Relativity and the evolution of the Galactic center S-star orbits

2.2 MB·English

by  Fabio Antonini

Draftversion January8,2013
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
RELATIVITY AND THE EVOLUTION OF THE GALACTIC CENTER S-STAR ORBITS
Fabio Antonini
CanadianInstituteforTheoretical Astrophysics,UniversityofToronto,60St. GeorgeStreet, Toronto,OntarioM5S3H8,Canada
David Merritt
DepartmentofPhysicsandCenterforComputational RelativityandGravitation,RochesterInstituteofTechnology, 85LombMemorial
Drive,Rochester, NY14623,USA
3 Draft version January 8, 2013
1
0 ABSTRACT
2 WeconsidertheorbitalevolutionoftheS-stars,theyoungmain-sequencestarsnearthesupermassive
n blackhole(SBH)attheGalacticcenter(GC),andputconstraintsoncompetingmodelsfortheirorigin.
a Our analysis includes for the first time the joint effects of Newtonian and relativistic perturbations
J to the motion, including the dragging of inertial frames by a spinning SBH as well as torques due
7 to finite-N asymmetries in the field-star distribution (resonant relaxation, RR). The evolution of the
S-star orbits is strongly influenced by the Schwarzschild barrier (SB), the locus in the (E,L) plane

] where RR is ineffective at driving orbits to higher eccentricities. Formation models that invoke tidal
A disruption of binary stars by the SBH tend to place stars below (i.e., at higher eccentricities than)
G the SB;some starsremainbelow the barrier,but moststars areable to penetrate it, after which they
are subject to RR and achieve a nearly thermal distribution of eccentricities. This process requires
.
h roughly 50 Myr in nuclear models with relaxed stellar cusps, or & 10 Myr, regardless of the initial
p distributionofeccentricities,innuclearmodels thatinclude adense cluster of10M⊙ blackholes. We
- find a probability of .1 % for any S-star to be tidally disrupted by the SBH over its lifetime.
o
Subject headings: blackholephysics-Galaxy:center-Galaxy:kinematicsanddynamics-stellardynamics
r
t
s
a
[ 1. INTRODUCTION initial conditions that correspondto the different forma-
tion models proposedin the literature. Evolving the ini-
ObservationsoftheGalacticcenter(GC)revealaclus-
2 tial conditions for a time of the order the lifetime of the
ter of about 20 stars, mainly main-sequence B stars,
v S-stars,andcomparingwiththe observeddistributionof
that extends outward to about a tenth of a parsec from
4 orbital elements, allows us to place constraints on both
the central supermassive black hole (SBH; Ghez et al.
9 the parameters of the nuclear cusp and the S-star origin
2008). These stars, usually referred to as “S-stars,” fol-
5
models.
4 low orbits that are randomly oriented and have a nearly
. “thermal” distribution of eccentricities, N(e)de ede
1 (Gillessen et al. 2009). The existence of such ∼young 2. GRAVITATONALENCOUNTERSNEARTHESBH
1
starssoclosetothe GCSBHchallengesourunderstand- Timescales of interest are of order 100 Myr, the main-
2
ing of star formation since the strong tidal field of the
1 sequencelifetimeofaBstar,orless. Suchtimesareshort
: SBH should inhibit the collapse and fragmentation of compared with two-body (non-resonant, NR) relaxation
v molecular clouds (Morris 1993). For this reason, it is times near the center of the Milky Way (e.g., Merritt
Xi usually assumed that the S-stars formed elsewhere and 2010; Antonini & Merritt 2012), hence we ignore NR
migrated to their current locations. However, the mi- relaxationin what follows and assume that orbital ener-
ar gration mechanisms proposed in the literature result in gies, i.e. semi-major axes a, are unchanged once a star
orbital distributions that differ substantially from what has been deposited near Sgr A∗.
is observed. Post-migration dynamical evolution due to Resonant relaxation (RR) (Rauch & Tremaine 1996;
gravitationalinteractionswithotherstarsorstellarblack Hopman & Alexander 2006) acts to change orbital ec-
holes (BHs) has been invoked to bring the predicted or- centricities in a time
bital distributions more in line with observations (e.g.,
Merritt et al. 2009; Perets et al. 2009; Madigan et al. L 2
c
2011; Zhang et al. 2012). TRR = tcoh, (1)
(cid:18) ∆L (cid:19)
TheS-starsapproachcloselyenoughtoSgrA*thatrel- | coh|
ativistic corrections to their equations of motion can be
the “incoherent RR time,” where L is the angular mo-
important. Inthis paper,we applyrecentinsightsabout c
mentum of a circular orbit having the same semi-major
how relativity interacts with Newtonian (star-star) per-
axis as the test star and t is the “coherence time,”
turbations near Schwarzschild and Kerr SBHs. Using coh
defined as the time for a typical field-star to change its
anapproximateHamiltonianformulationthatincludesa
orbital orientation; the latter is the shortest of the mass
post-Newtoniandescriptionoftheeffectsofrelativity,we
precession time (due to the distributed mass), the rela-
explore the evolution of the S-star orbits starting from
tivistic precession time (due to the 1PN corrections to
[email protected] the Newtonian equations of motion), and the time for
[email protected] RR itself to reorient orbital planes. For instance, in the
2 Antonini and Merritt
Figure 1. LocationoftheGalacticcenterS-starsonthe(a,e)plane,comparedwiththeSchwarzschildbarrier(dashedline,equation3),
and the curve along which frame-dragging torques compete with √N torques from the stars (dash-dotted lines, equation 5, with two
different values of the SBH spin χ). The three panels are for three models of the nucleus, as described in the text. The S-star data are
fromGillessenetal. (2009). Horizontal tick marks give the expected amplitude of eccentricity changes as an orbitprecesses inthe fixed
torquingfieldduetothefieldstars(equation 41fromMerrittetal. 2011).
casethatfield-starprecessionisdominatedbyrelativity, where again M• = 4 106M⊙ has been assumed. How-
×
ever, 2dRR itself ceases to be effective for orbits that
3 rg M• 2 P comesufficiently closetothe SBH,where draggingofin-
T (2)
RR≈π2 a (cid:18) m (cid:19) N(<a) ertial frames by a spinning SBH induces Lense-Thirring
precessionwithaperiodthatisshorterthanthetimefor
1/2 −2 −1
1.4 105 a m N yr 2dRR to randomize orbitalplanes. The condition for an
≈ × (cid:18)10mpc(cid:19) (cid:18)1M⊙(cid:19) (cid:18)103(cid:19) orbit to be in this regime is (Merritt & Vasiliev 2012)
where rg ≡ GM•/c2, P is the orbital period of the test 1 e2 3 a 3 . 16χ2 M• 2 (5)
star,N(<a)isthenumberoffield-starswithsemi-major − (cid:18)r (cid:19) N(<a)(cid:18) m (cid:19)
axeslessthana,misthemassofthefieldstars,andmpc (cid:0) (cid:1) g
isRmRillicpeaarsseesctso; Mbe•e=ffe4ct×iv1e0a6tMch⊙anhgaisnbgetehneaescscuemnterdic.ities wanitdhSχt≡hecSSB/(HGMsp•i2n)athnegudliamremnosimonelnetsusmsp.iWn oefdtehfienSeBaH,,
K
of stars whose orbits lie below (at higher eccentricities the “radius of rotational influence” of the SBH, as the
than) the “Schwarzschildbarrier” (SB), the locus in the value of a that satisfies equation (5) with e = 1; a
K
(a,e) plane where relativistic precession of the test star is roughly 1 mpc for the Milky Way assuming χ = 1
acts in a time shorter than the time for the field-star (Merritt et al. 2010).
torques to change L. The SB is defined approximately While the joint evolution of an ensemble of stars
by (Merritt et al. 2011) near a spinning SBH can only be convincingly treated
using an N-body code, Monte-Carlo algorithms have
1 e2 1/2 rg M• 1 . (3) beenconstructedthatfaithfully reproducethe eccentric-
− SB ≈ a m N(<a) ity evolution of single (test) stars due to the dynami-
(cid:0) (cid:1)
p cal mechanisms described above, assuming that N(a,e)
Orbitsabove(atlowerethan)the SBevolveinresponse for the field-star distribution is not evolving. In this
toRRbyundergoingarandomwalkine. Ifsuchanorbit paper, we use an algorithm similar to that described
“strikes” the SB, it is “reflected” in a time of order the by Merritt et al. (2011). The Hamiltonian that de-
coherence time and random-walks again to lower e, in a fines the test-star motion includes terms representing
time TRR, before eventually striking the SB againetc. the effects of the spherically-distributed mass (which re-
∼
Penetrationof the SB from above can occur but only on sults in precession of the argument of periastron), 1PN
a timescale that is longer than both the RR and NRR (Schwarzschild), 1.5PN (Lense-Thirring) precession due
timescales (Merritt et al. 2011). to relativity and dipole and quadruple order terms rep-
If a star should find itself below the SB, torques from resenting the torquing due to the finite-N asymmetry
the field stars are still able to change the orientation in the field-star distribution (which induces changes in
of its orbital plane (“2d RR”) even though changes in all the orbital elements). The direction of the torquing
eccentricityaresuppressed. Thetimescaleforchangesin field is changed smoothly with time and is randomized
orientation is in a time of t , as described in VB of Merritt et al.
coh
§
P M• 1 (2011).
T2dRR (4) Above the SB, where test-star precession times are
≈2π m⋆ N(<a) comparable with typical field-star precession times, the
p a 3/2 m −1 N −1/2 assumptionsunderlyingthe derivationofresonantrelax-
9.4 105 yr ationaresatisfiedandthealgorithmcorrectlyreproduces
≈ × (cid:18)10mpc(cid:19) (cid:18)1M⊙(cid:19) (cid:18)103(cid:19)
Relativity and the S-star orbits 3
Table 1
OriginmodelsfortheS-stars
pa TDsb(%)
BinaryDisruption
Burstscenarioc 5Myr 20Myr 50Myr 100Myr 200Myr
γ=0.5; m=1M⊙ 7.41×10−14 7.39×10−12 2.01×10−10 1.31×10−9 9.36×10−9 0
γ=7/4; m=1M⊙ 5.11×10−5 0.176 0.765 7.93×10−2 3.10×10−2 1.9
γ=2; m=10M⊙ 0.593 0.296 0.239 0.160 0.201 0.36
Migrationfrom
gaseousdiskc 5Myr 20Myr 50Myr 100Myr 200Myr
γ=0.5; m=1M⊙ 2.90×10−11 9.57×10−10 8.29×10−9 3.62×10−7 1.12×10−5 0
γ=7/4; m=1M⊙ 3.12×10−9 6.14×10−7 4.06×10−5 1.70×10−4 1.37×10−3 0.12
γ=2; m=10M⊙ 6.78×10−3 3.76×10−2 0.138 0.168 0.184 0.41
BinaryDisruption
Continuousscenariod 5Myr 20Myr 50Myr 100Myr 200Myr
γ=0.5; m=1M⊙ 7.21×10−13 6.41×10−13 5.32×10−13 1.33×10−13 1.267×10−13 0
γ=7/4; m=1M⊙ 3.14×10−5 0.147 0.645 0.819 0.660 0.16
γ=2; m=10M⊙ 8.07×10−2 0.108 0.410 0.499 0.310 0
aProbabilityvalueofthe2samplesKolmogorv-Smirnovtest
bPercentage ofstellartidaldisruptionsafter200Myr
cOrbitsinitializedatt=0
dOrbitsinitializedatrandomtimesbetween[0,200Myr]
Figure 2. Initial(redpoints)andfinal(blackopencircles,after200Myr)locationsofofthetestparticlesintheMonte-Carlointegrations.
Dashedanddot-dashedlinesaredefinedinFigure1;dottedlinesgivethetidaldisruptionradiusofa10M⊙ star. Starsthatinitiallyhave
largeeccentricitiesandlienear,buttotheleftof,theSBcanpenetratethebarrierandmovetotheright,wheretheyremain. Thesestars
endupwithanearlyuniformdistributionofangularmomenta. StarsthatareinitiallyrightoftheSBtendtoremainthere,thoughsome
barrierpenetration (fromrighttoleft)isobservednearacrit,thelimitingvalueofaforwhichtheSBexists.
4 Antonini and Merritt
theeccentricityevolutionpredictedbyRR,aswellasthe Figure 1 plots the S stars on the (a,e) plane, as well
“bounce”observedinN-bodyintegrationswhenanorbit as the location of the SB; the latter depends on the pa-
strikes the SB. For a test star that finds itself below the rametersdefiningthe nuclearcuspthroughequation(3).
SB,the Schwarzschildprecessiontime is shortcompared Dot-dashedlines inthe figure delineatethe regionwhere
with field-star precession times. In this regime, a test frame-draggingtorquesfromaspinningSBHwoulddom-
star precesses with period close to inate stellar torques, equation (4). Particularly for large
χ, many of the S stars lie close to this transition re-
P a
t = (1 e2), (6) gion,suggestingthat frame draggingcouldbe animpor-
GR
3 rg − tantinfluence ontheir orbitalevolution; for instance, by
inhibiting 2dRR. This figure does not give information
the 1PN apsidal precession time. During one preces-
about timescales, but we note that characteristic times
sional period, the field-star torques are nearly constant;
like T are functions of the nuclear parameters, which
as a result, the test-star’s angular momentum oscillates RR
can be important given the limited lifetimes of the S
withperiodt andwithapproximateamplitude(inthe
GR stars.
small-ℓ limit)
Inthetwomodelswithasteepcusp,theSBdelineates
∆ℓ 2ℓ2 A sini, (7a) the boundary of the S-star orbits, with only a few stars
A =≈ av 1D M⋆(<a) a . (7b) lwyhinicghbtehleowbatrhrieermeixniismtsu.mInath=esaecmrito≈del(sM,t•h/eme√xiNste)nrgcefoorf
D
3 N(<a) M• rg the SB is expected to strongly influence the eccentricity
p evolutionof at leastsome of the S stars; for instance, by
(Merritt et al. 2011). Here, ℓ2 =1 e2, M (<a) is the
⋆ limiting the maximum eccentricity attainable by a star
−
mass in stars at radii r a, and sini specifies the incli-
≤ that starts above the barrier. Furthermore, for some of
nation of the major axis of the torquing potential with
the nuclear models, Figure 1 shows that some of the S-
respecttotheorbit. Bythemselves,theseperiodicvaria-
stars can lie both above a and below the SB. Such a
tionsinedo notimply anydirectedevolutioninangular crit
location would be highly unlikely, in a time as short as
momentum,butrandomswitchingofthedirectionofthe
100 Myr, for stars that started above the SB, but is
torquingpotentialdoesresultinarandomwalkinatest ∼
reasonableifthestarswereplacedinitiallyonsuchorbits
star’sangularmomentum,allowingastarthatisinitially
via one of the mechanisms described below.
below the SB to approach it.
Dragging of inertial frames results in orbit-averaged 3. FORMATIONMODELS
ratesofchangeoftheargumentofperiastron,ω,andthe
Weconsideredtwomodelsforthe originofthe Sstars.
angle of nodes, Ω, of the test star according to:
dΩ 2G2M2 (1) Formation of the S-stars in binaries far from the
= • χ, (8a) center (r > 0.1 pc). In this model, the bina-
(cid:18)dt (cid:19) c3a3(1 e2)
FD − ries are scattered onto low-angular-momentum or-
dω 6G2M2 bits that bring them close enough to the SBH
= • cosiχ. (8b) thatanexchangeinteractioncanoccur,leavingone
(cid:18)dt(cid:19) −c3a3(1 e2)
FD − star on a tightly-bound orbit around SgrA* (Hills
In equations (8), the “reference plane” for Ω and for i 1998; Yu & Tremaine 2003; Antonini et al. 2010).
(the orbital inclination) is the SBH equatorial plane. The radius at which the SBH tidally disrupt a bi-
Sincelittleisknownaboutthedistributionofstarsand nary is typically a few tens of AU for main se-
stellar remnants near the Galactic center, we explored quence binaries, and the orbital eccentricity is ex-
a range of different models for the field-star distribu- pected to be large. The initial orbital inclinations
tion. Assuming a power-law density profile, the number of the S-stars will be either randomly distributed
of stars at radii less than r is if the binaries originated in an isotropic stel-
lar cusp (Perets et al. 2007; Perets & Gualandris
r 3−γ 2010)orhighlycorrelatediftheyformedinastellar
N(<r)=N , (9)
0.2(cid:18)0.2 pc(cid:19) disk (Madigan et al. 2009).
whereN N(<0.2pc). The parameters N ,γ,m (2) Formation of the S-stars in a disk at roughly their
0.2 0.2
≡ { }
then uniquely define the background distribution in current radius, either one of the known stellar
which the test particle orbits are evolved. In two mod- disks, or a pre-existing one. Formation in the disk
els, we set m = 1 M⊙, and we take either γ = 0.5 and wouldbe followedbymigrationto their currentlo-
N =8 104 (equation (5.247)in Merritt 2013), simi- cations. This model predicts initially small eccen-
0.2
×
lar to what is inferred from observations(e.g., Do et al. tricities and inclinations (Levin 2007).
2009),orγ =7/4andN =1.6 105 (equation(5.246)
0.2
×
inMerritt 2013),the expectedvaluesforadynamically-
relaxed population of stars (Bahcall & Wolf 1977). In
another model, we adopt N = 4.8 103, γ = 2 and
0.2
×
m = 10M⊙. This latter choice of parameters approxi-
mately reproduces the density of stellar BHs predicted
by collisionally relaxed models of a cusp of stars and
stellar remnants around SgrA* (Hopman & Alexander
2006).
Relativity and the S-star orbits 5
Figure 3. Cumulativedistributionofeccentricities fordifferentS-starformationmodelsafter5, 20, 50, 100and200Myr(linethickness
increases withtime). Opencircles givethe observed distribution; dot-red lineshows a“thermal”eccentricity distribution. Left andright
panels correspond to formation through capture following binary disruption, middle panels to formation in a stellar disk followed by
migrationthroughinteractionwithagaseous disk.
4. ORBITALEVOLUTION In all cases we set χ=1.
Stars were assumed to be tidally disrupted when
We start by assuming that N(a) is known: it is given
they approached the SBH within a distance r =
by the observed values of a. For each S-star (i.e. for t
eachvalue ofa), 100Monte-Carloexperiments werecar- 2R(M•/m)1/3(Antonini et al. 2011a),withm=10M⊙
ried out using the Hamiltonian model described above, and R=8 R⊙.
in each of the three nuclear models, for an integration Figure2comparesthe initialandfinal(after200Myr)
time of 200 Myr. The initial orbital eccentricities were a e distributions. In the two, steep-cusp models with
−
assigned from a thermal distribution, N(< e) e2, highinitial e,mostofthe starsstartoffto the left ofthe
over some specified range in e. In the case of m∝igra- SB, where evolution in angular momentum is strongly
tion from a gaseous disk we required e 0.5 initially. suppressed by the Schwarzschild precession. Neverthe-
When considering the binary disruption m≤odel, we only less, it can be seen that, after some time, an initially
considered orbits with initial eccentricities in the range eccentric population separates into two subpopulations:
0.93 e 0.99. In this case, we assume either that starsthataresofarleftwardoftheSBinitiallythattheir
the S≤-stars≤are brought to their current location at the eccentricities hardly evolve; and stars that either begin
sametime(burst scenario)orthattheyarriveatrandom rightwardof the SB, and remain there, or that are close
timesbetween0and200Myr(continuousscenario). The enoughinitiallytothebarriertocrossit. Thelatterstars
formerchoicecorrespondstoaburstofS-starformation, are subject to RR after crossing the barrier and end up
forinstanceinastellardisk(Madigan et al. 2009),while with a nearly thermal eccentricity distribution. A clear
the latter choice assumes that the S-stars form continu- “gap” between the two populations is evident in several
ously in the isotropic stellar cusp (Perets et al. 2007). of the frames of Figure 2; the gap extends from the SB
6 Antonini and Merritt
on the right, to a somewhat higher eccentricity on the The fraction of stars that would have been tidally dis-
left. The SB acts like a membrane that is permeable in rupted after 200 Myr of evolution was never larger than
one direction only, from left to right: having crossed the 1% (Table 1). As a comparison, in purely Newto-
∼
barrierfromleftto right,astarmovesquickly(ina time nian integrations, Perets et al. (2009) found that up to
of T )to a regionoflowere whereit remains 1. The 30%ofstarsweredisruptedafter20Myrofevolution.
∼ RR ∼
observed evolution below the SB is due to reorientation
ofthe torquingpotentialwhichresults inarandomwalk 5. CONCLUSIONS
inangularmomentum,preferentiallytowardlowere(see In this paper we studied the combined effects of New-
VB of Merritt et al. 2011); this mechanism is qualita- tonian and relativistic perturbations on the angularmo-
§
tivelysimilartoresonantrelaxationbutobeysadifferent mentum evolution of the Galactic center (GC) S-stars.
set of relations (Alexander & Merritt 2012). For the first time we have shown that the a e distri-
−
Ontheotherhand,wheneccentricitiesareinitiallylow, bution of the S-stars predicted by the binary disruption
asinthe bottompanels ofFigure2,they tend to remain model, in which the stars are delivered to the GC on
low,i.e.,abovetheSB.AsnotedinMerritt et al. (2011), high-eccentricity orbits, is consistent with the observed
the permeability of the SB tends to increase near a orbits even when relativistic effects are considered. In
crit
and this can be seen in Figure 2 as well. these formation models, most of the orbits lie initially
Time evolution of the cumulative distribution of ec- below the Schwarzschild barrier, the locus in the (a,e)
centricities is shown in Figure 3. (In this figure and plane where resonant relaxation is ineffective at chang-
in the analysis that follows, we do not include S- ing eccentricities. Contrary to this basic prediction, we
stars that likely belong to the disk(s) of O/WR stars, foundthatorbitsstartingsufficientlyclosetothe barrier
i.e. S66, S67, S83, S87, S96, S97). Eccentricity distribu- are sometimes able to penetrate it, diffusing above and
tionswerefoundtoapproachanearly“thermal”formin reachinganearlythermale-distribution;asmallfraction
a time of order T . Based on Figure 3, we see that the ofstarsremainconfinedbelowthebarrieratlowangular
RR
lifetime of the S-stars may, or may not, be long enough momenta (e & 0.95). A good match to observations is
for this to happen, depending on the nuclear model. We achievedafter 20Myrofevolutionifthedistributionof
∼
compared the results of the Monte-Carlo integrations fieldstarsattheGCfollowsanearlydynamicallyrelaxed
with S-star data by performing 2-sample Kolmogorov- form. Models thatinclude amass-segregatedpopulation
Simornov (K-S) tests on the e distributions (Table 1). of stellar BHs also generate after 10 Myr of evolution
∼
The best match to observations is attained after 20 Myr distributions that are marginally consistent with obser-
ofevolutioninstellar cuspmodelsinthecontinuoussce- vations.
nario with γ = 7/4 and starting from initially high ec- Based on the origin models considered here, the
centricities (K-S test p-values of 0.7). Integrations S-star obits can only be reproduced by postu-
≈
that include stellar BHs also generate orbital distribu- lating dynamically relaxed states (i.e., steep den-
tions which are in agreementwith observationsafter ap- sity cusps) for the GC. This result is interesting
proximately 5 and 20 Myr of evolution for initially high given that such models are currently disfavored by
and low eccentricity distributions respectively. observations (Buchholz et al. 2009; Do et al. 2009)
We tested the degree of randomness of the orbital and by some theoretical arguments (Merritt 2010;
planes using the Rayleigh statistic (Rayleigh 1919), Antonini et al. 2011b; Gualandris & Merritt 2012).
R
defined as the resultant of the unit vectors l , i =
i
1...Nmc, where li is perpendicular to the orbital plane DM was supported in part by the National Science
of the ith star and Nmc is the total number of Monte- Foundation under grant no. 08-21141 and by the
Carlodatapoints(i.e.,Nmc =1900). Sincetheteststars National Aeronautics and Space Administration under
were initialized with the same inclination, the orbits are grant no. NNX-07AH15G.
initially strongly correlated and N ; over a time
mc
R ≈
of order T2dRR, 2dRR tends to randomize the orbital REFERENCES
planes and approaches √N , the value expected for
mc
R
a random distribution. When the main contribution to
Alexander,T.,&Merritt,D.2012,inpreparation
dynamicalrelaxationcomesfromstellarBHs, reached
Antonini,F.,Faber,J.,Gualandris,A.,&Merritt,D.2010, ApJ,
R
valuesconsistentwithisotropyafter 20 and 50Myr 713,90
∼ ∼
for initially high and low eccentricity distributions re- Antonini,F.,Lombardi,J.C.,Jr.,&Merritt,D. 2011a, ApJ,
spectively. In the stellar-cusp model with γ = 7/4 and 731,128
Antonini,F.,Capuzzo-Dolcetta, R.,Mastrobuono-Battisti,A.,&
staring from initial high e, reached values consistent
R Merritt,D.2011b,ApJ,750,111
with isotropy after 100 Myr, while in all the other Antonini,F.&Merritt,D. 2012,ApJ,745,83
∼
stellar-cusp models we measured a significant departure Bahcall,J.N.,&Wolf,R.A.1977,ApJ,216,883
from randomness ( /N & 0.2) even after 200 Myr Buchholz,R.M.,Sch¨odel, R.,&Eckart,A.2009,A&A,499,483
mc
of evolution. We esRtimated separately for orbits that Do,T.,Ghez,A.M.,Morris,M.R.,Lu,J.R.,Matthews,K.,
R Yelda,S.,&Larkin,J. 2009,ApJ,703,1323
anytimewerebelowthedot-dashedlinesofFigure2and
Hills,J.G. 1988,Nature,331,687
found that this population had a distribution of orbital Hopman,C.,&Alexander,T.,2006,ApJ,L645
planeswhichwaslessconsistentwithbeingrandom. Evi- Ghez,A.M.,etal. 2008,ApJ,689,1044
dently,2dRRwassomewhatinhibitedbyframedragging Gillessen,S.,Eisenhauer,F.,Trippe,S.,Alexander,T.,Genzel,
for orbits with initially large eccentricities. R.,Martins,F., 2009,ApJ,692,1075
Gualandris,A.,&Merritt,D. 2012,ApJ,744,74
Levin,Y. 2007,MNRAS,374,L515
1 This behavior has been confirmed via direct N-body integra- Madigan,A.,Levin,Y.,&Hopman,C. 2009,ApJ,697,L44
tions. Madigan,A.,Hopman,C.,&Levin,Y. 2011, ApJ,738,99
Relativity and the S-star orbits 7
Merritt,D.,Gualandris,A,&Mikkola,S. 2009,ApJ,693,35 Morris,M.1993,ApJ,408,496
Merritt,D.,2010,ApJ,718,739 Perets,H.B.,Hopman,C.,&Alexander,T. 2007,ApJ,656,709
Merritt,D.,Alexander,T.,Mikkola,S.,&Will,C. 2010,PhRvD, Perets,H.B.,Gualandris,A.,Kupi,G.,Merritt,D.,&
81,062002 Alexander,T. 2009,ApJ,702,884
Merritt,D.,Alexander,T.,Mikkola,S.,&Will,C. 2011,PhRvD, Perets,H.B.,Gualandris,A.ApJ,719,220
84,4024 Rauch,K.P.,&Tremaine,S. 1996,NewAstronomy,1,149
Merritt,D.,2013,DynamicsandEvolutionofGalacticNuclei Rayleigh,L. 1919,Phil.Mag.,37,321
(Princeton: PrincetonUniversityPress),p.337. Zhang,F.,Lu,Y.,&Yu,Q. 2012,arXiv:1210.1901
Merritt,D.,&Vasiliev,E.2012,arXiv:1208.6274 Yu,Q.,&Tremaine,S. 2003,ApJ,599,1129