The distribution of old stars around the Milky Way's central black hole I: Star counts

2.4 MB·

by  E. Gallego-Cano

Astronomy&Astrophysicsmanuscriptno.GC_starcounts_Gallego (cid:13)cESO2017
January17,2017
The distribution of old stars around the Milky Way’s central black
hole: I. Star counts
E.Gallego-Cano1,R.Schödel1,H.Dong1,F.Nogueras-Lara1,A.T.Gallego-Calvente1,P.Amaro-Seoane2,andH.
Baumgardt3
1 InstitutodeAstrofísicadeAndalucía(CSIC),GlorietadelaAstronomías/n,18008Granada,Spaine-mail:[email protected]
2 InstitutdeCiènciesdel’Espai(CSIC-IEEC)atCampusUAB,CarrerdeCanMagranss/n08193Barcelona,Spain
InstituteofAppliedMathematics,AcademyofMathematicsandSystemsScience,ChineseAcademyofSciences,Beijing100190,
China
KavliInstituteforAstronomyandAstrophysics,Beijing100871,China
7
ZentrumfürAstronomieundAstrophysik,TUBerlin,Hardenbergstraße36,10623Berlin,Germany
1
3 SchoolofMathematicsandPhysics,UniversityofQueenslandSt.Lucia,QLD4068,Australia
0
2
Received;accepted

n
a
ABSTRACT
J

3 Context.Theexistenceofdynamicallyrelaxedstellardensitycuspsindenseclustersaroundmassiveblackholesisalong-standing
1 predictionofstellardynamics,butithassofarescapedunambiguousobservationalconfirmation.
Aims.InthispaperwerevisittheproblemofinferringtheinnermoststructureoftheMilkyWay’snuclearstarclusterviastarcounts,

] toclarifywhetheritdisplaysacoreoracusparoundthecentralblackhole.
A
Methods.WeusejudiciouslyselectedadaptiveopticsassistedhighangularresolutionimagesobtainedwiththeNACOinstrumentat
G theESOVLT.ThroughimagestackingandimprovedPSFfittingwepushthecompletenesslimitaboutonemagnitudedeeperthan
inprevious,comparablework.Crowdingandextinctioncorrectionsarederivedandappliedtothesurfacedensityestimates.Known
.
h young,andthereforedynamicallynotrelaxedstars,areexcludedfromtheanalysis.Contrarytopreviouswork,weanalysethestellar
p densityinwell-definedmagnituderangesinordertobeabletoconstrainstellarmassesandages.
- Results. WefocusonstarsintheRedClump(RC),withobservedmagnitudes K ≈ 15.5,andonstarswithobservedmagnitudes
o
K ≈18,whichweexpecttohavesimilarmeanagesandmassesthantheformer.TheRCandbrightergiantstarsdisplayacore-like
r
t surfacedensityprofilewithinaprojectedradiusR<0.3pcofthecentralblackhole,inagreementwithpreviousstudies,butshowa
s cusp-likesurfacedensitydistributionatlargerR.Thesurfacedensityofthefainterstarscanbedescribedwellbyasinglepower-law
a
atR < 2pc.Thecusp-likeprofileofthefaintstarspersistsevenifwetakeintoaccountthepossiblecontaminationofstarsinthis
[
brightnessrangebyyoungpre-mainsequencestars.Thedataareinconsistentwithacore-profileforthefaintstars.Weexcludefrom

1 ouranalysistheregionR < 1”/0.04pc,wherethedensityislessthanexpectedatallbrightnessranges,becausethisregionmaybe
v toocrowdedandalsohassufferedchangestoitsstellarpopulationbeyonddynamicalrelaxation.Finally,weshowthata3DNuker
6 lawprovidesaverygooddescriptionoftheclusterstructure.
1 Conclusions.WeconcludethattheobservedstellardensityattheGalacticCentre,asitcanbeinferredwithcurrentinstruments,is
8 consistentwiththeexistenceofastellarcusparoundtheMilkyWay’scentralblackhole,SagittariusA*.Thiscuspiswelldeveloped
3 inside the influence radius of about 3pc of SagittariusA* and can be described by a single three-dimensional power-law with an
0 exponent γ = 1.23±0.05. This corroborates existing conclusions from Nbody simulations performed in a companion paper. The
. apparentlackofRCstarsandbrightergiantsatprojecteddistancesofR (cid:46) 0.3pc(R (cid:46) 8”)ofthemassiveblackholemayindicate
1
thatsomemechanismhasalteredtheirdistributionorintrinsicluminosity.Weestimatethenumberofpossiblymissinggiantstoa
0
fewdozen.ThisvalueagreeswellwiththetheoreticalideathatapreviouslyexistinggaseousfragmentingdiscaroundSagittariusA*,
7
precursoroftheobservedyoung,massivestars,wasresponsibleforrenderinggiantstarsinvisible.
1
: Keywords. Galaxy:center–Galaxy:kinematicsanddynamics–Galaxy:nucleus
v
i
X
r1. Introduction can test observationally the existence of a stellar cusp. Such a
a stellar cusp is a prediction of stellar dynamics for the case of
Thisisthefirstoneofthreepapersaddressingthedistributionof
adynamicallyrelaxedclusterandhasbeenderivedandstudied
stars around SagittariusA* (SgrA*), the massive black hole at
by analytical, Monte Carlo and N-body methods (e.g., Bahcall
thecentreoftheMilkyWay.Theyarecloselyrelated,butfocus
& Wolf 1976; Lightman & Shapiro 1977; Murphy et al. 1991;
ondifferentmethodsandstellarpopulations.Inthisworkweuse
Baumgardt et al. 2004; Amaro-Seoane et al. 2004; Alexander
the method of star counts, while in our other paper (Schödel et
& Hopman 2009; Preto & Amaro-Seoane 2010). These consis-
al.,referredtoasPaperIIinthefollowing),weanalysethedif-
tenttheoreticalresultshave,however,notyetbeenconfirmedob-
fuselightfromtheunresolvedstellarpopulation.Finally,Baum-
servationally.Thereexistcurrentlyonlymeasurementsofabout
gardtetal.(PaperIII)presentnewNbodysimulationsthatcon-
twodozensystems,wherewecanactuallyresolvetheradiusof
firmtheagreementbetweenmodellingandobservations.
influence of the central black hole with several resolution ele-
ThedistributionofstarsaroundSgrA*isofgreatastrophys-
ments(see,e.g.,Table1inGültekinetal.2009).Thegreatdis-
icalinterestbecauseitistheonlymassiveblackholewherewe
Articlenumber,page1of14
A&Aproofs:manuscriptno.GC_starcounts_Gallego
tanceofmostextragalacticsystemsmeansthatwecanonlystudy (Yusef-Zadehetal.2012).Hence,wehaveonlyobservedthetip
the light density of hundreds to thousands, or even millions, of of the iceberg, which may not be representative for the overall,
starsperresolutionelement.Sincenuclearstarclustersareenti- underlyingstructure.
tieswithcomplexstellarpopulations,manyofwhichshowsigns This work continues similar efforts carried out by Genzel
ofrecentstarformation(seereviewby Böker2010),measured et al. (2003) and Schödel et al. (2007). As experiments are re-
lightdensitiesmayfrequentlybedominatedbyasmallnumber peated, both their accuracy and precision tend to increase be-
ofbrightstars,whicharegenerallytooyoungtobedynamically cause of factors such as improvements in observational tech-
relaxed.Thiscanleadtoambiguousorerroneousresults. niques, increasing know-how on data reduction and analysis,
Ontheotherhand,withitsdistanceofonlyabout8kpcfrom progressintheoryandinterpretation,andanincreasingamount
Earth,theGalacticCentre(GC)allowsustoresolvethestarsob- of data. The novel aspects of this work are, in particular, the
servationallyonscalesofabout2milli-parsecs(mpc),assuming stackingofhighqualityimageswithalargefield-of-view(FOV)
diffractionlimitedobservationsatabout2µmatan8-10mtele- toreachfaintercompletenesslimitsinthemostcrowdedregions
scope.AtthecentreoftheMilkyWay,a4×106M (e.g.,Boehle near SgrA*, an extension of high angular resolution data to a
(cid:12)
et al. 2016; Gillessen et al. 2016) massive black hole, Sagittar- largerfieldofabout1.5(cid:48)×1.5(cid:48),improvementsindatareduction
iusA*(SgrA*),liesembeddedina∼2.5×107M nuclearstar (rebinningoftheimages,removalofsystematicnoisefromde-
(cid:12)
cluster(Schödeletal.2014a,b).Therefore,theGCappearstobe, tector electronics), and analysis (improved PSF fitting with use
inprinciple,theidealtestcasefortheexistenceofstellarcusps. ofnoisemapsandaspatiallyvariablePSF).Finally,moreexplic-
Surprisingly,nounambiguousobservationalevidenceforthe itly than in previous work – and because our deeper data allow
existence(ornot)ofastellarcusparoundSgrA*hasbeenpre- us to do so – we focus on clearly delimited stellar brightness
sentedsofar.Whilethefirsthighangularresolutionobservations rangesinordertominimisethemixingofdifferentstellarpopu-
at an 8m telescope appeared to indicate the existence of a stel- lations.Weaddnewdataonthestellardistributionatprojected
lar cusp (Genzel et al. 2003; Schödel et al. 2007), it was later radii R (cid:38) 2pc from the literature (Schödel et al. 2014a; Fritz
realised that the star counts within about 0.5pc of SgrA* were et al. 2016) to facilitate the interpretation of the data and the
contaminated by a significant number of young, and therefore derivationofthe3Ddensitystructureofthestarsnearthemas-
dynamicallyunrelaxed,stars.Whenomittingtheyoungstars,the siveblackhole.
projectedstellardensityofgiantsappearsalmostflat,core-like,
within a few 0.1pc of SgrA* (Buchholz et al. 2009; Do et al.
2. Datareductionandanalysis
2009; Bartko et al. 2010). Observations of the stellar surface
brightness from old stars also appeared to indicate a possibly
2.1. Basicreduction
core-like structure (Fritz et al. 2016). These findings led to the
missing cusp problem and have given rise to a large number of We use H and Ks-band data obtained with the S27 camera
theoreticalpapers,tryingtoexplainitsabsence.Thehypotheses (0.027”pixelscale)ofNACO/VLT.TheAOwaslockedonthe
reach from a very long relaxation time (Merritt 2010), over the NIRbrightsupergiantGCIRS7thatislocatedabout5.5”north
destructionoftheenvelopesofgiants–thusrenderingthemin- of SgrA*. The data used are summarised in Table1. All data
visible–viastellarcollisions,whichcannotfullyexplaintheob- wereacquiredwithasimilar4pointditherpattern,roughlycen-
servations(Daleetal.2009)orafragmentedgaseousdisc,which tered on SgrA*, with the exception of the data from 11 May
can (Amaro-Seoane & Chen 2014). Other explanations involve 2011, which covered a shallow, but wider mosaic with a 4×4
theapparentstellarstructurearisingfromsubsequentepochsof dither pattern, centered on SgrA*. Preliminary data reduction
starformationand/ortheaccumulationofstellarremnantsnear was standard, with sky subtraction, bad pixel removal and flat
SgrA*(e.g.,Löckmannetal.2010;Aharon&Perets2015). fielding.Subsequently,asimpleshift-and-add(SSA)procedure
When evaluating the observational evidence, it is, however, was applied to obtain final images. A quadratic interpolation
ofutmostimportancetobeawareofitslimitations.Firstly,due with a rebinning factor of two was used because tests showed
to the extreme interstellar extinction toward the Galactic Cen- thatthisimprovedthephotometryandreducedresidualsinPSF
tre(e.g.,Nishiyamaetal.2009;Schödeletal.2010;Fritzetal. fitting,inparticularsincetheS27pixelscalebarelysamplesthe
2011), observations need to be performed in the near-infrared angularresolution,whichwasroughly0.06”FWHMforallim-
(NIR).AsecondrequirementforobservingtheGalacticCentre ages. Along with the mean SSA images we also created noise
(GC)istouseanangularresolutionashighaspossibletoover- mapsthatcontaintheerrorofthemeanofeachpixelintheSSA
cometheextremesourcecrowding.Hereweusetheadaptiveop- images.
tics(AO)assistedNIRcameraNACOinstalledattheESOVLT.
ImagingdataatHandK areusedtobeabletoestimateextinc-
S 2.2. Alignmentandstacking
tionand–bythesamemeans–toexcludeforegroundstars.
Because of these observational difficulties, our knowledge We treated each of the four pointings toward SgrA* indepen-
about the stellar population at the GC is limited to the bright- dently to avoid problems arising from camera distortions near
est few percent of stars: A few million-year-old hot post main the edges of the NACO S27 camera’s field (see Trippe et al.
sequence (MS) giants and MS O/B stars (the latter being al- 2008; Schödel et al. 2009). The final images from all epochs
readyatthefaintlimitofspectroscopiccapabilities),ontheone were aligned with the one from 09 Aug 2012 via a polynomial
hand, and, on the other hand, giants with luminosities equal to fitofdegreeone(IDLPOLYWARPandPOLY_2Dprocedures).
or higher than RC stars. In fact, the typical spectroscopic com- Theparametersofthelatterweredeterminedviaaniterativefit
pleteness reaches only about K = 15.5 stars and thus only usinglistsofdetectedstarsintheimage.Theimageswerecom-
S
half the RC (see Do et al. 2009; Bartko et al. 2010; Do et al. binedinasimplemeanandthecorrespondingnoisemapswere
2015).StudiesofthestellarsurfacedensityattheGChavesofar quadraticallycombined.Apossibleconcerninthisstackingpro-
beendominatedbyRCstarsandbrightergiants.Theonlyrecent cedure is the use of different observing epochs because of the
study that focused on the light from stars fainter than this limit largepropermotionsofthestarsintheGC.Atthedistanceofthe
didindeedfindacusp-likestructurewithin5”/0.2pcofSgrA* GC(hereassumedas8kpc,see,e.g.,Genzeletal.2010;Meyer
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Table1.Detailsoftheimagingobservationsusedinthiswork. 2.3. Patternremoval
Datea λ ∆λ Nb NDITc DITd The images from the individual epochs contained horizontal
central
[µm] [µm] [s] stripe patterns from the detector electronics. These horizontal
09May2010 1.66 0.33 4 64 2 stripescanbedetrimentalforsourcedetectionbecausetheymay
17May2011 2.18 0.35 4 9 2 either mask faint sources or be deblended into rows of stars
09Aug2012 2.18 0.35 8 60 1 by the PSF fitting program. It is therefore important to remove
11Sep2012 2.18 0.35 8 60 1 them. We proceeded as follows: We used the StarFinder pro-
12Sep2012 2.18 0.35 8 60 1 gramtodetectandsubtractrobustlydetectedpointsourcesfrom
eachimage(conservativesettingsoftheStarFinderparameters:
a UTCdateofbeginningofnight. min_correlation= 0.85 and deblend= 0) and to fit the diffuse
emission (from unresolved stars or dust and gas in the inter-
b Numberof(dithered)exposures stellarmedium).Thelatterwasfittedwithanangularresolution
of about 0.25”, a non-critical value that just needs to be large
c Number of integrations that were averaged on-line by the
enough to remove the variable background due to unresolved
read-outelectronics
stellar emission and small enough to roughly correspond to the
d Detectorintegrationtime.Thetotalintegrationtimeofeach sizeofdiffuse,uncresolvedstructuresinthemini-spiral(seePa-
observationamountstoN×NDIT×DIT. perII). While fitting of the diffuse background is important in
this procedure,the exact choiceof its variabilityscale is not. It
caneasilybechosentobeafactor2largerorsmaller.
Theresultingresidualimages,i.e.imageminusdiffuseemis-
sion minus point sources, were then dominated by small-scale
etal.2012),avelocityofabout40kms−1ontheplaneofthesky
(ontheorderafewpixelswidth)randomandsystematicnoise.
correspondstoapropermotionofonemilli-arcsecondperyear.
We determined the pattern of horizontal stripes induced by the
AdisplacementbyonepixeloftheNACOS27cameraperyear
electronicsthroughmediansmoothingeachrowofpixelswitha
thereforecorrespondstoavelocity>1000kms−1.Sinceweare
medianboxwidthofabout2.7”,correspondingto200pixels(in
notinterestedinhighprecisionastrometryorphotometry,thisef-
therebinnedimages).Thispatternwasthensubtractedfromthe
fectisthereforenegligibleforourdata,exceptpossiblyasmall
SSAimages.Wecouldthusremovemostofthesystematicnoise
numberofveryfastmovingstarswithin∼0.1”ofSgrA*,which
withoutintroducinganysignificantbiasonthepointsourcesor
isnotrelevanttotheproblemandscalesaddressedinthispaper. onthediffuseemissionbecausemoststarshadalreadybeensub-
tractedandbecausethemediansmoothingboxwasafactorofa
few to ten larger than the scales of the diffuse emission, of the
August 2012 Deep mean image size of PSF residuals, or of faint, unresolved sources. Finally,
afterhavingcleanedtheimagesofeachepoch,theywerecom-
bined to a deep mean image. This last step further reduced any
remaining systematics. To be conservative, we used the noise
mapsderivedfromtheuncleanedimages.Figure1showsdetails
of K -images to illustrate the effect of the systematic readout
s
noiseandtheimprovementafterremovingit.
2.4. Sourcedetection
PointsourceextractionwascarriedoutwiththePSFfittingpro-
5" grammeStarFinder(Diolaitietal.2000).Sincetheimagescover
areassimilartoorlargerthantheisoplanaticanglesattherespec-
tivewavelengths,carewastakentodealwiththespatialvariabil-
ity of the PSF. We parted the images into sub-fields of approx-
imately 10.5” × 10.5” size. Subsequently, ten of the brightest,
most isolated stars in each sub-field were used for an iterative
extraction of a local PSF (similar to what was done in Schödel
et al. 2010). Because of variable extinction and source density,
notallsub-fieldscontainedPSFreferencestarsofsimilarbright-
ness,whichwouldleadtoasystematicchangeofthezeropoint
acrossthefield.Also,nottakingintoaccounttheextendedsee-
inghalofromthelightthatcouldnotbecorrectedbytheAO,can
leadtoanenhanceddetectionofspuriousfaintstarsnearbright
Fig.1. Cleaningofhorizontalstripes(systematicreadoutnoise).Upper stars.AsremarkedbySchödel(2010),theseeinghaloisaffected
left:DetailofAugust2012Ks-bandimage.Lowerleft:Asupperleft, inaratherminorwaybyanisoplanaticeffects.Wethereforeused
butcleaned.Upperright:Detailofdeep,meancombinedKs-bandim-
the brightest star in the field, GCIRS7, to estimate the seeing
agewhentheinputimageshavenotbeencleaned.Lowerright:Detail
halo. The local PSFs were masked beyond radii of about 0.3”,
ofdeep,meancombinedKs-bandimageaftercleaningoftheinputim-
ages.Thedisplayedfieldislocatedabout12.0”westand1.7”northof uptowhichtheycouldbereliablydetermined.Thentheywere
SgrA*.Thecolourscaleislogarithmicandidenticalforallimages. matchedtotheseeinghalo(usingaleast-squaresfittodetermine
fluxoffsetsandnormalisationfactors).Thus,wecouldcreatelo-
calPSFsthatavoidedlargesystematicphotometriceffectsacross
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thefield.Sometests(similartowhatwasdoneinSchödel2010) artificialstar,thentheartificialstarwasconsideredasdetected.
showed that we could constrain the systematic photometric ef- This latter point is critical to avoid bias because the relatively
fectsfromthevariabilityofthePSFtoafewpercentacrossthe highdensityofartificallyintroducedstarswouldotherwiselead
field. to non-detection of real sources and thus an over-estimation of
In PSF fitting we have to walk a thin line between achiev- incompleteness.
ing an almost complete detection of sources while, at the same As mentioned above, to estimate the systematic errors in-
time,avoidingtopickupspuriousones,whichcanarise,inpar- ducedbyeitherthenon-detectionofrealsourcesorthedetection
ticular, close to bright stars or due to systematic effects from ofspurioussources,werepeatedthesourcedetectionandcom-
thedetectorelectronics.WevisuallyverifiedthattakingthePSF pleteness determination for the following combinations of the
seeing halos into account, along with the use of our SSA noise StarFinder parameters: min_correlation= 0.80,0.85,0.85,0.90
maps, effectively suppressed the detection of spurious sources and deblend= 0,0,1,1 (each value in the first list corresponds
near bright stars (see also Schödel et al. 2013). The PSF halos to the value with the same index in the second list). In Fig.2,
includeeffectssuchasdiffractionspikesandstaticspeckles.Our weshowthevaluesofcompletenessfortwoofthesecasesand
empirical noise maps seemed to deal well with suppressing the for different projected distance ranges from SgrA*. The dif-
detectionofspurioussourcesnearbrightstars. ferences between the different choices of parameters are gen-
Finally, since there can be no absolute certainty in the reli- erally small, on the order of a few percentage points, except
abilityofsourcedetection,wealsorepeatedlyanalysedtheim- for the faintest magnitudes, where the differences are some-
ageswithdifferentvaluesoftheStarFinderparametersthatdom- whatmorepronounced.Also,wecanobservetheexpectedgen-
inate the probability of source detection (for a fixed detection eral trend of less completeness for fainter magnitudes and in
threshold, which was chosen as 3σ in all cases). These param- the more crowded areas near SgrA*. Finally, Fig.2 also shows
etersaremin_correlationanddeblend.Fortheminimumcorre- thatthedifferencesofcompletenessbetweenthefourpointings
lationvaluewechose0.8−0.9,alwaysmoreconservativethan are small. For all cases, we found that source detection was, at
the standard value of 0.7, the default value of StarFinder. The all projected distances, at least 50% complete for magnitudes
keyworddeblendcanbesettodeblendclosesources.Whilede- Ks ≤18.5.
blendingcanbeveryuseful,itcanleadalsotothedetectionof
asignificantnumberofspurioussourcesinacrowdedfield.We
2.7. Extinction
includedmeasurementswithandwithoutsettingthiskeyword.
Weusedthe H−K photometryandtheintrinsicsmallcolours
s
of stars at these bands to create an extinction map for the en-
2.5. Photometriccalibrationandsourceselection tire field, with the same method as applied in Schödel et al.
(2010). We do not consider stars with H − K < 1.5 because
Finally, the photometry was calibrated with the stars s
we consider them as foreground stars. Neither do we consider
IRS16C, IRS16NW, and IRS33N (apparent magnitudes
starswithH−K >3.0becausetheymayeitherbebackground
K = 9.93,10.14,11.20 and H = 11.90,12.03,13.24, see s
s starsorintrinsicallyreddenedobjects(inanycase,theirnumber
Schödel et al. 2010). The uncertainty of the zero points was a
is very small, see Fig4 in Schödel et al. 2010). Median stellar
few percent. We note that for the purposes in this paper we do
colourswereobtainedfromtheindividualcoloursofthe20near-
not require any high accuracy/high precision photometry and
eststarsateachpositionandtheextinctionwasthencalculated
astrometry.
asinSchödeletal.(2010),assumingA ∝λ−2.2.
Almost all stars in the field have intrinsic colours −0.1 ≤ Ks
Ontheonehand,theextinctionmapwasusedtocorrectthe
H−K ≤0.3(see,e.g.,Doetal.2009;Schödeletal.2010).The individual stellar magnitudes for differential extinction. On the
meancolourduetoreddeningis H−K ≈ 2.1.Weexcludedall
other hand, we applied the methodology of Chatzopoulos et al.
stars with H − K < 1.5 as foreground stars. We also excluded
(2015b)tocomputethestellardetectioncompletenessvariation
spectroscopically identified young stars from our final star list
causedbyvariableextinction:Wemodeledtheluminosityfunc-
(Doetal.2009;Bartkoetal.2010).Subsequently,wecreatedan
tion (LF) by taking the product of a power-law stellar LF and
extinctionmap,byusingthe20starsnearesttoeachpoint.The
anerrorfunctionthatrepresentsthecompletenessfunction,asin
resulting map is similar, to within the uncertainties, to the one
expression (2) in Chatzopoulos et al. (2015b). The approxima-
presentedinSchödeletal.(2010).
tion of the LF with a power-law – which ignores the presence
of the RC bump – does not introduce any significant error be-
causeourdataaresensitiveenoughtoreachwellbelowtheRC
2.6. Crowdingandcompleteness
bump over the entire field and because including the RC bump
We determined the source detection completeness in the K - wouldonlyhaveaminoreffectasshownbyChatzopoulosetal.
s
imagesthroughthetechniqueofinsertingandrecoveringartifi- (2015b).First,wemeasuredtheobservedKLFforeachpointing
cialstars.Weusedamagnitudestepof0.5magandinsertedthe and each StarFinder parameter set (excluding a radius of about
starsona0.5”×0.5”grid.Withthisrelativelywidespacingwe 5”aroundSgrA*,wheretheKLFismoreincompletebecauseof
avoidedartificiallyincreasingthecrowding.Thegridwasshifted crowding). We computed the power law for each case. Finally,
severaltimestofinallyprobecompletenessonadense0.1”×0.1” we used these power-laws, combined it with the measured lo-
grid (as done by Schödel et al. 2007). We used the respective cal extinction and computed the corresponding local correction
local PSFs (see above). Subsequently, PSF fitting was carried factorsaccordingtoequation(5)inChatzopoulosetal.(2015b),
out with StarFinder and a source was considered as detected if but using the approximation of a single extinction screen, i.e.
it was found within a magnitude range of 0.5mag of the input no variability of A along the line-of-sight. Since we approxi-
Ks
magnitudeandwithinadistanceof0.054”oftheinputposition mate the LF with a power law, the equation takes on the form
(correspondingto2pixelsoftheS27cameraorroughlythean- p= L(−∆Ak)=10−γ∗∆Ak (seeChatzopoulosetal.2015b),where
gularresolutionofthedata).Ifarealstarofasimilarmagnitude pisthereductionfactorforthenumberoflocallydetectedstars,
was already present within this distance to the grid point of an ∆A isthedifferencebetweenthelocalextinctionandthemean
k
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E.Gallego-Cano etal.:ThedistributionofoldstarsaroundtheMilkyWay’scentralblackhole:I.Starcounts
a) b) c)
Fig. 2. Completeness of the star counts in the deep NACO K images. a) Completeness in pointing 1 for different projected distance ranges
s
fromSgrA*,formin_correlation= 0.80anddeblend= 0.b)Asina),butformin_correlation= 0.90anddeblend= 1.c)Completenessforall
four pointings and within 5” of SgrA*, for min_correlation= 0.80 and deblend= 0. The corresponding plots for other used combinations of
min_correlationanddeblendlookverysimilar.
extinctionoverthefield,andγisthepowerlawindexofthelu-
minosityfunction.Ifthelocalextinctionislowerthanthemean
extinction,then p > 1,andifthelocalextinctionishigherthan
themeanextinction,then p<1.
We apply the correction factor 1/p to each detected star. In
Fig.3weshowthe K LFforpointing1,formin_correlation=
s
0.8 and deblend= 0, along with the LF corrected for differen-
tial extinction, a power law fit to the stars 11 < K < 14.5 (
s
γ = 0.26±0.02,similar to the value obtained in Schödel et al.
(2010)),andthecompletenessfunction(blueline),asdefinedby
Chatzopoulos et al. (2015b). For the latter, we use m = 19.4
0
andσ = 0.2becausethesevaluesapproximateour K LFwell.
s
ThesevaluesaredifferentfromthoseusedinChatzopoulosetal.
(2015b) because our data are significantly deeper than theirs.
In Fig.4 the percentage reduction in observed stars versus pro-
jectedradiusisrepresentedforthedetectedstarsinpointing1,for
min_correlation= 0.8anddeblend= 0.Onecanseethatextinc-
tion is, on average, higher at larger distances from SgrA*, but
thattheeffectofdifferentialextinctiononcompletenessisrela-
tivelyminor,typically<10%andatmost20%.
Fig.3. TheKsLFinpointing1formin_correlation=0.8anddeblend=
2.8. Widefield 0ifshownasablackline.ThedottedredlineistheKsLFaftercorrect-
ingthemagnitudeofeachstarfordifferentialextinction.Thegreenlines
showsapowerlawfittothebrightstars11<K <14.5withpowerlaw
s
indexof0.26±0.02.Thebluelineshowstheeffectofthecompleteness
The2011dataareofexcellentquality,butrelativelyshallow.On functionaccordingtoequation(2)inChatzopoulosetal.(2015b).
theotherhand,thereare16pointingsthatcoverafieldofabout
1.5(cid:48)×1.5(cid:48),comparedtothesmallerfieldsofabout40”×40”cov-
ered by the other NACO observations. The 2011 observations
3. ThesurfacedensityofoldstarsintheGC
arethereforeideallysuitedtoextendthesensitivecentralobser-
vationswithhighangularresolution,albeitsomewhatshallower, 3.1. K luminosityfunction
s
dataouttolargerdistances.Fig.5.Inordertodealwiththedis-
tortionsoftheNACOS27camera(see,e.g.,Trippeetal.2008; WeshowtheK LF(KLF)determinedfromourdeepmosaicin
s
Schödeletal.2009)wealignedeachpointingoftheNACOmo- Fig.6.TheKLFscorrespondingtothefourdifferentStarFinder
saic with a reference frame created from positions measured in parameter settings are shown. The KLF derived in this work is
HSTWFC3observationsofthesamefield(Dongetal.,inprep.). aboutonemagnitudedeepercomparedtopreviouswork(Fig.10
WeapplyvariablePSFfittingasweexplaininsection2.4.Inthis in Schödel et al. 2007). This is a decisive advantage. When we
case,min_correlation=0.8anddeblend=0wereselectedforthe wanttoprobetheexistenceofadynamicallyrelaxedstellarcusp
StarFinderparameters. around SgrA*, we need to focus on stars that are at least sev-
Articlenumber,page5of14
A&Aproofs:manuscriptno.GC_starcounts_Gallego
other, fainter magnitude range with a high fraction of old stars
(17.5≤ K ≤18.5).Also,thestarsinthesetwobrightnessranges
s
havesimilarmasses,whichletsusexpectasimilarsurfaceden-
sitydistribution.
Thegoalofthisstudyistoinvestigatetheexistenceofastel-
lar cusp at the GC. This requires us to select stars old enough
tohaveundergonedynamicalrelaxation.Therelaxationtimeat
the GC is roughly a few Gyr (e.g., Alexander 2005, 2011). We
specifically exclude all spectroscopically identified early-type,
i.e. young and massive, stars from our sample (using the data
of Do et al. 2013). Unfortunately, spectroscopic stellar classi-
fication is limited to stars of about K ≤ 15.5 at the GC. For
s
fainter stars, we can only use their luminosity as a proxy for
their type. Fig.16 of Schödel et al. (2007) illustrates the LF,
mean masses, and old star fractions for stars of different mag-
nitudes at the GC, assuming a basic continuous star formation
model.Itshowsthatwecanprobeold((cid:38)1Gyr),low-massstars
in the range 15 (cid:46) K ≤ 16. This is the RC, which dominates
s
allpreviousstardensitymeasurements.Thefractionofoldstars
risesagainabove∼50%forstarsK >17.5,reachingpractically
s
100%atK ≈18.Therefore,inthisstudy,wefocusonthemag-
s
nituderanges15 ≤ K ≤ 16,theRC,and17.5 ≤ K ≤ 18.5,i.e.
s s
thefainteststarsaccessiblebyourdata.
Fig. 4. Relative detection frequency due to extinction versus pro-
jecteddistancetoSgrA*,inpointing1formin_correlation= 0.8and
deblend= 0.Thegreenlinerepresentsthemeanofthe p(%)consider-
ingdetectedstarsatthesamedistancefromSgrA*.Wecanobservethat
forclosedistancestoSgrA*p(%)ishigherthanforlargedistances,as
weexpected,becausetheextinctionnearSgA*islower.
Fig. 6. KLF for the deep K mosaic. The different colours corre-
s
spond to the different combinations of the values of min_correlation
N
anddeblend.

1 parsec E
3.2. Radialprofileofthenumberdensity
Fig.5. Wide-fieldmosaicoftheobservationsfrom11thMay2011.The
field-of-viewis1.5(cid:48)×1.5(cid:48).Thefieldofabout40”×40”thatcorresponds In order to analyse the surface density of stars at the Galactic
tothedeepimagingdataismarkedbyawhitesquare. Center,weassumethattheunderlyingspatialdistributionofthe
starsinthecentralparsecissphericallysymmetric.Althoughthe
nuclearclusterisflattened,asphericalapproximationshouldbe
eral Gyr old. As shown in the illustrative Figure16 in Schödel acceptable at projected radii R (cid:46) 2pc because the difference
et al. (2007), the only magnitude range where this was previ- between the density profiles along the orthogonal directions of
ouslypossiblewasaroundtheRC(15.25 ≤ K ≤ 16.25).Now, maximumdifferenceisonlyontheorderof10%−20%inthis
s
with the deeper data from our new analysis, we can probe an- region(seeSchödeletal.2014a;Fritzetal.2016).
Articlenumber,page6of14
E.Gallego-Cano etal.:ThedistributionofoldstarsaroundtheMilkyWay’scentralblackhole:I.Starcounts
a) b) c)
Fig.8. Azimuthallyaveragedextinctionandcrowdingcorrectedsurfacedensitiesvs.distancefromSgrA*.Straightdottedlinesillustratefitswith
asinglepower-law.a)Surfacedensityformin_correlation=0.85anddeblend=1.Thevalueofthepower-lawindex,Γ,forRCstars(redline)is
Γ=−0.36±0.04andforfainterstars(blackline)isΓ=−0.52±0.02.b)Surfacedensityformin_correlation=0.90anddeblend=1.Thevalue
oftheindexforRCstars(redline)isΓ=−0.34±0.04andforfainterstars(blackline)isΓ=−0.44±0.02.c)Surfacedensityprofilesforstarsof
17.5≤K ≤18.5forthedataobtainedwiththedifferentStarFinderparameters.
s
runs. For faint stellar magnitudes, we masked the regions with
completeness below 30%. We tested the effect of different
masks, with completeness< 30%,40%,50% respectively, and
found that the results did not vary significantly. Fig.8 shows
the radial surface density profiles for RC stars (K ≈ 15.5)
s
and fainter stars (K ≈ 18) for two different combinations of
s
StarFinder parameters (left panel: min_correlation= 0.85 and
deblend= 1;middlepanel:min_correlation= 0.9anddeblend=
1).Ascanbeseen,theresultingsurfacedensitiesareverysim-
ilar.TherightpanelofFig.8showsthesurfacenumberdensity
profilesforallfourdifferenceStarFinderparametersettings.Al-
though there are offsets between the densities, their shapes are
verysimilar.
Finally, we combined the surface density profiles obtained
withthefoursettingsoftheStarFinderparameters.Meandensi-
ties and standard deviations were computed and all uncertain-
ties were quadratically combined. The resulting final number
density profiles for the magnitude ranges 15 ≤ K ≤ 16 and
s
17.5 ≤ K ≤ 18.5 are shown in Fig.9. Simple power laws
s
were fit to the data at 1” ≤ R ≤ 20” (0.04pc ≤ R ≤ 0.8pc),
i.e. we deliberately left out the innermost data points from the
most crowded region around SgrA*. The corresponding power
law indices are Γ = −0.45 ± 0.03 for the faint stars and
cusp
Γ = −0.34 ± 0.04 for RC stars. There is a significant dip
Fig.7.Optimalbinning.WeshowtheRLPasafunctionofthenumber cusp
aroundR=0.2pc(5(cid:48)(cid:48))inthedensityprofileoftheRCstarsand,
ofbins.ThemaximumfortheRLPisreachedfor21bins(reddotted
possibly, an excess at R ≈ 7”. It can also be seen that a simple
line).
power law is a better approximation for the faint stars than for
theRCstars.
We computed the azimuthally averaged stellar surface den-
sitiesinannuliaroundSgrA*.Itisimportanttochooseanum-
3.3. Starcountsbeyond20”
berofbinssufficientlylargetocapturethemajorfeaturesinthe
datawhileignoringfinedetailsduetorandomfluctuations.Fol- Inordertostudythestellarnumberdensityinabroaderrangeof
lowingthestudiesofKnuth(2006)andWitzeletal.(2012)we distancesfromSgrA*weanalysedthelargemosaicimagefrom
firstdeterminethebestbinsize.ThedependenceoftheRelative the 2011 data. We did not apply any extinction and complete-
LogarithmicPosteriorProbability(RLP)onthebinnumberfor nesscorrectionsbecausetheeffectoftheextinctioncorrectionis
pointing1isshowninFig.7.ThemaximumfortheRLPforthe smallandcrowdingdoenotposeanyseriousproblematR>20”
starnumberisreachedfor21bins,andthebestbinsizeis1(cid:48)(cid:48).We and with the high angular resolution data used here. We did,
appliedthismethodologyforallpointings,withsimilarresults. however mask all the regions occupied by the dark clouds that
We computed extinction and completeness-corrected stellar can be seen show in Fig.5. Finally, the surface densities from
surfacedensitiesforthestarsdetectedinthedifferentStarFinder thewidefielddatawerescaledtothecompletenessandextinc-
Articlenumber,page7of14
A&Aproofs:manuscriptno.GC_starcounts_Gallego
Faint stars RC stars
Fig.10. Combineddeepfield(blue)pluswidefield(green)surfacedensityplots.
3.4. Theprojectedstellarsurfacedensity
As Figs.9 and 10 show, the projected surface number density
of the faintest stars in our sample (17.5 ≤ K ≤ 18.5) can be
s
describedwell(reducedχ2 = 2.8)byasinglepower-lawofthe
formΣ(R) ∝ R−Γ,whereΣisthesurfacenumberdensity,Rthe
projecteddistancefromSgrA*,andΓthepower-lawindex.We
findΓ=0.51±0.02forthefitshowninFig.10.
TheRCstarspresentadifferentpicture:Asinglepower-law
providesonlyagoodfittothedataatR (cid:38) 6”/0.24pc(reduced
χ2 = 3.9), and with a higher value of Γ = 0.68±0.03 than in
caseofthefaintstars(seeFig.10).Thisvalueagreeswellwith
what has been found by previous authors for RC and brighter
stars(seeSchödeletal.2007,andreferencestherein).Also,the
flatteningoftheRCsurfacenumberdensityinsideR≈8”agrees
verywellwithpreviousstudies(Buchholzetal.2009;Doetal.
2009;Bartkoetal.2010).Itisinterestingtonote,however,that
Γ=0.52±0.03ifweforceasingle-powerlawfit(reducedχ2 =
7.5) to the RC number density profile, in agreement with the
valuefoundforthefaintstars.
4. Discussion
4.1. Influenceofthecorrectionfactors
Fig.9.Azimuthallyaveragedextinctionandcrowdingcorrectedsurface
densityvs.distancefromSgrA*forallStarFinderdatasetsandforthe Whendeterminingthenumberdensityprofile,wehavetoapply
Ks-magnituderanges15−16(redline)and17.5−18.5(blackline).Blue correction factors to compensate the effects of variable stellar
linesrepresentsimplepower-lawfits,withthecorrespondingpower-law
crowdingandinterstellarextinction.Fig.11showsthemeasured
indicesindicatednexttothelines.
surfacedensityprofileforstarsofmagnitude17.5 ≤ K ≤ 18.5
s
without any correction applied (blue), after applying the com-
pleteness correction for crowding (red), and after applying the
tion corrected ones from the deep images in the overlap region completeness corrections for crowding and extinction (black).
fromR = 10(cid:48)(cid:48) −20(cid:48)(cid:48).Fig.10showsthecombinednumberden- As we can see, the completeness correction steepens the pro-
sityplotsforstarsinthemagnituderanges16.5≤ K ≤17.5and file somewhat. The extinction correction only introduces minor
s
(17.5 ≤ K ≤ 18.5(left)andRCstars(right),alongwithsimple changes because the azimuthal averaging compensates most of
s
power-lawfits. theeffectsofdifferentialextinctionacrossthefield.Insummary,
Articlenumber,page8of14
E.Gallego-Cano etal.:ThedistributionofoldstarsaroundtheMilkyWay’scentralblackhole:I.Starcounts
cending branch or main sequence (MS) stars of ∼2.5M . They
(cid:12)
couldalsobepre-MSstarsofafewsolarmassesorless(Luetal.
2013). From what is known about the star formation history of
theNSCweexpectthatthemajorityofstarsisold(∼80%ofthe
NSC’smasswereformed> 5Gyrago,accordingtoPfuhletal.
2011)andthatmostofthefaintstarsinoursamplearethusold,
(sub-)giants.However,sinceweknowthatastarformationevent
created on the order 104M of young stars in the region about
(cid:12)
0.5pcaroundSgrA*,contaminationbyasignificantamountof
pre-MSobjectsispossible.Wewilldiscussthispossibilityinthe
nextsubsection.
4.3. Possiblecontaminationbypre-MSstars
Fig.11. Meansurfacedensityprofileforstarswith(17.5≤K ≤18.5),
s
afteraveragingoverthefourrunswithdifferentStarfinderparameters.
Blue:Uncorrecteddata.Red:Datacorrectedforcrowding.Black:Data
correctedforcrowdingandextinction.
the applied correction factors, albeit necessary, do not signifi-
cantlyalterourresults.Thisshowsthatourdataarerobust.
Table2.Parametersusedintheestimationofthesurfacedensityprofile
ofpre-MSstarsandresultingcorrectedΓforthedensityprofileofstars
withmagnitudes17.5≤K ≤18.5.
s
ID ηa Nb Γc
1d 1.40 146 -0.41±0.03
2e 0.93 200 -0.41±0.03
3f 1.10 200 -0.39±0.03
4g 0.93 300 -0.37±0.03
Fig.12. Measuredextinctionandcrowding-correctedsurfacenumber
5h 1.10 300 -0.36±0.03 densityprofilesfor16.5 ≤ K ≤ 17.5(redline)and17.5 ≤ K ≤ 18.5
s s
(green line). The black line represents the surface density profile for
Notes. 17.5≤K ≤18.5aftercorrectionforthepossiblecontaminationbypre-
s
MSstars,usingη=0.93±0.09(Doetal.2013)andN =300,roughly
a Power-lawindexofthesurface-densityprofile. estimatedbyusingtheinformationinFig.15ofLuetal.(2013).The
dottedlinesaresimplepower-lawfitstothedataandtheircorrespond-
b Totalnumberofpre-MSstarsatR=0.8”−12.5”.
ingindicesareindicatedinthepanel.
c Power-lawindexofthesurface-densityprofileoffaintstars
(17.5≤ K ≤18.5)aftercorrectionforpre-MSstars, Our primary goal in this study is to determine the spatial
s
distributionoftheold,relaxedstellarpopulationattheGC.Care
d Valueoftheη-parameterfrom(Bartkoetal.2010). mustthereforebetakentoexcludeyoungandthereforeprobably
dynamicallyunrelaxedstars.Wehaveexcludedfromtheanaly-
e Valueoftheη-parameterfrom(Doetal.2013).
sisallspectroscopicallyidentifiedearly-typestarsfromDoetal.
f,h Intermediatevalueoftheη-parametertotest. (2013). Unfortunately, the limit of spectroscopic identification
of early-type stars with current instruments is K ≈ 16 in the
S
g Valueoftheη-parameterfrom(Doetal.2013). GC (with the exception of a few, very deep exposures of small
fieldsthatreached K ≈ 17.5,seePfuhletal.2011).Asargued
s
intheprevioussection,thestarformationeventthattookplace
afewMyragointhecentralR ≈ 0.5pcmeansthatwehaveto
considerexplicitlythepossibilitythatourstellarsurfacenumber
4.2. Tracerpopulations
densitiesarecontaminatedbypre-MSstars.
With an approximate magnitude of K = 18, the faintest stars To do this, we assume that the density profile of the young
s
in our sample are consistent with being (sub-)giants on the as- starscanbedescribedbyasimplepower-law(e.g.,Bartkoetal.
Articlenumber,page9of14
A&Aproofs:manuscriptno.GC_starcounts_Gallego
2010;Luetal.2013).Themainparameterstotakeintoaccount the NSC stellar population is well mixed and that its average
are: η, the index of the power law and N, the total number of properties(massfunction)donotchange.Wescaledthedataof
young stars in the magnitude range 17.5 ≤ K ≤ 18.5 and Fritzetal.(2016)tooursintherange0.5pc≤R≤0.9pctoour
s
within R ≤ 12”. We assumed different values of η and N from surfacenumberdensityforstars17.5≤ K ≤18.5,correctedfor
s
the literature, subtracted the resulting pre-MS number density thepossiblepresenceofpre-MSstarswiththeparametersfrom
profiles and then derived the power-law indices of the profiles model ID4 in Table2. Subsequently, we subtracted a constant
forthecorrectednumberdensityprofiles,assummarisedinTa- star density offset to take into account the contribution of the
ble2.InFig.12weshowthesurfacedensityprofileforstarsof starsthatdonotbelongtothenuclearclusterbuttootherstruc-
17.5 ≤ K ≤ 18.5bothwithandwithoutcorrectionofcontami- tures,suchasthenucleardiskortheGalacticBulge.Thissimple
s
nationbypre-MSstars.Ascanbeseenfromthedifferentvalues procedureispossiblebecausethescalelengthsofthelattercom-
in Table2 , taking the possible bias from pre-MS stars into ac- ponents are one to several orders of magnitude larger than the
countwitharangeofreasonablevalues,canreducethemeasured half-light radius of the NSC (see, e.g., Launhardt et al. 2002;
power-law index from Γ = −0.51±0.02 to Γ = −0.36±0.03. Bland-Hawthorn&Gerhard2016).
This value is the most conservative one from our estimates be- We used a 3D Nuker model (Lauer et al. 1995) as given in
causeitassumesthehighestcontaminationratebypre-MSstars, equation 1 of Fritz et al. (2016) and projected it onto the sky
and we will adopt it here as a measure for the projected sur- to fit the surface density profiles at R ≤ 20pc. The fits were
face density of old, faint stars. We note that in PaperII we find repeated several times, using different values for the subtracted
Γ = −0.28±0.03 for the surface density of even fainter, constantstardensityandfortheinnermostradialdatarangethat
diffuse
unresolvedstars.Thetwovaluesoverlapwithintheir2σuncer- wasincludedinthefit.Wealsoperformedafitwithoutthecor-
tainties. rectionforthepotentialpre-MSstars.Theresultsofourfitsare
The value of Γ = −0.36 ± 0.03 lies > 10σ away from a listed in Table3. While the parameters r (break radius) and β
b
flatcore.Finally,wealsonotethatthebestpower-lawfittothe (outer power-law index) vary significantly, γ (inner power-law
surface number density of stars in the range 16.5 ≤ K ≤ 17.5 index) shows consistent values for all fits, which is reassuring
s
resultsinaverysimilarΓ=0.39±0.03.Althoughthereissome becauseitisthemostimportantparameterforthiswork,where
uncertaintyastowhetherstarsinthisbrightnessrangearesuit- we are interested in whether there exists of stellar cusp at the
abletracersofarelaxedpopulation,itisreassuringtonotethat GC. Fit IDs 1 and 2 explore extreme values for the subtracted
thisvaluecoincideswiththepre-MScorrectedΓofthefaintest constantstardensity.Weusethemeananderrorofthemeanof
tracers.Sincethemagnituderange16.5≤ K ≤17.5wouldcor- theparametersoftheotherfourfitstoobtainorientativevalues
s
respondtomoremassive–andthisshorter-lived–pre-MSstars, fortheaveragebestNukermodel:r = 3.0±0.4pc(77±10”),
b
weexpectaweakerpre-MScontaminationinthisrange(seealso γ =1.29±0.02,β=2.1±0.1,andadensityatthebreakradius
Fig.5in Luetal.2013). of ρ(r ) = 3900±900pc−3. The best fit according to line 4 in
b
Table3isshowninFig.13.
TheNukerfitshowsthatthefaintstarsshowacusp-likedis-
4.4. 3Dprofile:Nukerfit
tribution around SgrA*. A flat core can be excluded with high
significance.Weexplicitlynotethatinthefitspresentedinthis
workweomittheregionR≤1”(0.04pc).Inthisregion,thestar
counts appear to drop slightly below the expected levels. How-
ever,thisregionisalsothemostcrowdedregion,whichmaylead
tostrongsystematicsinthestarcounts.Additionally,thestellar
population in the extremely close environment of SgrA* may
havebeenaltered,asisindicatedbythepresenceofthesocalled
”S-stars”, apparently B-type MS stars that appear concentrated
withinR < 1”ofSgrA*andmayhavebeendepositedthereby
individual scatter or capture events (see, e.g., Eisenhauer et al.
2005;Genzeletal.2010;Alexander2011).
4.5. DistributionofgiantstarsnearSgrA*
In agreement with previous work, we have found an unexpect-
edlyflatsurfacedensityforRCstarsandbrightergiantswithin
about 0.3pc of SgrA*. This may indicate a possible deficit of
giants in this region. Here we produce a 3D Nuker model fit
forthegiantstarsandtrytoconstrainthenumberofpotentially
Fig. 13. Nuker model fit (red line) to the surface density profile for missinggiants.Weproceededasinsection4.4.Atlargeradii,we
17.5≤K ≤18.5correctedforthecontaminationbypre-mainsequence used the data from Fritz et al. (2016) and subtracted a constant
s
starsintheregion0.8”-12.5”. densityof0.07starspersquarearcsecondfromthedata.Wepro-
ducedfitsforRCstars,usingasaproxyfortheselectionofRC
In order to convert the measured 2D profile into a 3D den- starsthattheRCbumpintheK LFiscomprisedapproximately
s
sitylaw,weneedtodealwithprojectioneffects,whichrequires between15≤ K ≤16,andforspectroscopicallyidentifiedgiant
s
ustoconstrainthesurfacedensityonscaleslargerthanwhatwe stars(K ≤ 15.5)fordifferentrangesofR.Theresultingbest-fit
s
could measure with NACO. For this purpose we use the data parametersaregiveninTable4.
fromFritzetal.(2016),whichtheyacquiredfromobservations As can be seen from the reduced χ2 values, the quality of
with NACO/VLT, WFC3/HST, and VIRCAM/VISTA. To com- thefitimprovesasweomitthecentralmostdatapoints.Overall,
binethesedatawithours,wehavetoassumethat,onlargescales, the surface densities of RC stars and spectroscopically identi-
Articlenumber,page10of14