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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.
Page 3
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 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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