Triaxial orbit-based modelling of the Milky Way Nuclear Star Cluster

2.1 MB·

by  A. Feldmeier-Krause

MNRAS000,1–14(2016) Preprint9January2017 CompiledusingMNRASLATEXstylefilev3.0
Triaxial orbit-based modelling of the Milky Way Nuclear
Star Cluster (cid:63)
A. Feldmeier-Krause,1,2† L. Zhu3, N. Neumayer3, G. van de Ven3, P. T. de Zeeuw,1,4
R. Sch¨odel5
1European Southern Observatory (ESO), Karl-Schwarzschild-Straße 2, 85748 Garching, Germany
2The University of Chicago, The Department of Astronomy and Astrophysics, 5640 S. Ellis Ave, Chicago, IL 60637, USA
3Max-Planck-Institut fu¨r Astronomie, K¨onigsstuhl 17, 69117 Heidelberg, Germany
7 4Sterrewacht Leiden, Leiden University, Postbus 9513, 2300 RA Leiden, The Netherlands
1 5Instituto de Astrof´ısica de Andaluc´ıa (CSIC), Glorieta de la Astronom´ıa s/n, 18008 Granada, Spain
0
2
Accepted2016December21.Received2016December19;inoriginalform2016July22
n
a
J
ABSTRACT
6
We construct triaxial dynamical models for the Milky Way nuclear star cluster us-

] ing Schwarzschild’s orbit superposition technique. We fit the stellar kinematic maps
A presented in Feldmeier et al. (2014). The models are used to constrain the su-
G permassive black hole mass M•, dynamical mass-to-light ratio Υ, and the intrin-
sic shape of the cluster. Our best-fitting model has M = (3.0+1.1)×106M , Υ =
. • −1.3 (cid:12)
h (0.90+0.76)M /L ,andacompressionoftheclusteralongtheline-of-sight.Our
p −0.08 (cid:12) (cid:12),4.5µm
results are in agreement with the direct measurement of the supermassive black hole
-
o mass using the motion of stars on Keplerian orbits. The mass-to-light ratio is consis-
r tentwithstellarpopulationstudiesofothergalaxiesinthemid-infrared.Itispossible
t
s that we underestimate M• and overestimate the cluster’s triaxiality due to observa-
a tional effects. The spatially semi-resolved kinematic data and extinction within the
[ nuclear star cluster bias the observations to the near side of the cluster, and may

1 appear as a compression of the nuclear star cluster along the line-of-sight. We de-
v rive a total dynamical mass for the Milky Way nuclear star cluster of MMWNSC =
3 (2.1±0.7)×107M withinaspherewithradiusr =2×r =8.4pc.Thebest-fitting
(cid:12) eff
8 modelistangentiallyanisotropicinthecentralr =0.5-2pcofthenuclearstarcluster,
5 but close to isotropic at larger radii. Our triaxial models are able to recover complex
1 kinematic substructures in the velocity map.
0
. Key words: Galaxy: center; kinematics and dynamics.
1
0
7
1
v: 1 INTRODUCTION ± 0.6)×106M(cid:12) (Ghez et al. 2008), (4.3 ± 0.39)×106M(cid:12)
Xi The Milky Way nuclear star cluster is the ideal object to (Gillessen et al. 2009), or (4.02 ± 0.20)×106M(cid:12) (Boehle
et al. 2016). Unfortunately, similar high-resolution observa-
study the dynamics of a stellar system around a supermas-
r tions are not yet possible in other galaxies.
a sive black hole. At a distance of 8kpc it is close enough
to resolve the individual stars, and measure discrete veloc-
Alreadyinthe1970stherequirementofacentralsuper-
ities in three dimensions. Modelling the stellar kinematics
massive black hole in the Galactic centre was discussed to
can constrain the mass distribution of the star cluster, and
explain observational data (e.g. Oort 1977). Several studies
reveal the presence of a central dark massive object. In the
used stellar radial velocities to constrain the mass distribu-
specialcaseofourownGalaxy,ititpossibletoobserveKep-
tionintheGalacticcentre(e.g.Rieke&Rieke1988;McGinn
lerianorbitsofstarsaroundadark,point-mass-likeobjectin
et al. 1989; Sellgren et al. 1990; Haller et al. 1996; Genzel
the Galactic centre. These observations constrain this dark
et al. 1996). Also stellar proper motions were used to study
object to be a supermassive black hole with a mass of (4.1
the Galactic centre mass distribution (Sch¨odel et al. 2009).
Several studies combined radial velocity and proper motion
data (Trippe et al. 2008; Do et al. 2013; Fritz et al. 2016).
(cid:63) BasedonobservationscollectedattheEuropeanOrganisation
ThemassdistributionwasderivedusingthesphericalJeans
for Astronomical Research in the Southern Hemisphere, Chile
(289.B-5010(A)). (1922) equations or the projected mass estimators of Bah-
† E-mail:[email protected] call&Tremaine(1981)forsphericalsystems.Thesestudies
(cid:13)c 2016TheAuthors
2 A. Feldmeier-Krause et al.
foundthatacentraldarkmassof2−5×106M isrequired (J2000.0coordinates,Reid&Brunthaler2004)withrespect
(cid:12)
to explain the observations. totheGalacticplane.Thispaperisorganisedasfollows:We
Together with the increase of observational data, also describe the kinematic and photometric data in Section 2.
the modelling became more advanced. Trippe et al. (2008) The dynamical models are introduced in Section 3. Section
includedtherotationofthenuclearstarclusterinthemod- 4 discusses the results, and Section 5 summarizes the main
elling, although the rotation velocity of their data was too conclusions.
high (Sch¨odel et al. 2009; Feldmeier et al. 2014). Feldmeier
et al. (2014) and Chatzopoulos et al. (2015a) studied the
Milky Way nuclear star cluster using axisymmetric Jeans
models. Chatzopoulos et al. (2015a) showed the advantages
of axisymmetric models over spherical Jeans models, which 2 DESCRIPTION OF THE DATA
cannot explain the observed asymmetry of the velocity dis-
2.1 Kinematic data
persionofpropermotionsparallelandperpendiculartothe
Galactic plane. The nuclear star cluster appears to be flat- The line-of-sight velocity distribution (LOSVD) provides
tenedinitslightdistribution(Sch¨odeletal.2014)aswellas constraintsonthedynamicalstructureofstellarsystems.To
inthekinematics(Chatzopoulosetal.2015a).Moststudies extractthisinformation,weusedthenear-infraredK−band
showed that the nuclear star cluster kinematics is in agree- spectroscopicdatacubeofFeldmeieretal.(2014),whichhas
ment with isotropy (Sch¨odel et al. 2009; Do et al. 2013; a pixel scale of 2.22arcsec pixel−1. We used the data cube
Chatzopoulos et al. 2015a), although the uncertainties are that was cleaned of foreground stars and bright stars. The
quitelarge(e.g.Doetal.2013).Allthesemodelsassumeda cleaneddatacubecontainsthelightoftheoldredgiantstar
constant mass-to-light ratio for the light distribution of the population.
cluster. WefittedtheLOSVDasinFeldmeieretal.(2014),i.e.
In this study we relax the assumption of axisymmetry on the stellar CO absorption lines (2.2902−2.365µm) with
and use triaxial orbit-based Schwarzschild (1979) models. the IDL routine pPXF (Cappellari & Emsellem 2004) and
Orbit-based models make no assumptions on the velocity the high resolution spectra of Wallace & Hinkle (1996) as
anisotropy of the stellar motions, as Jeans models do. Fur- templatestars.WeappliedthesamespatialbinningasFeld-
ther, the higher moments of the kinematics can also be in- meier et al. (2014), resulting in 175 spatial bins. Feldmeier
cluded (Rix et al. 1997), which is important to break the et al. (2014) fitted only the velocity V and velocity disper-
degeneracy of mass and anisotropy in dynamical models. sion σ. However, we fitted in addition also higher moments
Orbit-basedmodelsarecommonlyusedtoanalyseline- oftheLOSVD,inparticulartheGauss-Hermiteparameters
of-sight velocity data of other galaxies (e.g. van der Marel h andh .Weaddednoisetoeachofthe175integratedlight
3 4
etal.1998;Gebhardtetal.2000;Vallurietal.2005;vanden spectra in 100 Monte Carlo simulation runs and obtained a
Bosch et al. 2008), and are an excellent tool to detect and distributionforeachmomentoftheLOSVD.Themeanand
measure the masses of supermassive black holes and dark standarddeviationoftheMonteCarlodistributionaretaken
matter halos. For extragalactic systems, the data are usu- as measurement and 1σ uncertainty of the kinematics.
ally obtained from integrated light observations. Each data SincetheMilkyWaynuclearstarclusterisatadistance
point contains the accumulated kinematics of many stars, of only 8kpc, the spectroscopic observations are spatially
weightedbytheirrespectivebrightness.However,modelling semi-resolved.Brightstarscanberesolvedindividually,and
the dynamics of integrated light data may be prone to sys- contribute a large fraction of the flux. For that reason we
tematic uncertainties, and bias the results of the central usedthecleaneddatacubeofFeldmeieretal.(2014),where
black hole mass. Therefore, it is interesting to test dynami- bright stars were excluded. However, the kinematic maps
cal models on systems for which we know the central black still show stochastic shot noise. As a consequence, the dif-
holemassfromotherindependentmeasurements.TheMilky ferenceofthekinematicsinadjacentbinscanbehigherthan
Waynuclearstarclusterisagoodobjectforthiskindoftest. their uncertainties, which causes problems when we model
Also megamaser disc galaxies are useful to validate stellar thekinematics.Thestochasticnoisecanbemistakenforsig-
dynamical black hole measurements. Black hole mass mea- nal,andthismeansthebestfitwillbeachievedbymodelling
surementsfrommegamasersareveryprecisewithuncertain- the shot noise. To prevent this, we increased our kinematic
ties of only about 10 per cent. However, there is currently uncertainties (cid:15) such that the difference of the measure-
V
only one megamaser disc galaxy with a stellar dynamical ment in two adjacent bins (V −V ) is less than the sum
i j
black hole mass measurement (van den Bosch et al. 2016), of their uncertainties ((cid:15) +(cid:15) ). We did this for the ve-
Vi Vj
NGC 4258. Different dynamical studies found either a 15 locity V, velocity dispersion σ, h , and h , and find that
3 4
per cent lower or a 25 per cent higher black hole mass than it is required for about 68 per cent of the kinematic data
the maser measurement (Siopis et al. 2009; Drehmer et al. uncertainties. Additionally, we point-symmetrised the kine-
2015). matics using the procedure of van den Bosch & de Zeeuw
We use the triaxial orbit-based code by van den Bosch (2010). The median uncertainties or V, σ, h , and h are
3 4
et al. (2008) to model the light distribution and line-of- 24.6kms−1,18.4kms−1,0.15,and0.17.Ourresultingkine-
sight kinematics of the Milky Way nuclear star cluster. maticmapsareconsistentwiththemapsofFeldmeieretal.
We use the spectroscopic data cube constructed by Feld- (2014). We find rotation in the velocity map of approxi-
meieretal.(2014)forthekinematicdata,andderiveasur- mately 50kms−1 and an increase in the velocity dispersion
facebrightnessdistributionusingSpitzer 4.5µmandNACO fromabout65kms−1 towardsσ =135kms−1 atthecen-
max
H−band images. We assume a galactocentric distance of tre. The kinematic maps are shown on the top row of Fig.
8kpc(Malkin2012)andapositionangle31◦.40EastofNorth 1, the uncertainties are shown on the bottom row.
MNRAS000,1–14(2016)
Triaxial orbit-based modelling of the Milky Way Nuclear Star Cluster 3
V σ h h
3 4
150 150 150 150
100 100 100 100
Data 50 arcsec 50 arcsec 50 arcsec 50
0 0 0 0
-50 -50 -50 -50
km s-1 km s-1
-61 -31 0 31 61 36 61 86 110 135 -0.2 -0.1 0.0 0.1 0.3 -0.3 -0.1 0.0 0.2 0.3
150 150 150 150
s 100 100 100 100
e
nti c c c
ncertai 500 arcse 500 arcse 500 arcse 500
U
-50 -50 -50 -50
-100 -50 0 50 100 -100 -50 0 50 100 -100 -50 0 50 100 -100 -50 0 50 100
arcsec arcsec arcsec arcsec
km s-1 km s-1
5 19 33 47 61 8 19 31 43 54 0.06 0.13 0.20 0.27 0.34 0.08 0.14 0.20 0.26 0.32
Figure 1.Kinematicdata(toprow)andrespectiveuncertainties(bottomrow).ThecolumnsdenotevelocityV,velocitydispersionσ,
Gauss-Hermitemomentsh3,andh4.Whitepixelsareduetoexcludedbrightstars.
2.2 Imaging data and surface brightness the dark 20-kms−1-cloud M-0.13-0.08 (Garc´ıa-Mar´ın et al.
distribution 2011).
We used the MGE FIT SECTORS package (Cappel-
The light distribution of the nuclear star cluster traces the
lari 2002) to derive the surface brightness distribution. The
stellar density. We require the two-dimensional light distri-
Multi-Gaussian-Expansionmodel(Emsellemetal.1994)has
butionoftheredgiantstars,whichareourkinematictracers.
the advantage that it can be deprojected analytically. We
TheextinctionishighatopticalwavelengthsintheGalactic
measuredthephotometryofthetwoimagesalongthemajor
centre (A ∼30mag, Scoville et al. 2003; Gao et al. 2013),
V axisandtheminoraxis.Weassumedthatthecluster’smajor
therefore we used near- and mid-infrared images.
axisisalignedalongtheGalacticplane,asfoundbySch¨odel
Forthecentral40.4arcsec×40.4arcsec(1.6pc×1.6pc) et al. (2014), and constant. The centre is the position of
we used the high-resolution NACO H-band mosaic of SgrA*,whichistheradiosourceassociatedwiththeGalac-
Sch¨odel et al. (2009), which has a spatial scale of tic centre supermassive black hole. We fitted a scale factor
0.027arcsecpixel−1. We preferred the H band over the K to match the photometry of the two images in the region
bandinordertoavoidlightfromgasemissionlinesintheK wheretheyoverlap(16−27.8arcsec).Thenwemeasuredthe
band (Brγ and HeI, Paumard et al. 2004). Our kinematic photometryoneachimagealong12angularsectors,andcon-
tracersarecoollate-typestars,buttherearealsomorethan vertedtheNACOphotometrytotheSpitzer flux.Assuming
100 hot, young stars located in the centre of the cluster, four-foldsymmetry,themeasurementsoffourquadrantsare
withinaprojectedradiusr=0.5pc(∼12.8arcsec,Paumard averagedonellipticalannuliwithconstantellipticity.Using
etal.2006).Wemaskedouttheyoungstarsfromtheimage thephotometricmeasurementsofthetwoimages,wefitteda
with a 15 pixel radius. For the bright red supergiant IRS 7 setoftwo-dimensionalGaussianfunctions,takingthepoint-
we used a larger mask with a 30 pixel radius. Beyond the spread-function (PSF) of the NACO image into account.
central0.5pc,thenuclearstarclusterlightisdominatedby AcomparisonwiththesurfacebrightnessprofileofFritz
cool stars, and the contribution of young stars is negligible etal.(2016,theirFig.2)showedthatourprofileissteeperin
(Feldmeier-Krause et al. 2015). the central ∼30arcsec. A possible reason is the small over-
For the large-scale light distribution, we used Spitzer lap region of the Spitzer and NACO images, and that the
IRAC images (Stolovy et al. 2006). These images were cor- Spitzer fluxcouldbetoohighatthecentre.MaybethePAH
rected for dust extinction and PAH emission by Sch¨odel emission correction of the Spitzer image was too low. The
etal.(2014).Weusedtheextinctionandemissioncorrected mid-infrared dust emission is significant out to almost one
4.5µm image to measure the light distribution. The image arcmin.Fritzetal.(2016)usedNACOH-andK -bandim-
S
was smoothed to a scale of 5arcsec pixel−1, and extends agesinthecentralr=20arcsec.Outto1000arcsec(∼39pc)
over ∼270pc×200pc. We excluded a central circle with they used Hubble Space Telescope WFC3 data (M127 and
r=0.6pc (∼15.4arcsec) to avoid contribution from ionised M153filters)andpublicVISTAVariablesintheViaLactea
gas emission and young stars. In addition we masked out Survey images (H and K bands, Saito et al. 2012). We
S
the young Quintuplet star cluster (Figer et al. 1999), and lowered the intensities of the central Gaussians by scaling
MNRAS000,1–14(2016)
4 A. Feldmeier-Krause et al.
Table 1. The Multi-Gaussian-Expansion (MGE) fit parameters
r [arcsec]
for the 4.5µm Spitzer/IRAC dust extinction and PAH emission
1 10 100 1000
corrected image in combination with the NACO H-band mosaic 107
scaledtoSpitzer flux.Iscaled isthepeaksurfacebrightnessused Fritz et al. 2016
inthedynamicalmodelling,σMGEisthestandarddeviation,and SNpAitCzeOr ddaattaa
qMGE istheaxialratiooftheGaussiancomponents.Iunscaled is
thepeaksurfacebrightnessbeforescalingtoFritzetal.(2016).
-2c 106
e
s
Iscaled σMGE qMGE Iunscaled ar
(cid:2)104L(cid:12),4.5µmpc−2(cid:3) [arcsec] (cid:2)104L(cid:12),4.5µmpc−2(cid:3) pm
µ
03.28.46 110..74 00..3304 311624 L O •, 4.5 105
89.8 15.0 0.82 257
18.5 52.1 0.95 30.0 major axis
minor axis
17.0 98 0.36 29.3
7.1 154 0.95 7.4
0.7
4.8 637 0.36 4.9
3.2 2020 0.30 3.2 qproj 00..56
1.3 4590 0.81 1.3
0.4
0.1 1.0 10.0 100.0
r [pc]
ouraveragedprofiletotheone-dimensionalfluxdensitypro-
file of Fritz et al. (2016). As a result the amplitudes of the
inner Gaussians become smaller, but the outer Gaussians Figure2.UpperPanel:Surfacebrightnessprofilederivedfroma
(σ >100arcsec∼4pc)arenearlyunchanged.Welistthe dustextinctionandPAHemissioncorrectedSpitzer/IRAC4.5µm
MGE
image and NACO H-band mosaic for the centre, scaled to the
componentsoftheMulti-GaussianExpansioninTable1and
measurements of Fritz et al. (2016, blue crosses). The black full
plot the surface brightness profile in Fig. 2 (upper panel).
linedenotestheMGEfitalongthemajoraxis,andthereddashed
We also show the projected axial ratio q as a function
proj linealongtheminoraxis.Lowerpanel:Projectedaxialratioqproj
of radius in the lower panel of Fig. 2. Out to the central
asafunctionofr.
1pc,qproj isincreasingfrom0.4to0.7.Sch¨odeletal.(2014)
found a mean axial ratio of 0.71±0.02 for the nuclear star
cluster. This is in agreement with our maximum value of
perposition. The first step of Schwarzschild’s method is to
q . However, q decreases at larger radii, as the contri-
proj proj integratetheequationsofmotionforarepresentativelibrary
butionfromthenuclearstellardiscbecomesmoreimportant
ofstellarorbitsinagravitationalpotentialΦ.Thentheob-
and the light distribution therefore flatter.
servables for each orbit are computed, considering projec-
We note that there are three main differences with the
tion,PSFconvolutionandpixelbinning.Thenextstepisto
surface brightness distribution derived by Feldmeier et al.
findorbitalweightstocombinetheorbitssuchthattheyre-
(2014):(1)WeusedanH-bandinsteadofaK -bandNACO
S producetheobserveddata.Schwarzschildmodelsareapow-
image to avoid ionised gas emission; (2) We masked young
erful tool to recover the intrinsic kinematical structure and
starsintheNACOimagetomatchthedistributionofstars
the underlying gravitational potential (Schwarzschild 1979;
used as kinematic tracers; and (3) We scaled the central
van de Ven et al. 2008; van den Bosch & van de Ven 2009).
photometry to the flux density data of Fritz et al. (2016)
WereferthereaderforfurtherdetailstovandenBoschetal.
to avoid a possible overestimation of the central flux when
(2008) for implementation and van de Ven et al. (2008) for
scaledtotheSpitzer image.Allthreechangesinfluenceonly
verification of the triaxial Schwarzschild code.
the central part of the surface brightness distribution, as
ionised gas emission and light from young stars are only
important in the central parsec. 3.1.1 Mass model
The surface brightness profile is deprojected to obtain
We calculated orbits in the combined gravitational poten-
the three-dimensional spatial stellar light distribution. In
tialofasupermassiveblackholeΦ andthestarclusterΦ ,
general, the deprojection of a surface brightness profile is • (cid:63)
inferred from the imaging data. As we run triaxial models,
non-unique(e.g.Rybicki1987;Franx1988),andourdepro-
there are three intrinsic shape parameters, q, p, and u, for
jectionisonlyonepossiblesolution.TheMGEdeprojection
thecluster.Theshapeparameterscharacterisetheaxialra-
producessmoothintrinsicdensities,whichareinagreement
tios for the long, intermediate and short axes a, b, and c.
with the photometric observations (Cappellari 2002).
They are defined as q =c/a, p=b/a, and u=a(cid:48)/a, where
a(cid:48) is the length of the longest axis a projected on the sky.
Thus, u represents the compression of a due to projection
3 DYNAMICAL MODELS OF THE MILKY onthesky.Eachsetofaxialratiosreferstoasetofviewing
WAY NUCLEAR STAR CLUSTER angles(ϑ,φ,ψ,seealsovandenBoschetal.2008).Thesur-
facebrightnessdistributionisdeprojectedgiventheintrinsic
3.1 Schwarzschild’s method
shapeparametersq,p,u,andmultipliedwiththedynamical
Orbit-based models or Schwarzschild models are a useful mass-to-lightratioΥtogettheintrinsicstellarmassdensity
tool to model the dynamics of stellar systems by orbit su- (cid:37) . From Poisson’s equation ∇2Φ = 4πG(cid:37) one calculates
(cid:63) (cid:63) (cid:63)
MNRAS000,1–14(2016)
Triaxial orbit-based modelling of the Milky Way Nuclear Star Cluster 5
the gravitational potential. We did this for different values orbit sampling we refer to van den Bosch et al. (2008). In
oftheblackholemassM ,dynamicalmass-to-lightratioΥ, total, we have N × N × N = 35 × 14 × 14 = 6860
• E I2 I3
and different shape parameters. In total our model has five orbits.Eachorbitwasintegratedover200periods,andsam-
free parameters, M , Υ, q, p, and u. pledon100000pointsperorbit.Foreachorbitwestoredthe
•
Besidestheconsideredstellarpopulationandthesuper- intrinsic and projected properties. The projected orbits are
massive black hole, there are other components within the stored in a (x(cid:48),y(cid:48),v ) grid, with PSF convolution and pixel
z
nuclearstarcluster,whichweneglected.Wemeasuredady- size of the observed data taken into account. The velocities
namicalmass-to-lightratio,whichcombinesthestellarmass- were stored in 183 bins between −7.4 σ and +7.4 σ .
max max
to-lightratiowithothercomponents.Thesecomponentsare Thesenumbersguaranteeapropersamplingoftheobserved
the young stars, ionised gas, neutral gas, and dark matter. velocity profiles (Cretton et al. 1999).
The young stars are at a distance of about 0.5pc from the
supermassive black hole. The lower limit of the total mass
ofyoungstarsis12000M .However,thetotalenclosedex-
(cid:12)
tendedmassinthesameregionis∼106M (Ohetal.2009;
(cid:12)
Feldmeier et al. 2014), and the mass of the supermassive
black hole is 4×106M . The mass of the young stars is 3.1.3 Solving the orbital weight distribution
(cid:12)
thereforeprobablynegligible.Thehotionisedgashasamass
The model has to fit the kinematic data, the intrinsic and
of only a few 100M (Ferri`ere 2012), and cannot influence
(cid:12) theprojectedmassdistribution.Thefitwasdonebyfinding
the stellar dynamics significantly. The neutral gas in the
a linear combination of the orbits, and solving for orbital
circum-nuclear disc may contribute more mass, estimates
weights γ . Each orbital weight corresponds to a mass on
range from 104M (Etxaluze et al. 2011; Requena-Torres i
(cid:12) the respective orbit i, and the weights γ are therefore non-
etal.2012)to106M (Christopheretal.2005),thoughthis i
(cid:12) negative. We used the non-negative least-squares (NNLS)
isprobablytheupperlimit(Genzeletal.2010).Thecircum-
algorithm of Lawson & Hanson (1974) which was also used
nuclear disc extends over a distance of about 1pc to more
by Rix et al. (1997), van der Marel et al. (1998), and Cret-
than5pcfromthecentre.At5pc,thetotalenclosedmassis
ton et al. (1999). One of the fitting constraints is to make
∼107M (McGinnetal.1989;Feldmeieretal.2014).Wede-
(cid:12) surethatthemodelisself-consistent.Itisrequiredthatthe
cidedtoneglectthemassdistributionofthecircum-nuclear
orbit superposition reproduces the intrinsic and projected
discinourdynamicalmodels,sinceitisveryuncertain,and
aperture masses within two per cent, which is the typical
makes up only 0.1 to 10 per cent of the enclosed mass. The
accuracy of the observed surface brightness (van den Bosch
contributionofdarkmattertothenuclearstarclustermass
et al. 2008).
is also neglected. Linden (2014) showed that the fraction of
darkmatterinthecentral100pcoftheMilkyWayisabout
6.6percent,assumingthetraditionaldarkmatterprofileof
Navarro et al. (1996).
3.1.2 Orbit library 3.2 Constraining the input parameters
The orbit library should be as general as possible and rep- We ran 4899 models with different parameter combinations
resentativeforthegravitationalpotential.Weassumedthat of M , Υ, q,p,u. The black hole mass M was sampled in
• •
theorbitsareregularandthatthreeintegralsofmotion,E, logarithmic steps of 0.2 from 5.5 to 7.5, starting with 6.3
I , and I , are conserved. The orbit families consist of box (i.e. M ∼2×106M ). The mass-to-light ratio Υ was lin-
2 3 • (cid:12)
orbits,whichcancrossthecentreandhaveanaverageangu- earlysampledbetween0.1and2.0withstepsof0.04,witha
larmomentumofzero,andthreetypesoftubeorbits,which startingvalueof0.6(inunitsofM /L ).Thestarting
(cid:12) (cid:12),4.5µm
avoid the centre. The tube orbits are divided in short-axis- model had (q,p,u) = (0.29, 0.84, 0.99). We sampled differ-
tube orbits, which have non-zero mean angular momentum ent combinations of (q,p,u) with a step size of (0.01, 0.02,
(cid:104)L (cid:105) around the short axis, outer and inner long-axis-tube 0.01), with the boundaries 0.05 <q <0.29, 0.40 <p <0.99,
z
orbits, which have non-zero mean angular momentum (cid:104)L (cid:105) and0.70<u(cid:54)0.99.Whenfittingtheorbitalweights,weal-
x
aroundthelongaxis.Theorbitgridshouldsampletheentire readyappliedtheself-consistencycriteriatomakesurethat
phasespace.Ithastobedenseenoughtosuppressdiscrete- thephotometryisfittedwithahighaccuracyofatleasttwo
ness noise, but integration has to be done in a reasonable per cent for each model. The best fit of the five parameters
amount of computing time. M , Υ, q,p,u, however, was found by calculating the χ2 of
•
We followed van den Bosch et al. (2008) and sampled the kinematic moments. After the starting model we com-
the orbit energy E using a logarithmic grid in radius. Each puted different combinations of (M , Υ, q,p,u) in the grid.
•
energyEislinkedtotheradiusR bycalculatingthepoten- We used a χ2 analysis to find regions with good fits and
c
tial at (x,y,z) = (R , 0, 0). We sampled N = 35 energies started new models, with a finer sampling at low χ2. This
c E
calculated from R in logarithmic steps ranging from R = was done in several iterations, and it was not necessary to
c c
100.5 to R = 104.2, i.e. 3.16arcsec to 4.4degree or 0.12pc compute each point in the (M , Υ, q,p,u) grid.
c •
to616.5pc.Wenotethattheouterradiusisabout3.5times We measured four kinematic moments in 175 spatial
the outermost Gaussian σ of the MGE fit. We tested bins of the spectroscopic map in Sect. 2.1. In each bin we
MGE
lowervaluesoftheinnerradiusbutfoundconsistentresults. measured the velocity V, velocity dispersion σ, and the
For each energy, the starting point of an orbit was selected higherGauss-Hermitemomentsh ,andh .Thus,thenum-
3 4
from a linear grid over 14 values each. For details on the ber of observables is the number of spatial bins times the
MNRAS000,1–14(2016)
6 A. Feldmeier-Krause et al.
surface brightness V σ h h
3 4
150
100
c
se 50
c
ar
0
-50
km s-1 km s-1
-2.7 -2.1 -1.4 -0.7 0.0 -57 -29 0 29 57 36 61 86 111 136 -0.2 -0.1 0.0 0.1 0.3 -0.3 -0.1 0.0 0.2 0.3
150
100
c
se 50
c
ar
0
-50
-100 -50 0 50 100 -100 -50 0 50 100 -100 -50 0 50 100 -100 -50 0 50 100 -100 -50 0 50 100
arcsec arcsec arcsec arcsec arcsec
Figure3.Comparisonoftheobservedstellarsurfacebrightnessandkinematics(toprow)andthebest-fittingSchwarzschildmodel.The
columnsdenotesurfacebrightness,velocityV,velocitydispersionσ,Gauss-Hermitemomentsh3,andh4.
r [pc] number of kinematic moments, in our case N = 175 × 4 =
s -4 -2 0 2 4 700. We minimised the function
es 1.0
n
ht 0.8
-1m s]surface brig 0005...24600 χ2 =(cid:88)i1=751(cid:32)(cid:18)hV3,ii(cid:15)−−h(cid:15)3VV,hiiim3m,ioodd(cid:33)(cid:19)22++(cid:32)(cid:18)hσ4i,−i(cid:15)(cid:15)−hσσ4iim,him4o,diod(cid:19)(cid:33)2+2, (1)
k
V [ -50 where V,σ,h3, and h4 denote the kinematic measurements,
(cid:15) the respective measurement uncertainties, Vmod, σmod,
-100
hmod,andhmodthemodelkinematics,tofindthebest-fitting
150 3 4
-1s] model.
m 100
k
σ [ 50
0 3.3 Modelling results
0.4
3.3.1 The best-fitting model
0.2
h3 0.0 Our best-fitting parameters are M = 3.0×106M , Υ =
• (cid:12)
-0.2
0.90,q=0.28,p=0.64,u=0.99.Thiscorrespondstobest-
-0.4
fitting viewing angles ϑ = 80◦, ϕ = 79◦, ψ = 91◦. We show
0.4 the surface brightness map and the symmetrised kinematic
0.2 maps in Fig. 3. The upper row are the data, the lower row
h4 0.0
arethemapsofthebest-fittingmodel.Themisalignmentof
-0.2
thekinematicrotationaxiswithrespecttothephotometry,
-0.4
and the perpendicular rotating substructure at ∼20arcsec
-100 -50 0 50 100
(∼0.8pc) found by Feldmeier et al. (2014) are well repro-
r [arcsec]
ducedinthemodelvelocitymap.Wealsoshowthedataand
the best-fitting model in Fig. 4, the panels denote surface
Figure 4. The surface brightness and kinematics of the best-
brightness, V, σ, h and h . The surface brightness map is
fitting model (red filled diamond symbols) and the data (open 3 4
reproducedwithinonepercent.Thehighestandlowestval-
blacksquaresanderrorbars)asafunctionofradius.Datapoints
intheGalacticEastareplottedatnegativeradii,GalacticWest uesofV,h3,h4,andthelowestvaluesofσ inthedatahave
atpositiveradii.Fromtoptobottom:surfacebrightness,velocity higherabsolutevaluesthaninthebest-fittingmodel,butare
V,velocitydispersionσ,andhigherGauss-Hermitemomentsh3 consistent within their uncertainties. The best fit has χ2 =
andh4. 290.WithM=5fittedparametersandN=4×175=700ob-
servationalconstraints,thismeansχ2 =0.42.Thatχ2 is
red red
lessthanoneispartiallyduetothelargeuncertaintiesofthe
MNRAS000,1–14(2016)
Triaxial orbit-based modelling of the Milky Way Nuclear Star Cluster 7
Figure 5.Illustrationofthefittedparameterspace,χ2 wascomputedwithEq.1.Eachsymboldenotesamodel,thecolouredsymbols
aremodelswith∆χ2 <σχ2 =37.3,blackdiamondsaremodelswith∆χ2 >σχ2.The1σ,2σ,and3σ colourscorrespondingto∆χ2 =
5.9,11.3and18.2,aredenoted.Theblackcrossdenotesthebest-fittingmodel.
q p u
0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.50 0.60 0.70 0.80 0.90 0.94 0.95 0.96 0.97 0.98 0.99
2.52
χ2) 2.50
g(
o
l
2.48
2.46
2.52
χ2) 2.50
g(
o
l
2.48
2.46
0.50 1.00 1.50 5.8 6.0 6.2 6.4 6.6 6.8
ϒ log (M )
BH
Figure 6.Theχ2 valuesplottedagainstthefivefreeparameters(M•,Υ,q,p,u).χ2 wascomputedwithEq.1.Thebest-fittingmodel
isdenotedasblueasterisk,the1σ,2σ,and3σ confidencelevels,correspondingto∆χ2 =5.9,11.3,and18.2,aredenotedasbluelines.
Theredlinedenotesσ2 =37.3.
χ
MNRAS000,1–14(2016)
8 A. Feldmeier-Krause et al.
Table 2. The best-fitting model results and the 1σ and 3σ un- r [pc]
0.1 1.0 10.0 100.0
certainties,correspondingto∆χ2 =5.9and18.2. 109
McGinn et al. 1989
Lindqvist et al. 1992
parameter bestfit 1σ 3σ unit Deguchi et al. 2004
Trippe et al. 2008
Oh et al. 2009
M• 3.0 +−11..13 +−22..43 ×106 M(cid:12) 108 SFcehldömdeeli eert eatl .a 2l.0 210414
Υ 0.90 +−00..7068 +−10..1322 M(cid:12)/L(cid:12),4.5µm M ]O · CFrhitazt zeot paol.u 2l0o1s6 et al. 2015
pq 00..6248 +−+−0000....0100826 +−+−0000....0302062 ...... M [enclosed107
u 0.99 +0.0 +0.0 ...
−0.01 −0.05
106
kinematics, and the fact that the kinematic measurements
are correlated.
1 10 100 1000
We illustrate the distribution of χ2 for the 4899 mod- r [arcsec]
els in Fig. 5. We plot each combination of parameters. Red
colours denote low χ2, bluer, smaller symbols denote high Figure 7. Enclosed total mass in spherical shells as a func-
χ2. The black cross denotes the best-fitting model. The ob- tion of radius, in units of M(cid:12) and in logarithmic scaling. The
served projected flattening q of the surface brightness black line denotes the enclosed mass with Υ = 0.90 and M• =
profile constrains the viewingMGanEgles and thus also the in- 3.0×106M(cid:12), the grey shaded contours the 3σ uncertainty. The
horizontal line denotes a supermassive black hole with the mass
trinsic shape parameters. In particular, a flat q (cid:28) 1
MGE M• = 4×106M(cid:12). The vertical, dotted line denotes the outer
means that the stellar system is observed along one of the
edge of the kinematic data, the vertical, solid line the effective
principal planes (van den Bosch et al. 2008). We obtained radius.Wealsoplottheresultsfortheenclosedmassfromprevi-
aslowestvalueqMGE =0.30,thislimitsthepossibleprojec- ousstudies.WescaledthemassestoadistanceR0=8.0kpc,ifthe
tionsandputsanupperlimitontheparameterq.Likewise, studyassumedadifferentGalactocentricdistance:McGinnetal.
thevalueofu=0.99istheboundaryvalueofthegrid.The (1989,diamonds,assumedR0 =8.5kpc),Lindqvistetal.(1992,
values of q,p, and u denote the respective minimum values upward triangles, R0 = 8.5kpc), Deguchi et al. (2004, squares),
of the radially varying intrinsic axial ratios q , p and Trippe et al. (2008, x-symbol), Oh et al. (2009, leftfacing tri-
intr intr
angles), Sch¨odel et al. (2014, asterisk), Feldmeier et al. (2014,
u over the entire radial range of the photometry, i.e. the
intr blue dashed line), Chatzopoulos et al. (2015a, rightfacing trian-
nuclear stellar disc and the embedded nuclear star cluster.
TheupperleftpanelofFig.5showsthatforeachvalueofu, gle,R0 =8.3kpc),andFritzetal.(2016,downwardtriangle,R0
=8.2kpc).
thebest-fittingΥisapproximately0.90.Asimilarbehaviour
is found with q and p. There is only a slight increase of the
best-fittingΥwithhighervaluesofp.Atthesametime,the
r [pc]
best-fitting values of M• do not show a strong dependence 1 2 3 4 5
on q,p, or u (second row) in the allowed parameter ranges. 1.0
Theintrinsicshapeparametersdonotinfluenceourbestfit
for M , as this measurement is mostly made from the in- 0.8
•
nerbinsandtheouterbinscontributelittle.Theouterbins,
however,contributetotheintrinsicshapefit.Thesupermas- o 0.6
ati
sive black hole mass and the dynamical mass-to-light ratio s r
are correlated. For higher values of Υ, a lower M• fits the axi 0.4
data.
p=b/a
We show how χ2 depends on the different parameters
0.2
q=c/a
in Fig. 6. The best-fitting model, which has the lowest χ2,
T
is marked as blue asterisk symbol. The blue lines denote 0.0
the 1σ, 2σ, and 3σ confidence limits, corresponding to ∆χ2 20 40 60 80 100 120 140
= 5.9, 11.3, and 18.2. The red line illustrates the standard r [arcsec]
deviation of χ2 itself, i.e. (cid:112)2(N−M) = 37.3, where N =
700,andM=5.Thisvaluewasusedasconfidencelimitby Figure8.Intrinsicaxialratiosasafunctionofradiusr.Theblue
van den Bosch & van de Ven (2009). In Table 2 we list the dot-dashedlinedenotesq=c/a,theblacksolidlinep=b/a,the
greendashedlinethetriaxialityT =(1−p2)/(1−q2),theshaded
1σ and 3σ uncertainties.
regionsthe1σ uncertainties,thevertical,solidliner =4.2pc.
eff
3.3.2 Mass profile and intrinsic shape
(2014)andChatzopoulosetal.(2015a)assumedaxisymme-
We show the enclosed total mass as a function of the pro- try. Some of the studies used also different Galactocentric
jectedradiusinFig.7,greyshadedcontoursarethe3σ un- distances, so we scaled the masses using R = 8.0kpc. Our
0
certainties.Themasswascomputedwithinsphericalshells. results are in agreement with several studies in the central
Wealsoplottheresultsofvariousotherstudies.Moststud- 100arcsec. However, we obtain a lower enclosed mass than
iesassumedasphericalshapeofthecluster,Feldmeieretal. Trippe et al. (2008) and Oh et al. (2009), who used the
MNRAS000,1–14(2016)
Triaxial orbit-based modelling of the Milky Way Nuclear Star Cluster 9
Jeansequationforasphericalsystemtoobtaintheenclosed r [pc]
mass. At larger radii r ≈ 400arcsec (∼15.5pc) beyond the 1 2 3 4 5
reachofourkinematicdata,weobtainedahighermassthan
Lindqvist et al. (1992), but we are in agreement at r=750– 0.2
1600arcsec. Their data extend to larger radii, but their as-
0.1
sumptionofsphericalsymmetrydoesnolongerholdatsuch σ2 r
large radii. / 0.0
Theenclosedtotalmasswithinaspherewiththeradius σ2 t
r = 8.4pc, i.e. about two times the effective radius of the = 1 - -0.1
nuclearstarcluster,isMMWNSC=(2.1±0.7)×107M(cid:12),here β -0.2
we give the 3σ uncertainty.
The black hole influences the stellar kinematics only -0.3
at the centre of the nuclear star cluster. Out to r =
-0.4
53arcsec(∼2pc),thebest-fittingmassoftheblackhole(M• 20 40 60 80 100 120 140
=3.0×106M(cid:12)) is higher than the enclosed stellar mass of r [arcsec]
our best-fitting model. Merritt (2004) defined the radius of
-
r [pc]
influence of a black hole as the radius where the enclosed 1 2 orb3 4 5
stellarmassequalstwotimestheblackholemass.Withthis 100
definition and a black hole mass of 4×106M , we obtain Long-axis tube orbits
(cid:12)
rinfl = (104+−5269)arcsec, i.e. approximately (4.0+−21..21)pc. This %) 80 SBhooxr ot-rabxitiss tube orbits
resultisinagreementwithAlexander(2005),whofoundrinfl n (
=3pc.Thekinematicmeasurementsatlargerradiihavelit- ctio 60
tle influence on the black hole mass measurement, but are a
r
important to constrain the orbital structure and dynamical e f
massT-thoe-lisghhatpreatoifo.the nuclear star cluster is illustrated in bit-typ 40
r
Fig. 8. We show the intrinsic axial ratios q and p as a func- O 20
tion of radius r. The axial ratio q = c/a is low in the cen-
tre(q=0.3),increasestoq=0.8atr≈35arcsec,andthende-
creasestoq=0.6atr=150arcsec.Thebest-fittingshapepa- 20 40 60- 80 100 120 140
r [arcsec]
rameter,q=0.28,isapproximatelythecentralvalue.Theax- orb
ial ratio p=b/a is also low in the centre (∼0.65), increases
Figure 9. Top: Anisotropy β as a function of radius r. Neg-
to p=0.9 at r=40arcsec, and then decreases to p=0.75 at
ative values denote tangential anisotropy, positive values radial
r=150arcsec. Though, the uncertainty of p is rather high,
anisotropy. Bottom: Fraction of stellar mass per orbit type as a
and the decrease not significant. Also the best-fitting axial
functionofaverageorbitradiusr¯ .Thegreen,solidlinedenotes
orb
ratio, p=0.64, is close to the central value. We plot the tri-
long-axis-tubeorbits;theblue,dot-dashedlineshort-axis-tubeor-
axialityT =(1−p2)/(1−q2),itvariesbetween0.45and0.7. bits;thered,dashedlineboxorbits;thevertical,solidliner =
eff
The triaxiality increases from r=70arcsec to the outer ra- 4.2pc.
diusr=150arcsec.Thismaybeasignatureoftheincreasing
influenceofthetriaxialGalacticbulgeatlargerradii(Tsatsi
etal. 2017). However, giventhe large uncertaintyof T, this
increase is not significant. A constant T or a decreasing T
are not excluded by the large uncertainties.
thelongx-axis(bottompanel),andorbitswith(cid:104)λ (cid:105)(cid:54)=0are
x
contributed by long-axis-tube orbits. Short-axis-tube orbits
3.3.3 Internal dynamics
have(cid:104)λ (cid:105)=0.Long-axis-tubeorbitsaremostimportantin
x
Thebest-fittingmodelhastangentialanisotropyinthecen- the central 20–60arcsec and at larger radii r (cid:38) 80arcsec.
treofthecluster.Thevalueoftheanisotropyβ =1−σ2/σ2 Short-axis-tubeorbits,whichcontribute intotal more mass
t r
isnegative,whereσ isthetangentialvelocitydispersionand than long-axis-tube orbits, are most important at r = 60–
t
σ istheradialvelocitydispersion.Weshowtheanisotropy 140arcsec. We illustrate the distribution of the stellar mass
r
β as a function of radius in Fig. 9, top panel. We plot the on the different orbit types also in Fig. 9 (bottom panel)
mean anisotropy of the models within the 1σ uncertainty as a function of radius. Most stars (>50 per cent) are on
limit.Theuncertaintyofβ isabout0.1.Theplotextendsto short-axis-tube orbits, i.e. they orbit the minor axis. Long-
theouteredgeofthekinematicdataat150arcsec.Theclus- axis-tubeorbitscontributeabout40percenttotheluminos-
terkinematicsbecomesnearlyisotropicatradiir>70arcsec. ityandthusstellarmassinthecentral30arcsec.Theypro-
Weshowtheangularmomentumdistributionoftheor- ducetheperpendicularrotatingsubstructureatr≈20arcsec
bitsinFig.10.Thecoloursdenotethedensityoforbitspass- (∼0.8pc) found by Feldmeier et al. (2014). At larger radii,
ingradiusrwithmeanangularmomentum(cid:104)λ (cid:105)(toppanel) long-axis-tube orbits contribute only about 30 per cent to
z
or (cid:104)λ (cid:105) (bottom panel). The plot of (cid:104)λ (cid:105) denotes rotation thestellarmass.Boxorbitscontributelittlemassinthecen-
x z
abouttheshortz-axis.Orbitswith(cid:104)λ (cid:105)(cid:54)=0arecontributed tre(<10percent),buttheirrelativemassincreasestowards
z
by short-axis-tube orbits, while long-axis-tube orbits have largerradii.Atr =150arcsec(∼5.8pc),theycontribute20
(cid:104)λ (cid:105) = 0. On the other hand, (cid:104)λ (cid:105) denotes rotation about per cent.
z x
MNRAS000,1–14(2016)
10 A. Feldmeier-Krause et al.
- meieretal.(2014).Wederiveda3σlowerlimitfortheblack
0 1 2 rorb [3pc] 4 5 holeof0.7×106M(cid:12),andanupperlimitof5.4×106M(cid:12).We
1.0 briefly discuss the model degeneracies, possible reasons for
the different black hole mass measurements, and why our
results are closer to the direct measurement than the black
hole mass derived by Feldmeier et al. (2014).
0.5
4.1.1 Model degeneracies
λz 0.0 Somemodelparametersseemtobecorrelated.Thisbecomes
clear when looking at Fig. 5. The best-fitting value of p ap-
parently increases with increasing dynamical mass-to-light
6
0 ratioΥ(secondcolumnofthefirstrow).However,thevalue
0.
-0.5 / of p has little effect on M•, as can be seen in the second
00 column of the second row in Fig. 5 . With a higher value of
0. p,thebest-fittingM increasesslightly.Atlargerp,theχ2-
•
contoursofM andΥbroaden.Thismeansthatforamore
•
-1.0 oblateaxisymmetricclusterwithpclosertoone,M andΥ
•
0 20 40 60 80 100 120 140 are not as well constrained as with smaller values of p. The
-
r [arcsec] black hole mass M =4.1×106M can be obtained with a
orb • (cid:12)
highervalueofp=0.82andΥ=1.22,thismodeliswithinthe
-
r [pc] 1σ uncertainties.
orb
0 1 2 3 4 5
The dynamical mass-to-light ratio Υ is inversely corre-
1.0
latedwiththeblackholemass(fourthcolumnofthefirstrow
in Fig. 5). The higher Υ, i.e. the more massive the cluster,
the less massive is the black hole. This degeneracy is often
0.5 obtained in dynamical models. Valluri et al. (2004) found
that the degeneracy of M depends on how well the black
•
hole’ssphereofinfluenceisresolved.ThemeasurementofΥ
is better constrained when the data extend to larger radii,
λx 0.0 provided that Υ is constant over the entire field. We have
several kinematic data bins within the radius of influence
of the supermassive black hole, and our data extend to one
2
0.1 effective radius. This may not be sufficient to put strong
-0.5 / constraints on Υ. To get agreement with the measurement
0
0 of (4.02 ± 0.20)×106M (Boehle et al. 2016), we would
0. (cid:12)
require a lower value of Υ≈ 0.75, which is within the 1σ
uncertainties.
-1.0
0 20 40 60 80 100 120 140
-
r [arcsec]
orb 4.1.2 Influence of the surface brightness profile
Figure 10.Orbitdensitywithangularmomentumλz (top),i.e. The shape of the surface brightness profile is important to
rotation around the short axis, and λx (bottom), i.e. rotation estimate the mass of the supermassive black hole. The sur-
around the long axis as a function of average orbit radius r¯orb. face brightness profile has to represent the density of the
Dark,bluecolourindicateshigherorbitdensity.Theverticalline kinematic tracer. We excluded young stars and ionised gas
denotesr =4.2pc.
eff fromthesurfacebrightnessprofile,asthesecomponentscon-
tribute little mass compared to the cool, old stars we used
askinematictracers.Excludingthesecomponentsresultsin
4 DISCUSSION a lower surface brightness and stellar mass in the centre
compared to Feldmeier et al. (2014). The stellar mass we
4.1 Difference of the resulting black hole mass
obtain at r = 48arcsec (∼1.8pc) is 1.3×106M less. Our
(cid:12)
The currently best black hole mass estimate is (4.02 ± black hole mass is therefore higher, and closer to the direct
0.20)×106M ,derivedbyBoehleetal.(2016)fromKeple- measurement of M ≈ 4×106M . We ran the same ax-
(cid:12) • (cid:12)
rian stellar orbits around the supermassive black hole. Us- isymmetric models (Cappellari 2008), using the same kine-
ing axisymmetric Jeans models and the same spectroscopic maticdataasFeldmeieretal.(2014),butoursurfacebright-
data as our study, Feldmeier et al. (2014) found a lower nessdistributionfromTable1.Thebestfitisobtainedwith
valueofM =(1.7+1.4)×106M .Ourbestfitusingtriaxial M = (2.8+1.3)×106M , Υ = 0.89+0.12, and a constant
• −1.1 (cid:12) • −0.8 (cid:12) −0.19
Schwarzschild models is (3.0+1.1)×106M . This measure- anisotropyofβ =−0.3.Thisresultisinagreementwiththe
−1.3 (cid:12)
ment is consistent with the direct measurements of Boehle triaxialSchwarzschildmodels,andconfirmsthatthesurface
et al. (2016) within the 1σ uncertainty limit. The result is brightnessprofilehasastronginfluenceontheresultsofthe
also in agreement with the lower black hole mass of Feld- black hole mass and dynamical mass-to-light ratio.
MNRAS000,1–14(2016)