0.9
0.8
N (<|h|)
0.7
0.6
0.5
0.4
0.3
BD
DD
Thermal e
Lu et al. (2009)
Bartko et al. (2009)
0.2
0.1
0
0
0.2
0.4
0.6
|h|
0.8
1
1.2
Figure 4.6 – Cumulative |h| plots for the DD6 scenario and BD1 scenario with corresponding
values from a thermal eccentricity distribution as a comparison. Also plotted are the values for the
O/WR disk (Lu et al. 2009; Bartko et al. 2010). The DD results, though dependent on orientation
of the B-star disk plane, are qualitatively similar regardless of viewing angle.
mining the resulting |h| values and we plot DD1−4 in Figure 4.5. In DD1,2 ,
where the initial mean inclination angle between the two disks is hii < 90◦ ,
the B-star angular momentum vectors tend to overlap with those of the O/WR
94
Build-up of Stars in the Galactic Center
disk by the end of the simulation. In DD3,4 , where the initial mean inclination
angle between the two disks is 90◦ < hii < 180◦ , the two disks are still distinguishable from each other after 6 Myr. In all DD cases, a substantial fraction of
the B-stars are in an isotropic distribution at the end of the simulation.
We plot the cumulative |h| values in Figure 4.6 for representative examples
of each scenario, DD6 and BD1 , with a thermal distribution of eccentricities for comparison. The BD scenario produces more low |h| orbits than the
DD scenario. If we have the observed distribution for B-stars, we can perform a two-dimensional Kolmogorov-Smirnov (2DKS) test, a nonparametric
test which checks whether two data samples come from different distributions
(see e.g., Peacock 1983), against the simulated cumulative distributions. We
also present the cumulative |h| distributions for the O/WR stars from data in
Bartko et al. (2009) and Lu et al. (2009). Though the data sample given in
these two papers overlap, they are not identical and differences are observed;
the stars in the Lu et al. (2009) data sample have lower |h| values and hence are
more eccentric. The observational data include errors which are normally distributed, which tends to artificially “isotropize” their stellar distribution. This
issue will have to be addressed when considering real data from B-stars.
4.4 Summary
In this chapter we compare two formation scenarios of the B-stars in the GC;
the dissolved disk scenario based on the model proposed by Seth et al. (2006),
and the binary disruption scenario due to enhanced stellar relaxation from massive perturbers by Perets et al. (2007). We specifically investigate the change
in the B-star orbital parameters after dynamical gravitational interaction with
the O/WR disk over 6 Myr, concentrating on those at large radii where torques
from stellar orbits are weakest. We formulate a new, directly-observable statistic, h, which uses the position of stars on the sky (x, y) and their proper motion
(vx , vy ), and show that it is particularly sensitive to high-eccentricity orbits.
The results from our N -body simulations show that we can constrain the
formation scenario, in particular the high-eccentricity signature of the binary
disruption model. Non-parametric testing between the observed cumulative
distribution function of h and those simulated here may be used to compare
the two. Distinguishing between the scenarios based on the h-statistic is already possible with current observational techniques, and new data should
soon exclude one or both formation mechanisms.
If the dissolved disk scenario best fits the observations, this will inform us
about the build-up of stellar mass in the GC, and the h values of the stars at
large radii will help constrain the initial orbital parameters of the stars in the
disk. An additional feature to look for in this case is the outer cut-off in radius
of the B-stars. If the binary disruption scenario best fits the data, the formation
mechanism of the S-stars and the rest of the B-stars in the GC will be united.
Secular Stellar Dynamics
95
Acknowledgments A.-M.M. is supported by a TopTalent fellowship from
the Netherlands Organisation for Scientific Research (NWO), and thanks Monash
University for their hospitality during her visit. Y.L. is supported by a VIDI fellowship from NWO. We thank Clovis Hopman for useful discussions, and for
encouraging an economy of words in this chapter.
96
Build-up of Stars in the Galactic Center
4.A Statistical constraints on eccentricity from the h-statistic
1.5
1
1
0.8
0.5
e
j
0.6
0
0.4
-0.5
0.2
-1
-1.5
0
0.1
1
p (arcsec)
10
1.5
1
1
0.8
0.5
e
h
0.6
0
0.4
-0.5
0.2
-1
-1.5
0
0.1
1
p (arcsec)
10
Figure 4.7 – j (top) and h (bottom) values of stars as a function of projected radius, p, color-coded
according to orbital eccentricity, e. These plots illustrate the sensitivity of the h-parameter to higheccentricity orbits and demonstrate how it can be used to constrain the dynamical origin of stars
in the GC.
For a given eccentricity distribution we can plot the corresponding j and h
values and see how well they constrain the original eccentricity distribution.
We do so here for a thermal distribution of eccentricities, initializing one thousand B-star orbits with cumulative distribution function f (e) = e2 , and calculating their j and h values. Figure 4.7 shows both the j and h planes populated
by stars colored according to their orbital eccentricities. These plots visualize
the fidelity with which the h parameter, in contrast to the j parameter, differentiates between tangential vs radial orbits; near-radial orbits are confined to
Secular Stellar Dynamics
97
1
0.9
0.8
N (< e)
0.7
0.6
|h| < 0.1
|h| < 0.3
|h| < 0.6
|h| > 0.1
|h| > 0.3
|h| > 0.6
Thermal
0.5
0.4
0.3
0.2
0.1
0.02
0
-0.02
-0.04
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
e
Figure 4.8 – Cumulative plots of stellar eccentricity for different slices of |h|. The original distribution in this case has a thermal eccentricity profile, f (e) = e2 , also plotted for comparison. Below
are plotted the residuals from the simulated cumulative distribution functions and the fits to the
function in Equation 4.11.
low |h| values.
We plot the cumulative distribution of orbital eccentricities in specific |h|
bins in Figure 4.8. Our goal is to place statistical limits on the eccentricities
of stars using the h plot. For example, 50% of stars with |h| < 0.1(0.3) have
an eccentricity e > 0.87(0.84), while 90% of stars with |h| > 0.6 have an eccentricity e < 0.73. These statistics are dependent on the initial eccentricity
distribution. As neither the DD nor the BD scenario produce a thermal eccentricity distribution after 6 Myr, we cannot directly use these limits, but these
plots illustrate the usefulness of the h parameter and can used to constrain
older stellar populations which are expected to be isotropized.
We fit the cumulative e distributions shown in Figure 4.8 with the following
98
Build-up of Stars in the Galactic Center
Table 4.2 – Fits of e-distributions in Figure 4.8 to Equation (4.11)
|h|1
< 0.1
< 0.3
< 0.6
> 0.1
> 0.3
> 0.6
µ
σ2
β
γ
δ
ǫ
0.96
1.06
1.66
0.34
0.40
0.51
0.17
0.01
0.16
0.13
0.13
0.05
−1.06
4.22
5.43
5.75
6.95
-2.80
−1.22
0.15
0.09
0.29
1.35
−0.54
1.20
−0.16
−0.10
−0.45
−1.48
0.51
1.10
−4.09
−5.26
−6.52
−7.51
3.15
Notes
1
Absolute value of h corresponding to distributions in
Figure 4.8.
formula
x−µ
1
1 + erf √
f (x) =
2
2σ 2
p
+ βx + γ 1 − x2 + δ cos(x) + ǫ sin(x),
(4.11)
where µ and σ 2 are the mean and variance of a cumulative Gaussian distribution, β, γ, δ and ǫ are coefficients and x = e. The trigonometric terms are
chosen ad-hoc; they are necessary to fit the extreme distributions |h| > 0.6,
|h| < 0.1. The fitted values for each line in Figure 4.8 are listed in Table 4.2;
the residuals from subtracting the functions from the distributions are plotted
in the lower panel.
Secular Stellar Dynamics
99
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5
Secular Dynamical Anti-Friction in Galactic Nuclei
We identify a gravitational-dynamical process in near-Keplerian potentials
of galactic nuclei that occurs when an intermediate-mass black hole (IMBH)
is migrating on an eccentric orbit through the stellar cluster towards the
central supermassive black hole (SMBH). We find that, apart from the conventional dynamical friction, the IMBH experiences an often much stronger
systematic torque due to the secular (i.e., orbit-averaged) interactions with
the cluster’s stars. The force which results in this torque is applied, counterintuitively, in the same direction as the IMBH’s precession and we refer to
its action as “secular-dynamical anti-friction” (SDAF). We argue that SDAF,
and not the gravitational ejection of stars, is responsible for the IMBH’s
eccentricity increase as seen in previous N -body simulations. Our numerical experiments, supported by qualitative arguments, demonstrate that (1)
when the IMBH’s precession direction is artificially reversed, the torque
changes sign as well, which decreases the orbital eccentricity, and (2) the
rate of eccentricity growth is sensitive to the IMBH migration rate, with
zero systematic eccentricity growth for an IMBH whose orbit is artificially
prevented from inward migration.
Ann-Marie Madigan and Yuri Levin (2012)
To be submitted to The Astrophysical Journal
104
Secular Dynamical Anti-Friction
5.1 Introduction
R
ECENT studies on secular dynamics in near-Keplerian potentials (Rauch &
Tremaine 1996; Hopman & Alexander 2006; Eilon et al. 2009; Kocsis &
Tremaine 2011; Madigan et al. 2011) have focused attention on the long-term
dynamical evolution of stars and compact objects in galactic nuclei. Due to
persistent gravitational torques exerted by stellar orbits on each other as they
precess slowly around the supermassive black hole (SMBH), the angular momentum evolution of stellar orbits can proceed at a much higher rate than that
of energy evolution (resonant relaxation; Rauch & Tremaine 1996). In this
chapter we aim to extend such research to study secular dynamical effects on
the inspiral of a massive body, e.g., an intermediate-mass black hole (IMBH),
into the combined potential of an SMBH and nuclear stellar cluster.
As a body of mass M moves through a distribution of field stars of individual mass ma , the influence of accumulative encounters with the field stars
on its orbit may be described using the velocity diffusion coefficients in the
Fokker-Planck equation (e.g., Binney & Tremaine 2008). If M ≫ ma , the firstorder diffusion coefficient (or drift term, D[∆v|| ]) will be much larger than the
second-order coefficients (diffusion terms, D[(∆v⊥ )2 ]/v, D[(∆v|| )2 ]/v), by a
factor of order M/ma . The dominant effect of the lower mass stars is to exert
a frictional force on the massive body anti-parallel to its velocity v,
Z v
v
dv
2 2
2
2
(5.1)
= −16π G M ma ln Λ
dva va f (va ) 3 .
M
dt
v
0
This is Chandrasekhar’s dynamical friction formula (Chandrasekhar 1943; Tremaine
& Weinberg 1984) for
R an isotropic field star distribution normalized such that
the density n(r) = d3 va f (r, va ), and ln Λ is the Coulomb logarithm. The
formula accounts only for stars moving slower than the massive body, and it
neglects self-gravity of the stars themselves; Antonini & Merritt (2011) have recently updated the formula to include changes induced by the massive body on
the stellar velocity distribution and the contribution from stars moving faster.
The stars are deflected by the massive body into a gravitational ‘wake’ behind
it which results in an increase in stellar density, the amplitude of which is proportional to M (Danby & Camm 1957; Kalnajs 1971; Mulder 1983); hence
the gravitational force experienced by the massive body is proportional to M 2 .
This frictional force acts to decrease the kinetic energy and angular momentum
of the massive body and it sinks towards the centre of the potential.
The picture described above does not take into account the orbit-averaged
dynamics that is particular for near-Keplerian potentials and that has been
shown to play a central role in the angular momentum evolution of stars near
the SMBH. Indeed, it is natural to assume that some other form of dynamical
friction could be associated with the orbit-averaged potential created by the
IMBH, as this potential is rotating around the SMBH due to precession of the
IMBH’s eccentric orbit. The possibility of this secular-dynamical friction was already suggested by Rauch & Tremaine in 1996. In this chapter we explore this
Secular Stellar Dynamics
105
effect and find that, contrary to our experience with ordinary dynamical friction, the resulting torque comes from a gravitational force acting in the same
direction as that of the IMBH precession. We therefore refer to this effect as
“secular-dynamical anti-friction” (SDAF).
For the remainder of this chapter we will keep referring to the massive inspiraling body as an IMBH, though the reader should keep in mind that the
essential dynamics will hold for any massive body moving in a near-Keplerian
potential. As large mass ratio massive black hole binaries are expected to coalesce in at least some merging and non-merging galaxies (Milosavljević & Merritt 2003; Preto et al. 2011), the dynamics of such systems are important to
understand. In particular, IMBH inspirals into SMBHs will be a major source of
gravitational waves for space-based laser interferometers such as the proposed
New Gravitational Observatory (ELISA/NGO )1 (e.g., Miller 2005). Simulations of the inspiral of IMBHs (102 − 104 M⊙ ) through nuclear stellar clusters
embedding a massive black hole have shown an increase in eccentricity of the
IMBH (Baumgardt et al. 2006; Matsubayashi et al. 2007; Löckmann et al. 2008;
Iwasawa et al. 2011; Sesana et al. 2011; Antonini & Merritt 2011; Meiron
& Laor 2011). Here we show that the theory of SDAF can explain this phenomenon, and present results of N -body simulations set up to test this claim.
5.2 Secular dynamical anti-friction
Let us consider the combined gravitational potential of an SMBH, mass M• ,
embedded in a nuclear star cluster with a power-law number density distribution, n(r) ∝ r−α , and individual stellar masses m. The specific energy and
angular momentum of a stellar orbit can be written in terms of Kepler elements
as
GM•
,
(5.2)
E=−
2a
|J| = |r × v| = [GM• a(1 − e2 )]1/2 ,
(5.3)
where a is the semi major axis of the elliptical orbit, and e is the eccentricity.
The eccentricity vector, e, of the star is that which points from the occupied
focus of the orbit to the periapsis of the orbit and has a magnitude equal to the
scalar eccentricity of the orbit,
e=
1
(v × J) − r̂.
GM•
(5.4)
The orbit of a star in this potential is nearly Keplerian (i.e. closed) but e will
precess due to the additional (Newtonian) potential and general relativity. We
define the precession time, tprec , as a timescale over which an orbit precesses by
2π rad in its plane. As Newtonian precession acts with retrograde motion (i.e.,
1 https://lisa-light.aei.mpg.de/bin/view/
106
Secular Dynamical Anti-Friction
in the direction opposite of a star’s motion) and general relativistic precession
with prograde motion,
−1
1
1 (5.5)
tprec = cl − GR .
tprec
tprec
The general relativistic precession time is given by
a(1 − e2 )c2
P (a),
(5.6)
3GM•
p
wtih the orbital period P (a) = 2π a3 /GM• . The Newtonian precession time
due to the addition of the star cluster mass in an otherwise-Keplerian potential,
tcl
prec , can be written for α 6= 2 as
M•
cl
−1
P (a) ,
(5.7)
tprec = π(2−α)f (e, α)
N< m
tGR
prec =
where N< is the number of stars within semi-major axis a, and
∂
f (e, α) =
∂J
1
J
Z
π
0
J2
√
1 + 1 − J 2 cos φ
4−α
!
dφ
(5.8)
(Landau & Lifshitz 1969). Here J = J/(GM• a)1/2 = (1 − e2 )1/2 . For all values
of α, the Newtonian precession time of a stellar orbit increases with eccentricity. The difference in precession times for circular and highly eccentric stellar
orbits vary with α. We plot tprec in Figure 5.1, setting the quantity in brackets [M• /(N< m)P (a)] in Equation (5.7) equal to one, with f (e, α) as given in
Madigan et al. (2011). For a steep cusp, α = 1.75 (Bahcall & Wolf 1976), there
is a factor of ∼ 5 difference in precession time for a near-circular (e = 0.01)
and near-radial (e = 0.99) orbit, which increases to ∼ 7 for a shallow profile
α = 0.5. We plot Equation (13) from Merritt et al. (2011) and Equation (A11)
from Ivanov et al. (2005) (α = 1.5) as a comparison. Differences arise in the
normalization of the functions due to approximations made in deriving the analytical formulae in these papers; the monotonic dependence of the precession
time on orbital eccentricity persists. We note that the exact expressions for the
precession time in Ivanov et al. (2005) agrees with numerical evaluation of our
Equation (5.7).
We now introduce an IMBH of mass Mimbh on an orbit with non-zero eccentricity, eimbh, and semi major axis, aimbh , within the dynamical radius, rh , of
the SMBH, defined such that the mass in stars at rh equals that of the SMBH,
N (< rh )m = M• . As we are interested in secular dynamics2 , it is useful to
envisage the mass of the IMBH spread smoothly out along its orbit, such that
2 The term ‘secular’ refers to long-period dynamics, in which the dependence on the mean longitude of an orbit is dropped from the disturbing function (see e.g., Murray & Dermott 1999).
Secular Stellar Dynamics
Newtonian precession time
30
107
α = 1.75
α = 1.5
α = 1.0
α = 0.5
M11
I05
10
1
0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
e
Figure 5.1 – Dependence of the precession time (M• /(N< m)P (a) = 1) of a stellar orbit on orbital
eccentricity, e. Several values are plotted for different power-law density profiles, n(r) ∝ r −α . The
dependence given in Merritt & Vasiliev (2011) (M11) and Ivanov et al. (2005) (I05) are plotted
for comparison.
the local density at any segment is inversely proportional to its local velocity.
The position of furthest distance from the SMBH, the apoapsis of the orbit, will
therefore contain the most mass. Over a timescale longer than its orbital period
but much less than the precession time, it will exert a strong (specific) gravitational torque on the orbit of a star with a similar semi major axis (Gürkan &
Hopman 2007) on the order of
|τ | = |r × F| ∼
GMimbh eimbhe
δφ,
aimbh
(5.9)
where δφ is the angle between two orbits. The orbit of the IMBH will not
affect the energy (or equivalently, the semi major axis) of the stellar orbit as its
gravitational potential is stationary over this timescale. A star on a co-planar
orbit with the IMBH will experience a torque that is parallel (or anti-parallel)
to its angular momentum vector J. Hence the torque will act to change the
magnitude of J; equivalently, the eccentricity of the orbit. Let us consider the
stellar orbit with an eccentricity e ∼ eimbh and semi major axis a ∼ aimbh . We
place the orbit such that its eccentricity vector forms an angle δφ with that of
the IMBH.
If there existed no gravitational interaction (secular or otherwise) between
the IMBH and the star, they would precess in the background stellar potential
at the same rate. Their eccentricity vectors would rotate by 2π rad in one tprec
and δφ would remain constant. Switching on secular gravitational interactions
results in a strong secular encounter between the star and the IMBH. The star’s
orbit is pushed away from the IMBH, transferring negative or positive angular
108
Secular Dynamical Anti-Friction
Figure 5.2 – (Top) Schematic of orbits of IMBH (black, filled line) and prograde trailing (orange,
dashed line) and prograde leading (blue, dotted line) stellar orbits in the frame of reference rotating with the IMBH orbit. The velocity vectors of each are indicated at apoapsis and the direction
of precession of the IMBH is shown with a curved arrow at top left. (Bottom) Schematic of orbits
of IMBH (black, filled line) and retrograde trailing (orange, dashed line) and retrograde leading
stellar orbits (blue, dotted line) in the frame of reference rotating with the IMBH orbit.
momentum to the orbit of the IMBH, depending on its orientation. We elaborate on the dynamics in the following:
If the star is moving on a prograde orbit (in the direction of motion of the
IMBH) that trails behind the IMBH (see Figure 5.2: top), it will experience a
torque in the direction of its angular momentum vector J such that dJ/dt > 0.
Due to the resulting decrease in eccentricity, the stellar orbit will precess faster
(see Figure 5.1) and move away from the IMBH3 .
Conversely if the stellar orbit is leading, the torque exerted on it will be
3 This mechanism involves the same interplay between stellar torques and orbital precession
which results in an instability in an eccentric disk of stars near a SMBH (Madigan et al. 2009).
Secular Stellar Dynamics
109
anti-parallel to J. The stellar orbit will increase in eccentricity, precess more
slowly and in effect be pushed away from the orbit of the IMBH. In both cases,
d|δφ|/dt > 0.
For a star moving along a retrograde orbit (see Figure 5.2: bottom), its precession acts in the opposite direction to that of the IMBH. If the orbit is trailing
behind the IMBH, the torque will act to increase its eccentricity and hence decrease the differential precession rate between the two orbits. This increases
the timescale over which the stellar orbit experiences a coherent torque and
its eccentricity can increase to one. In this scenario the orbit can easily ‘flip’
through e = 1 to a prograde orbit, and be pushed away from the IMBH as
described above. If if does not flip, the retrograde orbit will precess past the
IMBH, experience a torque which will decrease its eccentricity and hence will
increase the differential precession rate between the two orbits. Once again, in
both cases the stellar orbit is pushed away from that of the IMBH, d|δφ|/dt > 0.
In contrast to Chandrasekhar’s dynamical friction, which acts to increase
the density of stars behind a massive body, the effect of SDAF is to push stellar
orbits away from that of the massive body. The varying strength of Chandrasekhar’s dynamical friction on different segments of a massive body’s orbit
causes evolution of orbital eccentricity, the sign of which is dependent on the
background stellar density profile (e.g., Gould & Quillen 2003; Levin et al.
2005). With SDAF, it is the relative number of prograde/retrograde stellar
orbits which determines the sign of the orbital eccentricity evolution of the
IMBH.
5.2.1
Increase in orbital eccentricity
The differential precession rate between the IMBH and co-planar prograde orbits at similar semi major axes is small in comparison with retrograde orbits, as
the latter precess in the opposite direction to the IMBH. Hence, the IMBH orbit
has a higher rate of strong interactions with retrograde orbits (on its trailing
side) than prograde orbits. Therefore, in contrast with Chandrasekhar’s dynamical friction where the frictional force acts against its orbital motion, the
net gravitational torque experienced by the IMBH orbit comes from a force
which acts along its direction of precession. Due to dynamical friction the
IMBH sinks towards the center of the potential, decreasing in semi major axis.
As it sinks, the retrograde orbits at similar semi-major axes donate negative
angular momentum to the IMBH orbit, and its orbital eccentricity increases.
5.2.2
Comparison with theories in the literature
The N -body simulations of Iwasawa et al. (2011) show an increase in the eccentricity of an IMBH as it spirals into an SMBH. With careful analysis of the
orbital properties of the field stars in their simulations, the authors attribute
the increase to secular chaos (and the Kozai mechanism), brought about by the
non-axisymmetric potential induced by the IMBH. This acts to bring the relative
110
Secular Dynamical Anti-Friction
number of orbits back into balance after prograde-moving stars are preferentially ejected by the IMBH (retrograde stars have larger relative velocities with
respect to the IMBH than prograde, and hence are ejected less frequently). The
retrograde orbits moving to prograde orbits extract angular momentum from
the orbit of the IMBH and its eccentricity increases.
Sesana et al. (2011) present both hybrid4 and direct N -body simulations
with stellar cusps of varying degrees of rotation, to examine the importance of
the relative number of prograde/retrograde orbits on the orbital evolution of
the IMBH. The authors propose a hypothesis in which the ejections of stars, via
the slingshot mechanism, is the cause of the evolution of the IMBH eccentricity.
Most retrograde orbits which end up being ejected from the potential, donate
their negative angular momentum to the IMBH and move to prograde orbits
before they escape. Again it is the preferential ejection of prograde orbits that
drives the increase of the orbital eccentricity of the IMBH.
With SDAF, it is the cumulative secular torques from retrograde stellar orbits that lead to the increase in orbital eccentricity of an IMBH, irrespective of
ejections of stars from the potential. SDAF makes testable predictions which
can distinguish itself from the above theories.
1. If the orbit of the IMBH is made to artificially precess in the opposite
direction to its true motion, its rate of change of angular momentum
should reverse sign as well.
2. If escaping stars are injected back into the cluster with the same parameters as before their interaction with the IMBH, the eccentricity evolution
of the IMBH should not qualitatively be affected.
3. Due to the ‘pushing away’ of co-planar stellar orbits, there will be a density anisotropy in the stellar distribution as a result of SDAF.
4. If dynamical friction is artificially slowed, there will be a lower rate of
axisymmetric flow of retrograde orbits to one side of the IMBH’s orbit
and hence a lower negative rate of change of angular momentum.
In the following section we present the results of numerical trials to test the
theory of SDAF and decrease of orbital angular momentum of an IMBH in a
galactic nucleus.
5.3 Numerical method and simulations
The simulations presented in this chapter are performed using a special-purpose,
mixed-variable symplectic N -body integrator as described in detail in Madigan
et al. (2011). We use a galactic nucleus model with parameters chosen to represent the Galactic center. At the center of the coordinate system, there is a
massive black hole with M• = 4 × 106 M⊙ (Ghez et al. 2003), and an IMBH
with a mass Mimbh within the range [103 − 104 ]M⊙ with 0.04 < a < 0.08 pc.
4 The hybrid model couples numerical three-body scatterings to an analytical formalism for
interactions between the cusp and the IMBH (Sesana et al. 2008).
Secular Stellar Dynamics
111
We simulate the nuclear star cluster as a smooth potential with a power-law
number density profile n(r) ∝ r−α ; this decreases the computation time and
lends a clear, direct interpretation to our results. Typically α ∈ [0.5, 1.75], and is
normalized such that at one parsec the stellar mass is M (< 1 pc) = 1 × 106 M⊙
(Schödel et al. 2007; Trippe et al. 2008; Schödel et al. 2009). To test our
theory of SDAF, we include an isotropically-distributed population of test stars
with individual masses m within the range [10−100]M⊙ which gravitationally
interact only with the IMBH, not with each other. In doing so we isolate the
effects of the IMBH on the stellar orbits. The orbits of the IMBH and the test
stars precess according to Equation (5.5), such that for each time-step, dt, they
rotate in their orbital plane by an angle |δφ| = 2π(dt/tprec ).
As the IMBH moves in a smooth stellar potential, it experiences no dynamical friction from it. To account for this we use the dynamical friction formulae
given by Just et al. (2011) to artificially decrease the semi major axis of the
IMBH’s orbit. Just et al. (2011) use a self-consistent velocity distribution function in place of the standard Maxwellian assumption and an improved formula
for the Coulomb logarithm to arrive at dynamical friction timescale for a nearKeplerian potential, which can differ by a factor of three from the standard
formula in Equation (5.1). The radial evolution of the IMBH (point-like object)
behaves, for α 6= 3/2, as5
2
2α−3
t
,
r = r0 1 −
tDF
(5.10)
where tDF is the dynamical friction timescale calculated from the starting position of the massive body r0 ,
tDF =
M•2
P
.
2π(3−α)χ ln Λ MimbhM<
(5.11)
χ is the fraction of stars with velocity smaller than that of the IMBH (circular
case),
α−3/2
Z 1 1
u2
2
χ(α) = √
du
(5.12)
u 1−
2
2B(3/2, α−1/2) 0
and Λ is given by
Λ=
(3 + α) M•
.
α(1 + α) Mimbh
(5.13)
We implement Equations (5.10) - (5.13) by adjusting the semi major axis
of the IMBH orbit at each time-step with the following transformation
rt → β rt−1
5 For
vt−1
vt → √ ,
β
α = 3/2 see similar equations in Just et al. (2011).
(5.14)
112
Secular Dynamical Anti-Friction
-0.5
0
0.5
2
4
t (Myr)
6
1
ey
0.5
0
-0.5
-1
1
0.8
e
0.6
0.4
0.2
0
0
8
Figure 5.3 – Example of a secular horseshoe orbit of a star of mass m = 1M⊙ , bound to a massive
black hole of mass M• = 4 × 106 M⊙ . The stellar orbit feels a gravitational torque from an
IMBH (mass Mimbh = 103 M⊙ ) and precesses due to general relativity and a background stellar
potential. (Top) The eccentricity vectors of the IMBH and the prograde-directed single star in the
frame co-precessing with the eccentricity vector of the IMBH. (Bottom) Eccentricity evolution of
the stellar orbit over 8 Myr during which it performs a few secular horseshoe orbits.
β = 1−
t
tDF (rt−1 )
2
2α−3
.
(5.15)
Essentially we are re-scaling the stellar ellipse, without changing its eccentricity. In the next subsections we present three examples of dynamics resulting
from SDAF.
5.3.1
Secular horseshoe orbits
Horseshoe orbits describe the motion of satellite bodies in the circular planar restricted three-body problem (for a comprehensive overview see Murray
& Dermott 1999) which librate on paths encompassing the L3, L4 and L5
Secular Stellar Dynamics
113
Lagrangian points. Such orbits map out horseshoe shapes in the frame corotating with two massive bodies and have been observed in the Saturnian
satellite system (Dermott & Murray 1981) and in the co-orbital Earth Asteroid
2002 AA29 (Connors et al. 2002).
Secular dynamical anti-friction results in secular horseshoe orbits6 . Secular horseshoe orbits differ from those in our above description in that it is the
eccentricity vector, as opposed to the guiding centre of the orbit which maps
out a horseshoe in the frame co-rotating with the IMBH (for close analogies in
planetary dynamics, see Michtchenko & Malhotra 2004; Pan & Sari 2004; Lithwick & Wu 2011). We illustrate this behaviour in Figure 5.3 where we show
the position of the eccentricity vectors of a IMBH and a star in the frame of
reference co-precessing with the eccentricity vector of the IMBH.
In this example M• = 4 × 106 M⊙ , Mimbh = 103 M⊙ , m = 1M⊙ , α = 1,
M (< 1 pc) = 1 × 106 M⊙ , aimbh = a = 0.0703 pc, eimbh = 0.7 and e = 0.69.
Both the IMBH and stellar orbits are on the xy-plane with an initial angle of
δφ = π/4 rad between their eccentricity vectors. This behaviour is transitory
due to non-secular gravitational interactions. The timescale over which the
star in this simulation performs a horseshoe in eccentricity space is ∼ 1.5 Myr,
or ∼ 1700 orbital periods.
5.3.2
Separation of potential into pro- and retro- grade orbits
As illustrated in Figure 5.3 the eccentricity vectors of prograde orbits which
have a small inclination angle with respect to an IMBH will tend to librate
about a point π rad opposite that of the IMBH – the secular analogue of the
Lagrangian L3 point. In contrast, the eccentricity vectors of retrograde orbits
(those which do not flip to prograde orbits) will, in a time-averaged sense, cluster around that of the IMBH before they are pushed away, as they precess more
slowly due to the increase in their orbital eccentricity. We find that the presence of a massive body, such as an IMBH, surrounded by less massive objects in
a near-Keplerian potential tends to segregate the lighter objects into a region
dominated by prograde orbits and one dominated by retrograde. This mechanism has the strongest effect for, but is not restricted to, co-planar orbits, as
scalar angular momentum changes result in eccentricity changes whereas vector angular momentum relaxation re-orients the stellar orbital plane.
In Figure 5.4 we plot an example of this segregation. The top figure shows
a schematic of eight octants in eccentricity vector space. We rotate the cluster
such that the eccentricity vector of the IMBH lies parallel with (1,1,1), and
follow the change in the number of massless stellar orbits with eccentricity
vectors in each octant. The middle plot shows the results for prograde orbits
and the bottom plot for retrograde. There is a clear increase in the number
6 Referred to as “windshield-wiper orbits” in Merritt & Vasiliev (2011) in the context of nonaxisymmetric star clusters.
114
Secular Dynamical Anti-Friction
of prograde stellar orbits in octant 8, opposite that in which the IMBH lies, as
expected. Furthermore, there is an enhancement of retrograde stellar orbits in
octant 1 as the theory predicts.
5.3.3
Increase in orbital eccentricity of IMBH
Here we examine two of the main testable predictions of SDAF: the eccentricity evolution of an IMBH with 1) artificially reversed apsidal precession, and
2) with a decreased migration rate from Chandrasekhar’s dynamical friction.
We present the results of short-term simulations performed to test these predictions, using ten realizations for each simulation to look at the statistical
results. In the following, M• = 4 × 106 M⊙ , Mimbh = 104 M⊙ , aimbh = 0.06 pc,
eimbh = 0.4, α = 1.5, M (< 1 pc) = 1 × 106 M⊙ . We integrate the orbits of
1000 test stars in each simulation, distributed from 0.006 pc < a < 0.12 pc, with
m = 20M⊙ .
In Figure 5.5 we show the evolution of orbital eccentricity of the IMBH
as a function of time in units of the initial orbital period, P (a = 0.06 pc) ≃
692 yr. Each line shows the mean eccentricity of ten simulations. Cprec is a
pre-factor placed in front of the apsidal precession angle, δφ, at each time-step.
The reference simulation with the correct precession rate Cprec = 1.00 shows
an increase in eccentricity as expected. Increasing the precession rate by a
factor of 1.25 does not however increase the rate of eccentricity growth, as the
IMBH orbit has less time to exert torques on the surrounding stellar orbits. In
the next two simulations we reverse the direction of apsidal precession of the
IMBH, keeping the precession period roughly constant in relation to the first
two simulations. The overall eccentricity growth of the IMBH is negative, as
predicted.
Next, we present the results of simulations wherein we change the migration rate of the IMBH (Figure 5.6). SDAF predicts that lower migration rates
lead to slower eccentricity growth as the IMBH interacts with retrograde orbits
at a lower rate. In this figure, Ctdf is a multiplicative pre-factor in front of the
dynamical friction timescale given in Equation (5.11). The semi-major axis of
the IMBH orbit shrinks from 0.06 pc to ∼ 0.024 pc over the simulation run for
Ctdf = 1. For Ctdf = 2, 5, 10 the eccentricity growth of the IMBH is suppressed
(though still positive), exactly as SDAF predicts.
In the simulation in which we half the friction timescale (Ctdf = 0.5), and
hence increase the migration rate by a factor of two, the eccentricity growth
rate does not increase. We attribute this to the shortening over the timescale
over which the IMBH exerts persistent torques on the surrounding stellar orbits.
Finally, we run simulations wherein we soften the IMBH’s gravity to such
an extent that it switches off entirely for a star coming within 0.01 pc of the
IMBH itself. We do so to confirm that secular dynamics are dominating the
eccentricity evolution of the IMBH orbit; these conditions inhibit the threebody scatterings and ejections which are the basis for the theories given in
Secular Stellar Dynamics
115
Iwasawa et al. (2011) and Sesana et al. (2011). We find it makes no significant
difference for the angular momentum evolution of the IMBH even though it
significantly reduces the number of stellar ejections (by a factor of ∼ 5).
5.4 Discussion
We have presented a new secular gravitational dynamical process in galactic
nuclei which acts to push low-mass stellar orbits away from that of massive
bodies. It can also act to decrease the orbital angular momentum of the massive body. There are two essential components to the decrease. The first is
SDAF and the net flow of retrograde orbits past that of the massive body which
extracts orbital angular momentum. The second is Chandrasekhar’s dynamical
friction force acting on the massive body which decreases its semi major axis
and provides a mechanism for net negative J change. Although we have told
this story in the context of an IMBH inspiral into the centre of a galaxy, the interactions described here will be valid for any massive body in a near-Keplerian
potential. Observational predictions of SDAF include higher mass objects with
increasingly larger eccentricities, and density anisotropy in the stellar distribution near the plane of the massive body.
Acknowledgments A.-M.M gratefully acknowledges the hospitality of Monash
University and the Aspen Center for Physics. We thank Fabio Antonini, Dan
Fabrycky, Clovis Hopman, Margaret Pan, Alberto Sesana and Scott Tremaine
for valuable discussions and/or comments on the text. A.-M.M is supported
by a TopTalent fellowship from the Netherlands Organisation for Scientific Research (NWO) and Y.L. by a VIDI grant from NWO.
116
Secular Dynamical Anti-Friction
2
1
6
1
4
0.5
ez
3
5
0
-0.5
1
8
0.5
-1
-1
7
0
-0.5
ex
0.8
0.5
ey
1 -1
1
2
3
4
5
6
7
8
0.6
0.4
(n - n0)/n0
-0.5
0
0.2
0
-0.2
-0.4
0
200
400
600
800
1000
600
800
1000
t (P)
0.8
1
2
3
4
5
6
7
8
0.6
(n - n0)/n0
0.4
0.2
0
-0.2
-0.4
0
200
400
t (P)
Figure 5.4 – Schematic of octants in eccentricity vector space (top). The unit eccentricity vector
of the IMBH lies parallel with (1,1,1) in octant 1 (black dot). Fractional change in number of
prograde (middle) and retrograde (bottom) stars in each octant in the frame rotating with the
IMBH. Time is given in units of initial orbital period of the IMBH, P (a = 0.06 pc). The increase
in the number of prograde stellar orbits in octant 8, and retrograde stellar orbits in octant 1, is as
predicted.
Secular Stellar Dynamics
0.46
117
Cprec
1.00
1.25
-1.25
-1.00
0.45
0.44
0.43
e
0.42
0.41
0.4
0.39
0.38
0.37
0
100
200
300
400
500
t (P)
600
700
800
900
Figure 5.5 – Mean eccentricity evolution (each curve is the average of ten simulation runs) of an
IMBH experiencing four different rates of apsidal precession. The first is the reference simulation
with the correct precession rate corresponding to the smooth background potential (red solid line).
In the second simulation, the IMBH has an increased precession rate by a factor of 1.25 (orange
dotted line). In the third simulation, the IMBH has an increased precession rate by the same
factor but the IMBH is made to precess in a reversed direction (green dotted line). In the fourth
simulation, the IMBH has a precession period corresponding to the reference simulation but again
in reversed direction (blue dotted line). Time is given in units of the initial orbital period P (a =
0.06 pc).
0.46
Ctdf
1.00
0.50
2.00
5.00
10.00
0.45
0.44
e
0.43
0.42
0.41
0.4
0.39
0
100
200
300
400
500
t (P)
600
700
800
900
Figure 5.6 – Mean eccentricity evolution (each curve is the average of ten simulation runs) of
an IMBH with five different migration rates due to dynamical friction. SDAF predicts lower rates
of eccentricity growth with increasing dynamical friction timescales. The parameter Ctdf is a
multiplicative pre-factor in front of the dynamical friction timescale given in Equation (5.11).
118
Secular Dynamical Anti-Friction
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Nederlandse samenvatting
D
E meeste, zo niet alle sterrenstelsels hebben met elkaar gemeen dat in
het centrum een massief zwart gat1 huist. Deze zwarte gaten zijn veel
zwaarder dan sterren, en hun massa’s lopen uiteen van miljoenen tot miljarden zonsmassa’s. Onze eigen Melkweg is geen uitzondering op deze regel. Zeer
gedetailleerde waarnemingen in infrarood hebben onomstotelijk de aanwezigheid van een zwart gat bewezen, dat ongeveer vier miljoen keer zo zwaar is als
de zon.
Het zwarte gat kan (per definitie) niet direct worden waargenomen. Maar
aangezien het centrum van de Melkweg op een afstand van slechts 8 kpc ligt2
is het mogelijk om de banen van individuele sterren waar te nemen. Door deze
banen is uit te rekenen wat de massa en dus (gegeven de afstand) de dichtheid
van het object is dat ze aantrekt. Hieruit volgt de conclusie dat dit object enkel
een zwart gat kan zijn.
De sterren die deze banen omschrijven vormen zelf een grote verrassing.
Vóór de waarnemingen was de gangbare mening dat zich geen sterren zouden
kunnen vormen in de buurt van een zwart gat, vanwege de zeer grote getijdenkrachten. Omdat het veel tijd kost voordat sterren van baan veranderen,
was de verwachting dat zo er al sterren in de buurt van een massieve zwart gat
zouden zijn, deze ouder dan een miljard jaar zouden moeten zijn. De sterren
rond het zwarte gat in het centrum van de Melkweg zijn echter zeer zwaar,
en omdat zware sterren maar kort leven (slechts enkele miljoenen jaren in het
geval van sommige sterren rond het zwarte gat) moeten ze dus ook vrij jong
zijn. Bovendien ontdekten Levin & Beloborodov (2003) dat sommige van deze
jonge sterren in de vorm van een schijf rond het zwarte gat bewegen, vergelijkbaar met de beweging van de planeten om de zon. Dit suggereert dat deze
groep sterren ook in die schijf zijn gevormd.
Dit proefschrift gaat over de dynamiek van sterren die zo dicht bij massieve zwarte gaten zijn dat de potentiaal wordt gedomineerd door het zwarte
gat zelf. In deze situatie bewegen sterren zich ongeveer in ellipsen rond het
zwarte gat, maar deze ellipsen worden over langere tijd gezien verstoord door
wisselwerking met andere sterren.
1 Een “massief” zwart gat is een pleonasme. In dit proefschrift nemen we dit pleonasme over van
het vakjargon, waarin het woord “massief” duidt op zwarte gaten die miljoenen malen zwaarder
zijn dan de zon.
2 1 kpc = 3,3 duizend lichtjaar.
120
Nederlandse samenvatting
Omdat de tijdschalen waarop verschillende dynamische effecten optreden
sterk uiteenlopen is het moeilijk om deze direct te simuleren met standaard
simulatiemodellen. Sommige sterren draaien in een jaar om het zwarte gat,
terwijl de banen pas in de loop van miljoenen of miljarden jaren veranderen
als gevolg van de verstoring door andere sterren. Voor het onderzoek in dit
proefschrift heb ik daarom een nieuwe N -body simulatie ontwikkeld, die het
feit uitbuit dat de baan van sterren rond het zwarte gat (dus zonder de verstoring van andere sterren) analytisch oplosbaar is. De wisselwerking met andere
sterren wordt vervolgens als verstoring van deze banen beschouwd.
Het effect van verstoringen op een baan in een “gladde” potentiaal leidt er
in veel andere situaties toe dat een ster een dronkaards wandel (random walk)
uitvoert. De reden is dat ontmoetingen met andere sterren willekeurig zijn,
zodat de kans op een afwijking van het “gladde” pad naar links en naar rechts
even groot zijn.
Rond een massief zwart gat geldt deze aanname echter niet in alle omstandigheden. Banen in de potentiaal van een zwart gat hebben de unieke
eigenschap dat ze geen precessie vertonen: de banen sluiten, en hun oriëntatie ten opzichte van de banen van andere sterren blijft gedurende zeer lange
tijd hetzelfde. Als gevolg hiervan is de wisselwerking met andere sterren veel
minder willekeurig dan in andere situaties, waardoor de banen veel sneller
veranderen dan anders. In dit proefschrift onderzoek ik wat de gevolgen zijn
van deze “seculiere effecten” voor de dynamica van sterren rond zwarte gaten.
In hoofdstuk 2 beschouw ik de evolutie van een excentrische schijf van
sterren, zoals die mogelijk is gevormd in het centrum van de Melkweg. Ik
laat zien dat deze configuratie instabiel is: wanneer een baan een iets hoger
dan gemiddelde excentriciteit heeft, dan groeit deze binnen relatief korte tijd
(ongeveer een miljoen jaar) naar een waarde van bijna 1. Omgekeerd, als de
excentriciteit iets lager is dan gemiddeld, dan wordt de baan snel cirkelvormig.
Deze instabiliteit is interessant vanuit theoretisch oogpunt, maar heeft ook consequenties voor het centrum van de Melkweg. Als sommige van de objecten in
de schijf dubbelsterren zijn, kan de baan zo excentrisch worden dat de dubbelster uiteen wordt gescheurd door de getijdenkrachten van het zwarte gat. In
dat geval blijft één ster om een zeer nauwe baan rond het zwarte gat draaien.
Omdat de processen snel verlopen zou dit de oorsprong van de waargenomen
jonge sterren rond het zwarte gat kunnen verklaren.
In hoofdstuk 3 wordt de lange termijn evolutie van sterrenbanen om een
massief zwart gat gevolgd totdat een toestand van steady state wordt bereikt.
Het is niet mogelijk om de evolutie te volgen met N -body simulaties, omdat
deze te lang zouden duren. Voor het volgen van de lange termijn evolutie heb
ik daarom een Monte Carlo code ontwikkeld. Een nieuw aspect aan deze code
is dat zowel de gebruikelijke dronkaards wandel wordt gemodelleerd als de
seculiere effecten (“resonant relaxation”, voor het eerst beschreven in Rauch &
Tremaine 1996), die over vele honderden banen gecorreleerd zijn.
121
De diffusiecoëfficiënten in deze code zijn gekalibreerd met N -body simulaties. Ik laat zien dat de seculiere effecten van groot belang zijn voor de steady
state van een systeem van sterren in de buurt van een massief zwart gat. Deze
evenwichtstoestand wijkt sterk af van de toestand die werd gevonden in het
klassieke artikel van Bahcall & Wolf (1976). In dat artikel divergeerde de sterdichtheid naarmate de afstand tot het zwarte gat kleiner werd, maar in de
steady state die tot stand komt uit het samenspel van seculiere effecten en sterren die als gevolg daarvan in het zwarte gat terecht komen blijven juist alleen
banen met een grote half-lange as over. Een andere consequentie is dat de
snelheidsverdeling van sterren in de evenwichtstoestand niet isotropisch is.
De vraag die ik in hoofdstuk 4 probeer te beantwoorden is wat de oorsprong is van de B-sterren in het centrum van de Melkweg. Eén scenario uit
de literatuur is dat deze de overblijfselen zijn van dubbelsterren die door de
getijdenkrachten van het zwarte gat zijn verscheurd. Een andere mogelijkheid
is dat een schijf van sterren langzaam “oplost”. Om uitsluitsel te geven volg
ik de evolutie van sterrenbanen die volgt uit deze twee scenarios. De resulterende constellaties zijn zeer verschillend van elkaar. Ik laat zien dat deze
verschillen ook duidelijk te zien zijn in termen van een nieuwe, direct waarneembare variabele. De al waargenomen banen zijn nog niet gepubliceerd,
maar in de nabije toekomst kan deze methode waarschijnlijk (minstens) één
van deze twee scenarios uitsluiten.
Hoofdstuk 5 gaat over het pad dat een object aflegt dat zwaarder is dan
sterren, maar lichter dan het massieve zwarte gat – meer concreet kan hier
gedacht worden aan een hypothetisch “intermediate mass” zwart gat, van duizend tot tienduizend zonsmassa’s. Algemeen bekend is dat zo’n object inspiraliseert als gevolg van dynamische wrijving. Simulaties laten zien dat daarnaast
de excentriciteit onderweg toeneemt. Ik laat met analytische argumenten zien
wat de oorzaak is van de groeiende excentriciteit. Deze blijkt opnieuw te zijn
gelegen in seculiere effecten, waardoor sterren gemiddeld gezien impulsmoment aan het object onttrekken, een effect dat ik “seculiere dynamische antiwrijving” noem. Ik voer nieuwe simulaties uit om aan te tonen dat dit effect
inderdaad verantwoordelijk is voor de toenemende excentriciteit.
Curriculum vitae
How strange it is to be anything at all.
Jeff Mangum
I
WAS born on January 16, 1982 in Dublin, Ireland. I went to ‘St. Louis High
School’ in Rathmines, where my mother taught maths. At the age of 16, I
walked into my first physics class to find the question ‘what is light?’ written
on the blackboard, and was hooked. And so, armed with the laws of reflection
and refraction, I began a path in science.
In 2000, I began my bachelor studies in Physics and Astronomy at the University of Ireland, Galway. I graduated in 2004 with a First Class Honours
degree and a thesis on exo-planetary transits. Following a summer research
project searching for optical emission from pulsars, I spent eight months traveling in Asia. I began the Master in Astronomy program at Leiden University in
2005, supported by a research grant from the University.
During this time, three fortunate and formative events occurred: I took a
course on the theory of general relativity taught by an inspiring new faculty
member, Yuri Levin; I attended an exciting talk by a new postdoc, Clovis Hopman, on the inspiral of compact stellar remnants into massive black holes, and I
won an NWO ‘Toptalent’ fellowship which allowed me to independently choose
the direction of my future PhD research. I chose to work with Yuri and Clovis
on stellar dynamics near massive black holes, and this thesis is largely a result
of these timely circumstances.
During my PhD, I presented my research at international conferences in the
Netherlands, Germany, Ireland, Israel, China, Australia and the United States,
and visited Monash University in Melbourne for four months. As of December
2011, I am a post-doc at Leiden Observatory and, having sent off my post-doc
applications for the fall 2012, am looking forward to the next step on this path.
Acknowledgments
People love chopping wood.
In this activity one immediately sees results.
Albert Einstein
A
LTHOUGH writing a thesis can take a long time, my years at Leiden have
flown by too fast. It was wonderful to be a student here and I’m grateful to
everyone in the astronomy department for their help and friendship over the
years, including the support staff who hold it all together.
Thanks to the old gang (now sadly dispersed) who made Leiden so much
fun: Nina, Sarah, Remco, Rik, Hendrik, Andreas, Olmo, Rob, Freeke, Niruj,
Tip and Craig (Boooth!). And a nerdy high-five to the first astro(awesome)ph club, especially to Ryan, Steven, Jeroen, Ernst and Marcel who showed us
how to make spiral arms in our coffee. I’m very grateful to the members of
our theory group – Evghenii, Anders, Richard, Atakan, Clovis – I always felt
privileged to be part of it. I’m also thankful for the friends I’ve made outside
the observatory: in particular to Jonathan and Pantelis.
Thanks to everyone at Monash for showing me around Melbourne and instilling in me a healthy respect for rubidium. Maarten, I thoroughly enjoyed
our blackboard discussions (. . . magnets, how do they work?), and Rutger, skydiving was just the beginning; we will indeed keep it up. Thanks to Sarah Levin
for delicious meals and exquisite company, both in Leiden and Australia.
Conor, from sharing your maths notes with me 11 years ago to being by
my side during these last few frenetic months, you’ve been amazing (thanks a
million). Amelia, I hope we continue to narrate our lives together. It wouldn’t
be half so much fun without you. Martin, thank you for your warmth and
friendship, and for all the wonderful times we’ve had. Olja, you’ve been the
best friend and officemate in the whole universe - thank you for everything.
Finally, thanks so much to my family back home. First and foremost to my
parents, Ev and Dave, for your love and support. Thanks to my brothers Ciaran
and Danny, to all my aunts and uncles, grannies, my niece and (especially!)
cousins for your interest and encouragement throughout the years. Cheers.
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