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arXiv:1010.1535v1 [astro-ph.GA] 7 Oct 2010

DRAFT VERSION OCTOBER 11, 2010

Preprint typeset using LATEX style emulateapj v. 2/16/10

SECULAR STELLAR DYNAMICS NEAR A MASSIVE BLACK HOLE

ANN-MARIE MADIGAN1, CLOVIS HOPMAN1AND YURI LEVIN2,1

1Leiden University, Leiden Observatory, P.O. Box 9513, NL-2300 RA Leiden,

2School of Physics, Monash University, Clayton, Victoria 3800, Australia

Draft version October 11, 2010

Abstract

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 paper, 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 a MBH as a result of

tidal disruptions. In our Galactic center, the size of the depression is about 0.2 pc, consistent with the size of

the observed “hole” in the distribution of bright late-type stars. We also find that the velocity vectors of stars

around a 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 “S-stars”

in the Galactic center. We find that in this scenario more highly eccentric (e > 0.9) S-star orbits are produced

than have been observed to date.

Subject headings: black hole physics — Galaxy: center — stellar dynamics —methods: N-body simulations

1. INTRODUCTION

The 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 Ke-

plerian, and it is useful, both as a mental picture and as a

computational device, to average the mass of the stars over

theirorbitsandconsidersecularinteractionsbetweenthese el-

lipses, rather than interactions between point particles. These

stellar ellipses precess on timescales of many orbits, due to

deviations from the Newtonian point particle approximation:

thereis typicallyanextendedclusterofstars aroundtheMBH,

andthereis precessiondueto generalrelativistic effects. Nev-

ertheless, 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 stochas-

tic evolution of the angular momenta of the stars than normal

relaxation dynamics. They called this process resonant relax-

ation (RR). RR is potentially important for a number of as-

trophysical phenomena. Rauch & Ingalls (1998) showed that

it increases the tidal disruption rate, although in their calcu-

lations the effect was not very large since most tidally dis-

rupted stars originated at distances that were too large for

RR to be effective (in §7.2 we will revisit this argument).

By contrast with tidally disrupted main-sequence stars, inspi-

raling 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 be-

come gravitational waves sources is therefore much larger

(Hopman & Alexander2006a). RR alsoplaysaroleinseveral

proposed mechanisms for the origin of the “S-stars”, a cluster

of B-stars with randomised orbits in the Galactic center (e.g.

Hopman & Alexander 2006a; Levin 2007; Perets et al. 2007,

see §7.3 of this paper).

madigan@strw.leidenuniv.nl

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 in-

tegrated for many precession times, the simulations are com-

putationallydemanding. 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 interac-

tion 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 Fokker-

Planck 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 2006a,b). The current formalism how-

ever is not directly applicable to the case when RR plays an

important role. At the heart of all current Fokker-Plank ap-

proaches is the assumption of a random-walk diffusion of

angular momentum, whereas RR is a more complex relax-

ation 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 sta-

tistical properties of RR. This model, calibrated with special-

purposeN-bodysimulations, 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 paper is as follows. In section §2, we

present an extensive suite of special purpose N-body simula-

tions, 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.

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2MADIGAN, HOPMAN AND LEVIN

In §3, we introduce the ARMA model for the data analy-

sis of the N-body simulations. This description accommo-

dates the random, non-secular (non-resonant) effects on very

short timescales, the persistent autocorrelations for interme-

diate timescales less than a precession time, and the random

walk behaviourfor verylong times (the RR regime). In §4 we

extend the ARMA model, using physical arguments, to a pa-

rameter space that is larger than that of the simulations. In §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 §6 as an input to Monte

Carlo (MC) simulations which study the distribution of stars

near MBHs. In §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 Galactic center. We summarize in §8.

2. N-BODY SIMULATIONS

We have developed a special-purpose N-body code, de-

signed to accurately integrate stellar orbits in near-Keplerian

potentials; see Appendix (A) for a detailed description. We

wrote this code specifically for the detailed study of RR. Such

a study requires an integrationscheme with a strict absence of

spurious apsidal precession and one which is efficient enough

to do many steps per stellar orbit whilst integrating many or-

bits to resolve a secular process. The main features of this

code are:

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

integrator (MVS; Yoshida 1990; Wisdom & Holman

1991; Kinoshita et al. 1991; Saha & Tremaine 1992).

The MVS integrator switches between Cartesian coor-

dinates (in which the perturbations to the orbit are cal-

culated) 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 soften-

ing.To resolve the periapsis of eccentric orbits

(Rauch & Holman 1999) and close encountersbetween

particles, we adapt the time steps of the particles. We

use the compact K2kernel (Dehnen 2001) for gravita-

tional softening.

4. Time symmetric algorithm. As MVS integrators gener-

ally lose their symplectic properties if used with adap-

tive time stepping, we enhance the energy conservation

by time-symmetrising the algorithm.

Our code is efficient enough to study the evolution of energy

and angular momentum of stars around MBHs for many pre-

cession times, for a range of initial eccentricities.

2.1. Model of Galactic Nucleus

We base our galactic nucleus model on a simplified Galac-

tic center (GC) template1. It has three main components. (1)

A massive black hole (MBH) with mass M•= 4 × 106M⊙

which remains at rest in the center of the co-ordinate sys-

tem. (2) An embedded cluster of equal-mass field stars m =

10 M⊙, distributed isotropically from 0.003pc to 0.03pc,

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 pre-

cession rate is determined by the smooth potential of the field

stars themselves (see Appendix C) and general relativity. The

precession of the field stars is important to account for, be-

cause 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 decoher-

ence 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 = 0.01pc and a specified initial eccentricity e.

Following Equation (17) from Hopman & Alexander

(2006a), 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

galaxy’s bulge, we define the radius of influence as

rh=GM•

σ2

= 2.31pc

?

M•

4 × 106M⊙

?1/2

.

(1)

The number of field stars within radius r is

N(< r) = Nh

?r

rh

?3−α

,

(2)

where we assume that the mass in stars within rhequals 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 a 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(< 0.03pc) = 1754.

We summarise the potential for our galactic nucleus model in

Table (1).

We evolve this model of a galactic nucleus for a

wide range in eccentricity of the test stars,

0.01,0.1,0.2,0.3,0.4,0.6,0.8,0.9,0.99. For each initial ec-

centricity, we follow the evolution of a total of 80 test stars,

both massless and massive, in a galactic nucleus. Typically

we use four realisations of the surrounding stellar cluster for

each eccentricity, i.e., 20 test stars in each simulation. They

have randomly-orientedorbits with respect to one another, so

that they experience different torques within the same cluster.

e=

1Due tothe effect of general relativistic precession, the system isnot scale-

free and the masses need to be specified.

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SECULAR STELLAR DYNAMICS NEAR AN MBH3

TABLE 1

GALACTIC NUCLEUS MODEL

Parameter

M•

m

α

rh

rmin

rmax

ateststar

N(< rmax)

Numerical Value

4 × 106M⊙

10 M⊙

1.75

2.31pc

0.003pc

0.03pc

0.01pc

1754

The simulations are terminated after 6000 orbital periods

(henceforth denoted by P = 2π?a3/GM•) at a = 0.01pc,

centricity e = 0.6, deep into the RR regime for all eccentric-

ities; see §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

initialeccentricities. Inaddition,theautocorrelationfunctions

of their angular momenta changes drops to zero before this

time; see Figure (4).

roughly equivalent to three precession times for a star of ec-

2.2. Illustrative simulations

In §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 simula-

tions for illustration purposes, in order to highlight some in-

teresting points and motivate our approach. We define energy

E of the test star as

E =GM•

2a

,

(3)

and dimensionless angular momentum2J as

J =

?

1 − e2.

(4)

Secular torques should affect the angular momentum evo-

lution, but not the energy evolution of the stars (in the pic-

ture of interactions between massive ellipses of the introduc-

tion, the ellipses are fixed in space for times less than a pre-

cession time, t < tprec, such that the potential and there-

fore the energy is time-independent). It is therefore of in-

terest to compare the evolution of these two quantities. An

example is shown in Figure (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 (2) we show the eccentricity evolution for a sam-

ple population of stars with various initial orbital eccentrici-

ties. There is significant eccentricity evolution in most cases,

but almost none for e = 0.99, the reasons for which we eluci-

date in §6.

To quantify the rate at which the energy E and angular mo-

mentum J change as a function of eccentricity e we calcu-

late 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).

2Throughout this paper, we use units in which angular momentum J, and

torque τ, are normalised by the circular angular momentum for a given semi

major axis, Jc=√GM•a. All quantities are expressed per unit mass.

-0.18

-0.16

-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0 1000 2000 3000

t [P]

4000 5000 6000

0 0.1 0.2

t [Myr]

∆E/E0

∆J/Jc

FIG. 1.— Evolution of angular momentum J and energy E, normalised to

the initial values, of a massless test star with semi major axis a = 0.01pc

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 tprecis ∼ 2000 orbits), whereas the E evolution displays a

much slower, more erratic, random walk.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000

t [P]

4000 5000 6000

0 0.1 0.2

e

t [Myr]

FIG. 2.— Eccentricity evolution of a sample of massless test stars in a

fiducial galactic nucleus model. We show four realisations 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.

We compute the following quantities

tE=

E2

?(∆E)2?∆t

J2

c

?(∆J)2?∆t

(5)

tJ=

(6)

where ∆E and ∆J are the steps taken in a time ∆t, and we

take the mean ?? over eighty test stars in each eccentricity

bin. We plot the resulting timescales as a function of eccen-

tricity in Figure (3), taking several ∆t values to get order-of-

magnitude estimates for these timescales. We find no signifi-

cant difference between the results for the massless and mas-

sive test stars. We note that tEis only weakly dependenton e,

whereas tJis of the same order of magnitude as tEfor e → 0

and e → 1, but much smaller for intermediate eccentricities.

It is in the latter regime that secular torques are dominating

the evolution.

Page 4

4 MADIGAN, HOPMAN AND LEVIN

10

100

1000

10000

0 0.1 0.2 0.3 0.4 0.5

e

0.6 0.7 0.8 0.9 1

tx [Myr]

x = E, ∆t = 4000

∆t = 2000

∆t = 1000

∆t = 500

x = J, ∆t = 4000

∆t = 2000

∆t = 1000

∆t = 500

FIG. 3.— The 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 (5) and (6). We find a weak trend for lower E relaxation times with

increasing eccentricity (from ∼ 400Myr to 200Myr). J relaxation occurs

on very short timescales (∼ 20Myr) at eccentricities of 0.8 < e < 0.9.

Stellar orbits with low eccentricities have very long J relaxational timescales

(∼ 1Gyr) as the torque on a circular orbit drops to zero. This timescale

increases again at very high eccentricities.

In the following sections, we will give a detailed model for

the evolution of angular momentum. Since the focus of this

paper is mainly on secular dynamics, we do not further con-

sider energy diffusion here, but in Appendix B we discuss

the timescale for cusp formation due to energy evolution, and

compare our results to other results in the literature.

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 (1), RR

cannot be modelled 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 consid-

ering 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 autocorrela-

tions 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 angularmomentaat a regularspac-

ingofoneperiod. This choiceis arbitrary,andbelowwe show

how our results generalise to any choice of time steps. The

normalised autocorrelation function can be written as

ρt=?(∆Js+t− ?∆J?)(∆Js− ?∆J?)?

?(∆Jt− ?∆J?)2?

where ∆Jsis the change in angular momentum between sub-

sequentmeasurementsat timesandit isassumedthatthetime

series is stationary. A typical autocorrelation function from

our simulations is shown in Figure (4). The fundamental fea-

ture of this autocorrelation function 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 autocorrelationfunctionas that

,

(7)

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

1 10 100

t [P]

1000 10000

0.0001 0.001 0.01 0.1

ρt

t [Myr]

ρ(data)

ρ(theory)

FIG. 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.

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. Second, a generated time

series with the same statistical properties as the time series

generated in the physical process can be used in Monte Carlo

simulations to solve for the long term evolution of RR.

3.1. The autoregressive moving average model ARMA(1, 1)

We now introduce the autoregressive moving average

(ARMA) model, which is often used in econometrics (see

e.g., Heij 2004). We show that this model can generate time-

seriesthatreproducetheautocorrelationfunctionandvariance

of ∆Jt. The model is ad hoc in that it does not have a direct

physical foundation. However, we will find physical interpre-

tations for the free parameters of the model in §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

∆1Jt= φ1∆1Jt−1+ θ1ǫ(1)

t−1+ ǫ(1)

t .

(8)

The label “1” in this equation refers to the fact that the data

have a regular spacing of one period. In this equation, φ1and

θ1are free parameters, and the random variable ǫ is drawn

from a normal distribution with

?ǫ(1)? = 0;?ǫ(1)

t ǫ(1)

s? = σ2

1δts

(9)

where σ1is a third free parameter, and δtsis the Kronecker

delta. For such a model, the variance of the angular momen-

tum step is

?∆J2

t? =1 + θ2

1+ 2θ1φ1

1 − φ2

1

σ2

1.

(10)

Page 5

SECULAR STELLAR DYNAMICS NEAR AN MBH5

The autocorrelation function (ACF) can be derived analyti-

cally (see Appendix D for details),

ρt= φt

1

?

1 +

θ1/φ1

1 + (φ1+ θ1)2/(1 − φ2

1)

?

(t > 0).

(11)

From this expression, we see that the decay time of the

ACF is determined by the parameter φ1, and θ1captures the

magnitude of the ACF. As an example, we plot the ACF in

Figure (4) with parameters typical for RR. We emphasise

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

∆Jt=

∞

?

k=0

ψkǫt−k

(12)

where

ψk= (φ1+ θ1)φk−1

1

(k > 0);ψ0= 1.

(13)

Oncethemodelparameters(φ1,θ1,σ1) arefoundforatime

step of one period, δt = P, the variance after N steps can be

computed. The displacement after N steps is given by

N

?

n=1

∆Jn=

N

?

n=1

∞

?

k=0

ψkǫn−k,

(14)

where (n − k) ≥ 0. We denote the variance of the displace-

ment after N steps by

V (N) ≡

??N

?

n=1

∆Jn

?2?

,

(15)

such that

V (N)=

N

?

n=1

∞

?

N

?

k=0

N

?

∞

?

m=1

∞

?

N

?

l=0

ψkψl?ǫn−kǫm−l?

=σ2

1

n=1

k=0

m=1

∞

?

l=0

ψkψlδn−k,m−l,

(16)

which can be decomposed as

V (N)σ−2

1

= N +

N

?

∞

?

m=2

m−1

?

N

?

n=1

ψm−n+

N−1

?

k=1

ψk(N − k)

+

k=1

m=1

min(k+m−1,N)

?

n=1

ψkψk+m−n.

(17)

This expression can be cast in a form without summations

by repeatedly using the properties of the independent normal

random variables ǫtin Equation (9).

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000 100000

0.0001 0.001 0.01 0.1 1

<variance after time t>

t [P]

t [Myr]

FIG. 5.— Variance in angular momentum J after time t [V(t); from Equa-

tion (18)] as a function of time in units of the initial orbital period P, with

φ1= 0.999, θ1= −0.988, 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”).

This results, after some algebra, in

V (N)σ−2

1

= N +φ1+ θ1

1 − φ1

?φ1+ θ1

?

2N − 1 +2φN

?2?

1− φ1− 1

1 − φ1

?

+

1 − φ1

N − 2φ1(1 − φN

1)

1 − φ2

1

?

.

(18)

Note that in the special case that N = 1, Equation (10) is

recovered. We plot this expression in Figure (5).

Three characteristics timescales are clearly discernible in

this figure. On a short timescale the angular momentum dif-

fusion is dominated by the usual two-body gravitational scat-

tering (i.e., non-resonant relaxation, hereafter NR). In this

regime the angular momentum deviation scales as√t. On an

intermediate timescale the orbital angular momentumevolves

duetothenearly-constantorbit-averagedseculartorquesfrom

the other stellar orbits, and thus the angular momentum de-

viation scales linearly with time. On a yet longer timescale

the secular torques fluctuate due to orbital precession, and the

√t scaling is reinstated, albeit which a much higher effec-

tive diffusion coefficient. Thus the ARMA-driven evolution

is in agreement with what is expected for a combination of

resonant and non-resonantrelaxation acting together; see also

Rauch & Tremaine (1996) and Eilon et al. (2009) for discus-

sion.

Forthe typicalvaluesofφ1and θ1that we find, these differ-

entregimescanbe identifiedin theexpressionforthe variance

V (N); Equation (18). For very short times, the expression is

dominatedbythe first term, whichcan thus be associatedwith

NR. The second term and third term are both proportional to

N2for short times (this can be verified by Taylor expansions

assumingNφ1≪ 1; thelinear termscancel), andas suchrep-

resent coherent torques; and they become proportional to N

for large time, representing RR. In Figure (5) we plot Equa-

tion (18) for typical values of the parameters of the model.

Equation (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.

Page 6

6MADIGAN, HOPMAN AND LEVIN

0

0.495

50

100

150

200

250

300

350

0.5

J

0.505

dN/dJ

0.1

1

3

10

FIG. 6.— Monte Carlo simulation of RR with the model parameters

φ1 = 0.999, θ1 = −0.988 and σ1 = 10−6(chosen for purely illustra-

tive purposes), and the time steps in units of the orbital period indicated in

the legend. After 104orbits, the distribution is very similar for all time steps

chosen, confirming that the diffusion in J is independent of the time step.

If the new process makes steps of N orbits each, we write

the new model as

∆NJt= φN∆NJt−N+ θNǫ(N)

?[ǫ(N)]2? = σ2

where the parameters [φ(δt),θ(δt)] are related to (φ1,θ1) by3

t−N+ ǫ(N)

t

;

N,

(19)

φ(δt) = φδt/P

1

;θ(δt) = −[−θ1]δt/P,

(20)

and in particular, the ARMA parameters for steps of N orbits

are

φN= φ(NP) = φN

Since after N small steps of one period,the varianceshould

be the same as after 1 single step of time N orbits, we can find

the new σ to be (see Equations 10, 18)

1;θN= θ(NP) = −[−θ1]N. (21)

σ2

N=

1 − φ2

N+ 2φNθNV (N).

N

1 + θ2

(22)

Once the ARMA parameters for steps of one orbital period

are found, they can be generalised to a new ARMA model via

Equation (19), so that the time step equals an arbitrary num-

ber of periods (not necessarily an integer number), and the

new parameters are found by use of Equations (21) and (22).

We will exploit these relations in Monte Carlo simulations

(§6), in which we use adaptive time steps. We have tested

the equivalence of the models by considering an ARMA dif-

fusion process over a fixed time interval but with varied time

steps. We show an examplein Figure(6) where we havetaken

a delta function for the initial J-distribution, and have plot-

ted the resulting final distributions after 104orbits for sev-

eral different time steps which vary by two orders of mag-

nitude. We find excellent agreement between these distribu-

tion functions which verify the generalisation of (φ1,θ1,σ1)

to (φN,θN,σN).

3For φ1we use the fact that the theoretical autocorrelation function of an

ARMA(1, 1) model decays on a timescale P/lnφ1. We have no interpreta-

tion for the functional form of θ1, which was found after some experiment-

ing. We confirm numerically that this form leads to excellent agreement with

simulations with arbitrary time; see Figure 6.

4. INTERPRETATION OF THE ARMA PARAMETERS AND

EXTENSION OF PARAMETER SPACE

The ARMA(1, 1) model presented here contains the three

freeparameters(φ1,θ1,σ1). Inthis section, weinterpretthese

parameters in the context of a stylised model of a galactic nu-

cleus. Our purpose in doing so is twofold. First, it relates the

ARMA model to the relevant physical processes that play a

part in the angular momentumevolution of stars near massive

black holes, 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 §5 we calibrate the parameters as

a function of eccentricity using N-body simulations for one

specific configuration - as specified in (§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 generalise 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 the-

oretical model of RR, but rather parametrize the model in a

simplified way that is useful for generalisation to other sys-

tems. We will use this model in our Monte Carlo simulations

(§6).

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/tNRafter one period, where

tNR= ANR

?M•

m

?2

1

N<

1

lnΛP

(23)

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 ra-

dius equal to the semi-major axis of the star, Λ = M•/m, and

ANR = 0.26, chosen to match the value of tEfrom the N-

body simulations; see Figure (3). Since this variance should

be approximately equal to σ2

1, it follows that

σ1= fσ

?m

M•

??N<lnΛ

ANR

.

(24)

fσis a (new) free parameter which, calibrated against our N-

body simulations, we expect to be of order unity (the devia-

tion from unity will quantify a difference in tEand NR com-

ponent of tJ). In our N-body simulations we find that σ1is

an increasing function of e; see Figures (12) and (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 fσas a function of e.

4.2. 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.

Page 7

SECULAR STELLAR DYNAMICS NEAR AN MBH7

The latter timescale is of importance as there exists at ev-

ery radius, an orbital eccentricity which has equal, but op-

positely directed, Newtonian precession due to the potential

of the stellar cluster, and general-relativistic (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 occurs is e ∼ 0.92 at 0.01pc. Its

coherence time, however, does not tend to infinity, as the sur-

rounding stellar orbits within the cluster continue to spatially

randomise.

The timescale for the Newtonian precession (i.e., the time

it takes for the orbital periapsis to precess by 360 degrees) is

given by

tcl

prec= π(2 − α)

(see Appendix 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

?

M•

N<m

?

P(a)f(e,α)−1,

(25)

f(e,α = 7/4)−1

?

The timescale for precession due to general relativity is given

by

prec=1

=0.975(1− e)−1/2+ 0.362(1− e) + 0.689

?

.

(26)

tGR

3(1 − e2)ac2

GM•P(a)

(27)

(Einstein 1916). The combined precession time is

tprec(a,e) =

????

1

tcl

prec(a,e)−

1

tGR

prec(a,e)

????

−1

.

(28)

As stated, thecoherencetimefora particularstaris themin-

imumofits precessiontime, andthatofthemedianprecession

time of the surrounding stars,

tφ= fφmin[tprec(a,e),tprec(a, ˜ e)],

where ˜ e is the median value of the eccentricity of the field

stars (so ˜ e =

?1/2 for a thermal distribution) and fφis a

We now have all the ingredientsto define the RR time. This

is the timescale over which the angular momentum of a star

changes by order of the circular momentum. The angularmo-

mentum evolution is coherent over a time of order tφ, and the

changein angularmomentumduringthat time is τtφ, whereτ

is torque exerted on the star’s orbit (see Equation (34) below).

Assuming that the evolution is random for longer times, this

leads to the definition

(29)

second (new) free parameter.

tRR(E,J) =

?

1

τtφ

?2

tφ.

(30)

Even if the coherence time is very long, the evolution of

the orbit may not be driven by secular dynamics. The sec-

ular torques on an orbit are proportional to its eccentricity

(Gürkan & Hopman2007), so if theeccentricityis verysmall,

secular effects are weak. As a result, the evolution is dom-

inated by two-body interactions, which have a vanishingly

short coherence time. Within our formalism, this effect can

be accounted for by multiplying tφby a function

S =

1

1 + exp(k [e − ecrit])

(31)

0

0.001

0.2

0.4

0.6

0.8

1

0.01 0.05

ecrit

a [pc]

FIG. 7.— Critical eccentricity ecritfor 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 dominate mechanism. The lower values

of ecritdecrease with distance to the MBH as RR becomes more effective (τ

and tφincrease with decreasing a). At a ∼ 0.004pc this value rises again in

our model as tNRdecreases rapidly as r−1/4. The higher values of ecritare

due to general relativistic effects which decrease tφtowards the MBH and

substantially reduce the effectiveness of RR. The lower values of ecritare

due to the fact that τ ∝ e.

which interpolates between the secular and two-body regime.

Here k > 0 is a new parameter which determines the steep-

ness of the transition (a third free parameter), and ecritis de-

finedas thecritical eccentricityat whichthe NR and RR times

are equal (Equations 23, 30),

ecrit(a,e) =

?

lnΛ

ANRA2

τ

?P

tφ

?1/2

.

(32)

Inserting relevant values into the above equation and solv-

ing numerically using a Newton-Raphson algorithm, we cal-

culate ecrit ∼ 0.3 at a = 0.01pc which is in good agree-

ment with the N-body simulations. We plot ecritas a func-

tion of a in Figure (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

(32); this is due to general relativistic effects which compete

with mass precessionand decrease the coherencetime to such

an extent as to nullify the effects of RR at high eccentrici-

ties. Theseresults are derivedfora simplifiedgalacticnucleus

model and will change for different assumptions on the mass

distribution; nevertheless we find the concept of ecrituseful

for understanding the relative importance of RR and NR at

different radii.

Since the theoretical autocorrelation function of an

ARMA(1, 1) model in Equation (11) shows that it decays on

a timescale P/lnφ1, we find that

φ1= exp

?

−P

Stφ

?

(33)

where we remind the reader that the subscript “1” is used to

emphasise that this is the value of φ for a time step of one or-

bital period. The function S appears in this equation because

even for long precession times, the relevant process may be

2-body interactions, which have a coherence time of zero.

Page 8

8MADIGAN, HOPMAN AND LEVIN

4.3. Magnitude of resonant relaxation steps: the parameter θ

For times less than tφ, there are coherent torques between

stellar orbits. Normalised to the circular angular momentum,

this torque is (Rauch & Tremaine 1996)

τ = Aτ

m

M•

√N<

P

e,

(34)

where the factor e was determined in Gürkan & Hopman

(2007) who find Aτ = 1.57. The expected variance after a

time tφis then

?∆J2

φ? = A2

τ

?m

M•

?2

N<

?tφ

P

?2

e2.

(35)

Alternativelywecanuse theautocorrelationfunction(see Ap-

pendix D for derivation) to find

?∆J2

φ? = σ2

1

?tφ

P

?2(θ1+ φ1)(θ1+ 1/φ1)

1 − φ2

1

.

(36)

Equating Equations (35) and (36) gives two values for θ1

θ1=1

2

?

−

?1

φ1

+ φ1

?

±

?

1

φ2

1

+ φ2

1− 2 +4(1 − φ2

1)τ2P2

σ2

1

(37)

?

Takingthepositiverootwe findanexcellentmatchtothedata.

For values of e > ecritthe first term1

recognising the Taylor expansion of −exp(−x) ≈ x − 1 we

simplify this expression to

2(1

φ1+ φ1) ≈ 1 and

θ1= −exp

?

−fθ

2

?

1

φ2

1

+ φ2

1− 2 +4(1 − φ2

1)τ2P2

σ2

1

?

(38)

where fθis 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 de-

terminable by our N-body simulations, which we calibrate

our model for use in MC simulations.

5. RESULTS: ARMA ANALYSIS OF THE N-BODY SIMULATIONS

We nowreturnto theN-bodysimulations,anduse thetime-

series of angular momenta that are generated to calibrate the

ARMA model. These parameters fully define both the res-

onant and non-resonant relaxation 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 §6.

When a test star has mass, there will be a back-reaction on

the field stars known as resonant friction, analogous to dy-

namical friction (Rauch & Tremaine 1996). In our simula-

tions, 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 for stars of all

eccentricities is O(10−8).

0.01

0.1

1

0.001 0.01 0.1 1

1 + θ1

1 - φ1

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

FIG. 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 text. From inspection of several in-

dividual 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 auto-

correlations for long times, similar to Figure (4). This is as expected from

Equation (11). Test stars with φ1,θ1≈ 0 have autocorrelation functions that

are close to zero everywhere, even for short times.

0.01

0.1

1

0.001 0.01 0.1 1

1 + θ1

1 - φ1

e=0.01

e=0.2

e=0.4

e=0.6

e=0.8

e=0.9

e=0.99

FIG. 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 − φ1and 1 + θ1cluster

around those for the massless cases.

In Figure (8) and (9) we show scatter plots of the model

parameters 1 − φ1and 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 speak-

ing, the relevant question is by how much the parameters

differ from (minus) unity.For intermediate eccentricities

(0.4 ? e ? 0.9), 1−φ1is much smaller than unity, indicating

that the angular momentum autocorrelations are indeed

persistent, and there are significant torques between stellar

orbits. However, the 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 − φ1and 1 + θ1confirm that although the autocorrelation

function 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.

Page 9

SECULAR STELLAR DYNAMICS NEAR AN MBH9

0.001

0.01

0.1

1

10

0 0.2 0.4 0.6 0.8 1

1 - φ1

e

massive

massless

FIG. 10.— Median value of 1 − φ1, plotted on a log scale, for both mass-

less 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 in-

crease, which in the interpretation of §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 is due to resonant friction.

0.001

0.01

0.1

1

10

0 0.2 0.4 0.6 0.8 1

1 + θ1

e

massive

massless

FIG. 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.

For very small and very large eccentricities, we find φ1≈

θ1≈ 0, so there are no persistent torques of significance. The

reasonfortheabsenceofcoherenttorquesonthe highandlow

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 very high eccentricity end,

thereare significanttorques,butthe coherencetimeis so short

duetogeneralrelativity,thatthepersistenceofthesetorquesis

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 − φ1and 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.

0.00016

0.00018

0.0002

0.00022

0.00024

0.00026

0.00028

0.0003

0 0.2 0.4 0.6 0.8 1

σ1

e

massive

massless

FIG. 12.— Data for the model parameter σ1as 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.

TABLE 2

MEDIAN VALUES OF 1 − φ1, 1 + θ1AND σ1FOR MASSLESS TEST STARS

AS A FUNCTION OF ECCENTRICITY.

e

1 − φ1

0.9916

0.9930

0.9913

0.5689

0.0064

0.0051

0.0043

0.0038

0.0468

1 + θ1

1.0037

1.0018

1.0034

0.6085

0.0240

0.0254

0.0314

0.0330

0.1556

σ1[10−4]

1.8530

1.8277

1.8658

1.9233

2.0274

2.1844

2.3792

2.4601

2.6255

0.01

0.1

0.2

0.3

0.4

0.6

0.8

0.9

0.99

In Figures (10) and (11) we compare the median values of

1 − φ1and 1 + θ1for 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 mas-

sive test stars are very small, such that we did not see a strong

feed-backeffectdueto resonantfriction,exceptforsmall stel-

lar eccentricities. Resonant friction acts to drag stars away

from circular angular momentum. The same deviation is seen

for the σ1model parameter in Figure (12).

From here on, we use only the results of the massless simu-

lations as we have a larger range of test star eccentricities and

have found the ARMA parameters (φ1,θ1,σ1) to be similar

in value. We refer the reader to Table 2 for exact quantities.

The ARMA model parameters for both very low eccentric-

ities (e ∼ 0.01) and high eccentricities (e ? 0.4) tend to clus-

ter together. For these cases one form of relaxation, either NR

or RR, dominates the form of the autocorrelation functions,

and taking mean values of (φ1,θ1) gives an accurate repre-

sentation of the population. However, near the ecritbound-

ary 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.

Page 10

10MADIGAN, HOPMAN AND LEVIN

0.001

0.01

0.1

1

10

0 0.1 0.2 0.3 0.4 0.5

e

0.6 0.7 0.8 0.9 1

1 - φ1, 1 + θ1

1 - φ1: data

fit

1 + θ1: data

fit

FIG. 13.— Median value of (1−φ1) and (1+θ1), compared with the theo-

retical values from (33, 38). 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 ecritto reconcile this difference and we find that this is

not important for our results in the next section.

0.00018

0.00019

0.0002

0.00021

0.00022

0.00023

0.00024

0.00025

0.00026

0.00027

0 0.1 0.2 0.3 0.4 0.5

e

0.6 0.7 0.8 0.9 1

σ1

data

fit

FIG. 14.— Median values of the model parameter σ1 (related to non-

resonant relaxation) and theoretical approximation, as a function of eccen-

tricity for massless test stars.

In Figures (13) and (14) we compare the experimental me-

dian values of (1 − φ1), (1 + θ1) and σ1to 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) = (0.105,1.2,60). The value

of fφshows that the coherence time tφis much less than the

precession time tprec; see Equation (29). Fitting fσas a func-

tion of e we find

fσ= 0.52 + 0.62e − 0.36e2+ 0.21e3− 0.29√e.

6. MONTE-CARLO SIMULATIONS AND APPLICATIONS

Our aims in this section are to (1) explore the long term

evolutionof stars arounda MBH, (2) investigatethe depletion

of stars around a 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

set-ups. Direct N-body simulations of secular dynamics are

challenging due to the inherent large range in timescales.

(39)

1

0.0001

10

100

1000

0.001 0.01 0.1

t [Myr]

a [pc]

tnr

trr e=0.1

e=0.2

e=0.4

e=0.6

e=0.8

e=0.9

e=0.99

FIG. 15.— RR and NR timescales in units of Myr as a function of semi

major axis a in our galactic nucleus model with m = 10M⊙for different

eccentricities.

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 Monte Carlo (MC) simula-

tionstostudylongtermsecularevolution. Forthispurposewe

have developed a two dimensional (energy and angular mo-

mentum) MC code. The main new aspect compared to earlier

work is that it implicitly includes evolution due to resonant

relaxation. The angular momentum diffusion is based on the

ARMA(1, 1) model presented in §3. For the energy evolution

we use a similar 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 power-law, 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 rel-

evant for the Galactic center (GC); see (2). We set the MBH

mass M• = 4 × 106M⊙, which affects the physics through

the GR precession rate and the loss-cone. We linearise the

system by considering the evolution of test stars on a fixed

background stellar potential. We assume a single-mass dis-

tribution 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 (2).

In Figure (15) we plot the NR and RR timescales for stars

with different orbital eccentricities as a function of semi ma-

jor axis for this model with m = 10M⊙. A cuspy mini-

mum arises in the RR timescales where general relativistic

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 =

0.007pc (see their Figure (1)), which falls precisely within

the range of values in our model (0.004 − 0.02pc). Figure

(16) shows the RR time as a function of eccentricity e for dif-

ferent stellar masses within 0.01pc. Models with different

enclosed masses will be used in §7.3.

Page 11

SECULAR STELLAR DYNAMICS NEAR AN MBH11

1

10

100

1000

0.2 0.3 0.4 0.5 0.6

e

0.7 0.8 0.9 1

trr [Myr]

N = 800

444

200

100

80

50

no GR

FIG. 16.— Resonant relaxation timescale trrfor varying stellar mass (Nm

where m = 10M⊙) within 0.01pc 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

Nm = 1000M⊙. At this radius, GR effects are important for RR and trris

not independent of N.

In reality, several stellar populations are present in the GC,

and our choice for a single mass model is therefore a sim-

plification. 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

self-consistent as a collection of interacting single mass parti-

cleswillrelaxtohavethisdistribution(Bahcall & Wolf1976).

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 (33, 38, 24). We then find the model parameters

(φN,θN,σN) for the time step δt, where N(= δt/P) ∈ R >

0, using Equations (20 - 22). Finally, the step in angular mo-

mentum is given by Equation (19)

∆NJt= φN∆NJt−N+ θNǫ(N)

?[ǫ(N)]2? = σ2

t−N+ ǫ(N)

t

;

N.

(40)

6.2. Energy

The energy of a star is given in units of the square of the

velocity dispersion at the radius of influence, v2

E = rh/2a. The step in energy during a time δt is

h, such that

∆E = ξE

?δt

tNR

?1/2

,

(41)

where ξ is an independent normal random variable with zero

mean and unit variance, and tNRis given by Equation (23)

with ANR= 0.26.

6.3. Initial conditions and boundary conditions

For our steady state simulations, we initialise stars with en-

ergiesE = 1 andangularmomentadrawnfroma thermal dis-

tribution, dN(J)/dJ ∝ J. We follow the dynamics of stars

betweenenergyboundaries0.5 = Emin< E < Emax= 104.

For our model this corresponds to 2.31pc

1.155 × 10−4pc. Stars that diffuse to E < Eminare reini-

tialised, but their time is not set to zero (zero-flow solution,

Bahcall & Wolf 1976, 1977). Stars with E > Emaxare dis-

rupted by the MBH and also reinitialised while keeping their

time. Stars have angular momenta in the range Jlc< J < 1,

where Jlcis the angular momentum of the last stable orbit.

Stars with J < Jlcare disrupted and reinitialised, while stars

with J > 1 are reflected such that J → 2 − J.

<a<

6.4. Time steps

In the coherent regime, the change in angular mo-

mentum during a time δt is of order δJ

Aτe(m/M•)√N<(δt/P). The changes in angular momen-

tum 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 Jcand J, i.e., compared to 1 − J. We

therefore define the time step to be

∼τδt∼

δt = ftτ(a,e)−1min[Jc− J,max(|J − Jlc|,Jlc)], (42)

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.

6.5. Treatment of the loss-cone

Stars with mass m and radius R⋆ on orbits which pass

through the tidal radius

rt=

?2M•

m

?1/3

R⋆

(43)

are disrupted by the MBH. The loss-cone is the region in an-

gular momentum space which delineates these orbits, and is

given by

?

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 periap-

sis. Only if floor(t/P) − floor[(t − δt)/P] > 0, do we con-

siderthestartohavecrossedperiapsisduringthelasttimestep

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 initialise a new

star, which starts at the time t at which the star was disrupted.

Jlc=

2GM•rt.

(44)

7. RESULTS

7.1. Evolution to steady state

We first consider a theoretical benchmark problem, that of

the steady state distribution function of a single mass pop-

ulation of stars around a MBH. With m = 1M⊙this solu-

tion appears after ∼ 100Gyr (roughly ten energy diffusion

timescales; See Appendix (B)). In order to highlight the dif-

ferences caused by RR relative to NR, we define the function

g(E,J2) ≡E5/4

J2

d2N(E,J2)

dlnEdlnJ2.

(45)

Page 12

12MADIGAN, HOPMAN AND LEVIN

1 10 100

E

1000 10000

0.001

0.01

0.1

1

J2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

FIG. 17.— Two dimensional distribution of stars, normalised such that for

an isothermal Bahcall & Wolf (1976) profile the distribution should be con-

stant (see Equation 45). Note that angular momentum is in units of the circu-

lar 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.

For a Bahcall & Wolf (1976) distribution without RR, this

functionis constant. In Toonen et al. (2009), simulations sim-

ilar to those presented here were used, except that angular

momentum relaxation was assumed to be NR. It was shown

that indeed g(E,J2) is approximately constant for all (E,J)

for the NR case; see Figure (A1) in that paper.

In Figure (17) we show a density plot of the function

g(E,J2) from our steady state simulations, which illustrates

several interesting effects of RR. At low energies, g(E,J2)

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 momen-

tum 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 calcula-

tions (Hopman & Alexander 2006a, see their Figure (2)). At

the highest energies considered (E > 100) most orbits accu-

mulate close to the loss-cone, and there are a few orbits pop-

ulated at larger angular momenta, with a “desert” in between.

To further illustrate the reaction of the stellar orbits to RR,

in Figure (18) we show the distribution of eccentricities of

stars in slices of energy space. As in Figure (17) we see

that for higherenergiesthe distributionis doublepeaked,with

most stars accumulating at the highest eccentricities, and an-

other peak at eccentricities around e = 0.2. This distribution

can be understoodas follows: RR is very effective at interme-

diate eccentricities, where torques are strong and the coher-

ence time is very long. As a result, the evolution of such ec-

centricitiesoccurson a shorttimescale. At higheccentricities,

the coherence time is short due to general relativistic preces-

sion, 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.

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

dN/de

e

E=3

E=10

E=30

E=100

E=300

E=1000

FIG. 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 cut-off at the loss cone.

1e-06

1e-05

0.0001

0.001

0.01

0.1

1 10 100

E

1000 10000

0.001 0.01 0.1 1

N0

-1 dNdisr/dlogE dt [Gyr-1]

r [pc]

Disruption rate per star

FIG. 19.— Disruption rate per star per logE, in units of Gyr−1, as a func-

tion of energy. The rate drops quickly as the stellar distribution is depleted

due to RR.

In figure (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 & Tremaine1999) yields that

thedisruptionratecontinuestorisewithdecreasingenergyfor

energies larger than the “critical energy4”, which in our case

is at E < 1. This is in accordance with Figure (19). As the

figure shows, the rate drops quickly; this is due to the deple-

tion of stars, and was also found (though to a lesser extent,

compare their Figure 4) in Hopman & Alexander (2006a).

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 self-consistently find

the steady state solution; we have shown that RR significantly

depletes stars at inner radii. To what extent this will affect our

steadystate solutionis beyondthescopeofthepaper. We note

the RR time does not depend on the stellar number density

until such energies and angular momenta that GR precession

becomes important.

4The critical energy is approximately the energy where the change in an-

gular momentum per orbit equals the size of the loss-cone.

Page 13

SECULAR STELLAR DYNAMICS NEAR AN MBH 13

7.2. A Depression in the Galactic Center

Early studies of integrated starlight at the Galactic center

(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; Figer et al. 2000; Genzel et al. 2003;

Figer et al. 2003; Schödel et al. 2007; Zhu et al. 2008). Re-

cent 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 a 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 0.16pc 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 stellar distribution

or simply a very shallow power law volume density profile.

Buchholz et al. (2009) show that the late-type density func-

tion 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 ∼ 0.2pc

is depleted of both bright and faint giants. These results are

confirmedby 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 nu-

clear star cluster at the center of our Galaxy.

cusp formation theory predicts a volume density profile

n(r) ∝ r−αof relaxed stars with α = 7/4 for a sin-

gle mass population (Bahcall & Wolf 1976) and has been

confirmed in many theoretical papers with different meth-

ods, including Fokker-Planck (e.g. Cohn & Kulsrud 1978;

Murphy et al. 1991), Monte Carlo (e.g. Shapiro & Marchant

1978; Freitag & Benz 2002)andN-bodymethods(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 concen-

trated.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 between two types of solutions put forward in

the literature to explain the discrepancy between cusp forma-

tion 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 mecha-

nismsincludestrongmasssegregation(Alexander & Hopman

2009) and envelope destruction of giants by stellar collisions

(see Dale et al. 2009, and references within). The second so-

lution 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 bi-

nary 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 dis-

turbance of the old stellar cluster (Trippe et al. 2008), and

Shen et al. (2010) findthat the Galaxyshows noobservational

sign that it suffered a major merger in the past 9 − 10Gyr.

Stellar

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

by a depression of stars carved out by resonant relaxation act-

ing together with tidal disruption. The underlying reason is

that stars move rapidly in angular momentum space and en-

ter 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 steady state is reached in longer than a

Hubble time, the observed depression forms on a shorter time

scale.

We perform simulations of stars, both for m = 1M⊙and

m = 10M⊙,initiallydistributedinaBahcall-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 domi-

nantstellar populationat small radii. We plotthe stellar distri-

bution function f(E), which scales as E1/4for BW cusp, at

various times during the simulation as the stars move in angu-

lar momentum space due to RR; see Figure (20). We include

a background potential as detailed in the previous section. In

∼ 1Gyr a depression is carved out of the population of stars,

significantly depleting the stellar population around the MBH

out to ∼ 0.2pc. 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?= 0) such that angular momentum

relaxation follows a random walk. In this case no depression

forms. 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 arounda MBH. We see the

same appearance of a depression when we substitute our non-

resonantparameters(bothenergyandangularmomentum)for

those of Eilon et al. (2009).

In Figure(21) we plot the numberdensity of stars as a func-

tion of radius from the MBH. We do this by sampling each

stellar orbit randomly in mean anomaly. Black lines indi-

cate the initial BW number density slope of −1.75, a slope

of −1 and that of +0.4. The slope begins to deviate, al-

beit gently, from the BW solution at ∼ 0.5pc, decreases to

−1 at ∼ 0.1pc before turning over and becoming positive at

∼ 0.01pc. We also present results from m = 1M⊙simula-

tions (see Figures (22) and (23)) although, due to mass seg-

regation, we do not expect low mass stars to dominate the

potential so close to a MBH. The major difference between

the two simulations is the timescale on which the depression

forms: 1Gyr and 10Gyr respectively.

We note that the potential in our model does not evolve in

responsetoRR. Asthedepressiondevelopsandthenumberof

stars, N, decreases, the energyrelaxationtimescale shouldin-

crease. As N is not important for the RR timescale for all en-

ergies but the largest (GR precession at small radii affects the

precession rate), this could further enlarge the depression as

the rate of stars flowing inwards towards the MBH decreases.

Indeed such a reaction seems to inevitably lead to a “hole”

in stars. One implication of this depression is that we ex-

pect no gravitational wave bursts to be observed from the GC

(Hopman et al. 2007). For the implications of the existence

of a depression of stars in the vicinity of a MBH on predicted

eventrates of gravitational waves inspirals of compact objects

(EMRIs) and bursts, see Merritt (2010).

Page 14

14MADIGAN, HOPMAN AND LEVIN

0.001

0.01

0.1

1

10

1 10 100

log(E)

1000 10000

0.001 0.01 0.1 1

f(E)

r [pc]

t = 0

250 Myr

1 Gyr

2.5 Gyr

FIG. 20.— Carving out of a depression in the distribution of stars around

a MBH for simulations in which m = 10M⊙. The top (broken orange)

line shows the stellar distribution at the start of the simulation. The thick

coloured lines show the results for RR while the corresponding (in colour

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 ∼ 0.1pc.

1

0.0001

10

100

1000

10000

100000

1e+06

1e+07

0.001 0.01 0.1

0.01 0.1 1 10

n(r)

r [pc]

r [’’]

t = 0

250 Myr

1 Gyr

2.5 Gyr

FIG. 21.— Number density of stars as a function of radius (in parsec and

arcsecond) from the MBH for m = 10M⊙simulations. The stellar distri-

bution begins to deviate from the BW solution at ∼ 0.5pc. The black lines

indicate a slope of −1.75 (top), −1.0 (bottom right) and +0.4 (bottom left).

7.3. Dynamical evolution of the S-stars

At a distance of ∼ 0.01pc from the MBH in the Galactic

center (GC), there is a cluster of young stars known as the

S-stars. This cluster has been the subject of extensive obser-

vational and theoretical work, for which we refer the reader

to Alexander (2005) and Genzel et al. (2010). The stars are

spectroscopicallyconsistent with being late 0- to early B-type

dwarfs, implying that they have lifetimes that are limited to

between ? 6 ∼ 400Myr (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 whilst the

lower constraint is a maximum age limit on a specific star

(S2/S0-2). 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.

0.001

0.01

0.1

1

10

1 10 100

log(E)

1000 10000

0.001 0.01 0.1 1

f(E)

r [pc]

t = 0

1 Gyr

5 Gyr

10 Gyr

FIG. 22.— Carving out of a depression in the distribution of stars around

a 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 precession.

1

0.0001

10

100

1000

10000

100000

1e+06

1e+07

0.001 0.01 0.1

0.01 0.1 1 10

n(r)

r [pc]

r [’’]

t = 0

1 Gyr

5 Gyr

10 Gyr

FIG. 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.

Several scenarios incorporating the formation of these

stars further out from the MBH, and transport inwards

through dynamical processes, have been proposed (e.g.

Gould & Quillen 2003; Alexander & Livio 2004; Levin2007;

Perets et al. 2007; Löckmann et al. 2008; Madigan et al.

2009; Merritt et al. 2009; Löckmann et al. 2009; Griv 2010).

Many leading hypotheses involve binary disruptions (Hills

1988, 1991). In this scenario a close encounter between a

binary system and a 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 rpof the binary’s orbit

(with combined mass mbinand semi major axis abin) must

come within the tidal radius

rt=

?2M•

mbin

?1/3

abin.

(46)

Page 15

SECULAR STELLAR DYNAMICS NEAR AN MBH15

The captured star will remain with a semi major axis acap

that scales as acap∼ (M•/mbin)2/3abin, 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 subse-

quentphaseofRR hasdriventhemtotheircurrentdistribution

(Hopman & Alexander 2006a; 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 eccen-

tricity distribution that is statistically indistinguishable to that

of the S-stars. They conclude that the S-star orbital param-

eters are consistent with a binary disruption formation sce-

nario. However, these simulations do not include general rel-

ativistic precession, which we have found to be important for

the angular momentum evolution of stars at these radii.

7.3.1. Simulations

We useourARMA code,whichincludesgeneralrelativistic

precession,to studythe evolutionofthe S-star orbitalparame-

ters within the binarydisruptioncontext. We initialise 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 dis-

tance to SgrA* of 8.0kpc (Eisenhauer et al. 2003; Ghez et al.

2008; Gillessen et al. 2009). For this distance, one arcsecond

corresponds to a distance of 0.0388pc. 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 orig-

inated from the disk with lower initial orbital eccentricities

than we simulate here. Secondly, 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 domi-

nate. We run the simulation one hundred times for each test

star to look statistically at the resulting distribution.

We simulate two different initial set-ups. The first, which

we call the continuousscenario, assumes that the tidal disrup-

tion of binaries is an ongoing process, as is most likely in the

case considered by Perets et al. (2007), where individual bi-

naries are scattered into the loss-cone by massive perturbers.

The same initial conditions also hold if low angular momen-

tum binaries are generated as a result of triaxiality of the sur-

roundingpotential. For this study we start evolving the stellar

orbitsat timesthatarerandomlydistributedbetween[0,tmax],

and continue the simulation until the time t = tmax. In our

second set-up, which we call the burst scenario, we consider

a situation in which S-stars 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 evolu-

tion simultaneously. A burst of S-star formation, over a short

period of a fewMyr, could for example result from an insta-

bility in an eccentric stellar disk (Madigan et al. 2009); we

note that the instability is even more effective for a depressed

cusp (Madigan 2010).

TABLE 3

S-STAR PARAMETERS

Model A

pa

tmax[Myr]

burstd

continuouse

burst

continuous

burst

continuous

610 20100

5e-4

0.02

.07

.16

.72

.55

0.012

2e-4

.40

.47

.23

.09

0.147

0.002

.29

.40

.32

.17

0.007

0.05

.24

.33

.46

.29

f > 0.97b

DRc

Model B

pa

burst0.002

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

continuous

burst

continuous

burst

continuous

f > 0.97b

DRc

a2D (a, e) Kolmogorov-Smirnov p-values

bFraction of S-stars with e > 0.97 at end of simulation.

cDisruption rate (fraction) of S-stars.

dStars initialised at t = 0

eStars randomly initialised between [0,tmax].

We use two different models. Model A is similar to the

galactic nucleus model discussed in §2. The stellar cusp has a

density profile n(r) ∝ r−αwith α = 1.75, and the number of

stars is normalised with N(< 0.01pc) = 222 and m = 10.

We chose this model such that it faithfully reflects the domi-

nant stellar content at a distance of 0.01pc, 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, 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(< 0.01pc) = 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 (16) for

an illustration of this effect where α = 1.75. Interestingly

for all values of N(< 0.01pc) ? 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.

7.3.2. Results

For both models we find the cumulative eccentricity distri-

bution of the stars at different times and compare them to the

observeddistributionfromGillessen et al. (2009); see Figures

(24) and (25). In Table (3) we present results from nonpara-

metric two-dimensional Kolmogorov-Smirnov (2DKS) test-

ingbetweentheobservedcumulativedistributionand thesim-

ulations. 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 to-

day.

Page 16

16MADIGAN, HOPMAN AND LEVIN

Model A: The best fit to the S-star observations are for the

burst scenario with tmax = 6 − 10Myr. Greater values of

tmaxproduce too many low eccentricity stars in our simula-

tions; 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 − 20Myr. In both cases the 2DKS p-

values≪ 1, suggestingthatnosimulateddistributionmatches

verywell the observed. Both scenarios predict a largenumber

(7 − 45%) of high eccentricity stars with e > 0.97.

Model B: The best match to the observations is the burst

scenariowith tmax= 20Myr, thoughthecontinuousscenario

has a relatively high p-value at a time of tmax= 100Myr. In

both cases a longer tmaxis 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 > 100Myr pro-

duce too many low eccentricity stars with respect to the ob-

servations. Simulations with tmax= 200− 400Myr have in-

creasingly smaller p-values for this reason. The small number

of observed low eccentricity S-stars could in principle con-

strain the time S-stars have spent at these radii (their “dynam-

ical lifetimes”). Although the S-stars can in theory have ages

up to ∼ 400Myr, we find that, in the binary disruption con-

text, their dynamical lifetimes at these radii are < 100Myr.

Furthermore, as noted by Perets et al. (2009), the lack of ob-

served low eccentricity S-stars is difficult to reconcile with

formationscenarios which bringthe stars to these radii on ini-

tially low eccentricity orbits. trris long and stars will retain

their initial values for long times.

For all values of tmaxthere is a clear discrepancy between

the lack of high eccentricity S-stars orbits observed and the

results from both formation scenarios and models. To fur-

ther illustrate this problem we show a scatter plot from our

simulations in Figure (26), of eccentricity e as a function of

semi major axis a after 6Myr. 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 the-

ory over-predicts the amount of high eccentricity orbits with

respect to the observations. Such orbits at these radii are ex-

pected; stars with high orbital eccentricities, although they

experience large torques (Equation 34), have short coherence

times (Equation 29) and hence long RR times (Equation 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).

An interesting solution to this problem is that there are S-

stars with higher orbital eccentricities but that their orbital el-

ements 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 0.01pc (see their Figure

(14)). Future observations will be able to reveal if this can

account for the discrepancy (Weinberg et al. 2005).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

N(<e)

e

Observations

Thermal

6 Myr

10 Myr

20 Myr

100 Myr

FIG. 24.— Cumulative distribution of eccentricities for the burst scenario

in which the stars’ initial time is zero. The red step function is the ob-

served cumulative eccentricity distribution of S-stars 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).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

N(<e)

e

Observations

Thermal

6 Myr

10 Myr

20 Myr

100 Myr

FIG. 25.— 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 dis-

tribution 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 ec-

centricity e are indicated in the legend. Thick (thin) lines are the results for

Model A (B).

Page 17

SECULAR STELLAR DYNAMICS NEAR AN MBH17

0

0.001

0.2

0.4

0.6

0.8

1

0.01 0.1 1

e

a [pc]

Simulation

S Stars

Disk Stars

FIG. 26.— Scatter plot of eccentricity, e, ofthe S-stars, asa function ofsemi

major axis, a, on a log scale for the Model B burst scenario. The small orange

dots represent the results of our simulations after 6Myr, 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.

8. SUMMARY

We have used several techniques to study the angular mo-

mentum 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 gen-

erate new, much-longer time series in our Monte Carlo code,

which enabled us to study the steady state distribution of stel-

lar orbits around a MBH, and the evolution of the angular

momentum distribution of possible progenitors of the S-stars.

We have shown that the steady state angular momentum

distribution of stars around a MBH is not isotropic when RR

is a dominant relaxation mechanism. For the Galactic cen-

ter this corresponds to the innermost couple of tenths of par-

secs. We note that Schödel et al. (2009) find a slight degree

of anisotropy in the velocity distribution of late-type stars in

the Galactic center within the innermost 6" (∼ 0.2pc). It is

of particular interest to note that close to the MBH, we find

that the angular momentum distribution is steeper than lin-

ear, which may have consequences for the stability of the sys-

tem. 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 flat-

tened). 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

developa “gravitational loss-cone instability”. This condition

is also satisfied for a range of energies. An assessment of the

importance of these instabilities is beyond the scope of this

work.

We have furthermore demonstrated the presence of a de-

pression in the distribution of stars near massive black holes

due to resonant tidal disruption. While it is tempting to in-

voke this mechanism to explain the “hole” in late-type stars

in the Galactic center, we stress that our fiducial galactic nu-

cleus model, a single-mass Bahcall-Wolf distribution of stars

around a MBH, is greatly simplified compared to reality. We

expect the Galactic center to be far more complex, with on-

going 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 merrit to ourapproach: evenwith

our simplified picture, we illustrate the importance of RR on

stellar dynamics close to a 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 unless there are a

number of high eccentricity S-stars whose orbital parameters

have not yet been derived. We have confirmed the result by

Perets et al. (2009) in that it is unlikely that the S-stars were

formedonloweccentricityorbits. Inaddition,we placeacon-

straint on the dynamical age of the S-stars of 100Myr from

the small number of low eccentricity orbits observed.

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 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.

APPENDIX

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 (C8)), and do not directly react to the gravitational potential from their neighbours. 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. Both test and field particles’ orbits precess due to general relativistic effects. This reduces the

force calculation from O(N2) → O(nN), where N is the number of field particles and n is the number of test particles in the

simulation.

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 (MVS) (Wisdom & Holman 1991; Kinoshita et al. 1991; Saha & Tremaine 1992),

so called due to the switching of the co-ordinate 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).

Page 18

18MADIGAN, HOPMAN AND LEVIN

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.

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 Hinteractionand produces a change of momenta in fixed Cartesian co-

ordinates. ThedriftstagedescribesmotionunderHKeplerandproducesmovementalongunperturbedKeplerellipses. Tointegrate

along these ellipses, we exploit the heavily-studiedsolutions to Kepler’s equation. This is a less computationallyefficient 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 paper we use a fourth-order integrator (Yoshida 1990) which is achieved by concatenating

second order leapfrog steps in the ratio 1:−21/3:1.

We use adaptive time stepping to resolve high eccentricity orbits at periapsis, and to accurately treat close encounters between

particles,henceourintegratoris notsymplectic. Toreducetheenergyerrorsincurredwiththeloss ofsymplecticity,we implement

time symmetry in the algorithm. 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

K2kernel (Dehnen 2001).

An additionalfeatureof the codeis the optionto implement“reflective”walls, which bouncefield particles backinto 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.

We have parallelised our code for shared-memory architecture on single multiple-core processors using OPENMP.

(A1)

B. ENERGY EVOLUTION AND CUSP FORMATION

Many different definitions of relaxation times are used in the literature. In §2.2 we considered the energy diffusion timescale

tE, defined as tE ≡ E2/DEEwhere DEE ≡ ?(∆E)2?/∆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) and Equation (5)) with a commonly

used reference time for relaxation processes defined in Binney & Tremaine (2008),

tr≡

0.34σ3

(Gm)2nln(M•/m),

(B1)

where for a density cusp with slope α,

σ2=

1

1 + α

GM•

r

(B2)

(Alexander 2003). In this expression for the relaxation time we set the Coulomb logarithm to Λ = M•/m (Bahcall & Wolf

1976). For the cusp model used in our N-body simulations, we find that

tE≡

E2

DEE

= 0.26

?M•

m

?2

1

N<

1

lnΛP = 0.56 tr,

(B3)

with tr= 611Myr, tE= 340Myr. Thus with this definition, the energy of a star changes by order unity in slightly less than a

relaxation time.

We compare our result for the energy relaxation time to that found by Eilon et al. (2009) using an entirely different inte-

grator. Their N-body simulations incorporate a 5th order Runge-Kutta algorithm with optional pairwise KS regularization

(Kustaanheimo & Stiefel 1965), without the need for gravitational softening. We insert their numerically derived value [see

their Table (1)] of ?α? ≡ ?αΛ/√lnΛ? into their expression for the energy relaxation time TE

compare with our Equation (23) with its own numerically derived parameter. We find an energy relaxation timescale which is a

factor of about two longer than in their paper. Our maximum radius at 0.03pc means we do not simulate several “octaves” in

energy space which could contribute to energy relaxation. The latter problem is not a strong effect however. In a Keplerian cusp,

the variancein the velocity changesof a star, (∆v)2, is not independentof 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 cut-off radius, r = 0.03pc, out to the radius of influence,

rh= 2.31pc, can account for at most a factor of two difference in tE.

We now consider the timescale for building a cusp due to energy diffusion. For this purpose, we use our MC code without

angularmomentumevolution. In figure(27), we show the distributionfunctionfor severaltimes, wherethe system was initialized

such that all stars start at E = 1. From this figure, we find that it takes about 10 tEto form a cusp. The result that it takes much

longer than tEis also implicitly found in Bahcall & Wolf (1976). They give the 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, one finds that

c2(E = 1) = 16/tBW

NR= (M•/m)2P/(Nα2

Λ), and

cusp, where tBW

cuspis the reference time in which Bahcall & Wolf (1976) find a cusp is formed.

Page 19

SECULAR STELLAR DYNAMICS NEAR AN MBH19

0.1

1

10

1 10 100

E

1000 10000

f(E)

α = 7/4

t = 1 tE

t = 3 tE

t = 10 tE

FIG. 27.— Stellar distribution function f(E) for single mass stars as a function of energy E for different times t = 1 − 10 tE. 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.

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 used equation (B1) and (B3) to calculate the cusp formation times in Table 1 of Preto et al. (2004) in terms of the energy

diffusion time. We found 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.

For a single mass (m = 1M⊙) GC model (see §2.1), we find that at the radius of influence, tE= 12Gyr, and a cusp forms in

∼ 120Gyr. We stress that the time for cusp formation is much longer than the relaxation time. This is in contradiction to what

is often claimed (e.g. Bahcall & Wolf 1976; Lightman & Shapiro 1977); one cause of the discrepancy is that in those models,

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. It thus appearsthat the GC cannothave reacheda steadystate (see also Merritt 2010). Eventhoughmass-segregation

may enhance the rate of cusp formation by a factor of a few (Preto & Amaro-Seoane 2010; Hopman & Madigan 2010), it seems

improbable that the GC has reached a steady state.

C. PRECESSION DUE TO POWER-LAW STELLAR CUSP

The presence of a nuclear stellar cluster in the vicinity of a 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

∂

∂LL

δφ =

?2

?π

0

r2δUdφ

?

,

(C1)

where L =

continue to use specific parameters in the following equations. The stellar mass within radius r is

?GM•a (1 − e2) is the unnormalised angular momentum of a Keplerian orbit (Landau & Lifshitz 1969). We

?r

M0, r0being proportionality constants which we normalise to the radius of influence, r0= rh, M0= M•. Thus the potential

felt by a massless particle due to this enclosed mass is

M(r) = M0

r0

?3−α

,

(C2)

δU = −

=GM•rα−3

(2 − α)

?r

0

F(r′)∂r′

h

r2−α

(α ?= 2).

(C3)

Inserting δU into Equation (C1) yields

δφ =GM•rα−3

(2 − α)

h

∂

∂L

?2

L

?π

0

r4−αdφ

?

.

(C4)

We apply a change of co-ordinates here, replacing L with normalised angular momentum J =

√GM•aJ.

√1 − e2, such that L =

Page 20

20MADIGAN, HOPMAN AND LEVIN

Expressing radius r in terms of Kepler elements, semi major axis a, eccentricity e and angle from position of periapsis φ,

r = a

1 − e2

1 + ecosφ= a

J2

1 +√1 − J2cosφ,

(C5)

yields

δφ =

2

(2 − α)

2

(2 − α)f(e,α)

?a

rh

?3−α

?N<m

f(e,α)

=

M•

?

,

(C6)

where m is the mass of a single star, N<is the number enclosed within radius r and

f(e,α) =

∂

∂J

?

1

J

?π

0

?

J2

1 +√1 − J2cosφ

?4−α

dφ

?

.

(C7)

The function f(e,α) is calculated numerically, and fit, for a given α. This returns a cluster precession time of

tcl

prec= π(2 − α)f(e,α)−1

?

M•

N<mP(a)

?

(α ?= 2),

(C8)

where P(a) is the period of an orbit with semi major axis a.

D. EQUATIONS OF ARMA(1,1) MODEL

We will use Equations (8) and (9) in our calculations; for clarity we repeat them here, dropping the label “1”,

∆Jt= φ∆Jt−1+ θǫt−1+ ǫt,

(D1)

?ǫ? = 0;?ǫtǫs? = σ2δt,s.

D.1. Variance

(D2)

?∆J2

t? = φ2?∆J2

t−1? + 2φθ?∆Jt−1ǫt−1? + 2φ?∆Jt−1ǫt?

+ θ2?ǫ2

t−1? + 2θ?ǫt−1ǫt? + ?ǫ2

t?

(D3)

Using ?∆J2

t? = ?∆J2

t−1?, expanding ∆Jt−1in the second and third terms, and applying ?ǫtǫs? = σ2δt,syields

?∆J2

which returns Equation (10):

t? =1 + θ2+ 2θφ

t?(1 − φ2) = σ2(2φθ + θ2+ 1),

(D4)

?∆J2

1 − φ2

σ2.

(D5)

D.2. Autocorrelation function

The autocorrelation function for the ARMA model as described by Equation (D1), is defined as

ρt=?∆Js+t∆Js?

?∆J2

t?

(t > 0).

(D6)

Expanding the numerator gives

?∆Js+t∆Js? = φ2?∆Js+t−1∆Js−1? + φθ?∆Js+t−1ǫs−1?

+ φ?∆Js+t−1ǫs? + φθ?∆Js−1ǫs+t−1?

+ φ?∆Js−1ǫs+t? + θ2?ǫs+t−1ǫs−1?

+ θ?ǫs+t−1ǫs? + θ?ǫs+tǫs−1? + ?ǫsǫs+t?,

(D7)

where ?∆Js+t∆Js? = φ2?∆Js+t−1∆Js−1?. Recursively expanding the ∆J terms, and again using ?ǫtǫs? = σ2

fourth and fifth terms to zero, and greatly simplifies the expression to

1δt,s, reduces the

?∆Js+t∆Js? =σ2φt(θ + φ)(θ + 1/φ)

1 − φ2

.

(D8)

Page 21

SECULAR STELLAR DYNAMICS NEAR AN MBH 21

Normalising by the variance returns Equation (11):

ρt= φt

?

D.3. Variance at coherence time tφ

1 +

θ/φ

1 + (φ + θ)2/(1 − φ2)

?

(t > 0).

(D9)

To calculate the variance at the coherence time, ?∆J2

φ?, we begin by calculating the variance after sometime t,

??

00

?t

??

t

?

n=0

∆J2

n

=

?t

?t

?τ(t1)τ(t2)?dt1dt2

= ?∆J2

t?

0

?t

0

ρ(t1−t2)dt1dt2.

(D10)

Here ρ(t1−t2)is the autocorrelation function of the torque at (t1− t2) > 0, normalised with ?∆J2

Equation (11),

t? to be comparable with

ρ(t1−t2)= φ(t1−t2)

?

1 +

θ/φ

1 + (φ + θ)2/(1 − φ2)

?

.

(D11)

From Equation (33) we can write

φ(t1−t2)= exp

?

−(t1− t2)

tφ

?

.

(D12)

Inserting this into (D10), writing

A = ?∆J2

t?

?

1 +

θ/φ

1 + (φ + θ)2/(1 − φ2)

?

(D13)

and taking an upper limit of t = tφyields

?∆J2

φ?=A

?tφ

?tφ

0

dt1

?tφ

0

exp

?

−(t1− t2)

tφ

?

dt2

=σ2

P

?2(θ + φ)(θ + 1/φ)

1 − φ2

.

(D14)

where we have approximated (e + 1/e − 2) ≈ 1.

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