Table Of ContentAstron.Nachr./ANxxx(xxxx)x,xxx–xxx
Weighing the cusp at the Galactic Centre
N.Mouawad1, A. Eckart1,S. Pfalzner1, R. Scho¨del1,J. Moultaka1, andR. Spurzem2
1 I.PhysikalischesInstitut,Universita¨tzuKo¨ln,Zu¨lpicherStr.77,50937Ko¨ln,Germany
5 2 AstronomischesRecheninstitut,Mo¨nchhofstr.12-14,69120Heidelberg,Germany
0
0
2 Received<date>;accepted<date>;publishedonline<date>
n
a
J
4
Abstract.Asstarsclosetothegalacticcentrehaveshortorbitalperiodsithasbeenpossibletotracelargefractionsoftheir
3 orbitsintherecentyears.PreviouslythedataoftheorbitofthestarS2havebeenfittedwithKeplerianorbitscorrespondingto
v amassiveblackhole(MBH)withamassofM =3-4 106M implyinganinsignificantcuspmass.However,ithasalsobeen
8 shownthatthecentralblackholeresidesinaBH1”dia×meters⊙tellarclusterofaprioriunknownmass.Inasphericalpotential
3 whichisneitherKepleriannorharmonic, orb∼itswillprecessresultingininclinedrosettashapedtrajectoriesonthesky.In
3 thiscase, theassumptionofnon-Keplerian orbitsisamorephysicalapproach. Itisalsotheonlyapproachthrough which
2
cuspmassinformationcanbeobtainedviastellardynamics ofthecuspmembers.Thispaper presentsthefirstexemplary
0
modellingeffortsinthisdirection.UsingpositionalandradialdataofstarS2,wefindthattherecouldexistanunobserved
4
extended mass component of several 105M forming a so-called ’cusp’ centered on the black hole position. Considering
0
onlythefractionofthecuspmassM wi⊙thintheapo-centeroftheS2orbitwefindasanupperlimitthatM /(M +
/ S2apo S2apo BH
h M ) 0.05.Alargeextendedcuspmass,ifpresent,isunlikelytobecomposedofsub-solarmassconstituents,butcould
p beSe2xappola≤inedratherwellbyaclusterofhighM/Lstellarremnants,whichwefindtoformastableconfiguration.
-
o
r Keywords:GalacticCentre–Stellardynamics–Stellarcluster–Cusp–BlackHole
t
s
a c0000WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim
(cid:13)
:
v
i
X 1. Introduction largeground-basedtelescopesoffertheopportunitytoobtain
r anunprecedentedview ofthe Galactic centre.Initially,with
a
Over the last decade, evidence has been found for the exis- speckle imaging techniques and lately with adaptive optics
tence of massive black holes(MBH) in the centresof many techniques, high angular resolution images on the Galaxy’s
nearby galaxies. With increasing observational data – 12 central cluster were obtained. This first set of observations
MBHs candidates detected until 1995 (Kormendy & Rich- wasable tomeasurestellar motionsonthe planeofthe sky,
stone 1995),andmorethan 37until2001(Kormendy2001, yieldingestimatesoftheprojectedvelocities(Eckart&Gen-
Ferrareseetal.2001)–itisarguedthatmostgalaxiesharbor zel1996;Ghezetal.1998),projectedaccelerations(Ghezet
nucleidominatedbyMBHswithmassesthatrangebetween al. 2000;Eckartet al. 2002a),andthree-dimensionalorbital
106and109.5solarmasses. motions(Eckartetal.2002a,Scho¨deletal.2002,2003;Ghez
et al. 2003b), which each provided a successively stronger
Locatedata distanceofonly 8kpcfromthesolarsys-
∼ caseforasuper-massiveblackholeatthecentreoftheMilky
tem(Reid1993;Eisenhauer2003),theGalacticCentre(GC)
WayanditsassociationwiththeunusualradiosourceSgrA*
istheclosestandthereforebestobjectforinvestigatingphysi-
(Loetal.1985).
calprocessesinthegalacticnucleusofatypicalspiralgalaxy.
It offers a unique “laboratory” for studying stars and gas More than a decadeof high-resolutioninfraredobserva-
in the sphere of influence on a super-massive black hole, tionsofpropermotionsintheGC,withtheESONewTech-
(e.g., Genzel, Hollenbach, & Townes 1994; Morris & Ser- nologyTelescope(NTT)andtheESOVeryLargeTelescope
abyn 1996; Mezger, Duschl, & Zylka 1996; Melia & Fal- (VLT)(Eckart& Genzel1996;Eckartetal. 2002a;Scho¨del
cke 2001), with a degree of detail that cannot be accessed etal.2002,2003),aswellaswiththeKecktelescope(Ghez
in anyother galacticnucleusin the foreseeablefuture.With et al. 1998,2000,2003b),have revealedat least 6 stars that
the now available high sensitivity and angular resolution, showsubstantialaccelerationduetothesuper-massiveblack
holeassociatedwithSgrA*andareonboundorbitsaround
Correspondenceto:[email protected] it. With a peri-centre of less than 0.6 mpc (15 mas) and an
2 MODELLINGTHESTELLARCLUSTERANDTHEENCLOSEDMASS
orbital periodof 15 years, S2 is the most striking case of assumptionofEckartetal.(2002a),Scho¨deletal.(2002),and
∼
these. Ghezetal.(2003a)thatthe M/Lratioat2µminthecuspis
AseriesofobservationswiththeNAOS/CONICAadap- aslowasintheouterstellarcluster(M/L=2M /L ).
⊙ ⊙
tiveopticssystem/near-infraredcameraattheESOVLTunit Consideringthesecircumstances,starswithshortorbitalpe-
telescope 4 that covered the peri-centre passage of the star riods,inparticularthestarS2,ofwhichorbitaldataarebest
S2 aroundSgr A*allowedScho¨deletal.(2002)to approxi- known,play a key role in exploringthe gravitationalpoten-
mateaKeplerianorbitandtomeasuretheencloseddarkmass tial.WeshowthatthepresentobservationaldataonS2cannot
downtoadistanceof 0.6mpcfromSgrA*.Withtheseob- discriminate between a Keplerian and a non-Keplerian po-
servations,theycould∼excludeaneutrinoballscenario(Mun- tential.Subsequently,we studythe influenceofan extended
yaneza&Viollier,2002)asanalternativeexplanationforthe distributionofdarkmassnearSgrA*takingintoconsidera-
darkmassconcentration.Theyexcludedaswell a clusterof tiontheconstraintssetbytheorbitofS2aswellasthelimits
darkastrophysicalobjects(Maoz1998)suchasneutronstars, set on the total enclosed mass at larger radii. In this study
leavingacentralsuper-massiveblackholeasthemostprob- relativisticeffectsareneglectedastheyaresecondordercor-
ableexplanation. rections.
UsingtheKeck10mtelescope,Ghezetal.(2003a)mea- In 2 we outline our modelling of the density distribu-
§
sured a Keplerian orbit for the star S2 which agreed with tion of the centralregionandshow that the star countsnear
the results of Scho¨del et al. (2002). Additionally, Ghez et SgrA*canbedescribedbyacompactPlummermodelcore.
al.(2003a)reportedthefirstdetectionofspectralabsorption Themethodusedtocomputeorbitsinnon-Keplerianpoten-
lines(bothBrγ(2.1661µm)andHeI(2.1126µm)),provid- tialsisexplainedin 3.In 4possibleorbitalmodelsforS2
§ §
ing the first line-of-sight velocity measurements of star S2. arepresented.Wediscusstheimplicationsofthelimitsonthe
These measurementsresolved the ambiguity on the inclina- cuspmassderivedfromthenon-Keplerianorbitmodellingin
tionoftheS2orbitindicatingthatitspositionwasbehindthe 5,anddrawourconclusionsin 6.
§ §
blackholewhenitpassedthroughitsperi-centre.Inaddition,
stellarrotationalvelocitiessuggestthatS2isanO8-B0dwarf
2. Modellingthestellarclusterandthe enclosed
starandthusamassive( 15M )youngstar( 10Myrs).
∼ ⊙ ≤ mass
FromdatatakenwithNAOS/CONICAandthenewNIR
integral field spectrometer SPIFFI at the ESO VLT, Eisen-
In this section it is described how the most recent observa-
haueretal.(2003)reportednewastrometricobservationsand
tionsoftheGCstarcluster(Genzeletal.,2000,2003)canbe
additionalspectroscopicobservationsofthestarS2,reducing
usedtomodelthemassdistributionfunctionascloseasafew
theuncertaintiesontheorbitalparameters.Theyalsogavethe
milli-parsecsfromSgrA*.Asurfacedensityprofileisfitted
mostaccurateprimarydistancemeasurementtothecentreof
tonumberdensitycountsandused,togetherwithdynamical
the Milky Way of 7.94 0.42 kpc, which is in agreement
± massestimators,todeducethe3dimensionalmassdistribu-
withearlierdeterminations(seeReid1993).
tionfunction.Byfollowingthisapproach,wehaveassumed
In this paper,we explorethe possibility thatthere exists
thatthevisiblestarstracethemassdistributionofthecluster.
acompact,continuousmassdistributioncomposedofseveral
undetectable faint stars or perhaps some more exotic mate-
rialinadditiontothepointmassofSgrA*.Inthiscase,the 2.1. Densitymodelofthestellarcluster
orbitofS2willnotfollowasimpleKeplerianorbit,butwill
rathershowperi-centre-shiftsthatresultinrosettashapedor- Genzel et al. (2003, see their Fig.7) determined the surface
bits.Thisideaismotivatedbytheobservationthatthestellar number density of stars in the central parsec by counting
densitydoesnotflattenout,butexhibitsasteeppeaktowards the sources in annuliwith increasingradiusaroundSgr A*.
thecentre,aso-calledcusp(Eckartet.al, 1995;Alexander, Within 2arc-secondsofSgrA*,theirdataindicateanex-
∼
1999;Genzelet al., 2003).In contrastto earlierstudies, the cessoffaintstarsabovethecountsexpectedinaclusterwith
main approach of this work is that the mass-to-light ratio, aflatcore,asithadbeenalreadysuggestedearlier(Eckartet
M/L, is not considered to be constant over the entire range al. 1995;Alexander1999).Theyconfirmedthe existenceof
oftheGCstellarcluster.Indeed,thereareindicationsthatthe a ”cusp”and showedthatit is centredata positionof (∆α,
stellar population varies with position and is not quite well ∆δ)=(+0.09”,-0.15”)fromSgrA*,withanuncertaintyof
±
mixed (Alexander, 1999). Furthermore, the exact composi- 0.2”.HereweusethesameK-bandcounts(K 18)together
≤
tionofthecuspisstillunknownandourcurrentunderstand- withtheearlierSHARP/NTTcounts(K 15),withouttaking
≤
ingofthestellardistributionintheGCisincomplete(Genzel intoourstudytheirhighermagnitudelimitH-bandcounts(H
et al. 2003). With currentobservations, low-mass stars (K 19).Incontrasttothe’by-eye’H-bandcounts,theK-band
21mag)cannotbeobservedinthesedensecuspregionsan≥d ≤counts,weredeterminedbyanautomatedprocedure,theyare
thetruevalueoftheM/Lratioisnotknown.Also,aspointed thereforepossiblyless biased,andcoverthe wholerangeof
out by Baumgardt et al. (2003), dynamical evolution of a thecluster,notonlythecentralarc-second.
densestellarclusterwillresultinastrongincreaseofM/Lra- Genzel et al. (2003) fitted a broken power-law to their
tiobysegregationofstellarevolutionremnantstothecentre. stellarcounts,withadistinctbreakat10arcsec.Inthissec-
Ontheotherhand,theapproximationthatthedynamicsinthe tion,weshowthatitisalsopossibletofitthenumberdensity
central region is Keplerian is directly related to the implicit countswithasuperpositionofseveralPlummermodels.The
2.2 Theenclosedmass 2 MODELLINGTHESTELLARCLUSTERANDTHEENCLOSEDMASS
Table 1. The parameters of the ’composite model’ used to
fit the GC stellar cluster. For each Plummer model we give
the value of the core radius R (in arcsec and in parsec) de-
i
ducedfromthesurfacedensityplot. M isthecorresponding
i
total mass scaled to the enclosed mass distribution (Fig. 2).
M isthecontributioninmassinsideaspherewhichhasa
apoi
radiusequaltotheapo-astronofS2.ρ isthecalculatedaver-
i
agedensityinM .pc 3.ThefirstPlummermodel(R =0.4”
− 1
⊙
or0.015pc)correspondstotheinnercuspcenteredatSgrA*.
Theadditionoftheremaining5modelsrepresentsthedensity
distributionoftheouterstellarcluster.
i R[arcsec] R[pc] M[M ] M [M ] ρ(0)
1 0.i4 0.i015 85i40 ⊙ 10a8p0oi ⊙ 6.i0 108
×
2 13 0.5 1.8 106 10 3.5 106
× ×
3 52 2.0 2.7 106 0.25 7.9 104
× ×
4 97 3.8 6.7 106 0.01 2.9 104
× ×
5 220 8.5 13.1 106 0.002 5.1 103
× ×
6 321 12.4 27.6 106 0.001 3.4 103
× ×
well;sixwasthesmallestnumbersofcomponentswithwhich
wewereabletorepresentthedata,i.e.thecuspplusthestellar
cluster.
Fig.1. Surface density of stars vs. distance from Sgr A*.
The downward pointing triangles represent the CON- 2.2. Theenclosedmass
ICA/NAOS data for K 18, the upward pointing triangles
s
≤ Similarly to eq.3, the mass distribution of the ’composite
are SHARP/NTT data for K 15 stars scaled up by a fac-
s
≤ model’isgivenby
torof5inordertomatchthefainterNAOS/CONICAcounts.
ThedataaretakenfromGenzeletal.(2000,2003).Thelong-
6 Mr3
dashedlineisthesuperpositionofthe5outermostPlummer M(r)=X i . (4)
(r2+R2)3/2
models (see, Table 1). The addition of the innermostPlum- i=1 i
mermodel(the’inner’cusp’)givesthe’compositemodel’.It
Undertheassumptionthatthevisiblestarstracethemass,
isrepresentedbythe3othercurves,wherethethick-straight
the M can be determined by fitting eq.4 to the dynamical
linecorrespondstothemiddlevaluesofthe’innercusp’pa- i
enclosedmassestimates.
rametersR andM ,thethin-straightlinetoitslowervalues,
1 1
InordertocalibratetheenclosedmassM(r)weonlyrely
and the short-dashed line to its upper values. and the short-
onasolidmeasureofthecentralmass(within0.001pcfrom
dashedlinetoitsuppervalues.
SgrA*) derived from the Keplerian orbit of the star S2 and
mass estimates of the entire stellar cluster at distances of
Plummer model has the advantage of being analytically in- severalparsecsfromthecenter.Attheselargeradiithepub-
tegrable.Itresemblesactualclusterswithcompactcoresand lished results of all differentstatistical mass estimators give
anextendedouterenvelope(Spitzer1975). veryconsistentresultswithin uncertaintiesthatare fullyac-
Inthismodelthe3dimensionalradialdensitydistribution ceptableforourstudy(Eisenhauer2003,Genzeletal.2000,
ρ(r)andtheprojectedmassdensityσ(r)aredescribedby: Serabyn&Lacy1985;Gu¨stenetal. 1987;Lacyetal.1991;
3M 1 Herbstetal.1993;Roberts&Goss1993).Bytyingourmass
ρ(r) = , (1)
4πR3(1+ Rr22)25 msitoyddelilsitnrigbuatsioanf,uintcdtoioensnooftraddeipuesntdootnhepostseslilbalrenaunmisboetrrodpeines-
4ρ(0)R 1
σ(r) = , (2) of the velocity field at radii of less than a parsec. It is only
3 (1+ Rr22)2 therewheretheanisotropiesareapparentlypresentatacon-
whereRisthecoreradiusandMthetotalmass. siderable level (Genzel et al. 2000, Scho¨del et al. 2003). In
Ourbestsurfacedensityfittothedataconsistsofasuper- particular we would like to point out that we are not fitting
positionof6differentPlummermodels,whichwillbetermed toenclosedmassestimatesatthesesmallradii.InFig.2these
inthefollowingas’compositemodel’: are only shown for completeness.The only assumption that
6 M R2 goes into our considerations is that of a constant M/L ratio
σ(r)=X i i . (3) asafunctionofdistance.Thisassumptionisrelaxedanddis-
i=1 π (r2+R2i)2 cussedinthefollowingsectionsofthepaper.
Thesixvaluesofthecoreradii,R,aregiveninTable1. Our best fit to these dynamical mass estimates corre-
i
FitswithlargernumbersofPlummermodelsarepossibleas spondsto a 2.9 106 M 0.3 106 M black hole plus the
× ⊙ ± × ⊙
3 MODELLINGTHEORBITOFS2
massdistributionofeq.4,ofwhichthe6totalmassesM,are
i
listed together with their correspondingdensities, ρ, in Ta-
i
ble1.Anoffsetinthevalueoftheenclosedcompactmasswill
not influence our modelling significantly because the shape
and massof ourclustercompositemodeldependsprimarily
ontheprofileofthestellarnumbercounts.
Thecombinationofthefiveoutermostcomponentsgives
a similar configurationas the flattened isothermalsphere of
core radius 0.3 pc derived by Genzel et al. (2003) (see
∼
massdensitiesbelow).Fig.1showsthatthereexistsamini-
corecenteredonSgrA*,whichwedenominatebythe’inner
cusp’.ItisrepresentedbytheinnermostPlummermodelwith
acoreradiusR =15.5mpc(cid:27)0.4”.
1
With an apo-centre of approximately 9 mpc (Scho¨del,
2004; Ghez et al., 2003a), the star S2 orbits in a region in-
side the core radius (R 15mpc) of this innermost model.
1
∼
Therefore its path can be mainly influenced, in addition to
the BH potential, by the ’inner cusp’ gravitational potential
(seecolomn5inTable1).Atthesedistancesfromthecentre,
thepresentnumberdensitycountshavelargeerrorbars(see
Fig. 1). Thelimits onthe coreradius, R , of our model,de-
1
ducedfromtheerrorbarsonthenumberdensitycounts,are Fig.2. Massdistributioninthe central10pc ofthe Galactic
foundtobe13.2mpcforthelowervalueRmin,and20.2mpc Centre for an 8 kpc distance (Reid et al. 2003; Eisenhauer
1
fortheupperoneRmax.Varyingthemass, M ,ofthecentral et al., 2003).The star point denotes the mass derived from
1 1
Plummer model component allows to study cases in which the orbit of S2. The error bar combines the orbital fit and
themass-to-lightratio, M/L,variesasafunctionofdistance astrometryerrors(Eisenhaueret al. 2003).Filled circles are
fromSgrA*.Asdescribedin 3and 4,exploringthepos- mass estimates from a parameterised Jeans-equation model
§ §
siblerangesforR andM thatfulfilltheconstraintsgivenby of the radial velocities of late type stars, assuming isotropy
1 1
theorbitalmeasurementsofstarS2allowstoderivedynami- (Genzel et al. 2000, Scho¨del et al. 2002). We omitted all
calinformationonthecuspmass.Inthefollowing,wemake massestimatorsatradiioflessthen1”sincetheyareeffected
use ofthismodelwhichfixesthedensityprofileofthecusp bythe surface mass densityof the cusp whichhad notbeen
tothestellarprofile. takenintoaccountinpreviousinvestigations.Filledupward-
InFig.2thelong-dashedlinecorrespondstoourbestfit, pointingtrianglesdenoteLeonard-Merrittprojectedmasses-
thethick-straightlineshowsonlytheenclosedmasswithout timators from a new NTT proper motion data set (Scho¨del
a BH obtained with our model of the extended cluster. The et al. 2002).Filled down-pointingtriangles denote mass es-
thin-straight and the short-dashed lines represent the above timatesfromtheionizedandneutralgasdynamics(Serabyn
discussedextremecasesofaninnercuspradiusR =Rmin = & Lacy 1985; Gu¨sten et al. 1987; Lacy et al. 1991; Herbst
1 1
13.2 mpc with M = 8540 and R = Rmax = 20.2 mpc with etal.1993;Roberts&Goss1993).Acomparaisonofrepre-
1 1
M =8820 .Notethatthes⊙eerrorsresultfromtheuncertain- sentativemassmodelstothemeasuredvelocitydispersionsis
ties of the⊙stellar numbercountsat the smallest distances to givenin Genzeletal. (1996,2000).The long-dashedline is
SgrA*(Fig.1). afittothemeasuredenclosedmassagainstdistance.Thatre-
The density of the ’inner cusp’ is a few hundred times sultsfromacombinationofthemassdistributions(see 2.2)
§
larger than that of the second Plummer model and it has a deduced from the composite model fit to the stellar cluster,
33timessmallercoreradius(seeFig.1andTable1).Our plus a 2.9 106 M black hole. The same model without a
∼modelimpliesdensityvaluesashighas1.68 108M pc 3at point mass×is repre⊙sented by the 3 other curves, they corre-
−
0.1”,and1.54 107M pc 3at1”.Thesevalu×esareco⊙mpara- spond to acceptable models of the stellar cluster within the
−
bletotheones×derived⊙byGenzelatal.(2003)fromabroken errorbars.
power-lawdensity profile(7 108 M pc 3 at0.1”and3 107
−
M pc 3at1”). × ⊙ ×
⊙ − thispurpose,wechoseafourthorderHermiteintegratorde-
Inthefollowingwewillusetheabovederivedmassdis-
rivedfromtheoneusedinhigh-accuracyN-bodysimulations
tributiontodeterminethepathofS2inresponsetothisnon-
(Aarseth1999,Spurzem1999,foranintroductionoftheHer-
pointmasspotentialplusthecentralblackhole.
mite scheme see Makino & Aarseth 1992). The advantage
of the Hermite scheme is that it allows a fourth order accu-
3. Modellingthe orbitofS2 rateintegrationbasedononlytwotimesteps.Thereforeitre-
quiresthe analyticcomputationof thetimederivativeofthe
In order to fit non-Keplerian orbits to the measured time- gravitational force. This is the point where the integrability
dependentpositionsofthestarS2intheabovedescribedtype of the Plummer model is convenient.We computed the tra-
of mass potential,one needsa high accuracyintegrator.For jectoryofS2aroundtheBH andthroughtheextendedmass
4 RESULTSOFNON-KEPLERIANFITS
Table2.Characteristicχ˜2 ofourfits.Thefirstcolumnlists and fitted to the data. In the following, we give a short
tot
theobtainedcharacteristicχ˜2 forwhichtheresultingorbits descriptionofthisparameterspace:
tot
areconsideredtobefittingorbits.The2nd columnexpresses
thetotalnon-reducedχ2tot for33degreesoffreedom(χ˜2tot = The total mass Mtot, together with the the three veloc-
χ2tot/33).When column3 lists the deviationof the obtained ities and three coordinates determine the orbital elements.
χ2 fromthelowestobtainedχ2 ,column4correspondstheir Fromtheimagingdata,skypositionsandpropermotionsare
tot tot
resultingσerrorbars. given in a certain error range for each epoch. For a chosen
initialepoch,thesesixinputvaluesarevariedwithintheirob-
χ˜2 χ2 ∆χ2 σ servationaluncertainties.Ourminimizationmethodisagrid
tot tot tot
0.68 22.5 0 <1σ scheme,which,althoughexpensiveincomputingtime(time
0.71 23.5 1 1σ N6,Nbeingthenumberofstepsperparameter),isarobust
0.80 26.5 4 2σ ∝methodwhichdoesnotfalleasilyonlocalminima.Choosing
0.95 31.5 9 3σ an appropriate N for each parameter, χ2 values were com-
1.17 38.5 16 4σ
putedanda6-Dgridwasconstructed.Ultimately,thelowest
1.44 47.5 25 5σ
χ2valuedeterminesthebestfittingorbit.
The black hole is treated as a point mass with the ex-
asgivenbyourmassmodel(see 2).Thepresenceofanex- tended component being centered on it. The lower central
tendedmassinducesretrogradep§ericentre-shiftswhichresult mass limit (Reid et al. 2003) and upper central mass limit
inopenrosettashapedorbits. (Ghezetal.2003;Scho¨deletal.2003)arewelldetermined.
Intotaltherearetenfitparameters-sixorbitalelements, Basedonthese earlierworks,thevalue ofthetotalcentral
thetotalcentralmassM ,thefraction f oftheextendedmass massisvariedbetween2.6 106M and4.8 106M ,consid-
to the total mass Mtot, atontdthe 2-D position P of the central eringsixdifferentvaluesof×Mtot.In⊙addition×,foran⊙yofthese
mass. All these values are known within a certain margin. giventotalcentralmassesvalues,Mtot,theeffectofvarious
We used the position measurements for S2, the dynamical extendedmassfractions, f,wasinvestigated(seeFig.4).
positionofSgrA*andthe 5line-of-sightvelocitymeasure- We used the position of Sgr A* as determined by Reid
mentsasdeterminedbyScho¨deletal.(2003),Eisenhaueret etal. (2003)withanuncertaintyof10mas.Thelocationof
al.(2003)andGhezetal.(2003a). thecentralmasswasvariedat14differentpositionswithin
A valid computedorbit has to fit both the measured ve- this error bar. Fig. 3 shows these positions along with their
locitiesandpositions.Therefore,tworeducedχ˜2 values,one obtained χ˜2tot values. These were fairly distributed around
for the positions, χ˜2 , andanotherforthe velocities,χ˜2 , the offset positionwhich is the deviationof the focusof the
pos vel
were estimated. While fitting the orbits, we considered the Keplerianorbitfromthecentralradiopositionthewaydeter-
χ˜2 values for the positions and velocities independently minedbyEisenhaueretal.(2003).Inordertochecktheaccu-
pos
becausetheχ˜2 valuesforthevelocitieswere,ingeneral,rel- racyofthefocusoftheKeplerianorbit,wecomparethetwo
positionsforthestarS2attheepoch2002.30calculatedwith
atively much smaller than for the positions. In some cases
theorbitalparametersgivenbyEisenhaueretal.(2003)and
they indicated that measurement errors were systematically
byGhezetal.(2003b).Fromthecorrespondingpositionsand
overestimated. This way, it was possible to attain, both in
uncertainties,anoffsetoftheprojectedorbitalfocusgivenby
velocities and in positions, a similar fit quality. Under this
condition,it is alwayspossible to definea commonχ2 by Ghez et al. (2003)fromthe oneby Eisenhaueret al. (2003)
tot
addingboththeχ2valuesobtainedfromthepositionsandthe of0.0 2.9masEastand0.8 2.0masNorthcanbederived.
± ±
velocities. The total reduced χ˜2, χ˜2 , is simply the χ2 di- Hence,wefindanexcellentagreementbetweenthetwoinde-
tot tot pendentresults.
videdbythenumberofdegreesoffreedomwhichisequalto
33 in our case. Table 2 lists obtained total reduced χ˜2 and In order to take into accountthe uncertaintyof the core
tot radius of the inner cusp, the orbital fits computed for our
non-reducedχ2 whichindicatederrors 5σ.Theseerrors
tot ≤ best fitting model value of R = 15.5 mpc, were repeated
wereestimatedbyapplyingadeviation∆χ2 fromthelow- 1
tot
onlyinthecaseofthefittingorbitswithanerror 2σ,for
est obtained χ2tot. In our following analysis we used orbital theextremecasesofRmin =13.2mpcandRmax =≤20.2mpc.
fitswithanerror 1σ. 1 1
≤
3.1. Influenceofmodelparameters 4. Results ofnon-Keplerianfits
In order to study the main effects of a varying cusp mass 4.1. Orbitswithlowestχ2 values
tot
on the orbit of the star S2 we investigated a representative
coverageof the parameterspace that includesthe fullrange OurresultsshowthattheavailabledataonthepathofS2can
of possible scenarios. In each single orbital run, the total be fitted very well with non-Keplerian orbits. However, at
central mass, M and its position as well as the extended presentthe orbitofS2 is notsufficientlywell determinedto
tot
mass fraction, f, were fixed; only the 6 orbital elements exclude a pure Keplerian orbit (see Fig. 6). In fact, in both
are varied. The three parameters, M , the central mass cases a minimumχ˜2 value of 0.7 was obtained.In other
tot tot
∼
position and f, were varied independently forming a fully words, it cannot be excluded that there exists an extended
representative3-Dgrid.Intotal 360orbitswere computed masscomponentinadditiontotheblackhole.Inthecaseof
∼
4 RESULTSOFNON-KEPLERIANFITS 4.1 Orbitswithlowestχ2 values
tot
Central Positions
CCCCCCCCCCCCCCCeeeeeeeeeeeeeeennnnnnnnnnnnnnnttttttttttttttteeeeeeeeeeeeeeerrrrrrrrrrrrrrr ooooooooooooooofffffffffffffff mmmmmmmmmmmmmmmaaaaaaaaaaaaaaassssssssssssssssssssssssssssss aaaaaaaaaaaaaaattttttttttttttt OOOOOOOOOOOOOOOffffffffffffffffffffffffffffffssssssssssssssseeeeeeeeeeeeeeettttttttttttttt PPPPPPPPPPPPPPPooooooooooooooosssssssssssssssiiiiiiiiiiiiiiitttttttttttttttiiiiiiiiiiiiiiiooooooooooooooonnnnnnnnnnnnnnn
a = 0.68
000000......444444 b=0.71 2.6 333333333333333...............555555555555555 5% 10% 15% 20% 25% 30%
c= 0.69 333333333333333
0%
d=0.69
c]c]c]c]c]c] 000000......222222 222222222222222...............555555555555555
pppppp
EC[mEC[mEC[mEC[mEC[mEC[m 0.82 0.89 χχχχχχχχχχχχχχχ222222222222222 tottottottottottottottottottottottottottottot 111111111111111 ...............222222222222222555555555555555 5σ
DDDDDD 000000 1.35 2.35
e e e e e e a 111111111111111
ativativativativativativ 0.72 d b 0.76 000000000000000...............555555555555555 < 1 σ
RelRelRelRelRelRel ------000000......222222 c
000000000000000
222222222222222...............555555555555555 333333333333333 333333333333333...............555555555555555 444444444444444 444444444444444...............555555555555555 555555555555555
0.84 CCCCCCCCCCCCCCCeeeeeeeeeeeeeeennnnnnnnnnnnnnntttttttttttttttrrrrrrrrrrrrrrraaaaaaaaaaaaaaalllllllllllllll MMMMMMMMMMMMMMMaaaaaaaaaaaaaaassssssssssssssssssssssssssssss (((((((((((((((111111111111111000000000000000666666666666666MMMMMMMMMMMMMMMsssssssssssssssuuuuuuuuuuuuuuunnnnnnnnnnnnnnn)))))))))))))))
0.69
------000000......444444 Offset Position 1.2 Fig.4.Plotoftheχ˜2tot valuesvs.thetotalcentralmass,Mtot,
calculated for differentextendedmass fractions, f, with the
point mass at the offset position. Values of the same f are
000000......444444 000000......222222 000000 ------000000......222222 ------000000......444444
fitted with splines. Crosses: f = 0; “x” symbols: f = 0.05;
RRRRRReeeeeellllllaaaaaattttttiiiiiivvvvvveeeeee RRRRRR......AAAAAA[[[[[[mmmmmmppppppcccccc]]]]]]
stars: f = 0.1; empty squares: f = 0.15; filled squares: 0.2,
Fig.3.Thedifferentpositionsofthecentralmassastheywere emptycircles: f = 0.25;andfilled circles: f = 0.3.Thetwo
consideredassociatedwiththelowestobtainedχ˜2 valueat horizontallinesdenotedby<1σand5σdelimittheregion
tot
eachgivenposition.Theoffsetpositioniscentredinthearea offittingorbits(χ˜2 1.44).
tot ≤
showingthelowestχ˜2 values.
tot
χ2
//////SSSSSSeeeeeeppppppaaaaaarrrrrraaaaaattttttiiiiiioooooonnnnnn ffffffrrrrrroooooommmmmm tttttthhhhhheeeeee OOOOOOffffffffffffsssssseeeeeetttttt PPPPPPoooooossssssiiiiiittttttiiiiiioooooonnnnnn
tot
Keplerian orbits and for different central mass position, fits 333333
for a 3.65 106M black hole shows the lowest χ˜2 value.
tot
× ⊙
If we assume an additional extended mass component, the
222222......555555
lowest χ˜2 values corresponded to fits for a total mass of
tot
3.65 106M ofwhich5%areextendedora4.1 106M total
mass×ofwhi⊙ch10%or15%areextended,or4.4×5 106M⊙ to- 222222
talmassofwhich20%areextended,aswellasa×4.8 10⊙6M 2tot
× ⊙ χ
totalmassofwhich25%areextended.Wecanalsoconsider 111111......555555 5σ
onlythefractionofthecuspmassM whichislocatedat
S2apo 4σ
distancesfromtheMBHsmallerthantheapocenteroftheS2
111111 3σ
orbit.Inthiscasethe25%limitcorrespondsto 2σ
1σ
M /(M +M ) 0.05.
S2apo BH S2apo ≤ 000000......555555
000000 000000......000000555555 000000......111111 000000......111111555555 000000......222222 000000......222222555555 000000......333333 000000......333333555555 000000......444444 000000......444444555555 000000......555555
Theorbitalparametersforthese 1σfittingorbitsaregiven RRRRRReeeeeellllllaaaaaattttttiiiiiivvvvvveeeeee SSSSSSeeeeeeppppppaaaaaarrrrrraaaaaattttttiiiiiioooooonnnnnn ffffffrrrrrroooooommmmmm tttttthhhhhheeeeee OOOOOOffffffffffffsssssseeeeeetttttt PPPPPPoooooossssssiiiiiittttttiiiiiioooooonnnnnn[mpc]
≤
in Appendix A together with the full parameter range that Fig.5.Thelowestobtainedreducedχ˜2 valuesinfunctionof
tot
producesorbitswithanerrorof 5σ.Withintheuncertain-
theseparationfromtheoffsetposition.Thesearefittedwitha
≤
ties the orbitalparametersforthe Kepleriancase agree with functionoftheform7.2x2+0.5x+0.7.
earlierwork(Scho¨deletal.(2003),Ghezetal.(2003),Eisen-
haueretal.(2003)).
at a 5 σ level of confidence the S2 orbit could fit the data
4.1.1. PositionofSgrA* forcuspmassesofupto70%ofthetotalmass.Thisresulted
in an upper limit of cusp masses as high as 8 106 M cor-
Onlyorbitswithfocusesata,b,c,dandoffsetpositions(see respondingtoanextendedcomponentof 1 1×06 M ⊙inside
Fig. 3 and Fig. 5) resulted with an error of 1σ. This gives ∼ × ⊙
≤ thesphereofaradiusequaltotheapocentreofS2.However,
aconstraintontheSMBHblackholepositionofonlyasep-
the total masses in these cases are in disagreementwith the
arationof 0.05mpc( 1.3mas)inR.AorinDec.fromthe
± ± dynamicalmassestimatesatlargeradii.Therefore,weregard
offsetpositions.
themasunrealisticbutmentiontheresultforcompleteness.
Large amounts of extended mass inside the S2 orbit
4.1.2. Highertotalcentralmassesandcuspmasses
wouldbepossibleifthemassdistributionfunctionatthecen-
Fig.4showsaplotofχ˜2 vs.differentdiscretecentraltotal treweresteeperthaninourmodelorifthecoreradiuswere
tot
massvaluesandforeachfraction f of theextendedmassat smallerthantheapo-centreofS2( 9mpc).Inthatcasewe
∼
theoffsetposition.Wealsoexaminedorbitswithtotalmasses found that mass densities – exceeding the currently derived
higherthan4.8 106M ,upto11.6 106M .Wefoundthat few 108 M pc 3 by three or fourordersof magnitude– are
−
× ⊙ × ⊙ ⊙
5 DISCUSSION
CCCCCCCeeeeeeennnnnnnttttttteeeeeeerrrrrrr ooooooofffffff MMMMMMMaaaaaaassssssssssssss aaaaaaattttttt OOOOOOOffffffffffffffssssssseeeeeeettttttt PPPPPPPooooooosssssssiiiiiiitttttttiiiiiiiooooooonnnnnnn CCCCCCCeeeeeeennnnnnnttttttteeeeeeerrrrrrr ooooooofffffff MMMMMMMaaaaaaassssssssssssss aaaaaaattttttt OOOOOOOffffffffffffffssssssseeeeeeettttttt PPPPPPPooooooosssssssiiiiiiitttttttiiiiiiiooooooonnnnnnn CCCCCCCeeeeeeennnnnnnttttttteeeeeeerrrrrrr ooooooofffffff MMMMMMMaaaaaaassssssssssssss aaaaaaattttttt OOOOOOOffffffffffffffssssssseeeeeeettttttt PPPPPPPooooooosssssssiiiiiiitttttttiiiiiiiooooooonnnnnnn
7777777 7777777 7777777
MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM ===============================================================================================================================================================================3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333...............................................................................................................................................................................66666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666665555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555 MMMMMMMMMMMMMMMMMMMMM =====================444444444444444444444.....................111111111111111111111 MMMMMMMMMMMMMMMMMMMMM =====================444444444444444444444.....................888888888888888888888
tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooottttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt tttttttttttttttttttttooooooooooooooooooooottttttttttttttttttttt tttttttttttttttttttttooooooooooooooooooooottttttttttttttttttttt
6666666 1992.23 fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff===============================================================================================================================================================================0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000...............................................................................................................................................................................0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 6666666 ∆α 1992.23 fffffffffffffffffffff=====================000000000000000000000.....................111111111111111111111 6666666 ∆α 1992.23 fffffffffffffffffffff=====================000000000000000000000.....................222222222222222222222555555555555555555555
5555555 5555555 5555555
c]c]c]c]c]c]c]
mpmpmpmpmpmpmp 4444444 4444444 4444444
C[C[C[C[C[C[C[ 1999.47 1999.47 1999.47
EEEEEEE 3333333 2003.45 3333333 2003.45 3333333 2003.45
DDDDDDD
e e e e e e e
ativativativativativativativ 2222222 2222222 2222222
elelelelelelel
RRRRRRR 1111111 1111111 1111111
0000000 0000000 0000000
-------1111111 Line of nodes2002.33 Major axis -------1111111 Line of nodes 2002.33 Major axis -------1111111 Line of nodes2002.23 Major axis
3333333 2222222 1111111 0000000 -------1111111 -------2222222 -------3333333 -------4444444 3333333 2222222 1111111 0000000 -------1111111 -------2222222 -------3333333 -------4444444 3333333 2222222 1111111 0000000 -------1111111 -------2222222 -------3333333 -------4444444
RRRRRRReeeeeeelllllllaaaaaaatttttttiiiiiiivvvvvvveeeeeee RRRRRRR.......AAAAAAA[[[[[[[mmmmmmmpppppppccccccc]]]]]]] RRRRRRReeeeeeelllllllaaaaaaatttttttiiiiiiivvvvvvveeeeeee RRRRRRR.......AAAAAAA[[[[[[[mmmmmmmpppppppccccccc]]]]]]] RRRRRRReeeeeeelllllllaaaaaaatttttttiiiiiiivvvvvvveeeeeee RRRRRRR.......AAAAAAA[[[[[[[mmmmmmmpppppppccccccc]]]]]]]
--66 88 550000
MMMMMMttttttooooootttttt======444444......111111 MMMMMMttttttooooootttttt======444444......111111 MMMMMMMMMMMMMMMMMMttttttttttttttttttooooooooooooooooootttttttttttttttttt==================444444444444444444..................111111111111111111
Relative R.A[mpc] with ErrorbarsRelative R.A[mpc] with Errorbars ----02024242 ffffff======000000......111111 Relative DEC[mpc] with ErrorbarsRelative DEC[mpc] with Errorbars 02460246 ffffff======000000......111111 Velocity(Km/s)Velocity(Km/s) ----1111--55505000000000000000 ffffffffffffffffff==================000000000000000000..................111111111111111111
44 --22 --22000000
11999922 11999944 11999966 11999988 22000000 22000022 22000044 11999922 11999944 11999966 11999988 22000000 22000022 22000044 22000066 22000022..44 22000022..66 22000022..88 22000033 22000033..22 22000033..44 22000033..66
TTiimmee iinn YYeeaarrss TTiimmee iinn YYeeaarrss TTiimmee iinn YYeeaarrss
Fig.6.Threeexemplaryorbitsdeterminedbyourfitting.Upperleftpanel:Keplerianorbitwith3.65 106M pointmass.Up-
permiddlepanel,non-Keplerianorbitswith4.1 106M totalmass,thereof10%extended.Upperri×ghtpan⊙el,non-Keplerian
orbitswith 4.8 106M totalmass, thereof25%×extend⊙ed.Here the centralmass is atthe offset position,0.09mpceastand
× ⊙
0.12mpcsouthfromthenominalradiopositionofSgrA*.ThelowerpanelsshowthevelocityintherelativeR.A,therelative
Dec., and along the line-of-sightas a function of time for the case of 4.1 106M total central mass with a 10% extended
component.Thiscaseisrepresentativefortheothercases.Thedirectionof×∆αthe⊙pericentre-shiftisshownbyanarrow.
unlikely to form a stable configurationbecause stellar colli- expect power-law indecies in the range 3/2 7/4 (Bahcall
≤
sionswillbecomeofincreasingimportance(seesection5.2). & Wolf 1977; Murphy et al. 1991). To study the influence
of the denstiy profile on the pericentre-shift, we performed
fittings of the S2 orbit considering a power-law profile of
4.1.3. Coreradiusandslopeofthedensityprofile
a slope equal to 1.4. The pericentre-shift induced by the
Inthecaseoftheoffsetposition,foralltheorbitswitherrors same amount of mass inside S2’s apocentre but different
2 σ, the fitting procedure was repeated for the extreme fractions of mass, f, did not exceed the one induced by a
≤cases of Rmin=13.2 mpc and Rmax=20.2 mpc of the inner Plummermodeldensityprofilebymorethan12%.Thus,by
1 1
cuspradius.Evenwiththesedifferentcoreradii,thecase of covering a wide range of slopes 1.4 α 5, we found that
≤ ≤
the offset position is found to give the lowest χ˜2 values. fortheshallowermassdistribution,thepericentre-shiftnever
tot
Also the fit quality remains unchanged at χ˜2 0.7. We exceeds12%ofourinitiallyderivedvalue,thusourestimates
tot
also considerthat the compactcomponentcan ha≃ve a much indicate an upper limit on the mass that could be present
smaller core radius than the stellar profile, possibly well insidetheS2apocentre,andconsequentlyonthemassofthe
within the S2 apocentre. In this case, following the result cusp.
of Rubilar & Eckart (2001, see their Fig.5), we deduce
that for the same extended mass fraction, f, the induced
pericentre-shiftis smaller for a bigger core radius. Thusfor 5. Discussion
a core radius that does not exceed a value 15.5 mpc, our
estimationonthemassofthecuspisanupperlimit. In the previous sections it was demonstrated that the mea-
suredpositionsandline-of-sightvelocitiesofS2arenotsuffi-
Using a series of Plummer model density profiles, cientlywelldeterminedtoexcludeaKeplerianorbit.Assum-
impliesavalueofthecuspslopeofα 5.Asmentionedin ingasimpleKepleriancase,thecentralgravitationalpotential
≤
2,Genzeletal.(2003)fittedthestellarnumberdensitywith bydefinitionisentirelydominatedbythepointmassassoci-
§
abrokenpower-lawofaslopeequal1.4 1.Fromtheoretical atedwithSgrA*andnodynamicalconstraintscanbederived
±
models of cusps around massive black holes, one would onanyextendedmasscomponentduetoe.g.thesurrounding
5 DISCUSSION 5.1 ExtendingtheK-bandLuminosityFunctiontoFaintStars
stellarcluster.Morerevealing,however,isthemorephysical Table3.Extinctionfreemass-to-lightratios, M/L(2µm),for
assumptionofnon-Keplerianorbits.Thepresenceofastellar differentstellartypesinluminosityclassVatthedistanceof
cuspshowsthat–atleasttoa certainextent–thereissome theGalacticCentre.
extendedmasspresentneartheblackhole.Thispotentialcan
M/L(2µm)[M /L ] K[mag] Mass[M ] Spectraltypes
be modelled by a central black hole plus an extended mass 1.29 ⊙ ⊙ 16.425 1.60 ⊙ F0
component.Weassumedthatthecusphasthesameshapeas 1.93 17.005 1.40 F5
the observed stellar number density counts which we fitted 2.73 17.695 1.05 G0
byaPlummerdensitymodel(see 2).We alsotookintoac- 5.02 18.665 0.79 K0
§
countthatthetotalenclosedmassatlargerradii(1pc)would 5.36 18.915 0.67 K5
notexceedavalueof 4.8 106M (seesection3.1). 11.77 20.065 0.51 M0
∼ × ⊙ 9.23 20.065 0.4 M2
Havingshownthatthecurrentdataallowforanextended
16.66 20.915 0.33 M3
mass component higher than the one of 8540 M inferred
⊙ 38.49 22.315 0.21 M5
from stellar number density counts which were scaled to
87.55 23.815 0.12 M7
match the velocity distribution at large radii (see 2), we
§ 166.43 25.265 0.06 M8
will discuss its nature in sections 5.1 and 5.2. The value
§ §
of 8540M , representingthe totalmass ofthe ’innercusp’,
⊙
isequivalenttoamassof3100M insidethecoreradiusR
1
⊙ or15.5mpc)brighterthanagivenmagnitudecanbedirectly
(15.5mpcor0.4”).Itisbasedontheassumptionthatthera-
estimatedfromthe(extrapolated)KLF.
tioofthestellarnumbercountstototalstellarmassdoesnot
varywithradiusandenvironmentintheGalacticCentre,i.e. The KLF gives the number of stars per surface area per
that the mass-to-light ratio M/L(2µm) is constant (see also magnitude.Inordertocalculatethetotalnumberofstarsand
Genzel et al., 2003).It is verylikely that this assumption is thetotalstellarmasspresentwithinthesphericalvolumeen-
unjustified. Genzel et al. (2003)find that the stellar popula- closedbythecoreradiusofR= 0.4 ofourcentralPlummer
′′
tion in the cusp differs to a certain degree from the popula- modelcomponent,wede-projectedtheKLF.Table4liststhe
tionofthesurrounding,large-scalecluster.Also,effectssuch resulting numbers of stars, the total cluster mass, M , the
Cl
as mass segregation and stellar collisions might work very averagestellarmassinthecluster,M ,andthecorrespond-
aver
effectively in the dense environment of the cusp and M/L ing M/L (2µm) for clusters given by KLFs with slopes of
wouldshowhighervaluestowardsthecentre(Baumgardtet β = 0.21andβ = 0.35andfordifferentcut-offmagnitudes,
al. 2003). Therefore, we consider that the mass-to-light ra- betweenK=20(M =1M )andK=28(M =0.06M ).
min min
tio,M/L(2µm)ofthe’innercusp’,canbedifferentfromthat M iscalculatedfromtheob⊙servedKLFusingA =3andt⊙he
Cl K
of the outer stellar cluster. In the following,two hypotheses M/L(2µm)valueslistedinTable3.
willbediscussed:Theexistenceofaclusteroffaintlow-mass
ThenumberslistedinTable4showthatthemassofstars
stars,notyetdetectablewiththeresolutionandsensitivityof
presentwithinR=0.4”deducedfromaβ = 0.21slopeKLF
current instruments, and the existence of a cluster of heavy
cannotbehigherthan800M ,evenafterextrapolationtothe
darkobjectslikestellarblackholesandneutronstars.
⊙
faintest magnitudes (see Table 3 for a list of K-magnitudes
and0.1051correspondingMS-stars at the GC). Therefore,a
5.1. ExtendingtheK-bandLuminosityFunctiontoFaint
clusterwithaKLFofβ=0.21cannotexplainamassof3100
Stars
M within R= 0.4 , estimated from direct number density
′′
WeusetheK-bandluminosityfunction(KLF)forK 18mag co⊙untswithanM/L(2µm)=2M /L (see 2).
≤ ⊙ ⊙ §
within a projectedradiusof 1.5 fromSgr A* as it was de-
′′ M/L (2µm) convergesto a value 2 M /L for β = 0.35,
terminedbyGenzeletal.(2003-seetheirFig.11).Notethat ⊙ ⊙
and a magnitude limit between 26 and 27, in this case we
the KLF is not corrected for extinction or for the distance
find the value of 3100 M required by our mass distribu-
modulus.Theauthorsfitted the KLFwith a power-lawwith ⊙
tionmodellingin 2.Lowermagnitudelimitswouldincrease
aslopeofβ=0.21 0.03,(β=dlogN/dK).Sincethisresultis §
± M/L(2µm)further,andthemassoftheinnercuspcalculated
basedonanumberofroughly60starswithin1.5 ofSgrA*,
′′ from our Plummer model would be underestimated. Stellar
theobservedstarsthemselvescannotaccountforasignificant
types corresponding to the required M/L (2µm) ratios are
extendedmasscomponent.Itisthereforenecessarytoextrap-
listedinTable3.
olatetheKLFatthefaintend.However,weassumethatthe
observedstarstracethemasscarryingpopulationwhichcan Table4alsoshowsthatforamain-sequencestellarpopu-
currentlyonlybeaccessedtheoreticallybyextrapolatingthe lationM/L(2µm)doesnotexceed4.0M /L ,evenifwecon-
KLFtowardsitsfaintend. siderveryfaint/low-massstars(K=28)⊙and⊙asteepfaint-end
FortheextrapolationoftheKLFtwodifferentslopesare KLF(β=0.35).Thisvaluecorrespondstoacusprepresent-
considered,β= 0.21anda steeper one,β =0.35,whichfits ingonly0.2%ofthetotalmass(seeTable5).Weconludethat
the Bulge KLF (Alexander& Sternberg,1999) as well as a the amountof extendedmassallowedinside the coreradius
model of an old ( 10 Gyr) stellar cluster of solar metal- ofthe’innercusp’wouldbefartoolargetobeexplainedby
∼
icity (Zoccali et al. 2003). In the following we concentrate astellarpopulationofMSstars,andthereforewouldrequire
ontheregionwithinR oftheinnerPlummermodelcompo- anothertype of mass carriers. We show in 5.3 that such a
1
§
nent. The total number of stars in the inner cusp (R 0.4 configurationispossible.
′′
≤
5.3 IstheCuspDominatedbyDarkandMassiveObjects? 5 DISCUSSION
Table4.Foreach’observed’K-magnitudelimit(notcorrectedforextinction)wegivethenumbersofstars N,clustermass
M ,averagestellarmassesM ,andmass-to-luminosityratio,M/L(2µm),withinthecentral0.4 forhypotheticalPlummer
Cl aver ′′
modeltypeclusterswithKLFsextrapolatedbytwopower-lawslopesofβ = 0.21andβ = 0.35.M isthemassinsidethe
Cl
core radius R of the innermostPlummer model component.The derived M/L values include an extinction correctionfor
1
A =3.
K
β=0.21 β=0.35
Magnitudelimit N M [M ] M [M ] M/L(2µm)[M /L ] N M [M ] M [M ] M/L(2µm)[M /L ]
Cl aver Cl aver
K 28 3500 800 ⊙ 0.9 ⊙ 1.0 ⊙ ⊙ 44500 5200 ⊙ 0.1 ⊙ 3.7 ⊙ ⊙
≤
K 27 2200 710 1.4 0.9 20000 3700 0.15 2.8
≤
K 26 1350 620 2.3 0.8 9000 2400 0.3 1.8
≤
K 25 820 525 3.7 0.7 4100 1500 0.8 1.1
≤
K 24 510 450 6.0 0.6 1920 1000 1.6 0.8
≤
K 23 310 390 10 0.5 950 700 3.3 0.5
≤
K 22 190 325 16 0.4 510 480 6.0 0.4
≤
K 21 120 235 26 0.4 320 380 9.7 0.3
≤
K 20 70 193 42 0.3 230 270 11.2 0.2
≤
Table5.Mass-to-lightratiosM/L(2µm),forthedifferentoff- achieveenergyequipartitionwiththefainterolderstellarpop-
set position fitting orbits obtained from our model calcula- ulation.Theyare hencedynamicallyun-relaxed.Also,there
tions.TheerrorsonM/Laremodeldependent.Fromscaling areindicationsforaradialanisotropyofthestarsinthecusp
to theenclosedmassestimateinFig.2we estimateanerror whichmightbeun-relaxed(Scho¨deletal.,2003),inspiteof
of5%.Columns3,4,and5givethecuspmassinsidetheor- theexpectedshortrelaxationtimeinthisdenseenvironment.
bitofS2forcoreradiiofrespectively20.2mpc,15.5mpcand Because of this generallack of theoreticalunderstandingof
13.2mpc. theclusternearSgrA*,weconsiderthataclusteroffaint/low
massstarsshouldnotberuledoutentirelyfromthepossible
M f M M M M/L(2µm)
tot S2apo S2apo S2apo interpretationsoftheinnercusp.
[106M ] R = R = R = [M /L ]
1 1 1
⊙ 20.2mpc 15.5mpc 13.2mpc ⊙ ⊙
3.65 0.0 0.001 0.001 0.001 2.00 5.3. IstheCuspDominatedbyDarkandMassive
3.65 0.05 0.011 0.021 0.031 42.74
Objects?
3.65 0.10 0.022 0.044 0.062 85.48
4.10 0.05 0.013 0.024 0.033 48.01 Inthissection,weconsiderorbitalfitswitherrorsof 1σthat
4.10 0.10 0.025 0.049 0.071 96.02 ≤
result in high cusp masses yet are in good agreement with
4.10 0.15 0.038 0.071 0.104 144.03
the enclosed mass measurements. The evaluation presented
4.10 0.20 0.052 0.098 0.140 192.04
in 5.1showsthatsuchaheavycuspisunlikelytoconsistof
4.45 0.20 0.049 0.105 0.155 227.7 §
starsonly.Herewestudywhethersuchacuspcouldconsist
4.80 0.20 0.061 0.108 0.157 224.82
of heavier mass carriers like stellar black holes or neutron
4.80 0.25 0.069 0.135 0.190 269.32
4.80 0.30 0.089 0.164 0.237 337.24 stars.
5.3.1. Presenceofstellarremnantsinthecenterdueto
5.2. StabilityofaClusterofLow-MassStars dynamicalfriction
For a multi-mass stellar distribution, high mass stellar rem- Whenthecompactremnantsofmassivestarsarethemselves
nants (stellar black holes and/or neutron stars) are expected significantlymoremassivethan the normalfield stars in the
tomigratetothecentreasa consequenceofdynamicalfric- GalacticCentre,aswouldbethecaseofblackholeremnants,
tion.Onewouldexpectthat,withinaHubbletime,thesecom- than they are susceptible to inward migration as a conse-
pact objects show a higher concentration toward the centre quenceofdynamicalfriction.Theresultingmasssegregation
than the lighter ones (Morris et al., 1993; Miralda-Escude´ can lead to a pronounced concentration of compact objects
& Gould, 2000),which should be transferredby that mech- within the central stellar core within a Hubble time (Mor-
anisms to orbits at greater distances from the centre of the ris, 1993). N-Body simulations of globular clusters showed
cluster.Thisarguesagainsttheexistenceofaclusteroflow- thatthecombinationofstellar evolution(productionofstel-
massstarsintheinnercusp. larmassblackholes,neutronstars,whitedwarfs,andofbina-
Ontheotherhand,thepossibilityofaclusteroflow-mass riesincludingsuchobjects)andstellardynamicswillalmost
stars cannotbeexcludedandwe arefar fromunderstanding certainlyleadtoastrongincreaseofM/Linthecentralparts
thepropertiesofthestarsinthecusp.Thereisforexamplethe ofthenuclearstarcluster.Blackholesandneutronstarssink
unexplainedpresenceof massiveyoungstars, e.g. MS O/B- to the centre and may coalesce (Gu¨rkan et al. 2003). Such
typestarsclosetotheblackhole(Genzeletal.,1997;Eckart, highly detailed self-consistent simulations of the dynamical
Ott & Genzel, 1999; Figer et al., 2000; Gezari et al., 2002; episodes are, however, not yet possible on the scale of the
Ghez et al., 2003).These stars havenothadenoughtime to Galaxy.N-Bodysimulations(evenwithcutting-edgespecial
5 DISCUSSION 5.3 IstheCuspDominatedbyDarkandMassiveObjects?
purposes computers like GRAPE-6 (Makino 2001)) cannot formationof a dense cluster also at the center. Here, we as-
followtheevolutionofagalacticnucleusoveraHubbletime sume that these stars would end their lives in the formation
ifrelaxationisappreciable(Freitag&Benz,2002). of neutron stars or stellar black holes with masses between
Under certain assumptionsand for some range of initial 1.5M and10M (onaverage7M ),wealsoconsidertheir
parameters,lessrealisticbutmoreefficientMonteCarlonu- lifetim⊙es to be .⊙107 yrs. If after a⊙single lifetime about40
merical simulations of the evolution of the GC (Freitag & ofsuchstarsformstellarremnants,throughouttheageofthe
Benz, 2002) showed that the SBHs sink to the center on a Galaxy( 1010yrs),itispossibletoaccountforabout105M
∼ ⊙
shorttimescaleofafewgigayears,settleintoacentrallycon- neededtoexplainouruppermasslimitofthe’innercusp’.
centrated distribution and dominate the stellar mass there. Considering the two above described scenarios - while
Similarly, Murphy Cohn & Durisen (1991, see their figure these do not represent the complete history of the galactic
8b)showedthatdensitieshigherthan109 M pc 3 couldre- nucleus-itis,however,afairconclusionthatstrongtomod-
−
side in regions as close as few mpc from th⊙e central SBH. eratedynamicallycausedM/LvariationsprevailattheGC.
OtherstudiesbyMorris(1993)andMiralda-Escude´&Gould Ourattemptsinthisworkisnottoexplaintheformation
(2000)estimatethat104 - 105 SBHs, duetodynamicalfric- of a high density cluster of stellar remnants at the GC. We
tion,wouldhaveaccumulatedatdistancesoflessthanabout wouldratherlike toanalysesucha configurationif existant.
1pcfromthecenter.Theselatterworkscanonlyaccountfor Inthefollowingweinvestigateifsuchahypotheticallyhigh
about102-104 M withinseparationsofabout20mpcfrom M/L configuration of stellar remnants can form a stable
⊙
the centre which represents only 1% to 10% of our upper configuration.
mass limit derived from the orbit of S2. However, the cen-
traldensityoftheSBHclusterdependsonvariousuncertain
quantities:theSBHmassfunction,thestellarIMFandforma-
5.3.3. Stabilityofaclusterofstellarremnants
tion rate, theremnantprogenitormasses, andthe dynamical
ageoftheGC.Morris(1993)arguedthatwithinawiderange Rauch&Tremaine(1996)studiedtheconfigurationofacen-
ofassumptionsabouttheIMF,andabouttheminimumstel- tralmassiveblackholeplusanextendedmassdistributionM
larmasscapableofproducingablackholeremnant,thetotal ofradiusRconsistingofobjectswithmassmintermsofits
mass of remnantsconcentratedinto the inner few tenth of a non resonantrelaxationtimetnr.Undertheassumptionthat
parsec,wouldbeabout0.4-5106 M .Iftheblackholerem- M <−< M , Rauch & Tremainreel(1996) derive how tnr de-
⊙ BH rel
nants were to achieveequipartitionwith the field stars, they pendson M , M,andtheorbitaltimescalet attheouter
BH orb
wouldformaninnercorewitha radiusofasize assmallas edgeof the cluster. If (a) the stellar orbitshave randomori-
50 mpc. These would coalesce or form a quasi-stable clus- entationsandmoderateeccentricities,(b)thedensityofstars
terofstellarmassblackholes.Evenifacatastrophicmerger isapproximatelyuniformwithinR,and(c) M / M =
cluster BH
of stellarremnantsdidoccuratsomepointattheage ofthe 10 2-10 5thenRauch&Tremaine(1996)findthat
− −
galacticnucleus,thecontinuousinfluxofmassivestellarrem-
nantswouldensurethataconcentratedpopulationofthemis M2
tnr BH t , (5)
presentwithinthestellarcore.Thesefindingsapply,however, rel ∼ m2NlnΛ orb
intheabsenceofaninitialcentralblackhole.Toourknowl-
where lnΛ is the Coulomb logarithm 13 in this case.
edge,asimilarstudyofaclusterwithalreadyapre-existing ∼
This situation should - to first order - be applicable to the
super-massive black hole, as in the centre of our Galaxy, is
BlackHole/cuspscenarioattheGalacticcentre.Condition(a)
notyetavailable.
isprobablyfulfilledwiththepossibleexceptionthatthestars
in the cusp mighthave fairly higheccentricities(Scho¨delet
al. 2003). Assuming a Plummer model as a cusp descrip-
5.3.2. Insituformationofstellarremnants
tionfulfillscondition(b).Weconsiderthecasewhereaperi-
In the case of the Galactic Centre, complex dynamical centre-shiftoftheorderof1◦ isinduced.Asanexample,we
episodes have taken place. A numerous variety of young choosethe case of a cusp massof 15%with a total massof
early-type, bright and massive stars exist at distances of 4.1 106 M . Thiscuspmassis wellbelowtheupperlimit
× ⊙
10 mpc - 400 mpc from the centre. There exist a dozen derived in section 4.1. In the case of a cusp mass of 15%,
of bright O/B main sequence stars within about 40 mpc of M/MBH isoftheorderof10−2,ifwerestrictourselvestothe
Sgr A*, these are fast moving S-stars (mass 20 M ) of regionwithinthe coreradiusR1.Thisiscloseto whatisre-
whichthestarS2isanexample.Thereexistsal∼so 30⊙more quiredbycondition(c).
∼
massive (30 M . mass . 100 M ) very bright early-type Here, M = 3.5 106 and m is the average mass
BH sr
stars,theso-cal⊙ledHestarsexhibiti⊙ngHe/HIemissionlines. of the stellar remnants.×A value of m = 5 M is roughly
sr
⊙
The existance of these two types of stars at these regions consistentwithacompositeclustermadeofneutronstars(m
from the center is still not understood. These could have 1.5M )andstellarblackholes(5M <m<25M ).We
formed there or would have migrated there due to different ∼thenfind⊙thattnr 106 t ,ifweass⊙umethatmost⊙ofthe
rel ∼ × orb
in-spiralingprocesses(Genzeletal. 2003;Ghezetal.2003; mass inside the core radius is present in the form of stellar
Gerhard et al. 2001, Krabbe et al. 1995).If such stars were remnants of average mass m . Considering that the core
sr
alwayspresentinthecenteroftheGalaxy,duetostellarevo- radiusof 15-20mpc will define t we find tnr to be about
orb rel
lution,theirremnantswouldcontributeveryefficientlytothe 2 107years.
×
