THE NUCLEAR CLUSTER OF THE MILKY WAY: TOTAL MASS AND LUMINOSITY^*

To derive the proper motions and determine the stellar density profile in the center (R${}_{\mathrm{box}}\approx 20^{\prime\prime}$), we use adaptive optics images with a resolution of $\approx 0\_\_AMP\_\_farcs;080$. In the central parsec, we use the same NACO/VLT images (Lenzen et al. 2003; Rousset et al. 2003) as described in Trippe et al. (2008) and Gillessen et al. (2009). We add images obtained in further epochs since then, in the 13 mas pixel⁻¹ scale matching the Gillessen et al. (2009) data set and in the 27 mas pixel⁻¹ scale extending the Trippe et al. (2008) data set. The images are listed in Appendix A.

To obtain proper motions outside the central parsec, we use adaptive optics images covering a larger field of view. These are four epochs of NACO/VLT images, one epoch of MAD/CAMCAO at the VLT (Marchetti et al. 2004; Amorim et al. 2006), and one epoch of Hokupa'a+Quirc (Hodapp et al. 1996; Graves et al. 1998) Gemini North images (see Appendix A). Most images cover the Ks band, but some are obtained with the H band or narrower filters within the K band. The VLT images are flat-fielded, bad pixel corrected, and sky subtracted. In the case of the Gemini data, we use the publicly available images.⁴ These images are combinations of reduced images with nearly the same pointings.

To obtain the structural properties of the nuclear cluster outside of the central ${R}_{\mathrm{box}}=20^{\prime\prime}$, we use two additional data sets. Here, high resolution is less important, while area coverage and extinction correction are the keys.

• 1.  
  Closer to the center, we use Hubble Space Telescope (HST) WFC3/IR data.⁵ The central ${R}_{\mathrm{box}}\approx 68^{\prime\prime}$ around Sgr A* are covered by the filters M127, M139, and M153. We use the images in M127 and M153 in our analysis. We optimize the data reduction compared to the pipeline in order to achieve Nyquist-sampled final pixels. We use MultiDrizzle to combine the different images in the same filter, with a drop size parameter of 0.6 with boxes and a final pixel size of 60 mas. These choices achieve a high resolution while still sampling the image homogeneously enough to avoid pixels without flux. We do not subtract the sky background from the images, since it is difficult to find a source-free region from where one could estimate it. Since we use only point source fluxes, this does not affect our analysis. We change the cosmic removal parameters to 7.5, 7, 2.3, and 1.9 to avoid the removal of actual sources. Due to the brightness of the GC sources, cosmics are only of minor importance. The final images have an effective resolution of 015.
• 2.  
  On a larger scale, we use the public VISTA Variables in the Via Lactea Survey data obtained with VIRCAM (Saito et al. 2012). From data release 1, we use  central tile 333 in the H and Ks bands. The data contain flux calibrated, but not background subtracted, images. The resolution is about 1''. These images cover more than one square degree around the GC. We only use ${R}_{\mathrm{box}}\approx 1000^{\prime\prime}$. In the center, the crowding is severe and nearly all of the sources are saturated in the Ks band. This is not a limitation, since we can use the higher-resolution images from NACO and WFC3/IR.

To obtain the spectra of the stars, we use data cubes obtained with the integral field spectrometer SINFONI (Bonnet et al. 2003; Eisenhauer et al. 2003a). We use data with combined H+K-band (spectral resolution 1500) and K-band (spectral resolution 4000) grating. The spatial scale of the data varies between the smallest pixel scale ($12.5\;$ mas pixel⁻¹ × $25\;$ mas pixel⁻¹) and the largest scale ($125\;$ mas pixel⁻¹ × $250\;$ mas pixel⁻¹). The spatial resolution of the data is correspondingly between 70 mas and 2''. It has been matched at the time of the observations to the stellar density, which increases steeply toward Sgr A*. Thus, we can detect many sources in the center and can also sample large areas at large radii. We apply the standard data reduction SPRED (Schreiber et al. 2004; Abuter et al. 2006) for the SINFONI data, including detector calibrations (such as bad pixel correction, flat-fielding, and distortion correction) and cube reconstruction. The wavelength scale is calibrated with emission line lamps and is finetuned with atmospheric OH lines. Finally, we correct the data for the atmosphere by dividing with a late B-star spectrum and multiplying with a blackbody of the same temperature.

In this work, we combine proper motions obtained from four different data sets: central field, extended field, large field, and outer field (see Figure 1). Trippe et al. (2008) used the extended field. In the central field, stellar crowding is high and a different analysis is needed than that used out. Also, the number of epochs and their similarity decrease from the inside moving out, which requires a more careful, separate error analysis further out. Here, we provide a short overview of the data and methods used. The details are explained in Appendix A.

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• 1.  
  In the central ($R\leqslant \;2^{\prime\prime}$), we use the same method for astrometry as in Gillessen et al. (2009, see our Appendix A.1). We now track stars out to ${r}_{\mathrm{box}}\approx 2^{\prime\prime}$. With this increased field of view, we more than double the number of stars compared to Gillessen et al. (2009). We use 79 stars with a median magnitude of ${m}_{\mathrm{Ks}}=15.45$. Some of the old, late-type stars have significant accelerations (Gillessen et al. 2009), but the curvatures of their orbits are not important compared to their linear motions, and thus the linear motion approximation is sufficient. Due to the sample size, the Poisson error of the dispersions dominates all other dispersion errors.
• 2.  
  In the radial range between a box with a radius of 2'' and 20'' (extended field), we expand slightly on the data and method used by Trippe et al. (2008). We do not change the field of view and the selection of well-isolated stars compared to Trippe et al. (2008). The number of stars with velocities has increased a bit due to the addition of new images. In total, we have dynamics for 5813 stars in this field. The median magnitude is ${m}_{\mathrm{Ks}}=15.76$. We obtain a proper motion dispersion of ${\sigma }_{1{\rm{D}}}=2.677\pm 0.018$ mas years⁻¹ using all of the stars and averaging the two dimensions. The error includes only Poisson noise, which is likely the dominating error, because we measure identical dispersion values and errors for the bright and faint halves of the sample (see Appendix A.2).
• 3.  
  Outside a box radius of 20'' (large field), nearly no proper motions were available, with the exception of a small area in Schödel et al. (2009) which the authors did not use for their analysis. From the images with sufficiently good AO correction, we obtain the velocities for 3826 stars (see Appendix A.3). The median magnitude is ${m}_{\mathrm{Ks}}=15.30$. For the proper motion dispersion, we obtain ${\sigma }_{1{\rm{D}}}=2.330\pm 0.019$ mas years⁻¹. The comparison of the dispersion for fainter and brighter stars shows that the error on the dispersion (after subtracting the velocity errors in quadrature) of the fainter stars is $2\sigma$ larger than for the brighter stars. Since this is barely significant, the errors again include only Poisson noise.
• 4.  
  To expand our coverage of proper motions to 78'', we use the outer field. These data do not cover the full circle around Sgr A*, but only a selected field. We have two epochs for this field: Gemini data from 2000 and NACO data from 2011 May 29. In this field, we exclude stars with ${m}_{\mathrm{Ks}}\gt 15.6$ because we obtain a higher velocity dispersion for them than for brighter stars (see Appendix A.4). In this way, we select 633 proper motion stars with a median magnitude of ${m}_{\mathrm{Ks}}=14.79$. From these stars, we obtain ${\sigma }_{1{\rm{D}}}=1.918\pm 0.038$ mas years⁻¹ using both dimensions together. We again assume that the Poisson error dominates over other error sources for the stars selected.

Since we can resolve the stellar population in the GC, we measure the velocities of single stars and obtain dispersions by binning stars together. We use two different sources for the velocities (see also Appendix B).

• 1.  
  Within $R\lt 95^{\prime\prime}$ we use our SINFONI data to measure the radial velocities of the late-type giants in the GC (see Appendix B.1 for the details). We obtain radial velocities for 2513 stars. The median velocity error of these velocities is about 8 km s⁻¹, which we obtain by comparing independent measurements for multiple covered stars. The line-of-sight velocity dispersion of these stars is ${\sigma }_{z}=102.2\pm 1.4$ km s⁻¹. The velocity errors have less than 1σ influence on the measured dispersion. When we take into account the uneven distribution in l* (Figure 1) with binning (to avoid the influence of rotation), we obtain 6.1 ± 3.8 km s⁻¹as the total radial motion of the nuclear cluster. By definition it should be 0. This indicates that our velocity calibration is also probably correct. In conclusion, it seems likely that Poisson errors dominate the dispersion uncertainty for our radial velocity sample, and we only include those in the final analysis.
• 2.  
  Outside of 110'' we use radial velocities from the literature (Lindqvist et al. 1992b; Deguchi et al. 2004) out to ≈3000'' (Appendix B.2). Both of these works used maser radial velocities. We match the two samples and use each star only once. As a side product of this matching, we confirm that the typical velocity uncertainty is less than 3 km s⁻¹ as stated in Lindqvist et al. (1992b) and Deguchi et al. (2004). Due to the big position errors in these radio data, it is difficult to find corresponding IR stars. We therefore exclude from the combined list the 11 stars that overlap spatially with the areas in which we obtained spectra, with the aim of avoiding using stars twice. Overall, we use 261 radial velocities outside the central field.

The radial velocity star sample is not identical with the proper motion star sample. The majority of the faint proper motions stars have no radial velocities and many of the outer radial velocity stars have no proper motion coverage. The radial velocity stars have a median magnitude of ${m}_{\mathrm{Ks}}=13.66$, while the proper motion stars are in the median 1.85 magnitudes fainter. We thus use the common (e.g., van de Ven et al. 2006; Trippe et al. 2008; van der Marel & Anderson 2010) approach of using different stars for proper motions and radial velocities. Otherwise, we would have only 1840 stars in both samples, with an inhomogeneous spatial distribution. The difference in magnitude is likely not a problem because the stars in both samples are giants. Their different luminosities are mainly caused by different evolutionary stages, not by different ages and masses. Thus, mass-dependent effects, like mass segregation, affect both samples in the same way and we can safely use all 10,351 proper motion and all 2774 radial velocity stars.

In order to probe the gravitational potential of the GC out to more than 100'', we need to use a stellar population in dynamical equilibrium extending over a sufficiently large radial range. The late-type population in the GC is suited for this task. A small fraction of the stars in our sample does not belong to this population, and is therefore excluded as far as possible from our analysis. We obtain the following three exclusion criteria (see Appendix C for the details).

• 1.  
  The young stars follow different radial profiles (Bartko et al. 2010) and have different dynamics. We thus exclude young stars from our sample. These are the early-type stars, the WR-, O-, and B-stars (Paumard et al. 2006; Bartko et al. 2009, 2010) and the red supergiant IRS7, which has the same age as the WR/O-stars of around 6 Myr (Blum et al. 2003; Pfuhl et al. 2011). These stars are the most important contamination, especially close to Sgr A*. Due to missing spectra, we cannot clean our proper motion sample completely from these stars. However, we choose the selection criteria (Appendix C.1) such that in all of the radial ranges, no more than about 4% of the stars are young.
• 2.  
  We also exclude stars which, due to their low extinction, belong to the Galactic disk or bulge, similar to what is done in Buchholz et al. (2009; see Appendix C.2). Since in some areas images in a second filter are missing, we cannot totally clean our sample of foreground stars. Since these stars are not clustered, however, this pollution is not important anywhere. Integrated, maybe 1% of the GC sample stars are foreground stars.
• 3.  
  Extreme outliers in velocity are visible in some subsamples (Appendix C.3). About 1% of the proper motion stars outside of the extended field are outliers probably caused by measurement flukes. In the maser sample, about 5% are outliers (Lindqvist et al. 1992b; Deguchi et al. 2004 and Section 5.5). (See Appendix C.3 for our procedure of outlier identification.) Their high velocities probably indicate that they belong to another population. The other samples are free from obvious outliers.

To derive the light properties, we use three different data sources (Section 2). In the case of the star density, for all areas we use the data set with the highest resolution. In the case of the flux density, we omit the WFC3/IR data because it is obtained in different filters.

We use the following steps to derive the stellar distribution.

• 1.  
  We exclude very bright foreground stars and GC clusters like the Arches.
• 2.  
  We correct the star counts for completeness if necessary (see Appendix D).
• 3.  
  We exclude stars younger than 10 Myr from our sample.
• 4.  
  We correct for extinction using two NIR filters. The resulting map is still patchy in some areas because the optical depth is too high for correction.
• 5.  
  We create masks to exclude emission from these areas. The masks are defined such that the maps appear smooth and symmetric in $| {l}^{*}|$ and $| {b}^{*}|$. Since only a few pixels are masked out, the overall bias is small.
• 6.  
  The areas excluded are masked out for two-dimensional fitting. They are replaced with the average of the other areas at the same $| {l}^{*}|$ and $| {b}^{*}|$ for the creation of radial profiles and for visualization.

Not all of these steps are necessary for all of the data subsets; the procedures are described in detail in Appendix D.

In Figure 2, we present the profiles obtained. The profiles obtained from the integrated flux and stellar number counts are similar but not identical within the errors. On the one hand, in the case of the flux profile, the assumption of screen extinction is a simplification. On the other hand, due to the high source density in the GC, the magnitudes of the point sources are less reliable than the extended flux in the GC. To be conservative, we use both profiles to fit the mass in our Jeans modeling (Section 5.4) and we include the scatter between the obtained masses in the mass error budget. Splitting the number-counts-based profile into $| {l}^{*}|$ and $| {b}^{*}|$ components (Appendix D.5 and Figure 3) yields profiles similar to the model of Launhardt et al. (2002).

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We now fit the density profile assuming spherical symmetry. We relax this assumption in Section 4.3. For the Jeans modeling, we need a space density profile. Nevertheless, here we first parametrize the projected density in order to compare our data set with the literature before we fit the space density. We often use power laws for the first comparisons. The power law of the projected density is as ${\rm{\Sigma }}(R)={R}^{-{\rm{\Delta }}}$, and the power law of the space density as $\rho (r)={r}^{-\delta }$. Δ and δ are our definitions for the power-law slopes. We use ${\delta }_{L}$ for the flux and star counts and ${\delta }_{M}$ for mass.

4.2.1. Nuker Models

The usually projected surface densities in the GC were fit with (broken) power laws (Genzel et al. 2003). The Nuker profile is a generalization to two slopes (Lauer et al. 1995):

$$\begin{eqnarray}&&{\rm{\Sigma }}(R)={\rm{\Sigma }}({R}_{b}){2}^{(\beta -{\rm{\Gamma }})/\alpha }{\left(\displaystyle \frac{R}{{R}_{b}}\right)}^{-{\rm{\Gamma }}}{\left[1+{\left(\displaystyle \frac{R}{{R}_{b}}\right)}^{\alpha }\right]}^{({\rm{\Gamma }}-\beta )/\alpha }.\end{eqnarray} \tag{ 1 }$$

Here, R_b and ${{\rm{\Sigma }}}_{b}$ are the break radius and the density at the break radius. The exponent Γ is the inner (usually flatter) power-law slope, and β is the outer (usually steeper) power-law slope. The parameter α is the sharpness of the transition; a large value of α yields a very sharp transition, that is, essentially a broken power law. Using $\alpha =100$ (fixed) our fits can be compared with the literature.

Due to the break at 220'' (Figure 2), we restrict the Nuker fits to $r\lt 220^{\prime\prime}$ (Table 1). Rows 1 and 2 in Table 1 give our fits for stellar number counts and the flux profile, respectively. Row 1 can be directly compared to the literature. Buchholz et al. (2009) conducted the largest area (r < 20'') study so far. They obtained ${\rm{\Gamma }}=-0.17\pm 0.09$ and $\beta =0.70\pm 0.09$ for ${R}_{b}=6\_\_AMP\_\_farcs;0$. This fit is broadly consistent with our data, although for this radial range we obtain no clear sign of a power-law break. The break radius (6'') of Buchholz et al. (2009) is smaller than ours, although these authors do not cite an error. The binned data of Buchholz et al. (2009) look similar to our data. The same is true for the data in Bartko et al. (2010), who do not attempt to fit the profile. These works find a very weak increase in density with radius inside of $\approx 5^{\prime\prime}$, and then a somewhat stronger decrease in density with radius further out. This is also the case in Do et al. (2009, 2013a). Do et al. (2013a) used data out to 14'' and find a single power law with a slope of ${{\rm{\Delta }}}_{L}=0.16\pm 0.07$.

Table 1.  Nuker Fits of the Surface Density Profile

No.	Data Source	Radial Range	α	${\chi }^{2}$/dof	Rb$(^{\prime\prime} )$	Σ(Rb)	β	Γ
1	star density	$R\lt 220^{\prime\prime}$	100	26.66/25	23 ± 3	0.86 ± 0.09	0.771 ± 0.018	0.277 ± 0.054
2	flux density	$R\lt 220^{\prime\prime}$	100	74.11/50	24 ± 2	71.2 ± 4.6	0.934 ± 0.016	0.304 ± 0.030
3	star density	$R\lt 90^{\prime\prime}$	100	11.43/20	13 ± 3	1.11 ± 0.15	0.645 ± 0.040	0.186 ± 0.084
4	star density	$R\lt 220^{\prime\prime}$	1	15.28/25	21 ± 14	0.84 ± 0.28	0.972 ± 0.089	0.059 ± 0.146
5	flux density	$R\lt 220^{\prime\prime}$	1	50.71/50	10 ± 3	110 ± 20	1.048 ± 0.040	−0.163 ± 0.148

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Our data seem to be consistent with a break radius for the late-type stars around 20''. When counting stars regardless of their age, a smaller break radius of about 8'' is found (Genzel et al. 2003; Schödel et al. 2007). For our large radial coverage, a single power law obviously fails to fit our data, even when we restrict our data to the range for which we have spectral classifications ($R\lt 90^{\prime\prime}$). The best-fit broken power law then has ${\chi }^{2}$/dof = 11.43/20 (Row 3 in Table 1), while a single power law yields ${\chi }^{2}$/dof $=\;53.12/22$.

This indicates that the transition between the two slopes is softer than for a broken power law. We also test whether a very small α (very soft transition) can be excluded. However, there are nearly no differences in the ${\chi }^{2}$ for small α; all α have nearly the same probability. Therefore, we give only upper limits for α: 1.15 and 0.65 for the flux and star count data, respectively. Figure 4 shows our data together with some fits.

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The older literature used single power-law profiles to describe their flux density profiles. Between 20'' and 220'', our data can be fit relatively well by single power laws of ${{\rm{\Delta }}}_{L}=0.765\pm 0.018$ (stellar density) and ${{\rm{\Delta }}}_{L}=0.915\pm 0.015$ (flux density). These slopes again show that the two data sets are not consistent. Becklin & Neugebauer (1968), Allen et al. (1983), and Haller et al. (1996) obtain a slope of ${{\rm{\Delta }}}_{L}=0.8$, while Philipp et al. (1999) obtain a flatter slope of ${{\rm{\Delta }}}_{L}=0.6$.

In contrast to these data, the flux profiles from Graham & Spitler (2009) ⁶ and Schödel et al. (2011) do not follow a single power law, but instead they flatten at about 80'', indicating that the bulge already dominates there. Likely this bright bulge is an artifact caused by a missing sky subtraction. The floor level in the profiles of Graham & Spitler (2009) and Schödel et al. (2011), which is much higher than the bulge floor in the COBE/DIRBE data of Launhardt et al. (2002), fits typical K-background observations on Earth. Similar to Vollmer et al. (2003), for 2MASS data, we subtract the light level toward dark clouds as sky. In contrast to our work, Becklin & Neugebauer (1968), Allen et al. (1983), Haller et al. (1996), Philipp et al. (1999), Graham & Spitler (2009), and Schödel et al. (2011) do not use two-color information to correct for extinction. The single power-law fit of Catchpole et al. (1990) to extinction-corrected star counts with a slope of ${{\rm{\Delta }}}_{L}=1.1$ in the radial range from 140'' to 5700'' is not well comparable to our data set due to the different radial ranges.

4.2.2. γ-Models

For our Jeans modeling (Section 5.4), we need a parametrization of the space density profile ${\rho }_{{\rm{N}}}(r)$. This is connected to the observable surface density profile ${\rm{\Sigma }}(R)$ by the following projection integral:

$$\begin{eqnarray}&&{\rm{\Sigma }}(R)=2{\displaystyle \int }_{R}^{\infty }{\rho }_{{\rm{N}}}(r)r{dr}/\sqrt{{r}^{2}-{R}^{2}}.\end{eqnarray} \tag{ 2 }$$

We use the spherical γ-model (Dehnen 1993; this is equivalent to the η-model of Tremaine et al. (1994) under the transformation $\gamma =3-\eta$):

$$\begin{eqnarray}&&{\rho }_{{\rm{N}}}(r)=\displaystyle \frac{3-\gamma }{4\;\pi }\displaystyle \frac{L}{{r}^{\gamma }}\displaystyle \frac{a}{{(r+a)}^{4-\gamma }}.\end{eqnarray} \tag{ 3 }$$

Here, L is the total flux of the star counts. a is the scale of the core of the model. The density slope is $-\gamma$ within the core and −4 at $\infty$. The γ-model has a known positive distribution function which we employ in Chatzopoulos et al. (2015). More complex profiles, for example a three-dimensional Nuker model, can also fit the data. The projected profiles are nearly independent of the parametrization in that case, e.g., for the star counts the ${\rm{\Delta }}{\chi }^{2}$ between the γ-model fit and the Nuker fit is only 1.3. However, the Nuker model has more degrees of freedom. Furthermore, complex profiles contain poorly constrained parameters (e.g., α in Nuker) for our data set; they overfit the data.

The GC light profile has two breaks (Figure 4). The breaks in the profiles around 200'' are probably a sign of a two-component nature of the nuclear light distribution, as suggested by Launhardt et al. (2002). They call the inner component the nuclear (stellar) cluster and the outer component the nuclear (stellar) disk, in analogy to other galaxies. In contrast, Serabyn & Morris (1996) assumed that the central active star-forming zone inside the inactive bulge of the Milky Way consists of a single component, that is, a central stellar cluster of $R=100$ pc.

Due to the breaks, we cannot fit the full data range with one γ-model. Instead, we use two independent γ-models. The use of two models is the main reason that we cannot use the Nuker model since many parameters are then ill-determined. Still, the central slope of the outer component is also difficult to determine for γ models from the data and we fix it to be flat by setting ${\gamma }_{\mathrm{outer}}=0$. A smaller value would correspond to a central depression, and values $\gt 0.5$ can create profiles in which the outer component dominates again at very small distances. We obtain the fits presented in Rows 1 and 2 in Table 2.

Table 2.  γ-Model Fits of the Surface Density Profile

No.	Data Source	Radial Range	${\chi }^{2}/{\rm{dof}}$	Linner	ainner	${\gamma }_{\mathrm{inner}}$	Kouter	aouter
1	star density	all R	29.1/55	6.73 ± 0.88 · 104stars	194 ± 33''	0.90 ± 0.11	$7.05\pm 1.49\cdot {10}^{6}$ stars	3396 ± 458''
2	flux density	all R	220.3/204	3.42 ± 0.14 · 106 mJy	117 ± 10''	0.76 ± 0.08	$5.48\pm 0.38\cdot {10}^{8}$ mJy	3711 ± 158''
3	star density	$R\lt 220^{\prime\prime}$	18.8/26	28.8 ± 3.4 · 104stars	626 ± 93''	1.14 ± 0.06	⋯	⋯
4	flux density	$R\lt 220^{\prime\prime}$	96.87/51	14.52 ± 0.83 × 106 mJy	511 ± 44''	1.21 ± 0.04	⋯	⋯

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The two best-fitting profiles for stars and light are similar (Figure 5) to each other, but again are not consistent within their errors as already noted during the Nuker profile fits. The central slope of the cluster is consistently ${\gamma }_{\mathrm{inner}}=0.83\pm 0.12$. However, the small error is a consequence of the functional form that we use. Other functions like Nuker(r) yield a range of inner slopes that appears to be consistent with the uncertainty reported by Do et al. (2009).

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The outer γ-model is needed inside of 220''. Rows 3 and 4 in Table 2 show fits with a single component. The resulting ${\gamma }_{\mathrm{inner}}$ are unrealistically large ($\gt 1$) when compared with Do et al. (2009). To quantify the mutual consistency of the two data set, we fit the two data sets with the best single γ fit of the other, allowing only the scaling to change. The star density data give ${\chi }^{2}/{\rm{dof}}$ = 70.0/28; the flux density data give ${\chi }^{2}/{\rm{dof}}$ = 194.0/53.

The space density models in the literature do not describe our data well, even when we restrict the comparison to the inner 220'' (see Figure 5). The density model of Schödel et al. (2009) is a bad fit to both of our data sets (${\chi }^{2}/{\rm{dof}}=\;83.37/28$ and ${\chi }^{2}/{\rm{dof}}=\;494.67/53$, for star and flux density, respectively). The reason is the combination of a large break radius with a relatively small outer slope. It thus overestimates the density further out and underestimates it in the center. The models of Trippe et al. (2008; ${\chi }^{2}/{\rm{dof}}=117.46/28$ and ${\chi }^{2}/{\rm{dof}}=\;142.71/53$) and especially Genzel et al. (1996; ${\chi }^{2}/{\rm{dof}}=\;323.04/28$ and ${\chi }^{2}/{\rm{dof}}=\;165.99/53$) especially fit our flux data better. Still, their average ${\chi }^{2}/{\rm{dof}}$ is worse than our average ${\chi }^{2}/{\rm{dof}}$ when we fit both data sets with the same parameters The reason for this is that the core radii of Genzel et al. (1996) and Trippe et al. (2008) are small. The different profiles of Do et al. (2013b), which are very similar to each other, do not fit our data at all because the outer slope of the cluster is much too steep and the core too large. These discrepancies with our fit are not surprising, since none of these works used as large a radial coverage as we do. Schödel et al. (2009) did not fit the density model; it is a fixed input to their modeling. They were guided by recent literature. Trippe et al. (2008) only fit dynamic data. Do et al. (2013b) fit density data together with dynamic data, but only inside 12''. That covers essentially only the core. This shows that it is important to use density data over a scale larger than the core of the nuclear cluster since otherwise the properties of the outer profile, which have influence on the core parameters, cannot be determined accurately.

Our profiles show that the axis ratio (q = b/a) of the nuclear cluster increases with radius (see Figure 3). We measure the flattening binwise (see Appendix D.5 and Table 3). In the innermost bin, $| {l}^{*}| \lt 68^{\prime\prime}$, $q=0.80\pm 0.04$. There is some indication in the data (Figure 3) that the flattening is smaller around 40'' than around 20''. That dip is consistent with noise. We do not find a relevant systematic error source (see Appendix D.6). The flattening increases further outside of our field, as it is visible in large-scale IRAC data and in the work of Launhardt et al. (2002). They model the inner ${r}_{\mathrm{box}}=2^\circ$ of the GC and obtain an axis ratio of 0.2 at around ${l}^{*}=3100^{\prime\prime}$.

Table 3.  Flattening Profile

$| {l}^{*}|$ Range	q = b/a
0''–68''	0.80 ± 0.04
68''–130''	0.63 ± 0.03
130''–248''	0.58 ± 0.05
248''–473''	0.45 ± 0.04
473''–1030''	0.32 ± 0.03

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We can use our two-dimensional data to distinguish between the nuclear cluster and nuclear disk, where the outer disk component is used to estimate the background for the cluster in the following manner. In the central 68'', we use the density profiles shown in Figure 3 in the corresponding quadrants. Further out, we use the VISTA data. Figure 6 shows the resulting map. For finding an empirical description, we use GALFIT (Peng et al. 2002). In the first fits, we fix the centers to the known location and enforce alignment of the inner component with the Galactic Plane, but not of the outer component. (Because we measure the density only in two sectors, inside of 68'' alignment with the Galactic Plane is enforced by construction.) In this fit, the outer component is aligned within $1.1\pm 1\_\_AMP\_\_fdg;3$. The uncertainty of that angle is obtained through four trials; in each trial, we mask out the lower/upper half in l/b. In contrast to many other parameters, that angle is robust. Thus, the misalignment is not significant and we fix the angle to be 0^◦. Our result is consistent with Schödel et al. (2014), who for both the nuclear disk and the nuclear cluster obtain alignment with about 2° uncertainty.

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We fit the data with Sérsic (1968) and Nuker profiles. Which of the profiles is used is not important, as similar sizes and flattening are obtained with both. However, it is difficult to disentangle the inner and the outer components. Small changes in ${\chi }^{2}/{\rm{dof}}$ result in large differences in most parameters. Mainly because the flattening has a local minimum around $40^{\prime\prime}$, a free fit results in a large Sérsic index for the outer components. In that case, the outer component contributes relevantly again in the center, which reduces the flattening of the inner component. Then the nuclear cluster has $q\approx 0.91$. However, it is physically unlikely that the outer component is more important in the very center than slightly further out. Since in the case of our γ-profiles (Section 4.2.2) the outer component does not contribute relevantly in the center, we assume that the inside 68'' measured flattening is identical to the flattening of the nuclear cluster as a whole. To obtain this flattening, for the nuclear disk we use a Nuker profile with a flat core, $\alpha =0.6$, $\beta =3$, ${\rm{\Gamma }}=0$. This set of parameters approximately models a projected γ-model in its outer and inner slope (Section 4.2.2). The fit obtained for the disk is $q=0.264$. In the center, we use a Sérsic model to enable comparison with recent literature. Under these assumptions, for the inner component we obtain $q=0.80\pm 0.05$, $n=1.46\pm 0.05$, ${R}_{e}=127\pm 10^{\prime\prime}$, and an integrated count uncertainty of 10%. In contrast to many other parameters, the Sérsic parameter is well constrained. In all cases it is between 1.4 and 1.6. In the very center, the outer component contributes 13% of the star counts of the inner component. The outer component dominates outside about 104''.

Our half-light radius of 5.0 pc is slightly larger than the preferred value of Schödel et al. (2014; R${}_{e}=4.2\pm 0.4$). However, when the outer Sérsic is also free, in their two Sérsic fits they obtain $6\pm 0.2$ pc. Our axis ratio is somewhat larger then their value of $q=0.71\pm 0.02$ but is within the uncertainties. Our Sérsic index n is smaller than theirs of n = 2 ± 0.2. Since they were not able to use the central parsec in their fit, it is likely that our fit is better in that region. The very center is important for the Sérsic index, and thus our n is probably better. Overall, from comparing our and their different fits, it is probable that both works underestimate the systematic error in component fitting. The main reason for that fact is the existence of two not clearly separated components in the GC and the high extinction toward the region.

To obtain the Ks luminosity in the GC we integrate the flux of the old stars (Figure 2). The total extinction-corrected flux within $R\lt 100^{\prime\prime}$ is $1052\pm 200\;$ Jy. The absolute error of 20% contains the uncertainty of the extinction law toward the GC (Fritz et al. 2011: 11%), the calibration uncertainty (7%), and 14% for the differences between the stellar density and flux density profiles. We obtain the latter from the scatter between the star counts profile and the flux profile scaled to each other.

To estimate the total luminosity of the nuclear cluster, we need to estimate its size. We again use different methods to estimate the systematic error. First, we use two-dimensional decomposition of the star counts in the nuclear cluster and nuclear disk (Section 4.3).⁷ The fraction of star counts from the inner component is 67% and the fraction from the nuclear cluster, which is within 100'', is 44%. That leads to a total luminosity of 1599 Jy. Using instead the one-dimensional γ-decomposition of Section 4.2.2 of the counts leads to 5058 Jy. The γ-model fit of the flux implies 3420 Jy (see Table 2). The main reason for the different fluxes is  the different models employed: a Sérsic can, when it is (as in this case) close to exponential, decay quickly outside its characteristic radius, while a γ-model decays only with its fixed power law of −3.

We obtain absolute luminosities using ${R}_{0}=8.2\;{\rm{kpc}}$ and ${M}_{\odot }({Ks})=3.28$. Thereby, we use 3420 Jy as the best estimate for the total flux and the other two values for the error range. We obtain ${7.4}_{-3.9}^{+3.5}\times {10}^{7}\;$L${}_{\odot }$ for the total nuclear cluster, consistent with the estimate of $6\pm 3\times {10}^{7}\;$L${}_{\odot }$ of Launhardt et al. (2002) and  $4.1\pm 0.4\times {10}^{7}\;$L${}_{\odot }$ at 4.5 μm of Schödel et al. (2014).

The young O(B) stars in the center, which are not included in our sample, add 25 Jy in the Ks band. Although they are irrelevant for the light in the Ks band, this is different for bolometric measurements: the bolometric luminosity of the young stars is about ${{\rm{L}}}_{{UV}}\approx {10}^{7.5}\;{L}_{\odot }$ and ${M}_{\mathrm{bol}}\approx -14.1$ (Mezger et al. 1996; Genzel et al. 2010), and thus is larger than what we obtain for the old stars, ${M}_{\mathrm{bol}}\approx -13.4$ within $R\lt 100^{\prime\prime}$. Also, the young stars are concentrated on a volume more than $\approx 1000$ times smaller than the old stars.

The cluster could be anisotropic. One type of anisotropy is radial anisotropy, which would manifest itself as a difference between the dispersions in the tangential and radial directions (Leonard & Merritt 1989; Schödel et al. 2009).

We obtain an estimate of the anisotropy from the proper motions from ${\beta }_{\mathrm{pm}}^{\prime }(R)=1-{[{\sigma }_{\mathrm{tan}\mathrm{PM}}(R)/{\sigma }_{\mathrm{rad}\mathrm{PM}}(R)]}^{2}$, where ${\sigma }_{\mathrm{tan}}$ and ${\sigma }_{\mathrm{rad}}$ are the radial and tangential dispersions of the proper motions, respectively. ${\beta }_{\mathrm{pm}}^{\prime }$ has the advantage that it follows directly from measured properties without modeling and does not depend on R₀. Uneven angular sampling of stars, together with flattening which causes ${\sigma }_{{l}^{*}}\gt {\sigma }_{{b}^{*}}$ (Figure 10), can mimic anisotropy in the following way. Consider a radial bin that is only covered close to the b*-axis. Then, ${\sigma }_{\mathrm{tan}}\approx {\sigma }_{{l}^{*}}$ and ${\sigma }_{\mathrm{rad}}\approx {\sigma }_{{b}^{*}}$. Since ${\sigma }_{{l}^{*}}\gt {\sigma }_{{b}^{*}}$, ${\beta }_{\mathrm{pm}}^{\prime }\lt 0$ implies tangential anisotropy. The arguments also hold for uneven angular sampling. Outside of the center, Do et al. (2013b) had only covered the Galactic Plane close to the b* axis. As expected, ${\sigma }_{\mathrm{tan}}\gt {\sigma }_{\mathrm{rad}}$ in these data. That is the reason that their fit prefers ${\beta }_{\infty }\lt 0$.

To avoid a spurious influence of the flattening on the anisotropy, we first restrict the analysis here to r < 40'' for full (not necessarily even) angular coverage. Second, to even out the density fluctuations, we obtain the dispersions by taking the average of the dispersions in two angular bins: one within $\phi \lt 45^\circ$ of the Galactic Plane and the other within $\phi \gt 45^\circ$. To even out the coverage fluctuations, we give the two bins equal weight. We obtain ${\beta }_{\mathrm{pm}}^{\prime }=-0.040\pm 0.022$ (Figure 7). The scatter between the bins is consistent with the Poisson errors. Similar to what was found by Schödel et al. (2009), we see that ${\sigma }_{\mathrm{tan}}$ is somewhat larger in the center. However, ${\chi }^{2}/{\rm{dof}}=\;33.71/39$ shows that this is not significant. Schödel et al. (2009) suggested that the increase in the center could be due to pollution from early-type stars, which are on average on more tangential orbits (Genzel et al. 2000; Bartko et al. 2009, 2010). We test this hypothesis by using only stars with late-type spectra. Integrated over the full field, this yields ${\beta }_{\mathrm{pm}}^{\prime }=0.045\pm 0.049$. That value is smaller than for all stars, but is not significantly different.

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Between 2'' and 5'', ${\beta }_{\mathrm{pm}}^{\prime }=-0.463\pm 0.170$ when using all stars and ${\beta }_{\mathrm{pm}}^{\prime }=-0.134\pm 0.207$ when using only late-type stars. Since these two values are again consistent, pollution is probably not important in this radial range.

The anisotropy parameter β is defined in 3D coordinates (r) (Binney & Tremaine 2008). β can only be estimated in full modeling, which also needs to account for projection effects. In such a model, R₀ would also need to be fit and it would also be necessary (Chatzopoulos et al. 2015) to use a flattening model to account for the different effects of flattening on v_z, ${v}_{{l}^{*}}$, and ${v}_{{b}^{*}}$. That is beyond the scope of this paper. The deprojected anisotropy parameter β is more different from 0 than ${\beta }_{\mathrm{pm}}^{\prime }$ (van der Marel & Anderson 2010). Due to the core-like density profile, the difference between ${\beta }^{\prime }$ and ${\beta }_{\mathrm{pm}}$ can be large, especially in the center. Because of projection effects, full modeling is required to constrain the radial dependency of the anisotropy (van der Marel & Anderson 2010). While the overall radial anisotropy is small, other deviations from prefect isotropy exist; see Section 5.2.

In Section 4.3,  we determined the major and minor axes of the distribution of stars in the nuclear cluster. Here, we use the new kinematic data to find the rotation axis of the system. First, we use the proper motion data.

Trippe et al. (2008) interpreted the difference between the velocity dispersions ${\sigma }_{l}$ and ${\sigma }_{b}$ (Figure 10) as a sign of rotation. However, ${\sigma }_{l}\gt {\sigma }_{b}$ is globally required for any axisymmetric star cluster flattened parallel to the Galactic plane. This is independent of whether the extra kinetic energy along the Galactic plane is due to net rotation or to higher in-plane velocity dispersions; reversing Lz for any orbit does not change ${\sigma }_{l}$. Furthermore, spherical clusters with rotation (Lynden-Bell 1960) do not show ${\sigma }_{l}\gt {\sigma }_{b}$. Therefore, the difference between ${\sigma }_{l}$ and ${\sigma }_{b}$ is ultimately due to the flattening, even though most of the flattening of the nuclear cluster is in fact generated by additional rotational kinetic energy (Chatzopoulos et al. 2015).

However, in any case, we can follow the approach of Trippe et al. (2008) to find the kinematic major axis of the nuclear cluster. To this end, we bin the motions in angle (their Figure 7). The pattern is sinusoidal but with more variation close to the peaks. Thus, for fitting we use the following function:

$$\begin{eqnarray}&&f(\theta )=a+b\times {(| \theta -\phi | )}^{c}.\end{eqnarray} \tag{ 4 }$$

Here, theta is the angle relative to the line to the east, ϕ is the position of the maximum, a is the constant floor, and b is the peak parameter; a big b implies that the maximum has a smaller width than the minimum. We obtain $b=3.97\pm 0.49$ and $\phi =58.8\pm 1\_\_AMP\_\_fdg;2$ consistent with the Galactic plane ($\phi =58\_\_AMP\_\_fdg;6$).

Finally, we use the radial velocities, whose gradient in the mean radial velocity as a function of $| {l}^{*}|$ (Trippe et al. 2008) can only be explained by rotation. In fitting the Galactic plane, we aim to find the angle for which the radial velocity is constant along the coordinate $x^{\prime\prime}$, which we obtain from the rotation. We assume cylindrical rotation since we ignore $y^{\prime\prime}$. The angle is the rotation axis rotated by $90^\circ$. The advantage of that angle compared to the rotation axis is that it is not necessary to fit at the same time for the possible complex rotation curve because vertical to it the velocity is identical everywhere. We obtain $53.7\pm 4\_\_AMP\_\_fdg;0$ broadly consistent with the Galactic plane.

Due to the finding of Feldmeier et al. (2014) that th radial velocity field is not simply a function of ${l}^{*}$ and ${b}^{*}$ as expected, we divided our velocities into radial bins (Figure 8). We can mostly confirm their findings. At medium radii (between 24'' and 90''), the radial velocity plane follows an angle of $45.1\pm 4\_\_AMP\_\_fdg;2$. The value is 3.2σ different from the Galactic Plane. That agrees well with the measurement of Feldmeier et al. (2014). Further out and further in, the rotation axis aligns with the Galactic plane within the partly large error. We do not find deviations of ${\sigma }_{z}$ in these bins.

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Although it is likely that the rotation is not a function of ${l}^{*}$ only, we assume that such is the case here to ease comparison with most of the literature. For illustration, we bin the radial velocity data (Section 3.2) in a less crowded way (Figure 9).

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Inside of 27'', our velocities are consistent with the velocities of Trippe et al. (2008), since our data set is largely identical. Outside of 27'', our new SINFONI radial velocities are on average smaller than the velocities of Trippe et al. (2008), who used only a subset of the velocities in McGinn et al. (1989), a problem already pointed out by Schödel et al. (2009). Surprisingly, our new high-resolution data do not yield the average of the velocities reported by McGinn et al. (1989), but agree roughly with the lower end of the values found. These lower-end values agree with the velocities of single bright stars by Rieke & Rieke (1988).

The maser data of Lindqvist et al. (1992b) and Deguchi et al. (2004) agree with the lower-velocity data of McGinn et al. (1989) and our CO bandhead velocities. Schödel et al. (2009) suggested that the differences in the radial velocities in the literature could be a sign of two populations in the GC. However, in our data, we find no sign for any population dependence in the rotation pattern. It is possible that the difference in the radial velocities in McGinn et al. (1989) and Rieke & Rieke (1988) is an indication that the velocity calibration of these old CO bandhead measurements was more difficult than assumed at the time. Overall, we are confident that our smaller rotation of the cluster compared to Trippe et al. (2008) is correct and is not population dependent.

For simplicity, we choose to bin our data. The loss of information (Scott 1992; Merritt & Tremblay 1994; Feigelson & Jogesh Babu 2012) is small, since we use a large amount of data and the variations between the bins are smaller than the errors. We assume symmetry relative to the Galactic plane, as supported by most observations (see Section 5.2). There might be small deviations in the radial velocities. However, since we fit second moments and not velocities and dispersions, the impact is very small. An edge-on flattened system has different symmetry properties in proper motion and in radial velocity. This is another reason for our different binnings which is explained in Appendix E. The same bins are also used by Chatzopoulos et al. (2015). We have also tried different binnings, which bin only in r and bin proper motions and radial velocities together. Our test includes bins of nearly equal size in r, log(r), and nearly equally populated bins. The Jeans masses usually vary by less than the formal fitting errors, sometimes slightly more. Since the systematic errors are much larger than the fitting error (and thus are also larger than the binning error) it is therefore unnecessary to include binning terms in the errors. In Figure 10, we show the binned dispersion data in all three dimensions (l${}^{*}$, ${b}^{*}$, z).

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It is obvious from the dispersion difference (Figure 10) in the two proper motion axes that the nuclear cluster is not a spherical, isotropic system, which would have dispersions in all directions. As shown in Chatzopoulos et al. (2015), this difference is caused by the flattening of the NSC; their Figure 10 shows how ${\sigma }_{l}\gt {\sigma }_{b}\gt {\sigma }_{z}$ over the whole range of radii for their favored axisymmetric models. While we cannot completely include anisotropy, it is very likely minor given our constrains from Section 5.1 and the fact that an isotropic rotator fits the data well (Chatzopoulos et al. 2015). Thus, anisotropic spherical modeling is at most only a small improvement compared to isotropic spherical modeling. Therefore, we use the simplest kind of Jeans modeling (Binney & Tremaine 2008) assuming isotropy and spherical symmetry for mass and light. That simple model is mainly used for illustration of what can be derived from the data. In a simple model, tests for other effects are also easier and faster, in contrast to more complex models.

Two variants of isotropic Jeans modeling were used in the past for the GC.

• 1.  
  Genzel et al. (1996) and Trippe et al. (2008) parametrized the deprojected dispersion.
• 2.  
  Schödel et al. (2009) used a direct mass parametrization.

Here, we follow Schödel et al. (2009). Since the existence of the SMBH is shown by orbits (Schödel et al. 2002), it is advantageous when it can be parametrized directly.

5.4.1. Relation of Dispersion and Mass

To relate the measured dispersion to the mass, we use the following equation derived from the Jeans equation (Binney & Tremaine 2008; Schödel et al. 2009):

$$\begin{eqnarray}&&{\sigma }_{P}^{2}(R)/G=\displaystyle \frac{{\displaystyle \int }_{R}^{\infty }{dr}\;{r}^{-2}{({r}^{2}-{R}^{2})}^{1/2}n(r)M(r)}{{\displaystyle \int }_{R}^{\infty }{dr}\;r{({r}^{2}-{R}^{2})}^{-1/2}n(r),}\end{eqnarray} \tag{ 5 }$$

and one needs to choose a parametrization for M(r). The assumption which has the fewest degrees of freedom is a constant mass-to-light ratio, or a theoretically predicted profile of that ratio (Lützgendorf et al. 2012). The GC data sets have been rich enough to leave the shape of the extended mass distribution as a free parameter (Schödel et al. 2009). Given that the presence of the central SMBH is well established in the GC, one can add the SMBH mass explicitly, (M_•) to M(r), as done in Schödel et al. (2009).

5.4.2. Projection

It is necessary to deproject the observed radii R into true 3D radii r, for which one needs to know the space tracer density distribution n(r) (see Section 4.2). This is done using two projection integrals, namely, Equation (2) and the following integral:

$$\begin{eqnarray}&&{\rm{\Sigma }}(R){\sigma }_{P}{(R)}^{2}=2{\displaystyle \int }_{R}^{\infty }n(r)\sigma {(r)}^{2}r{dr}/\sqrt{{r}^{2}-{R}^{2}}.\end{eqnarray} \tag{ 6 }$$

5.4.3. Averaging the 3D Dispersions

5.4.4. Errors of Dispersions

5.4.5. Tracer Distribution Profiles

5.4.6. Mass Parametrization

5.4.7. Fitting

We fit simultaneously the surface density and the projected dispersion data. To the surface density, we fit the density model, i.e., the double γ-model (Section 4.2.2). The dispersion data depends on both the mass model and the density model, which in that case is the tracer model. Since we fit both at once, both model components are constrained by both data sets. When we use the power-law model for the mass, the fit values of the density model are consistent with the fits, which only use the surface density (Table 2). This confirms that our fits converge well. For constant M/L, the parameters for the best-fitting density model differ often by more than 1σ from the ones when we fit only the surface density. The reason for this is that in the second case, the density model depends more on the dispersions because, due to constant M/L, the influence of the dispersion on the density model is grater than in the other case. Usually, the density model is less concentrated in the full fit than in the density-only fit. That is an expression of the low dispersion problem in the center (Section 6.1). However, the density parameter differences are not very big and do not influence the obtained masses relevantly. In one of our fits, the central slope (${\gamma }_{\mathrm{inner}}$) of the inner component of the tracer density is slightly (by only 0.02) smaller than 0.5. This is problematic because a slope smaller than 0.5 is not possible in the spherical isotropic case when a central point mass is dominant (Schödel et al. 2009). To retain self-consistency in our simple isotropic models, we fix the smaller slope to 0.5 in that case. The fix has no influence on the obtained masses. Also, the inner slope of the outer component is not well constrained and we usually fix it to 0. Neither restriction has any relevant influence on the masses obtained. In the fitting, we optimize ${\chi }^{2}$ (Press et al. 1986):

$$\begin{eqnarray}&&{\chi }^{2}=\displaystyle \sum _{i=1}^{n}\;{\left(\displaystyle \frac{{x}_{i}-{\mu }_{i}}{{\sigma }_{i}}\right)}^{2}.\end{eqnarray} \tag{ 9 }$$

Here, ${\mu }_{i}$ is the function which we optimize; it consists of two functions, i.e., one for density and one for the dispersion. The density function consists of two terms of Equation (3) projected with Equation (2), while the dispersion function consists of Equation (6) using one of our two types of extended mass parameterizations (power law or constant M/L). In both cases, the dispersion function is projected with Equation (5). x_i and ${\sigma }_{i}$ are the values and errors of the observables which are obtained in bins. We assume Gaussian errors.

We present the fits corresponding to the various choices of how to set up the Jeans model of the nuclear cluster in Table 4. We do not show the cluster properties since they do not differ by much. The robustness of the results can be assessed by comparing the different fits. For the fitting itself, we first use the routine mpfit (Markwardt 2009). Second, we also fit the data using Markoff-Chain Monte Carlo (MCMC) simulations following the method of Tegmark et al. (2004) and Gillessen et al. (2009). Our modification is that we start at the previously found minimum. We never find a better minimum than the starting point. The errors are usually rather similar with both methods. There is usually some asymmetry, i.e., the errors are larger on the positive side. For mass properties, that asymmetry is small. It is larger for some of the less well-defined tracer density parameters. Such asymmetry cannot be found by mpfit. In Table 4, we give as the best value the median value of the accepted MCMC values. As the error we give half of the difference between the 84.1% and the 15.9% quantiles. Such errors encompass the central 1σ range.

Table 4.  Jeans Model Fits

No.	Mass	Tracer	Tracer	Tracer	Dynamics	${M}_{\bullet }$	${M}_{100^{\prime\prime} }$	${\delta }_{M}$	${\chi }^{2}$/dof
	Model	Model	Source	Range	Range	$({10}^{6}{M}_{\odot })$	$({10}^{6}{M}_{\odot })$
1	power law	double γ	stars	all	all R	2.26 ± 0.26	9.28 ± 0.48	0.92 ± 0.04	185.43/182
2	M/L = const	double γ	stars	all	all R	4.37 ± 0.13	5.17 ± 0.24	⋯	276.16/184
3	power law	double ${\boldsymbol{\gamma }}$	stars	all	$10^{\prime\prime} \;\lt \;$R < 100''	4.17	5.81 ± 0.26	1.21 ± 0.05	145.83/137
4	power law	double γ	flux	all	$10^{\prime\prime} \;\lt$ R < 100''	4.17	6.17 ± 0.23	1.25 ± 0.04	349.51/286
5	M/L = const	double γ	stars	all	$10^{\prime\prime} \;\lt$ R < 100''	4.17	5.62 ± 0.17	⋯	156.97/138
6	M/L = const	double γ	flux	all	$10^{\prime\prime} \;\lt$ R < 100''	4.17	6.81 ± 0.16	⋯	342.58/287
7	power law	double γ	stars	all	R$\;\lt \;100^{\prime\prime}$	4.17	4.98 ± 0.29	1.33 ± 0.05	199.69/163
8	power law	double γ	flux	all	R$\;\lt 100\;^{\prime\prime}$	4.17	5.35 ± 0.23	1.37 ± 0.04	435.79/312
9	M/L = const	double γ	stars	all	R$\;\lt \;100^{\prime\prime}$	4.17	5.48 ± 0.16	⋯	195.82/164
10	M/L = const	double γ	flux	all	R$\;\lt \;100^{\prime\prime}$	4.17	6.70 ± 0.18	⋯	414.04/313
11	power law	double γ	stars	all	R > 10''	4.17	6.14 ± 0.18	1.12 ± 0.03	182.36/157
12	power law	double γ	flux	all	R > 10''	4.17	6.57 ± 0.17	1.12 ± 0.03	399.37/306
13	M/L = const	double γ	stars	all	R > 10''	4.17	5.58 ± 0.15	⋯	240.18/158
14	M/L = const	double γ	flux	all	R > 10''	4.17	6.65 ± 0.16	⋯	433.36/307
15	power law	double γ	stars	all	all R	4.17	5.59 ± 0.17	1.19 ± 0.03	247.59/183
16	power law	double γ	flux	all	all R	4.17	6.03 ± 0.17	1.19 ± 0.03	507.08/332
17	M/L = const	double γ	stars	all	all R	4.17	5.47 ± 0.14	⋯	277.14/185
18	M/L = const	double γ	flux	all	all R	4.17	6.56 ± 0.15	⋯	500.64/333
19	power law	single γ	stars	$R\lt \;220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	5.91 ± 0.21	1.32 ± 0.08	126.27/108
20	power law	single γ	flux	$R\lt \;220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	6.27 ± 0.18	1.36 ± 0.07	207.12/132
21	M/L = const	single γ	stars	$R\lt \;220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	6.23 ± 0.16	⋯	130.58/108
22	M/L = const	single γ	flux	$R\lt 220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	6.92 ± 0.15	⋯	205.30/134
23	power law	Schödel+ 2009	stars	$R\lt 220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	5.79 ± 0.25	1.16 ± 0.05	195.97/110
24	power law	Trippe+ 2008	stars	$R\lt 220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	7.22 ± 0.17	1.20 ± 0.04	453.40/110
25	power law	Genzel+ 1996	stars	$R\lt 220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	6.75 ± 0.18	1.12 ± 0.04	237.05/110
26	M/L = const	Schödel+ 2009	stars	$R\lt \;220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	4.66 ± 0.09	⋯	218.72/111
27	M/L = const	Trippe+ 2008	stars	$R\lt \;220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	7.56 ± 0.15	⋯	444.73/111
28	M/L = const	Genzel+ 1996	stars	$R\lt 220^{\prime\prime}$	$10^{\prime\prime} \;\lt \;$ R $\;\lt \;100^{\prime\prime}$	4.17	6.19 ± 0.12	⋯	243.69/111
29	M/L = const	double γ	stars	all	R$\;\lt \;27^{\prime\prime}$	3.37 ± 0.16	7.31 ± 0.42	⋯	91.93/120

Note. Jeans model fitting of our dynamics and density data, assuming different mass and tracer models, and different selections for the data. For the fitted surface density (tracer), we use two data sources and sometimes restrict ourselves to a subset of the available data range (Columns 3 and 4). The dynamics data consist of $\langle {v}^{2}\rangle$ in all three dimensions. We restrict it partly radially. The mass model includes in all cases a central point mass. ${M}_{100^{\prime\prime} }$ is the nuclear cluster mass within 100''. If no error is given for a parameter, then it is fixed. The literature tracer models are from Trippe et al. (2008), Schödel et al. (2009), and Genzel et al. (1996). The bold marks the most reliable fits.

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Initially, we fit with a free SMBH mass (Rows 1 and 2 in Table 4). However, especially for the power-law mass model, which has one parameter more, i.e., the cluster shape, the resulting mass of the SMBH is smaller than the direct mass measurements by means of stellar orbits (Ghez et al. 2008; Gillessen et al. 2009). We discuss the reason for the small SMBH mass in Section 6.1. The direct cause for the too-small SMBH mass is that models with a realistic black hole mass (M${}_{\bullet }=4.17\times {10}^{6}{M}_{\odot }$) predict a larger dispersion within $r\lt 10^{\prime\prime}$ than we measure. That implies that our model is in some aspect incomplete near the black hole. In the following, we fix the central mass to ${M}_{\bullet }=4.17\times {10}^{6}\;{M}_{\odot }$, corresponding to our fixed distance of ${R}_{0}=8.2$ kpc, and we neglect the small distance-independent uncertainty of 1.5% (Gillessen et al. 2009).

5.4.8. Results

The fits that best meet our assumption (Table 4) are in rows 3, 4, 5, and 6. They use a range of $10^{\prime\prime} \lt R\;\lt$ 100'' for the dynamics data. The surprisingly low dispersion in the center impacts the cluster mass in the cases of the fits in rows 7 and 8. These cases use $R\lt 100^{\prime\prime}$ for the dynamics data (i.e., including the central region) and allow the cluster shape to vary with the power law. All of the other fits are less influenced by the central dispersion and yield masses ${M}_{100^{\prime\prime} }$ between 5.5 and $6.8\times {10}^{6}\;$M${}_{\odot }$. The fact that the mass does not strongly depend ($\delta M\lt 15$%) on whether we include the central 10'' shows that the influence of the too-small SMBH mass on the cluster mass is much smaller than the effect of the neglected flattening (see Section 5.4.8).

Figure 11 illustrates the fits of rows 1, 3, and 5. In this figure, one can see that our models with the right SMBH mass have higher dispersions than the measurements at $R\lt 10^{\prime\prime}$. The models are probably not correct there. In Figure 12, we show all of the correlations of the fit in row 5. Some (outer) tracer density parameters are strongly correlated with each other. The cluster mass itself shows only weak correlations with other parameters. That explains the small fit errors for the cluster mass.

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Outside of $100^{\prime\prime}$, the fits are also not very good. The reason for that is probably the following.

• 1.  
  We do not have density data outside of 1000'', and hence cannot constrain the tracer profile there. The outer slope of the tracer profile is thus not well constrained.
• 2.  
  The fixed outer slope in the γ-model does not allow for variability in the outer slope.
• 3.  
  The assumption of spherical symmetry is a poor approximation outside of $\approx 300^{\prime\prime}$.

Therefore, the mass outside the central $\approx 100^{\prime\prime}$ is less reliable.

Apart from the fits in rows 7 and 8, the selection of the range for the dynamics data is less important than the choice of the tracer source for the obtained mass. The latter is particularly true when choosing a constant mass-to-light ratio as the mass model. Choosing the stellar number counts or the flux as the tracer results in a difference of up to 20% in ${M}_{100^{\prime\prime} }$ (compare rows 5 and 6 in the table). The smaller value occurs for the star density profile. The main reason for this is that in this profile, the inner component has a larger core radius than in the case of the flux profile. With a smaller core, a larger part of the mass causeing the dispersion needs to be close to the black hole. In Figure 13, we show the cluster mass probability distribution of rows 3 to 6, i.e., those which all use the same, likely the best, dynamic data. One can also see that there is a clear anticorrelation between the cluster mass and cluster mass slope. Still, the best values of both are well defined, in contrast to Schödel et al. (2009), thanks to our larger radial coverage.

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Looking at the ${\chi }^{2}$ values in the table seems to indicate that the fits using the flux density as a tracer profile are worse than the fits using the stellar number counts. However, the difference is due to the density data, where because of our error calculation (Appendix D) the ${\chi }^{2}$ of the flux density data is worse (Section 4.2). Since we identify no reason to prefer one of the two data sets (Section 4.1), we give of both of them equal weight in the following.

From the sample, we exclude rows 7 and 8 because the obtained slopes ${\delta }_{M}$ are significantly bigger than the slopes of the stars counts and flux profiles. That is unrealistic and is caused by the dispersion problem in the center. Thus, we consider a sample that contains the 14 fits in rows 3–6 and 9–18, which we give equal weight. The average mass is

$$\begin{eqnarray}&&{M}_{100^{\prime\prime} }=(6.09\pm 0.53)\times \;{10}^{6}\;{M}_{\odot }.\end{eqnarray} \tag{ 10 }$$

As in Gillessen et al. (2009), we calculate the systematic error from the scatter between the different fits. These errors dominate over the smaller fit errors. This method of error calculation has the advantage that it uses realistic uncertainties for the extended mass and star distribution. In the case of the mass, it explores the presence and absence of a dark cusp. In the case of the star distribution, it enables a realistic assessment of the uncertainty in the profile which a single profile cannot provide. Using only rows 3–6 yields a very similar result. According to Table 4, the main uncertainty is whether the star counts or the flux are used. They have different density profiles which results in different masses. By comparison, the other assumptions (like the mass profile) play a minor role.

The cluster mass slopes are between 1.12 and 1.25 for the cases used. The mean is ${\delta }_{M}=1.18\pm 0.06$, where the error is again dominated by the scatter between the different fits. This number is in reasonable agreement with the tracer profile, which has a slope of 1.18 (stellar number count based) or 1.06 (flux based) in the range $50^{\prime\prime} \lt R\;\lt$ 200''.

In order to check how important our two-component model is, we also fit single γ-models to our data (Rows 19–22). To ensure that a single component model is a reasonable fit to the data, we restrict the range of the tracer density data inside the break to the nuclear disk, i.e., $R\lt 220^{\prime\prime}$. With this restriction, the masses are slightly larger than before. The main reason for this is that the core of the inner γ-component, which is now the only γ, is larger in this case. Using the literature profiles of Genzel et al. (1996), Trippe et al. (2008), and Schödel et al. (2009), the mass range is larger, from $4.66\times {10}^{6}$ to $7.56\times {10}^{6}\;{M}_{\odot }$. The main reason for the mass trend between them is the effective outer slope. When the slope is steep, as in Trippe et al. (2008), the expected dispersion at large radii gets smaller.

Our data set would allow us to determine the distance to the GC, R₀, by means of a statistical parallax (Genzel et al. 2000; Eisenhauer et al. 2003b; Trippe et al. 2008). To be accurate in R₀, it is necessary to have a model that accounts for differences between ${\sigma }_{l}$, ${\sigma }_{b}$, and ${\sigma }_{z}$. Our spherical model cannot provide that and we defer this to the work of Chatzopoulos et al. (2015), where we present self-consistent flattened models.

We now obtain the mass uncertainty due to the distance uncertainty. We use the latest results for the orbit of the S-stars of ${R}_{0}=8.2\pm 0.34\;{\rm{kpc}}$ (Gillessen et al. 2013). Due to the small distance-independent SMBH mass error of 1.5% (Gillessen et al. 2009), which is vanishingly small compared to our other uncertainties, it is sufficient to derive the mass dependency on distance. To include the distance error in the error budget, we repeat the Jeans modeling for R₀ of 7.86 and 8.54 kpc. Thus, we assume that the distance error is Gaussian, which seems to be approximately true (see Figure 15 in Gillessen et al. 2009). We perform full ${\chi }^{2}$ adding in Chatzopoulos et al. (2015). We determine the SMBH mass from the ${M}_{\bullet }-{R}_{0}$ relation of Gillessen et al. (2009): ${M}_{\bullet }\propto {R}^{2.19}$. We find following mass-distance relation:

$$\begin{eqnarray}&&{M}_{100^{\prime\prime} }=6.09\cdot {10}^{6}\times {\left(\displaystyle \frac{{R}_{0}}{{M}_{\odot }}8.2\mathrm{kpc}\right)}^{3.83}.\end{eqnarray} \tag{ 11 }$$

The distance error of Gillessen et al. (2009) leads to following expression:

$$\begin{eqnarray}&&{M}_{100^{\prime\prime} }=(6.09\pm 0.53{| }_{\mathrm{fix}{R}_{0}}\pm 0.97{| }_{{R}_{0}})\times {10}^{6}{M}_{\odot }.\end{eqnarray} \tag{ 12 }$$

Thus, the distance-induced error is larger than the other considered errors. However, we do not consider all error sources in our work. Our model does not include anisotropy and flattening. The inclusion of both can introduce important mass changes, see, e.g., van der Marel & Anderson (2010), D'Souza & Rix (2013). Since we detect signs of flattening but not of anisotropy, flattening is more important. We use a flattened model in Chatzopoulos et al. (2015). Compared with that work, it can clearly be seen that the cluster mass is 47% bigger when flattening is included. Thus, it is important to include flattening in nuclear cluster models. Still, our relative errors are useful as an estimate of some systematic uncertainties. While the distance-dependent error may improve with better distance from, e.g., the statistical parallax (Chatzopoulos et al. 2015), it is more difficult to reduce contributions of the light and mass profile uncertainty. They cause an error of about 8.5% in our modeling which would be somewhat similar in more complex models. That error is larger than the statistical error of 3.5% of Chatzopoulos et al. (2015), which also includes the distance uncertainty. It is, however, smaller than their systematical error of 10.1%. Their systematic error does not include the tracer profile uncertainty, which is dominant in our modeling, but is dominated by the flattening uncertainty. Overall, our tests show that the uncertainty caused by the tracer profile is not the dominant error contribution.

Finally, we obtain an estimate for the total mass of the nuclear cluster by using the inner components of the fits with constant M/L. From the average and scatter, we obtain

$$\begin{eqnarray}&&{M}_{\mathrm{NC}}=(4.22\pm 0.50{| }_{\mathrm{fix}{R}_{0}}\pm 0.67{| }_{{R}_{0}})\times {10}^{7}{M}_{\odot }.\end{eqnarray} \tag{ 13 }$$

Since the outer cluster slope is more uncertain than the mass at 100'', the mass uncertainty is now larger. The full uncertainty is larger still. It is not present here because the outer slope is fixed in γ-models. The true total mass uncertainty is probably similar to the case of the luminosity (Section 4.4) around ±50% of the mass. In contrast, the extended mass within 100'' is not affected by that uncertainty because our data constrain the density profile within $\sim 100^{\prime\prime}$ well.

The Jeans modeling only used the first two moments of the velocity distributions, but they contain information beyond that. In particular, stars with high velocities in the wings of the distributions are interesting. The unbiased space (three-dimensional) velocity for each star is (Trippe et al. 2008)

$$\begin{eqnarray}&&{v}_{3{\rm{D}}}=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}+{v}_{z}^{2}-\delta {v}_{x}^{2}-\delta {v}_{y}^{2}-\delta {v}_{z}^{2}}.\end{eqnarray} \tag{ 14 }$$

Since the three-dimensional velocity is, like the dispersion positive by definition, we subtract the errors to get unbiased velocities. The errors of ${v}_{3{\rm{D}}}$ do not depend on the value and, in particular, there is no indication (especially inside 20'') that the higher velocity stars are caused by measurement problems. We divide the sample into radial bins of 32 stars each, and determine in each the maximum and the median 3D velocity (Figure 14). The maximum velocity cannot be described by a single power law. Inside of 75, it follows a power-law slope of −0.47 ± 0.04, which is close to a Keplerian slope of −0.5. Other high quantiles, like the second-fastest star, follow similar slopes in the center. Outside of 75, these slopes appear to be consistent with the single slope of the median velocity of around −0.19.

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5.5.1. Minimum Binding Mass

We now calculate for all stars the minimum binding mass, i.e., we assume that each star is in the plane of the sky and on a parabolic orbit around a point mass. In each radial bin, we can thus determine the highest mass (Figure 15). Inside of 83, the escape mass is close to the SMBH mass. The average escape mass there is $3.67\times {10}^{6}\;$M${}_{\odot }$. The most significant mass larger than that of the SMBH occurs for the star S111 with $4.33\pm 0.05\times {10}^{6}\;$M${}_{\odot }$, as was already noted by Trippe et al. (2008) and Gillessen et al. (2009). High escape masses indicate that the stars are close to the SMBH not only in projection, but also in three dimensions. These fast stars can be used as additional constraints on the late-type density profile close to the SMBH in advanced dynamic modeling.

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5.5.2. A Distance Estimate

The number of stars that appear unbound depends on the assumed distance R₀. Stars whose velocity is dominated by the radial velocity are bound for a large R₀, while stars with a large proper motion are bound for small R₀. In the following, we restrict the used data to $r\leqslant 8\_\_AMP\_\_farcs;45$, i.e., the area dominated by the SMBH. Our sample contains nine stars, which are unbound for some distance between 7.2 and 9.2 kpc. From these stars, we calculate ${\chi }^{2}$ as a function of R₀, summing up in squares the significances of the differences from the minimum binding mass minus the mass inside that radius. We only consider stars when they appear to be unbound, however, not when the minimum binding is smaller than the mass inside that radius. The mass is dominated by the SMBH, for which we use ${{\rm{M}}}_{\bullet }=(3.95\pm 0.06)\times {10}^{6}{({R}_{0}/(8.0\mathrm{kpc})}^{2.19}$M${}_{\odot }$ (Gillessen et al. 2009). That means that from that paper, we use only the distance-independent mass error, not the distance-dependent component. For the extended mass, we use our two preferred models, a power-law and a constant M/L case (see Section 6.4). The mass of the SMBH dominates, while the extended mass adds at most ${10}^{5}\;{M}_{\odot }$. We add the error on the SMBH mass by exploring the Gaussian 1.5% error. In the case of the extended mass, we give the two cases equal weight and get the error from the difference. Minimizing ${\chi }^{2}$ as a function of R₀ yields ${R}_{0}={8.53}_{-0.15}^{+0.21}$ kpc. The main constraint comes from two stars: S111 and a star with high proper motion, Id 569 (Figure 16).

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This estimate relies on some assumptions.

• 1.  
  We assume that all stars are bound. This is justified, since the chance to see an escaping star from a Hills-event (Hills 1988) is very low (Yu & Tremaine 2003; Perets et al. 2007). Also, the Hills mechanism, which causes stars to move faster than the local escape velocity at $\gtrsim 10^{\prime\prime}$, is unlikely to play a role (as suggested by the break around 8'' visible in Figure 14).
• 2.  
  The method relies on the mass-distance scaling of the SMBH mass from stellar orbits. Using the relation from Ghez et al. (2008), we obtain ${R}_{0}={8.38}_{-0.25}^{+0.23}$ kpc.
• 3.  
  We assume that we have correctly debiased the velocities. Since the total velocity error of the two most important stars is less than 1.5% of their velocity, this assumption seems to be noncritical.

A lower limit on the SMBH mass independent of the ${M}_{\bullet }-{R}_{0}$ can be obtained by using the smallest recent R₀ measurement of 7.2 kpc (Bica et al. 2006; Genzel et al. 2010). For that distance, S111 gives $3.6\times {10}^{6}{M}_{\odot }$, which is consistent with the orbit-based estimates and higher than most Jeans-model estimates.

5.5.3. Fast Stars at $R\gt 10^{\prime\prime}$

5.5.4. Velocity Histograms

In Figure 17, we show the velocity histograms for the three dimensions. In the case of the proper motion samples, we show $R\gt 7^{\prime\prime}$. For the radial velocities, we show two bins, $7^{\prime\prime} \lt R\lt 100^{\prime\prime}$ and $R\gt 100^{\prime\prime}$. In the latter bin (consisting of the maser stars), we have subtracted the rotation.

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We see deviations from Gaussian distributions in many aspects.

• 1.  
  The central part of the distributions in l* and b* are not as peaked as in Gaussians (Figure 17). In l*, the distribution has a flatter peak than a Gaussian (Trippe et al. 2008; Schödel et al. 2009). In b*, the distribution has a slightly, but significantly ($\gt 5\sigma$), steeper peak than a Gaussian. The flatter peak in l* is actually a signature of the flattening of the cluster, as shown in Chatzopoulos et al. (2015).
• 2.  
  The varying ratio of maximal and median velocity (Figure 14) is indicative of non-Gaussian wings within 7''. Trippe et al. (2008) did not see that because they did not radially subdivide their sample. Furthermore, our sample contains 12 more stars in the central 2'' with ${v}_{3{\rm{D}}}\gt 460$ km s⁻¹ than that from Trippe et al. (2008). In that work, the only such fast star was S111. The high-velocity wings in all three dimensions are mainly caused by the presence of the SMBH. Therefore, because of the higher number of fast stars, S111 is less unique than discussed in Trippe et al. (2008).
• 3.  
  From the 274 stars in the $R\gt 100^{\prime\prime}$ bin, roughly 13 are in the high (radial) velocity wings of the distribution (Figure 17). Most of these outliers were already noted by Lindqvist et al. (1992a) and Deguchi et al. (2004).

Table 5 gives an overview of the unusually fast stars ($R\leqslant \;20^{\prime\prime}$, further out, extremely fast stars are not reliably measured) in our sample. ID 787 is the star which Schödel et al. (2009) called a runaway candidate. We find a proper motion that is 3.8σ smaller, and derive a rather normal radial velocity from its late-type spectrum, making this star considerably less noteworthy. The difference in proper motion is probably due to our better distortion correction at the edge of the field of view (Section 6.1). The star with the highest 3D velocity is the giant ID 4258, but it is (projected) close to another bright late-type star, such that the radial velocity error might well be larger than indicated. The outlier fraction appears to be smaller than in the maser sample (Figure 17), but at smaller radii the dispersion is higher, and thus the identification of outliers is more difficult there.

Table 5.  Fast Stars

ID	R.A. $(^{\prime\prime} )$	decl. $(^{\prime\prime} )$	${v}_{{\rm{R}}.{\rm{A}}.}$ ( mas years−1)	v${}_{\mathrm{Decl}.}$ ( mas years−1)	vz (km s−1)	v3D (km s−1)	vesc (km s−1)	Comment
770	−1.11	−0.91	−2.78 ± 0.02	−7.72 ± 0.01	−741 ± 5	807 ± 5	792	S111
569	0.13	3.08	13.22 ± 0.04	−3.6 ± 0.05	−92 ± 6	540 ± 7	541
4	5.68	−6.33	2.96 ± 0.09	2.58 ± 0.08	−312 ± 14	347 ± 14	326	IRS 9
4258	2.66	13.61	−8.48 ± 0.15	−1.67 ± 0.09	−315 ± 45	458 ± 46	255	fastest v3D, $R\gt 7^{\prime\prime}$
899	−8.37	−12.21	−1.98 ± 0.15	11.03 ± 0.24	⋯	435 ± 11	247	fastest v2D, $R\gt 7^{\prime\prime}$
903	14.11	7.45	−1.58 ± 0.04	−0.98 ± 0.13	379 ± 24	385 ± 24	238	only high vz
787	−6.8	18.35	−5.96 ± 0.13	7.36 ± 0.2	−91 ± 11	379 ± 15	215	runaway candidate
								(Schödel et al. 2009)

Note. Extraordinarily fast stars sorted by their projected distance from Sgr A*. The errors of the positions are smaller than 5 mas. v_esc assumes ${R}_{0}=8.2$ kpc.

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Row 1 in Table 4 shows that fitting for the SMBH mass with a power-law profile yields a mass which is much smaller than the estimates from stellar orbits (Ghez et al. 2008; Gillessen et al. 2009). We now look closer at that problem. In order to reduce the rotation or flattening influence in the results, we use only the data in the central 27'' where the assumption of spherical symmetry is approximately fulfilled. The same effect also occurred in earlier works (Genzel et al. 1996; Trippe et al. 2008). However, recently, Schödel et al. (2009) and Do et al. (2013b) obtained higher masses from Jeans modeling of old stars.

From the three-dimensional motions of 248 late-type stars in the central 12'', Do et al. (2012) obtained  masses between ${M}_{\bullet }={3.77}_{-0.52}^{+0.62}\times {10}^{6}{M}_{\odot }$ and ${M}_{\bullet }={5.76}_{-1.26}^{+1.76}\times {10}^{6}{M}_{\odot }$, which are consistent with the orbit-based estimates considering the R₀ of the fits. They obtained higher values with anisotropic spherically symmetric Jeans modeling than with isotropic spherically symmetric Jeans modeling. These fits obtain isotropy in the center, but tangential anisotropy further out. The latter is probably spurious (see Section 5.1). The lowest value is obtained in isotropic modeling.

The break radius of Do et al. (2013b) is large and inconsistent with our data (see Figure 5). We test its influence on the obtained mass. We therefore select from our data those bins inside 12'' and fit the SMBH while fixing all other parameters to fit number 1 of Do et al. (2013b). We obtain ${M}_{\bullet }=3.77\pm 0.09\times {10}^{6}{M}_{\odot }$, which is fully consistent with Do et al. (2013b). (Our error is smaller because we had to fix most parameters.) Thus, our motions are consistent with the motions of Do et al. (2013b). Also, the binning has no relevant influence. The SMBH mass increases to about $4.5\times {10}^{6}{M}_{\odot }$ when fitting our profiles within 220'' with a single Nuker profile. In this and the previous fit, the cluster mass is zero. When the cluster is included with constant M/L, we obtain ${M}_{\bullet }=3.33\pm 0.00{| }_{\mathrm{tracer}\mathrm{profile}}\pm 0.34{| }_{\mathrm{dispersion}}\times {10}^{6}{M}_{\odot }$ with the Do profile and ${M}_{\bullet }=2.86\pm 0.10{| }_{\mathrm{tracer}\mathrm{profile}}\pm 0.23{| }_{\mathrm{dispersion}}\times {10}^{6}{M}_{\odot }$ with our density data. Note that when checking whether the tracer profile is important, only the error caused by the tracer profile is relevant. Thus, the reason for the high mass of Do et al. (2013b) is twofold: the tracer profile and the extended mass. In conclusion, not only is high-quality dynamic data important, but also good data for the surface density profile. Furthermore, all components including the extended mass need also to be fit in the inner 12'' due to projection effects.

From isotropic Jeans modeling, Schödel et al. (2009) obtained  ${M}_{\bullet }=3.55\times {10}^{6}\;{M}_{\odot }$, which is larger than our estimate and those of earlier works. For their assumed distance of ${R}_{0}=8$ kpc, the SMBH mass of Gillessen et al. (2009) is within the 90% probability interval of Schödel et al. (2009), but in their anisotropic modeling it is excluded by 99%. It is interesting to clarify the differences between our analysis and that of Schödel et al. (2009) using their publicly available data set.

• 1.  
  Using the Schödel et al. (2009) data, their tracer profile, and our two mass parameterizations, we get results consistent with what these authors found. For example, with the power law and ${R}_{0}=8\;$ kpc we obtain ${M}_{\bullet }=3.33\pm 0.56\times {10}^{6}{M}_{\odot }$. With our normal $R=8.2\;{\rm{kpc}}$, the mass increases to ${M}_{\bullet }=3.59\pm 0.60\times {10}^{6}{M}_{\odot }$. Hence, the details of the modeling only play a minor role.
• 2.  
  Using our dynamical data within 27'' and the tracer profile of Schödel et al. (2009), we retrieve ${M}_{\bullet }=2.88\pm 0.48\times {10}^{6}{M}_{\odot }$, repeating the previous fit with ${R}_{0}=8.2\;{\rm{kpc}}$. That decrease in mass is not because we also add radial velocities, quite the opposite: when we use only our proper motion, the mass decreases slightly to ${M}_{\bullet }=2.63\pm 0.60\times {10}^{6}{M}_{\odot }$. When we fit our density data inside the break with a single Nuker profile, in which $\gamma =0.5$ but the rest is free, we get ${M}_{\bullet }=2.34\pm 0.60\times {10}^{6}{M}_{\odot }$. Thus, the use of the Schödel et al. (2009) tracer profile increases the SMBH mass by about $0.5\times {10}^{6}$M_⊙. The use of different proper motion data sets changes the mass by about $0.7\times {10}^{6}$M_⊙. Therefore, differences in the tracer profiles are, aside from the dispersion data, the main reason for the differences in the mass obtained between our work and Schödel et al. (2009).

We now investigate the most important reason, namely, the higher dispersions in the data of Schödel et al. (2009).

• 1.  
  The largest relative differences occur within $R\lt 2\_\_AMP\_\_farcs;5$. There, Schödel et al. (2009) still have a few early-type stars contaminating their sample, since they used the spectroscopy-based, slightly outdated star list in Paumard et al. (2006) and the photometric identifications of Buchholz et al. (2009). Four certainly early-types are contained in their sample. For example, the well-known fast, early-type star S13 (Eisenhauer et al. 2005), for which an orbit is known, is part of their sample. In contrast, in the central 25, we use only those stars which we positively identify spectroscopically as late-type stars. This issue is critical in the center, since early-type stars are more concentrated toward the SMBH than late-type stars, and thus they show a higher dispersion in the center. Inside of  12, we obtain $\sigma =9.44\pm 1$ mas years⁻¹ using the data of Schödel et al. (2009) and $\sigma =6.92\pm 0.69$ mas years⁻¹ from our proper motions. Removing the early-type stars from the Schödel et al. (2009) sample yields a central dispersion consistent with our value.
• 2.  
  At R > 15'', differences occur again in the proper motion data. This could be caused by differences in the distortion correction, which is more important at large radii. An imperfect distortion correction enlarges dispersions artificially, and thus a smaller measured value is more likely to be correct.

Overall, we believe that our dispersions are more reliable than those of Schödel et al. (2009). This is also supported by the fact that we obtain an extended mass of ${M}_{100^{\prime\prime} }=5.65\pm 2.0\times {10}^{6}\;{M}_{\odot }$, which is consistent with our best value, when we restrict our dynamics data to the range R < 27'' (for fixed SMBH mass with the count profile and constant M/L) in contrast to Schödel et al. 2009 (Section 6.2).

Why do our and other attempts of Jeans modeling fail to recover the correct SMBH mass? The direct cause is that our measured dispersion within 10'' is smaller than the dispersion in our (isotropic) models, which obtain the correct black hole (see Figure 11). What is the reason for the dispersion difference, and thus the low mass?

• 1.  
  Neglecting anisotropy is probably not the reason. While Do et al. (2013b) obtain higher SMBH masses in that case, that is mainly caused by the R₀–${M}_{\bullet }$ relation. Do et al. (2013b) obtain ${R}_{0}=8.92\;{\rm{kpc}}$ in their free fit, a distance larger than all of the recent measurements (Genzel et al. 2010). While the data of Schödel et al. (2009) are problematic in some radial ranges, like the very center, we agree about anisotropy: we both get that the cluster shows slight (not significant) tangential anisotropy in the inner part. Central tangential anisotropy decreases mass estimates, see, e.g., Genzel et al. (2000). Therefore, Schödel et al. (2009) obtain a slightly smaller SMBH mass with anisotropic modeling. We therefore assume that the inclusion of anisotropy likely would decrease the obtained SMBH mass slightly.
• 2.  
  The binning and dispersion errors are probably not the reason. Our different binnings obtain all masses between $3.25\times {10}^{6}$ and $3.38\times {10}^{6}$ ${M}_{\odot }$ for the SMBH using the count profile, a constant mass-to-light ratio, and dynamics within 27''. That also makes it less likely that we underestimated the dispersion errors, since in each of the binnings we use a slightly different bias correction, but the results are consistent. The errors are easy to calculate, especially in the center, which is most relevant for the SMBH mass, since measurement errors are significantly smaller than Poisson errors.
• 3.  
  For checking whether or not we can recover the correct SMBH mass for another R₀, we use, as previously, the SMBH mass–R₀ relation of Gillessen et al. (2009) for another R₀. Even for an unrealistically large value of ${R}_{0}=9\;$kpc, the SMBH mass–R₀ relation is not recovered. Thus, the SMBH mass underestimation is largely independent of the distance.
• 4.  
  The power-law extended mass parametrization yields a smaller mass than when using a constant M/L: ${M}_{\bullet }=(2.68\pm 0.51)\times {10}^{6}\quad {M}_{\odot }$ for a free power-law slope and ${M}_{\bullet }=(2.98\pm 0.17)\times {10}^{6}\quad {M}_{\odot }$ for a Bahcall–Wulf cusp of ${\delta }_{M}=1.25$. Meanwhile, a constant mass-to-light ratio yields ${M}_{\bullet }=(3.37\pm 0.16)\times {10}^{6}\quad {M}_{\odot }$. Hence, a core in the stellar mass instead of a Bahcall–Wulf cusp may be part of the solution.
• 5.  
  Trippe et al. (2008) and Genzel et al. (2010) argued that a central core-like structure introduces a bias toward low SMBH masses. This, however, is only applicable when a tracer profile without a core is used for core data. If the correct profile contains a core and is modeled as such, then the central core only increases the error on the central mass. In our models, such an uncertainty is included via the free density profile, yet the SMBH mass falls outside of the error band. In Figure 18, it is visible that the correlation betweeen ${\gamma }_{\mathrm{in}}$ and the SMBH mass is weak. It is not possible to obtain the correct mass by changing γ within our model. Still, the fact that our flux and star count profiles are not consistent indicates that their errors are probably underestimated. Neither produces the right mass, but none of them may be the true profile. A profile with a large enough core radius might produce the correct mass. Another possibility is that our functional form is inadequate. Perhaps a profile works with a large radial transition region, in which the profile has a constant slope, somewhat steeper than in the center.
• 6.  
  Introducing a flat true core is unlikely to solve the issue. It is not only inconsistent with the apparent isotropy, but it also underpredicts the number of late-type stars with orbits.
• 7.  
  Also, the flattening influences the SMBH mass, both through the tracer profile and by reducing the dispersion in most dimensions (see Chatzopoulos et al. 2015). Also, well inside of 27'', the dispersion is different in l and b, indicating that flattening is also important there. We test that case by fitting the data within 27'' with the flattened 2-integral model of Chatzopoulos et al. (2015). We obtain ${M}_{\bullet }=(3.54\pm 0.18)\times {10}^{6}\quad {M}_{\odot }$. This error also includes the distance uncertainty.

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In summary, we have tested several potential solutions to the mass bias. For some, we have shown quantitatively that they alone cannot correct the bias. Others would require modeling that goes beyond this work. Likely, several together are necessary for a solution. The absence of a dark cusp and another, more complex tracer profile are perhaps the most likely solution. It could be that maximum likelihood modeling is needed for our discrete data set (D'Souza & Rix 2013), in particular, for properly incorporating the information the fast stars carry.

Overall, most M_cluster values from the literature are similar to our values. Comparison with other works requires some care, however, since different values for R₀ have been used, and hence we scale to ${R}_{0}=8.2$ kpc. We correct the masses in the literature to our distance using a mass scaling with exponent 1 for purely radial-velocity-based masses, and exponent 3 for purely proper-motion-based ones. We extrapolate from our best estimate ${M}_{100^{\prime\prime} }$ in two ways: on the one hand, we use a broken power law with break radius 100''. The inner slope is ${\delta }_{M}=1.232$ and the outer is ${\delta }_{M}=1.126$. (This implies that inside of 100'' we use the average ${\delta }_{M}$ of rows 3 and 4 of Table 4 and outside the average ${\delta }_{M}$ of rows 11 and 12. The first approach uses dynamics between 10 and 100'', while the second uses dynamics outside of 10''.) On the other hand, we use one of the preferred fits with constant M/L (Row 5 in Table 4). Most other mass profiles are similar to one of these cases. We quantify the quality of the comparisons only when errors are given in the literature and there is not more than one value in a source without a clear preference for one. We show all masses in Figure 19.

6.2.1. Schödel et al. (2009)

6.2.2. Trippe et al. (2008)

6.2.3. Inner CND

The gas in the in CND can be used to obtain a mass estimate. Serabyn & Lacy (1985) used the [NeII] 12.8 μm emission from gas streamers at the inner edge (including the western arc) of the CND to obtain its rotation velocity. They found $(3.9\pm 0.8)\times {10}^{6}{M}_{\odot }$ at 1.4 pc, which is 1.7σ and 2.4σ less than our constant M/L and power-law estimates. Their mass is smaller than the mass of the SMBH. The projected rotation velocity of the inner CND of $100$ km s⁻¹ is well determined (Genzel et al. 1985; Serabyn & Lacy 1985; Guesten et al. 1987; Jackson et al. 1993; Christopher et al. 2005), and hence the reason for the discrepancy has to be in the model assumed. Conceptually, the mass derivation of Serabyn & Lacy (1985) was simple: they assumed circular motion of all the gas in one ring with an inclination of 60°–70°.

In the HCN J-0 data of the inner CND in Guesten et al. (1987), the velocities in the southern and western parts follow the model of Serabyn & Lacy (1985), but not in the northern and eastern parts. The latter can be described by a less inclined ring ($\approx 45^\circ$) with an intrinsic rotation velocity of 137 ± 8 km s⁻¹. This velocity results in a total mass of $(6.1\pm 0.7)\times {10}^{6}{M}_{\odot }$, which is somewhat larger than our estimates. Using an average inclination between Serabyn & Lacy (1985) and Guesten et al. (1987) would hence yield a mass estimate very similar to ours.

However, Zhao et al. (2009) find an inclination consistent with that of Serabyn & Lacy (1985). Furthermore, Zhao et al. (2009) show that a single orbit, even when slightly elliptical, cannot fit all of the gas streamers in the western arc, and hence the inner CND is likely more complicated than assumed by Serabyn & Lacy (1985).

Another possibility for solving the discrepancy between our result and the result of Serabyn & Lacy (1985) is that the assumption of Serabyn & Lacy (1985), that the line width is due to nongravitional processes, may be wrong. If the line width is due to asymmetric drift (Binney & Tremaine 2008), then the true circular velocity ${v}_{{\rm{c}},\mathrm{true}}$ would be higher than the measured velocity ${v}_{{\rm{c}},\mathrm{obs}}$ (Kormendy & Ho 2013):

$$\begin{eqnarray}&&{v}_{{\rm{c}},\mathrm{true}}^{2}={v}_{{\rm{c}},\mathrm{obs}}^{2}+x\times {\sigma }^{2}.\end{eqnarray} \tag{ 15 }$$

The factor x depends on the disk profile and other not well-known parameters of the inner CND. It is between 1 and 3. From $\sigma =35\pm 5$ km s⁻¹ (Serabyn & Lacy 1985) follows for x = 3 an increase in the mass by 30% to $(5.1\pm 1.04)\times {10}^{6}\;{M}_{\odot }$. This would be consistent with our constant M/L estimate of $5.27\times {10}^{6}\;{M}_{\odot }$.

6.2.4. Outer CND

6.2.5. Rieke & Rieke (1988)

6.2.6. McGinn et al. (1989)

6.2.7. Lindqvist et al. (1992a)

6.2.8. Deguchi et al. (2004)

6.2.9. Genzel et al. (1996)

6.2.10. Chatzopoulos et al. (2015)

6.2.11. Works Using the Light Distribution

The distribution of old stars does not show the expected central cusp (Bahcall & Wolf 1976) with the inner ${\delta }_{L}=1.25$ slope (Buchholz et al. 2009; Do et al. 2009; Bartko et al. 2010). The stars follow a shallower slope in the center, which can either be a true core (${\delta }_{L}=3$) or a shallow cusp (${\delta }_{L}\gt 2$, Do et al. 2009). Also, here we call the second case "core," since it has much less light in the center than a cusp. By comparing our power-law mass models (with a central cusp) and the constant M/L models (with a core) we can constrain the central mass distribution.

• 1.  
  Given that a core can reduce the low dispersion bias, the Jeans modeling of our data favors a core over a cusp. The same is found by Schödel et al. (2009), who used an unrealistically large, fixed break radii for light and especially mass. Our results show that the preference for a smaller slope in the center is not only due to the large break radius in Schödel et al. (2009).
• 2.  
  The presence of a warped disk of young stars (Bartko et al. 2009; Löckmann & Baumgardt 2009) yields an additional constraint. Ulubay-Siddiki et al. (2013) found that with an isothermal cusp of M($r\lt 0.5$ pc) $=\;{10}^{6}\;$M${}_{\odot }$ and a flattening of $q=0.9$, an initial warped disk is too quickly destroyed and would not be observable today. For constant M/L, we obtain a smaller mass of M($r\lt 0.5$ pc) $\approx \;0.15\times {10}^{6}\;$M${}_{\odot }$, which reduces the torque by a factor of five, allowing the disk to survive longer. This conclusion is not firm, since the warping need not be primordial, and reduced central flattening might also yield less warping (Kocsis & Tremaine 2011).
• 3.  
  The most reliable mass measurement around 1.4 pc apart from our analysis, that is, Serabyn & Lacy (1985), is more consistent with a core than with a cusp.

Overall, it appears likely that the mass profile shows a central core, but better modeling including a solution of the mass bias is necessary. Another route for constraining the mass profile worth follow up is using direct accelerations of stars at radii between 1'' and 7'' with GRAVITY (Eisenhauer et al. 2008).

Our preference for a core in the mass profile is interesting for theories that aim to explain the missing light cusp. In some theories, the resolvable stars (giants) form a core while the dark components (remnants or main-sequence stars) form a mass cusp. Thus, our result means that mass segregation is less likely to solve the riddle of the missing light cusp. Keshet et al. (2009) derive a similar conclusion from different arguments. Another mechanism that destroys a light cusp, but not a mass cusp, is the destruction of giants by collisions (Dale et al. 2009), and hence our results also disfavor that model. One model that could work is presented by Merritt (2010), who propose that a relatively recent binary black hole (merger) ejected stars. This process would yield a core both in the light and mass profiles. Still, one would need to fine-tune the timing of such a model, since the early-type stars are concentrated toward the center, but the giants and even the younger red (super-) giants with ages down to 20 Myr (Blum et al. 2003; Pfuhl et al. 2011) are not.

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Another way of expressing our results is the cumulative mass profile. In Figure 20, we show three cases.

• 1.  
  We use a constant M/L model with a best-fitting inner slope of −0.81 (Row 5 in Table 4, model A).
• 2.  
  We use a constant M/L model with the shallowest possible inner slope of −0.5 (and otherwise row 5 in Table 4, model B).
• 3.  
  We use a power-law model with a slope of ${\delta }_{M}=1.232$ (model C), the average⁸ of rows 3 and 4 in Table 4.

We normalize these three models to have an overall best-fit value of ${M}_{100^{\prime\prime} }=6.09\times {10}^{6}\;{M}_{\odot }$. The tabulated values of the profiles can be found in Appendix F. The largest mass difference between M/L = const and the power-law model is reached at about 35'' with $0.59\times {10}^{6}\;{M}_{\odot }$. Since this number exceeds our mass error, it is possible to detect dynamically the cusp, provided that the low dispersion bias can be solved.

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Extrapolating the constant M/L models down to the regime of the S-stars S2 and S55/S0-102 (Meyer et al. 2012) yields a mass there which is smaller than that of the individual stars. In this regime, the S-stars dominate, and taking into account their Salpeter-like IMF (Bartko et al. 2010) or a potential mass-segregated cusp of stellar remnants (Freitag et al. 2006; Hopman & Alexander 2006) would make the dominance even stronger. For power-law models, it is not clear whether the old stars dominate the mass in the central arcsecond.

Outside the central arcsecond, the masses of the disks of early-type stars (Bartko et al. 2009; Lu et al. 2009) may be important. We derive the spatial distribution from Figure 2 in Bartko et al. (2010) and globally deproject to space distances with a factor of 1.2 the given projected distances. We estimate the total ZAMS mass of that population to be $1.5\times {10}^{4}\;{M}_{\odot }$ (Bartko et al. 2010), dominated by the O stars given the top-heavy shape of the IMF. Across the literature (Paumard et al. 2006; Bartko et al. 2010; Lu et al. 2013), the disk mass is uncertain by a factor 2.5. At $r=2^{\prime\prime}$, the O stars might be comparable in mass to the old stars if the constant M/L model is correct.

Further out, the only other component is the CND. Most publications (Genzel et al. 1985; Mezger et al. 1989; Etxaluze et al. 2011; Requena-Torres et al. 2012) agree that its mass is a few ${10}^{4}\;{M}_{\odot }$, and thus is irrelevant compared to the old stars. However, the mass of $\;\approx {10}^{6}\;{M}_{\odot }$ found by Christopher et al. (2005) would be about one-third of our mass estimate at that radius.

Our results also yield a new estimate for the sphere of influence of the SMBH (Alexander 2005), i.e., the radial range within which the extended mass is equal to the mass of the SMBH. The new estimate is completely consistent with previous values (Alexander 2005; Genzel et al. 2010). Using models A and B, we get ${r}_{\mathrm{infl}}=76.3\pm 5\_\_AMP\_\_farcs;5=3.03\pm 0.25$ pc.

We now obtain the mass-to-light ratio from the fits to the flux and dynamics with constant M/L. The mass-to-light ratio is directly calculated from the output of these models. We obtain ${\text{}}M/L=0.51\pm 0.12\;{M}_{\odot }/{L}_{\odot ,\mathrm{Ks}}$ using ${M}_{\mathrm{Ks}\odot }=3.28$ (Binney & Merrifield 1998). The error consists of 19% for the light, 10% for the mass, and 8% for the distance uncertainty. The latter number follows from the distance uncertainty of 4.1% (Gillessen et al. 2013) multiplied by an exponent of 1.83, which is the scaling M/L with distance given that M scales like 3.83 with distance (Section 5.4). Our M/L is consistent with the values in Pfuhl et al. (2011) and Launhardt et al. (2002). Our error, however, is smaller thanks to the smaller mass error. Using the improved mass-to-light ratio we can further constrain the IMF of the old stars.

First, we use M/L to determine the IMF slope α; we assume the respective star formation histories for the old stars (older than 10 Myr) from Pfuhl et al. (2011). Their results also appear to hold at the larger radii of interest here (Blum et al. 2003). With the accuracy of M/L from Pfuhl et al. (2011), it was already possible to discard extreme top-heavy IMFs ($\alpha \gt -0.6$), like those measured for the O- stars in the GC (Bartko et al. 2010). Also, with our accuracy, the IMF slope of $\alpha =-0.85$ (Paumard et al. 2006) is excluded.

Second, we concentrate on variants of the usual −2.3 slope below one solar mass, these are the IMFs of Salpeter (1955), Kroupa (2001), and Chabrier (2003). To check the result on the dependence of the used stellar population model, we use the models of Bruzual & Charlot (2003) and Maraston (2005). In the case of the Maraston (2005) models, we obtain ${\text{}}M/L=0.75{M}_{\odot }/{L}_{\odot ,\mathrm{Ks}}$ and ${\text{}}M/L=1.28{M}_{\odot }/{L}_{\odot ,\mathrm{Ks}}$ for Kroupa and Salpeter, respectively. For the Bruzual & Charlot (2003) models, we get ${\text{}}M/L=1.04{M}_{\odot }/{L}_{\odot ,\mathrm{Ks}}$ and ${\text{}}M/L=0.76{M}_{\odot }/{L}_{\odot ,\mathrm{Ks}}$ for Salpeter and Chabrier, respectively. The differences in the stellar population models cause  differences of up to 20%. (The stellar population model dependences increase with wavelength: at 4.5 μm (IRAC2), as used by Schödel et al. 2014 and Feldmeier et al. 2014, the model of Bruzual & Charlot 2003 obtains ${\text{}}M/L=0.71{M}_{\odot }/{L}_{\odot ,\mathrm{IRAC}2}$ for Chabrier and the model of Bruzual & Charlot (2003) obtains ${\text{}}M/L=1.01{M}_{\odot }/{L}_{\odot ,\mathrm{IRAC}2}$ for Kroupa.) Using the star formation history of Blum et al. (2003) instead of that of Pfuhl et al. (2011) causes 18% smaller M/L in the Ks band.

Recently, deviations from the Chabrier IMF were found for different stellar systems. On the one hand, elliptical galaxies seem to have a Salpeter, or more bottom-heavy, IMF as concluded by van Dokkum & Conroy (2010) and Conroy & van Dokkum (2012) using population modeling of the integrated spectra, and as concluded  by Cappellari et al. (2006, 2012) from advanced dynamic modeling. A Salpeter IMF is already in strong tension with our ratio. A more bottom-heavy IMF of, e.g., a slope of −2.8, has an even larger M/L (a factor of 1.57 more in M/L compared to Salpeter for the star formation history assumed in Cappellari et al. 2012 and Conroy & van Dokkum 2012), and is thus firmly excluded.

It seems that (metal-rich) globular clusters have M/L smaller than Kroupa (Kruijssen & Mieske 2009; Bastian et al. 2010; Sollima et al. 2012). This was found in the case of M31 by Conroy & van Dokkum (2012) using population modeling of integrated spectra. In addition, Strader et al. (2011) found the same by using mass and light measurements. To be precise, Strader et al. (2011) measured for solar metallicity ($[Z]=0\pm 0.3$) ${\text{}}M/L=0.39{M}_{\odot }/{L}_{\odot ,\mathrm{Ks}}$. This is already lower than our measurement, although the GC contains more medium old stars than these globulars. In an attempt to include these younger stars, we reduce in our Kroupa and Chabrier models the mass of the stars $\gt 5$ Gyrs to achieve ${\text{}}M/L=0.39{M}_{\odot }/{L}_{\odot ,\mathrm{Ks}}$ for these ages. We do not change the populations for the younger stars. This model has the motivation that it describes a possible formation scenario of the nuclear cluster in which the majority of the old stars originate in big metal-rich globular clusters (Capuzzo-Dolcetta & Miocchi 2008) and later stars that formed locally are added to it. This model results in ${\text{}}M/L=0.31{M}_{\odot }/{L}_{\odot ,\mathrm{Ks}}$, deviating by 2σ from our measurement. This argument against a globular cluster origin for the nuclear cluster is weakened by the fact that when globulars arrive in the GC, they likely have a different M/L due to mass segregation. In the case of M31, it is argued that the low M/L is mostly due to a different IMF and not only due to evolutionary effects (Strader et al. 2011). However, the preferred loss of low-mass objects due to internal mass segregation and the outer tidal field certainly exists (Bastian et al. 2010; Sollima et al. 2012) and should be further enhanced close to the GC. The problem is that while mass segregation first reduces the M/L, M/L is enhanced in late stages during the late-remnant-dominated stage of old populations (Marks et al. 2012; Sollima et al. 2012). Thus, without detailed modeling of mass segregation, the expected $M/L$ of a globular-cluster-dominated nuclear cluster is uncertain.

The GC shows a normal ratio of diffuse light to total light (Pfuhl et al. 2011). This ratio measures the ratio of main-sequence stars to giants and decreases monotonically from a bottom-heavy to a top-heavy IMF. Thus, it best excludes a cluster dominated by remnants and giants—a possible state for globular clusters inspiraling to the GC.

The Chabrier IMF common in both the Galactic disk and bulge (Zoccali et al. 2000; Bastian et al. 2010) is close to the M/L of the moderately old and old stellar populations of the GC. However, the deviation from it is 1.7σ when using the most recent star formation history determination (Pfuhl et al. 2011). That deviation vanishes when the bias to small masses due to our spherical modeling is corrected (see Chatzopoulos et al. 2015). In that case, the ratio is ${\text{}}M/L=0.76\pm 0.18\;{M}_{\odot }/{L}_{\odot ,\mathrm{Ks}}$, which matches well the Kroupa/Chabrier IMF. Therefore, a globular cluster origin for the nuclear cluster (Antonini et al. 2012; Gnedin et al. 2014) is somewhat disfavored by its M/L and L_diffuse/L_total.

We now compare the nuclear cluster of the Milky Way with nuclear clusters in other galaxies. The literature about the masses of other nuclear clusters (Walcher et al. 2005; Barth et al. 2009) is sparse and biased toward brighter nuclear clusters. We therefore mainly use the light for comparison, for which ample data from HST imaging are available (Carollo et al. 2002; Böker et al. 2004). Böker et al. (2004) studied late-type spirals (Scd to Sm) while Carollo et al. (2002) concentrated on early-type spirals (Sa to Sbc). For color correction we use ${}^{\prime }{{\rm{H}}}^{\prime }-\mathrm{Ks}=0.15$ and ${\rm{I}}-\mathrm{Ks}=1.1$ for the results of Carollo et al. (2002) and Böker et al. (2004), respectively. We compare the obtained magnitudes with our GALFIT decomposition for the nuclear cluster (Section 4.3) of the double γ-model fitting (Section 4.2.2). In addition, we use the cumulative flux as a function of radius. The latter is not affected by decomposition uncertainties (Figure 21).

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The size of the nuclear cluster of the Milky Way is typical. It is smaller than many clusters in Carollo et al. (2002) and larger than most clusters in Böker et al. (2004). This simply might be a consequence of the fact that the galaxy type of the Milky Way is broadly between the two samples.

In Figure 21, it can be seen that the nuclear cluster of the Milky Way has an unusually high surface brightness. Not only is its characteristic brightness high, but the cumulative brightness profile also lies above nearly all other clusters. What is the reason for this?

• 1.  
  The young ($\approx 6$ Myr) and moderately old ($\approx 200$ Myr) stars in the GC are likely not the reason for this offset. They are subdominant compared to the old stars in the Ks band (Blum et al. 2003; Pfuhl et al. 2011, and Section 4.4). Furthermore, many nuclear clusters (Rossa et al. 2006; Walcher et al. 2006) contain significant fractions of young or moderately old stars. Thus, the M/L of the GC is probably typical.
• 2.  
  Extinction is probably not an issue, but we do correct for it. The spirals in the sample of Carollo et al. (2002) and Böker et al. (2004) are seen more or less face-on. Furthermore, the mean IR color in Carollo et al. (2002) yields a small extinction of ${A}_{160W}\approx 0.2$. Also, the spectroscopic study of Rossa et al. (2006) obtained small extinctions for the same samples.
• 3.  
  The higher resolution of our data is also not the reason, since we compare the characteristic brightness within 4 pc, a size which can be resolved in most of the galaxies of Carollo et al. (2002) and Böker et al. (2004).
• 4.  
  The nuclear cluster of the Milky Way has a projected mass density of $\mathrm{log}{{\rm{\Sigma }}}_{e}=5.38\pm 0.17$ [M${}_{\odot }$/pc²]. This is larger than for all 11 of the nuclear clusters in the sample of Walcher et al. (2005).

We conclude that the main reason for the high surface light density is the high stellar mass density in the nuclear cluster compared to other galaxies.

For some nearby galaxies, a more detailed comparison is possible. For example, M33 shows a central star density which may be larger than that in the nuclear cluster of the Milky Way (Lauer et al. 1998). M33 does not contain an SMBH (Gebhardt et al. 2001; Merritt et al. 2001), and the old stars—if present at all—are outshined by young stars. Some nuclear clusters show flattening and rotation; in the case of the metallicity, the flattening is 16% (Lauer et al. 1998) combined with some rotation (Kormendy & McClure 1993). The large nuclear cluster in NGC4244 (R${}_{e}\approx 10\;$pc) is visually flattened and has ${v}_{\mathrm{rot}}/\sigma \approx 1$ (Seth et al. 2008). NGC404 is  similar (Seth et al. 2010). Thus, the flattening of the Milky Way nuclear cluster covers a large range of possible shapes.

Carollo (1999) finds that nuclear clusters with ${M}_{V}\lesssim -12$ are typically associated with signs of circumnuclear star formation, like dust lanes and an HII spectrum. In this respect, the cluster of the Milky Way with ${M}_{V}\lesssim -13$ seems typical for its nuclear disk. Bright nuclear clusters are preferentially observed together with nuclear disks, which might be explained by a physical coupling of the two systems. Since the GC is dominated by old stars, this connection is likely not only valid for the recent star formation event but also for the older population.

In the center ($R\leqslant \;2^{\prime\prime}$) the crowding is strong. Thus, it is difficult to fit the stars on the images with simple Gaussians. Instead, we extract the point-spread function (PSF) from the images and deconvolve the image with the Lucy-Richardson algorithm, as in Gillessen et al. (2009). Compared to that work, we expand the time baseline and enlarge the field of view from about 1'' to 2''.

We fit the astrometric data points and their errors for all of the stars by linear fits. We rescale the ${\chi }^{2}/{\rm{dof}}$ to 1 as in Gillessen et al. (2009). Some of the stars have significant accelerations (Gillessen et al. 2009). However, to avoid complicating our analysis by including acceleration for a very small fraction of the stars, for all of the stars we calculate only the linear motion. The velocity errors are small with an average 0.038 mas years⁻¹ compared to the velocity dispersion of 5.23 mas years⁻¹. Furthermore, due to the large number of data points (more than 100 for many stars), the errors can be determined well from rescaling. For the stars with significant acceleration, the errors are not strictly correct, since the error calculation assumes random errors. However, since the cubic deviation from linear motion is nearly never significant (Gillessen et al. 2009), the derived velocity is close to the velocity at the mean time of the observing interval. The derived error is slightly too big for this definition, as the scatter is mainly caused by non-random accelerations.

All of the fit errors, even the largest of 0.71 mas years⁻¹ for the velocity of −19.09 mas years⁻¹ for S38, are too small to have any influence compared to the large dispersion of ${\sigma }_{1{\rm{D}}}=5.25$ mas years⁻¹ and its associated Poisson error of 0.30 mas years⁻¹. (${\sigma }_{1{\rm{D}}}$ is, as in the following, the average dispersion using all 1D velocities in x and y.) Therefore, we neglect the error uncertainties for proper motions in the central two arcseconds in our analysis.

This data set is an update of that of Trippe et al. (2008). This field extends out to 20''. We exclude stars that are within the central field (Appendix A.1).

Compared to Trippe et al. (2008), we add new epochs (see Table 6 for a list of all images) and a new conversion to absolute coordinates aligned with Gillessen et al. (2009). The positions of the stars in these images are obtained by fitting Gaussians to distortion-corrected images. The distortion correction is applied in the same way as in Trippe et al. (2008). Like them, we also use the following linear transformation:

$$\begin{eqnarray}{x}^{\prime } & = & {a}_{0}+{a}_{1}\times x+{a}_{2}\times y\\ {y}^{\prime } & = & {b}_{0}+{b}_{1}\times x+{b}_{2}\times y.\end{eqnarray} \tag{ 16 }$$

This transformation contains also crossterms. Thus, it also automatically corrects for linear effects like differential atmospheric distortion. Only sources with no close neighbor on the images are included in the data set. When a source consists of two very close neighbors, it can be included in our data set. In such cases, the velocity is a flux-weighted average, which reduces the absolute velocities and therefore on average also the dispersion. In the center, we have also deconvolved images available over 10 years. Checking these deconvolved images shows that source confusion affects less than 1/10 of the sources in the extended field. Further out, the source density is smaller, and thus source confusion introduces no relevant dispersion bias in the extended field data set.

We change the procedure of outlier rejection and error calculation compared to Trippe et al. (2008). For the first fit, for each star we use the position uncertainty of the Gaussian on the image and rescale the ${\chi }^{2}/{\rm{dof}}$ of the fit to 1. We then calculate for each image the residual distribution of all of the stars compared to the fit. The width including 68.3% (≡W1S) of all entries is often quite different from 1σ. To change this, we iterate the fitting. In the iteration, we rescale the errors on each image by the W1S in σ. We repeat this progress a second time, and then the variation of the W1S between the different images is less than 0.1σ. Other factors are then more important, like magnitude with up to 0.2σ deviation. However, we ignore these effects because the sign of the deviations is randomly distributed over our 26 images. For the final fit, we exclude all 5σ outliers. Outliers between 2.5 and 5σ we reject by drawing random numbers to ensure that the distribution of all residual approximates a Gaussian distribution. In Table 6, we provide the median error for all of the images used. The final residual distribution is approximately Gaussian up to $| 3.7| \sigma$, see Figure 22.

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Assuming only Poisson errors, the derived dispersion is ${\sigma }_{1{\rm{D}}}=2.677\pm 0.018$. The error on the dispersion in this field is so small that other error sources could be important. The median and average velocity errors are 0.150 and 0.208 mas years⁻¹, respectively. They are so small that it does not matter which of them is used. However, this assumes that they are correctly determined. One possible check is to measure the dependence of the dispersion on magnitude (Clarkson et al. 2012). For this test, we divide our data into radial bins to avoid the radial trend diluting a signal, and measure the dispersion in these bins for the brighter and fainter stars, using the same threshold magnitude in all of the bins (see Figure 22). Taking the average of these bins, the difference between the fainter and brighter stars is $\delta {\sigma }_{1{\rm{D}}}=0.007\pm 0.041$ mas years⁻¹. The error is the scatter divided by $\sqrt{{N}_{\mathrm{bins}}}$, which is consistent with Poisson errors. The consistent dispersions imply that our stars are bright and isolated enough that magnitude-dependent errors like photon or halo noise (Fritz et al. 2010a) do not notably contribute. Thus, Poisson errors are likely the most important error source of the dispersion and we use only them.

For deriving proper motions between 20'' and 45'', we use the images listed in Table 7. (The last image of this table is used only for deriving proper motions in the outer field; see Appendix A.4.)

Table 7.  Images Used for Derivation of Proper Motions in the Large and Outer Field

Telescope/Instrument	Epoch	R.A. $(^{\prime\prime} )$	decl. $(^{\prime\prime}$)
Gemini/Hokupa'a	2000 Jul 02	−13 to 31	−6 to 75
VLT/NACO	2004 May 06	−37 to 36	−38 to 35
VLT/NACO	2006 Apr 29	−20 to 22	−16 to 25
VLT/MAD	2008 Aug 21	−37 to 35	−48 to 27
VLT/NACO	2011 May16	−44 to 44	−44 to 44
VLT/NACO	2011 May 29	0 to 28	45 to 73

Note. The field of view of the first two data sets is not approximately rectangular.

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As our master image, we choose the image from 2011 May 16. This image is the largest in most directions and has, thanks to 04 DIMM seeing, the best resolution. The overlap between its 16 different pointings is too small to obtain a reliable distortion correction in the manner of Trippe et al. (2008). The NACO distortion is, however, stable within the instrument's interventions (see Fritz et al. 2010a). We therefore use the distortion solution for the distortion correction that is valid during this epoch. Apart from this higher-than-linear distortion, it is also necessary to align the scales and pointings of the images by linear transformations. For this, we also follow the procedure of Trippe et al. (2008).

We again use the full linear transformation (Formula 16). On this final distortion-corrected and aligned image, we search for stars and fit them with two-dimensional Gaussians. We thereby exclude stars with close neighbors to reduce the influence of neighboring seeing halos (Fritz et al. 2010a) on the position of the target stars. We then translate the pixel positions of the stars to arcseconds by using the known positions and motions of bright stars in the central 20'' using our extended field sample.

For the NACO images in the other epochs, we removed the distortion from all single pointing images with the distortion solution valid during this time. For the image from 2004 May 6, the overlap of the different pointings is too small for a reliable alignment of the different pointings. In contrast, the pointing of the images from  2006 April 29 overlap enough to apply a reliable alignment. Consequently, in most epochs we have many, mostly single pointing images of different parts of the GC. The images overlap partly. On all of the images, we identify 10 bright stars from the master image for preliminary alignment of the images. We then search these images around the expected star positions from 2011 within a radius of 3 pixels for the local maxima. When a maximum is fittable by a Gaussian similar to the PSF core of the image, we treat the star as identified on this single pointing image. We then use all of the stars identified on the single pointing image to obtain the cubic transformation to the master image. This transformation is defined in the following way:

$$\begin{eqnarray*}x^{\prime} & = & {a}_{0}+{a}_{1}\times x+{a}_{2}\times y+{a}_{3}\times {x}^{2}+{a}_{4}\times {xy}+{a}_{5}\times {y}^{2}\\ & & +{a}_{6}\times {x}^{3}+{a}_{7}\times {x}^{2}y+{a}_{8}\times {{xy}}^{2}+{a}_{9}\times {y}^{3}\end{eqnarray*}$$

$$\begin{eqnarray}y^{\prime} & = & {b}_{0}+{b}_{1}\times x+{b}_{2}\times y+{b}_{3}\times {x}^{2}+{b}_{4}\times {xy}+{b}_{5}\times {y}^{2}\\ & & +{b}_{6}\times {x}^{3}+{b}_{7}\times {x}^{2}y+{b}_{8}\times {{xy}}^{2}+{b}_{9}\times {y}^{3}.\end{eqnarray} \tag{ 17 }$$

The primary purpose of applying this cubic transformation is to correct unknown distortions along with other, mostly linear effects (Fritz et al. 2010a). The number of stars used for the transformation depends on the field of view of the detector and the density of well-detected stars. For most epochs, on average we have about 4500, and at least 1000, stars. For the single NACO pointings of 2004 May 6, there are on average about 836 stars, and at least 660 stars. Only for the smaller Gemini field of view do we need to use fewer stars. On these image, there are on average 644 stars, and on one only 58 stars. Thus, all of the images have enough stars to average out the influence of the intrinsic motion of the stars on the cubic transformation. We use the median of all of the detections in each epoch as the position of the star in that epoch.

To obtain the velocity errors, we start with the same position error for all of the epochs and one that is three times smaller for the master image because it is the best image and is mostly corrected for distortion. We rescale the errors such that the reduced ${\chi }^{2}$ has the expected value. The errors obtained in this way vary a great deal at each magnitude. In the magnitude bins, the error of the stars at the 75% quantile of the error distribution is 3.4 times larger than the error at the 25% quantile of the error distribution. A broad fit error distribution is also expected for the perfectly Gaussian distribution of position errors in our case of fitting only three to five data points. Thus, obtaining the errors by rescaling is not possible. Since a clear correlation of the errors with magnitude is visible (see Figure 23) we calculate the median error in magnitude bins and use it as the error for most of the stars. We use the rescaled error only if the rescaled error is more than four times larger than the median error. We exclude from our sample those stars that in at least one dimension have errors larger than 10 mas years⁻¹. When the error is so large, the velocity cannot be reliably determined. For the other stars we use the same error for both dimensions. The faintest stars have an error three times larger than bright stars (Figure 23). This behavior is expected, since many errors are more important for fainter stars, see, e.g., Fritz et al. (2010a). Due to saturation effects, the errors increase somewhat again for very bright stars. Another possible error source is that the remaining distortion could be larger in the outer parts of the field than in the center. To test this, we compare the errors inside and outside the central $r=20^{\prime\prime}$ box (see Figure 23). We find no major difference between the errors inside and outside this box: the errors inside have a median 23% bigger than the errors outside, possibly due to the higher source density there.

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In principle, it is possible to calculate velocities from only two epochs. In this case, however, it is not possible to calculate the errors from the fit to the data points. Thus, great care is necessary. To test the reliability of velocity measurements using only two images, in the central part of the field of view we compare robust dispersion measurements (the median deviation) using all epochs and using only the master epoch (2011) and a single other epoch. We find that the dispersions obtained from two epochs are compatible with the dispersion from most data pairings with the master epoch. The only exception is the 2008 MAD image. The dispersion from this image is about 16% greater than the dispersion when using all of the images. The reason for the greater dispersion is probably the small time baseline between 2008 and 2011, together with the fact that two different instruments (with partly unknown distortion) are used for this velocity measurement. Through comparison to the dispersion using all of the data, it follows that the velocity error is 40% of the dispersion for the 2008–2011 pair. If we could be sure that the error has this value, then we could use the dispersion after correcting for the biasing velocity error. However, the errors due to distortion, and maybe other factors, are probably not constant over the field of view. Even if the errors would increase only to 50%, this would enlarge the dispersion values by 8%. This error-induced uncertainty is much larger for this data pair than for the other data pairs. We therefore exclude those stars only detected in 2011 and 2008 from the analysis. The stars detected only in two other data sets (due to the covered fields they are mostly in the 2000 and 2004 data sets) show no higher dispersion. We include them in our analysis.

As in the extended field, we further test if the dispersion depends on the magnitude (see Figure 23). For this test, we exclude stars with only two measurements. For this comparison, subtracting the errors is more important than would be the case closer to Sgr A*, since the median error is 0.464 mas years⁻¹. For the subtraction, we use $1.2\times$ the median error, since this is between the median error (which is too low because larger errors contribute more to the overall error) and the average (which is affected by few unrealistically large errors).

The half with ${m}_{\mathrm{Ks}}\gt 15.3$ has, after velocity error subtraction by 0.076 ± 0.037 mas years⁻¹, larger errors than the brighter half. Thus, the dispersion my increase by 2σ with magnitude. This is barely significant. Therefore, we do not include a systematic dispersion error uncertainty for this data set. We then compare this data set with the extended field data set and exclude all sources within this inner field. In total, we have 3826 late-type stars in the large field.

Of the two epochs of this field (Gemini in 2000 and NACO in 2011), the NACO data are of higher quality, and so we use them as the master images. We remove distortions with the distortion solution. We use STARFINDER (Diolaiti et al. 2000) to obtain first estimates for the stellar positions with a single PSF extracted from the full field. In a second step, we fit these stars with two-dimensional Gaussians and keep only those stars which are well fit by a Gaussian. We also fit the same stars  on three other NACO images in other filters but with the same pointing. The position in 2011 is the average of the four filters and the error is the scatter. The error in this epoch is negligible compared to the error in the Gemini epoch. The Gemini data of 2000 consist of 12 reduced images covering 3 pointings. Thereby, one pointing creates the overlap between the other two pointings, thus allowing estimates for distortion effects. We use 10 stars on each Gemini image to establish a first linear transformation to the NACO data. We then search in a radius of three pixels around the expected star positions for maxima and fit them with two-dimensional Gaussian functions. All of the stars that are found to be offset by less than 3 pixels from the NACO epoch position are used for a full cubic transformation (Equation (17)) of each Gemini image to the NACO image. These transformations use on average 470 stars and at least 133 stars.

For the calculation of errors, we use stars that are on more than one pointing. First, we calculate the average position of the different images in one pointing. Second, we obtain the final position and its error by using the average and the scatter of the different pointings. Since errors obtained from the scatter of two or three images have a large scatter, we construct a magnitude-dependent error model. Therefore, we bin the stars by magnitude to obtain the median error. We interpolate these data by a curve that has three components: an error floor for the brightest magnitudes, photon noise at intermediate magnitudes, and sky/read-out noise at the faintest level (Fritz et al. 2010a). If a star has an error of more than three times the expected error, then this larger error is used. The floor of the error model results in a total velocity error of 0.090 mas years⁻¹. This is much less than the dispersion of 2 mas years⁻¹. Thus, magnitude-independent errors like distortion are not important in the outer field.

We now test down to which magnitude the dispersions are reliable. For this test, we split the stars into three magnitude bins (see Figure 24). In the brightest bins, we use bright stars with ${m}_{K}\lt 14.9$ for which the luminosity does not limit astrometric accuracy. The upper magnitude edge of the second bin we choose such that the dispersion in this bin is identical to the dispersion in the brightest bin. This results in ${m}_{K}=15.6$ and a dispersion difference of −0.087 ± 0.075 mas years⁻¹ between the median and the brightest bin. The fainter stars have a 0.864 ± 0.078 mas years⁻¹ larger dispersion than the brightest stars. The high dispersion in this bin is partly due to the difficulty of cleaning it of outliers. We exclude the faint stars from the analysis and henceforth use only the 633 stars in the two brighter bins. Therefore, the magnitude distribution has a sharp cutoff in that field, in contrast to the other proper motion fields (Figure 25). There, the selection is very complicated; it depends not only the magnitude of a star but also on the magnitude of the neighboring stars. However, the magnitude difference between the different proper motion fields is smaller than the magnitude difference between the proper motions sample and the radial velocity sample. Also, the magnitude difference between proper motions and radial velocities is not important (Section 3.2). The total dispersion of both dimensions together is ${\sigma }_{1{\rm{D}}}=1.919\pm 0.038$ mas years⁻¹.

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We extract the star spectra from the data cubes using on and off spaxels. For measuring the radial velocities from the SINFONI data, we use the strong and sharp CO band head features between 2.29 and 2.37 μm. We cross-correlate the spectra with a template and correct the radial velocities to the LSR. The statistical error on the radial velocity is obtained by the uncertainty of the best crossmatch in the correlation. This error is typically larger than 4 km s⁻¹. The median velocity error obtained in this way is 7 km s⁻¹. For a subset of more than 300 stars, we have available velocities extracted from two or more entirely independent different data sets, which allows us to check the velocity error. The median velocity difference is 8.4 km s⁻¹, which is consistent with our error estimate.

Compared to the total dispersion of ${\sigma }_{z}=102.2$ km s⁻¹, the discussed uncertainty of the velocities is small and negligible. However, this uncertainty does not include systematic problems due to calibration errors or template mismatch. A possibile way to check our velocity accuracy is through comparison with LSR. No relative motion is expected between the average motion of the solar neighborhood and the nuclear cluster in the radial direction. The determination of the LSR is difficult in the direction of Galactic rotation (Dehnen & Binney 1998; Schönrich et al. 2010; Bovy et al. 2012), but here only the radial component U matters. We use the LSR calculation of the ATCA array⁹ , which uses U = 10.25 km s⁻¹, consistent with the recent values of 10.5 km s⁻¹ (Bovy et al. 2012) and 11.1 ± 1.2 km s⁻¹ (Schönrich et al. 2010). The nuclear cluster is rotating in the Galactic plane (McGinn et al. 1989; Lindqvist et al. 1992a; Genzel et al. 1996; Trippe et al. 2008). Since our coverage is not homogeneous, we cancel the rotation by using stellar number count weighted bins in $| {l}^{*}|$ (see Figure 26). The mean radial velocity over all bins is 6.1 ± 3.8 km s⁻¹, which is consistent with 0. Reverting the argument, assuming that the calibration is correct, we can conclude that the nuclear cluster is moving less than 15 km s⁻¹ radially relative to the LSR.

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In conclusion, it seems likely that Poisson errors dominate the dispersion uncertainty for our radial velocity sample, and we only include those errors.

In the central 100'', we ignore the literature for single-star radial velocities (Sellgren et al. 1987; Rieke & Rieke 1988; Genzel et al. 1996; Figer et al. 2003) because our sample is much larger than all of the previous samples. (The exception is Trippe et al. (2008), but we incorporate all of their velocities corrected by a constant velocity to account for the different CO template used.) Although our sample has 2513 velocities, it only covers $R\lt 4$ pc. Since this is less than the size of the nuclear cluster, we scan the literature for old star velocities further out. The sample of Rieke & Rieke (1988) contains 11 CO-based velocities in this area. However, we do not use them due to the small number and also because IR spectroscopy at this time may have had difficulties in calibration as indicated by the rather different dynamics in Rieke & Rieke (1988) and McGinn et al. (1989; Section 5.2).

Instead, we use the maser-based velocities from Lindqvist et al. (1992b) and Deguchi et al. (2004). Lindqvist et al. (1992b) used the VLA for detecting OH masers in a blind Galactic Center survey. This survey covered the GC out to ≈3000'' from Sgr A*. Stars with $| v| \gt 217$ km s⁻¹ were not detectable due to the limited spectral range. Since only very few stars have a velocity close to this value, probably very few stars were missed. Lindqvist et al. (1992a) subdivided the masers into two classes according to their age (as estimated from the the expansion velocity v_exp of the masers), but found that both follow the same projected rotation curve with the same sign and orientation as the Galactic rotation. We thus use both populations.

Deguchi et al. (2004) targeted large-amplitude variables within 950'' from Sgr A* as SiO masers with the 45 m Nobeyama radio telescope. The beam of the observations was 40''. Their sample probably contains multiple identifications, as there are a suspiciously large number of close neighbors (R $\;\lt \;40^{\prime\prime}$) that have velocity differences of less than 5 km s⁻¹. We clean the sample and also remove stars already present in the Lindqvist et al. (1992b) sample. As a side product of this matching, we confirm that the typical velocity uncertainty is less than 3 km s⁻¹, as stated in Lindqvist et al. (1992b) and Deguchi et al. (2004), and is therefore irrelevant. Due to the big position errors for the masers, it is difficult to find the corresponding IR stars. We therefore exclude from the combined list the 11 stars that overlap spatially with the areas in which we obtained spectra, with the aim of avoiding using stars twice. Overall, we use 274 radial velocities outside the central field. After the exclusion of outliers (see Section 3.3), 261 stars remain in our final maser sample.

Young stars are predicted to be unrelaxed (Alexander 2005) in the nuclear cluster. For the really young stars, the early-type stars, the WR-, O-, and B-stars (Paumard et al. 2006), and the red supergiant IRS7, the unrelaxed state is confirmed by the observed dynamics (Genzel et al. 2003; Bartko et al. 2009, 2010; Gillessen et al. 2009; Madigan et al. 2014). Surprisingly, the intermediate-age stars, with ages of about 20–200 Myr that constitute about 10% of all late-type stars (Pfuhl et al. 2011), share in dynamics and radial distribution the properties of the majority of the stars, which are more than 5 Gyrs old. Therefore, we do not exclude them. Hence, our selection criterion is a simple spectroscopic one: we exclude the early-type stars and IRS7 and use all late-type stars.

Since the ratio of young to old stars is a strong function of radius (Krabbe et al. 1991; Buchholz et al. 2009; Bartko et al. 2010; Do et al. 2013a), our selection criterion for old stars is radius dependent. In the central arcsecond, we have spectroscopically identified more early-type stars than late-type stars (see also Figure 1 of Gillessen et al. 2009). Since the KLF slope of the young stars in the central arcsecond (Bartko et al. 2010) is identical to that of the late-type stars (Buchholz et al. 2009; Pfuhl et al. 2011), the majority of the stars without spectral identification are probably young there. Outside the central arcsecond, the fraction of young stars decreases but remains high for $R\leqslant 2^{\prime\prime}$ (Buchholz et al. 2009; Do et al. 2009; Bartko et al. 2010). We therefore include for $R\leqslant 2^{\prime\prime}$ only those stars in our sample that are spectroscopically confirmed late-type stars. Therefore, as always for star classifications, we use our SINFONI spectra for spectral classifications.

Outside of $R\geqslant 2^{\prime\prime}$, there are fewer early-type than late-type stars (Buchholz et al. 2009; Do et al. 2009; Bartko et al. 2010; Pfuhl et al. 2011). In this radial range, we include all stars for which we did not record an early-type spectrum. Since we do not have spectra for all of the stars, early-type stars can be in the sample. Our selection is not radius independent because we have better spectral coverage close to the center (Bartko et al. 2010): at 2'', we have spectral identifications for about 50% of the stars, for which we have dynamics. This fraction decreases to 13% at 20''. Furthermore, our spectroscopic completeness decreases toward fainter stars. In conjunction with the top-heavy IMF for young stars in this radial range (Bartko et al. 2010; Lu et al. 2013), this means that the pollution of our sample with young stars is reduced compared to what would follow from the overall fractions. Quantitatively, we calculate the number of young stars that we still expect in our sample by a simple completeness correction, multiplying in each radius and magnitude bin the number of unidentified stars with the locally measured early-type fraction. We determine that we include about 140 early-type stars in the extended sample of 5864 stars. The total pollution fraction of $\approx 2.4$% increases only slightly to $\approx 3.7$% in the inner radial half. For $r\gt 20^{\prime\prime}$, we even find only six early-type stars, which is less than 1% of all stars with spectra. Thus, we are confident that our dispersions (or higher moments) are not biased in a significant way.

We find in our spectra that there are very few early-type stars outside of 20''. Surprisingly, Nishiyama & Schödel (2013) found from narrow-band imaging a rather high early-type candidate fraction of ≈7% for bright (m${}_{\mathrm{Ks}}\lt 12.25$) stars. At $R\gt 0.5$ pc, they found 35 stars. We have spectra for 24 of these stars, and see clear late-type signatures in 22 of them (see Figure 27). Only two of our spectra show early-type signatures. Thus, the contamination fraction is around 90%, and not 20% as claimed by Nishiyama & Schödel (2013). The main problem in their analysis is probably that they misidentify stars from the population of warm giants (Pfuhl et al. 2011).

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Foreground and background stars should obviously not be used to constrain the GC potential. Similar to Buchholz et al. (2009), we use color to exclude foreground stars. The selection criterion for foreground stars is ${\rm{H}}-\mathrm{Ks}\lt 1.3$, except in regions with unusual extinction where we use a local value. In total, we exclude 97 stars as foreground stars. For many stars no H-band magnitude is available. These stars are included in our sample, since from the more than 7000 stars with H-band information we can estimate that only $\approx 1$% of all stars belong to the foreground. Since foreground stars are not clustered, this fraction is rather constant over the field of view. Color selection does not work for the background. Most of the red stars in the GC are red due to intrinsic redness as shown by a detailed spectroscopic analysis, e.g., Fritz et al. (2010b). Due to the faint magnitudes of background stars, this is even less of a problem than the foreground.

Here, we describe how we reject some high-velocity stars from our sample. Only 63 of over 10,000 stars are excluded.

Mainly due to our mostly automatic procedure for finding and fitting stars it is possibly that extreme proper motions are not real. This problem does not affect the motions in the central and extended field. In these two fields all target stars selected on a single image have a reliable velocity due to the high number of epochs used. In contrast outside of 20'' we have at most five epochs for motions and the velocities might be affected by outliers. The velocity histogram for stars at $R\gt 20^{\prime\prime}$ shows an excess of stars at v_2D above 0.8 v_esc compared to the velocity distribution obtained from stars between 7'' and 20''. In order to avoid such a bias to high velocities, we look on the images of each star and exclude stars which are not well identified, also considering v_2D. This procedure excludes 30 stars in the extended field and makes the shape of the velocity histogram similar to the inner sample. Using the same procedure we exclude 20 stars in the outer field. Since only two epochs are available the fraction of excluded stars is with $\approx 3$% higher. This procedure reduces the dispersion significantly by 0.147 ± 0.020 mas years⁻¹ in the large field and by 0.172 ± 0.041 mas years⁻¹ in the outer field. Afterward, the velocity histogram for the outer stars looks more similar to the one in the radial range from 7'' to 20'' (Section 5.5). Since it is possible that we exclude somewhat too few or too many stars in this exclusion procedure, the tails of the velocity distribution are still unreliable. Thus, we do not analyze the wings of the velocity distribution in the large and outer fields.

In the maser sample, there are some surprisingly fast stars (see Lindqvist et al. 1992b; Deguchi et al. 2004, and Figure 17. After spatial binning of these stars (Section 5.3), we inspect the binwise velocity histograms and exclude stars that are apparent outliers. We exclude 13 of 274 stars. The scatter in mean velocity and dispersion of the bins is notably reduced by this procedure. Since with single, and partly even double, epoch radio observations a high fluke velocity is difficult to produce, these stars are probably truly fast stars from another population of stars.

For measuring stars from the VISTA data, we extract point sources from the central ${R}_{\mathrm{box}}=1000^{\prime\prime}$ in H and Ks using STARFINDER (Diolaiti et al. 2000). We divide the field of view into nine subimages: a central image and a ring of eight images around the central image. We use standard parameters of STARFINDER and PSF with a size of 15 pixels. Since we do not use the faintest stars, the exact parameters do not have a relevant influence on the result. The magnitudes obtained are adjusted to the VISTA catalog by matching bright, unsaturated stars. Essentially all stars outside the central 20'' are old stars (see Section 3.3). Thus, we do not need to exclude young field stars from the VISTA data, except the Arches and Quintuplet clusters.

For source counts completeness is a concern. To measure the source confusion, we insert in the central Ks image (R${}_{\mathrm{box}}=344^{\prime\prime}$) artificial stars, using the PSF extracted with STARFINDER. The PSFs are separated by 20 pixels to avoid artificial confusion. In this way, we create six images with artificial stars with $11\;\lt \;$ Ks $\;\lt \;16$ in steps of 1 mag. to cover all of the unsaturated magnitudes for which significant numbers are detected. In all of these images, we use STARFINDER to detect the inserted stars in the same way as for the original image. The resulting completeness maps have a resolution of 20 pixels. Locally, this may not be sufficient for a good map. However, since we are mainly interested in the global radial and azimuthal profiles outside the very center, the number of artificial stars used (10,201) is sufficient for our purposes. Outside the central field, the confusion is smaller. In the ring fields, we measure confusion in the same way as for Ks = 14 and Ks = 16, and then extrapolate to the other magnitudes using the multi-magnitude completeness curve in the center which best matches the completeness at Ks = 14 and Ks = 16.

We correct for extinction using stellar colors. The H-Ks color is nearly independent of stellar type (Cox 2000): H-Ks = 0.29 for ${M}_{\mathrm{Ks}}=-6.26$ giants and H-Ks = 0.15 for ${M}_{\mathrm{Ks}}=-3.46$ giants. We use a color of H-Ks = 0.2 for all stars. For the extinction correction, we use the extinction law toward the GC from Fritz et al. (2011), which implies ${A}_{\mathrm{Ks}}=({A}_{{\rm{H}}}-{A}_{\mathrm{Ks}})/0.753$ for VISTA filters. If possible, we use the ${\rm{H}}-\mathrm{Ks}$ of each star for extinction correction. In this way, we can better account for extinction variations in the line of sight than if we used the mean extinction. We use this method for Ks-sources with an H counterpart within two pixels and no other source within ${R}_{\mathrm{closest}}+1$ pixels. The vast majority of all bright Ks sources have such an H counterpart. For the other sources, we use the local extinction distribution of the matched sources in bins ${R}_{\mathrm{box}}=12\_\_AMP\_\_farcs;7$. The primary result of this procedure is stellar density maps with pixels of ${R}_{\mathrm{box}}=12\_\_AMP\_\_farcs;7$. We made these maps for different extinction-corrected magnitudes. The brightest magnitude not significantly affected by saturation is, after extinction correction, ${m}_{\mathrm{Ks},\mathrm{excor}}=9$. For our final density map, we use stars fainter than this magnitude and brighter than ${m}_{\mathrm{Ks},\mathrm{excor}}=10.5$ to exclude magnitudes that are severely incomplete in the center. Using this faint magnitude cut, the completeness is 52% at $r\approx 25^{\prime\prime}$ for the VISTA data. The use of stars fainter than ${m}_{\mathrm{Ks},\mathrm{excor}}=9$ also has the advantage of avoiding stars brighter than the tip of the giant branch which are partly younger than most old stars in the GC. These younger and brighter stars are more concentrated in Sgr A* on a large scale (Catchpole et al. 1990).

The result of this procedure is a star density map (Figure 28). Symmetry relative to the Galactic plane and the concentration toward the GC is visible on this map. A more complicated symmetry is also physically unlikely because the Galactic center is so small compared to its distance that any bar like structure would appear symmetric along its axis, independent of the orientation of the bar to the line of sight. Some Ks-dark clouds are still visible on the map. We mask out map pixels which show too low a density compared to their neighbors and others pixels at the same $| {b}^{*}|$ and $| {l}^{*}|$. In areas where completeness is not an issue at fainter magnitudes, we also base the masking on density maps for fainter stars to reduce the influence of small number statistics. At some radii, the masked areas are mostly at ${l}^{*}\approx 0$. At these radii, a calculation of the radial profile from the unmasked area would result in a density biased to ${b}^{*}\approx 0$. To avoid this, in our final map we replace the counts in the masked pixels by the average of the unmasked pixels at the same $| {b}^{*}|$ and $| {l}^{*}|$ (see Figure 28). To obtain a radial profile at higher resolution, we also binned the stars more finely by a factor of five. Therefore, we still use the larger binning for masking and completeness correction since on smaller areas Poisson noise dominates.

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To obtain error estimates for the general radial profile and the profiles in l* and b*, we first calculate the lowest possible error from Poisson statistics. We obtain another error estimate by calculating the difference between the upper and lower halves and the left and right halves of the maps, respectively. This second estimate is typically 4.4%, is larger than the first, and thus is used for most radial bins. With these, the reduced ${\chi }^{2}$ of log-polynomials of fourth degree is smaller than 1. This shows that the errors of neighboring bins are correlated. To estimate the flattening, we measure the density along the major and minor axes using data with less than 38'' separation from these axes. (Within 66'', where this method obviously smooths out the flattening, we do not use the VISTA data to estimate it.) For these profiles, we obtain the non-Poisson errors by comparing the two sides and obtain an error of 12.5% which we scale up by a factor $\sqrt{2}$ for points that have data only on one side. At large separations, polynomial fits yield a ${\chi }^{2}/{\rm{dof}}$ larger than 1. We therefore scale up the errors there to obtain ${\chi }^{2}/{\rm{dof}}=\;1$.

For WFC3/IR data we use the final Multidrizzle product images in M127 and M153. We use STARFINDER (Diolaiti et al. 2000) with standard parameters and a PSF box 2'' in diameter. The fluxes obtained with this PSF are converted to magnitudes using the zeropoints for a 04 PSF and an infinitely large PSF. We convert the pixel coordinates of both star lists in Galactic center coordinates by using bright stars that are detected in WFC3/IR data and in the NACO data. We match sources in the two filters and use only those sources which are detected in both filters. To be precise, this means that the closest neighbor in M127 is within 1 pixel and the second closest is at least 2.3 pixel further away. We use the colors in the following way to apply the extinction correction on a per-star basis: A${}_{{\rm{M}}153}=2.12\;\cdot ({m}_{{\rm{M}}127}-{m}_{{\rm{M}}153}-0.374)$. The factor is obtained from the −2.11 power-law extinction in Fritz et al. (2011). The giant color 0.374 is obtained from observed giant spectra, published in Rayner et al. (2009), approximately K4III.

From these extinction-corrected stars, we construct stellar density maps in galactic coordinates using 2'' pixels. We use stars brighter than ${m}_{{\rm{M}}153\mathrm{excor}}=12.5$, which also have ${m}_{{\rm{M}}127}-{m}_{{\rm{M}}153}\gt 2$ in order to exclude foreground stars. With this brightness cut, more than 87% of the unextincted stars, which typically have ${m}_{{\rm{M}}153,\mathrm{excor}}\lt 12.5$, have matches in M127. In the very center ($R\lt \;8^{\prime\prime}$), with WFC3 we recover 90% of the ${m}_{{\rm{M}}153\mathrm{excor}}\lt 12.5$ stars found in the higher-resolution NACO data. This completeness increases to 98% at 20''. Since we use WFC3 data only outside of 20'', we do not need to correct for completeness. Saturation is no worry, since only very few stars are saturated and many more fainter stars are included in the sample. As in the VISTA data, NIR dark clouds can locally block all GC light (see Figure 29). We use a density map of all stars with ${m}_{{\rm{M}}153\mathrm{excor}}\lt 15$ to identify these clouds and mask them out. This second density map saturates in the center due to crowding but is superior in areas of low star density, such as these dark clouds.

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Early-types are not important in the region of the map we use ($R\;\gt \;20^{\prime\prime}$; see Section 3.3). For visualization, as in the case of the VISTA data, we replace the masked-out areas with their symmetric partners, see Figure 29. There it can be seen that the cluster is, after extinction correction, much less flattened than at first sight. To quantify the weak flattening, we measure the stellar density separately in l* and b* using stars in halves with $| {l}^{*}| \gt | {b}^{*}|$ and $| {l}^{*}| \lt | {b}^{*}|$ respectively. As the error, we use the larger of the following: either Poisson statistics or the average difference between the density at $-{l}^{*}$ and $+{l}^{*}$ and $-{b}^{*}$ and $+{b}^{*}$ which is 7.8%. We obtain the full radial profile from the average of the two profiles in l* and b*.

In the center, we use our NACO data. We extract stars from an NACO image with STARFINDER (Diolaiti et al. 2000). We calibrate the magnitudes of these stars locally on the source list of (Schödel et al. 2010). We again correct for extinction. In the NACO field, the use of an extinction map is superior to a star-by-star color since the extinction does not vary much in the line of sight. Thus, a map that uses the local median reduces the noise. We use the extinction map of Schödel et al. (2010) scaled by a factor of 0.976 to align the map with the extinction of Fritz et al. (2011). We select stars with ${m}_{\mathrm{Ks},\mathrm{excor}}=12.65$. With this magnitude cut, we still have some stars in the central arcsecond and are, at the same time, complete. In imaging, we are essentially complete until this magnitude, we thus do not apply a completeness correction. Dark clouds are no issue in the central 20'', we therefore do not apply masks. In the NACO data, the exclusion of early-type stars is important; see Appendix C.1 for the detailed procedure. For the density profile, we use from 0 to 20'' the late-type fraction of the spectroscopic type stars with ${m}_{\mathrm{Ks},\mathrm{excor}}=12.65$ to convert the density profile of all stars to the late-type density profile. To obtain estimates for the flattening in the center, we also separately calculate the late-type stellar density in the same two halves of the data as for the WFC3/IR data. The three profiles from different instruments are aligned at their transition radial ranges.

We now describe how we obtain the extinction-corrected flux of the GC. We use VISTA and NACO data.

In case of the VISTA data, we subtract the median counts toward some dark clouds in the H and Ks bands as sky. We bin the flux in H and Ks, again in pixels of ${R}_{\mathrm{box}}=12\_\_AMP\_\_farcs;7$. This flux is then used for extinction correction, assuming a somewhat bluer color of ${\rm{H}}-\mathrm{Ks}=0.15$ since bright, cool giants are less important in total flux than in source counts of bright stars, especially when, as in our case, the brightest stars are saturated. In the extinction-corrected flux map, the dark clouds are better visible than in the stellar density map, since few bright blue foreground stars can bias the extinction estimate toward values that are too low. We therefore construct a new mask for the flux and apply it to the data. To obtain a profile with finer radial sampling, we bin the flux finer but still use the same extinction and mask maps. The final profile is similar but not identical to the stellar density profile. In the very center of the VISTA image, saturation affects many sources.

We use a flux calibrated NACO image, from which the sky is already subtracted, and the extinction map described in Appendix D.3 above to obtain the light profile. In the center, a few bright, young stars dominate the total flux. We find in the star list used in Appendix D.3 the spectroscopic young stars (early-type stars and the young red supergiant IRS7) and subtract their total flux. Their flux is only relevant in the inner 10''. Due to the top-heavy IMF (Bartko et al. 2010; Lu et al. 2013), the fact that we do not have spectra for all stars has a negligible influence on the total flux.

Finally, we use the transition region around 20'' to align the VISTA flux to the NACO flux. We estimate the errors on the resulting profile by fitting step-wise, low-order log-polynomials to the data and setting ${\chi }^{2}/{\rm{dof}}=\;1$. As expected for the noise, which is dominated by a few bright stars, the error decreases with radius.

For the binning of the density in $| {l}^{*}|$ and $| {b}^{*}|$, we use two different schemes dependent on the radius (see Figure 30).

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For the NACO and WFC3 data, the number counts in $| {l}^{*}|$ and $| {b}^{*}|$ are measured using the angles which are closer to $| {l}^{*}|$ and $| {b}^{*}|$, respectively. Using this method, we use all of the available information and as such are able to measure the small flattening at $R\lt 68^{\prime\prime}$. Further out, the signal is high enough that the flattening can be measured directly. There, for the density along the major and minor axis, we use only the data that is closer than 39'' to the respective axis.

Since our data are of too low S/N, it is not possible to measure the flattening directly by fitting constant density ellipses to it. For our measure, we use Galfit in bins. The bins are elliptical, which we adapt iteratively to the flattening in the bins. We fit a single Sérsic to the data in each bin. The flattening is not sensitive to the radial (Sérsic or other) functional form. We use the usual mask and fix the flattening to align with the Galactic plane. The innermost bin covers the NACO and WFC3 area: $| {l}^{*}| \lt 68^{\prime\prime}$. The fit errors, the dependence on the used functional form, and the dependence on the used $| {b}^{*}|$ range together add to the total errors.

The axis ratio is 0.80 ± 0.04 within $| {l}^{*}| \lt 68^{\prime\prime}$. In this radial range the ratio is rather constant with a slightly higher value around 20'' than elsewhere. The flattening increases outside of 68''. Within 68'' we obtain the flattening from from NACO and WFC3/IR data. Outside of this range we use VISTA data and there the flattening increases out to the end of our field of view. Due to this change of the data source where the flattening increases it is natural to ask the question if the flattening measured is affected by the data source used. We concentrate thereby on the WFC3/IR data range since that radial range is the most important range. For both transition regions of WFC3/IR, the inner (to NACO) and the outer (to VISTA) we obtain consistent flattenings in the different data sets. The signal of the VISTA data is too low that this comparison is constraining.

We test now whether one of the steps, which we use for WFC3/IR data, inhibits a known systematic uncertainty, which is relevant for flattening compared to the statistical uncertainty. A priori it may be worrying, that the WFC3/IR data is obtained at a smaller wavelength (1.53 μm) compared to the NACO and VISTA data (2.16 μm). The stellar color varies more at wavelength shortward of the Ks band. However, compared to other effects like extinction the importance of stellar color does not increase and can change q at most by 0.005.

At 1.53 μm, the extinction is stronger than in the Ks band by a factor 2.05. Extinction can have two different effects on stars. First, it can be weak, such that it can be corrected by two-color information using a known extinction law. Second, the extinction can be so strong that the stars are invisible in all our bands. The slope uncertainty of 0.06 in the extinction power law (Fritz et al. 2011) propagates to a systematic error of 0.005 in q. As an independent test of the extinction law, we use the red clump magnitude in our data like Schödel et al. (2010). We obtain an extinction law steeper by 0.12. A steeper extinction law decreases the flattening for extinction distribution. The change in q is however only 0.01, for the observed in the extinction law.

As another test whether extinction issues are important we analyze H/Ks data, namely, good seeing (04 FWHM) ISAAC data. Figure 5 of Schödel et al. (2007) is based on such data. No clear elongation in the Galactic plane is visible after extinction correction, as also stated in their text, confirming our WFC3/IR result. To get a more quantitative estimate we also analyze ISAAC images ourselves. We use therefore mainly newer ISAAC images, which were also used by Nishiyama & Schödel (2013). Precisely we use data in the filters IB171 and IB225, which are intermediate band filters close but not exactly in the middle of the H and Ks bands. We then proceed as in case of the other data to obtain a star based density map corrected from effects like extinction. We obtain from these data $q\approx 0.83$ consistent with the WFC3/IR data. Finally, we do not use these data because the ISAAC detector shows at GC fluxes a relevant change in the bias (Nishiyama & Schödel 2013) which is difficult to correct.

The blocking effect of extinction is more difficult to quantize. When we vary the blocked area by a reasonable amount, estimated per eye, q can be changed at most by 0.02. There are two possibilities how blocking dust clouds in the Galactic plane can cause relevant changes in the flattening without being detectable in the star density map. In the first case, large dark clouds are slightly behind the GC, and block all light from behind, but not from front. In that cause the clouds decrease the flattening slightly, but do not cause obvious dark patches. In the second case, the clouds are very small clumps undetectable at the resolution of our star density maps. No special location in the line of sight is required in this case. Both solutions require too much designed structure to be likely. The first solution would also result in particular dynamics which are not evident in our data. The second solution requires very dense rather small clumps: for example if the clouds are at the upper limit of possible diameter of $2^{\prime\prime} =0.08$ pc and have an extinction of four times the usual one, that are ${A}_{\mathrm{Ks}}=10$, a density of 10⁶ cm⁻² follows from 1 ${A}_{\mathrm{Ks}}\approx 3\times {10}^{22}\;{\mathrm{cm}}^{-2}\;$ N_H (Fritz et al. 2011). Such dust clouds should also be obvious in the far-IR/sub-millimeter data.

Rotation and flattening of the GC cluster are expected to occur in the direction along the Galactic plane, and hence we choose proper motion bins reflecting that. We use two-dimensional circular coordinates $(R,\phi )$, where R is the distance from the center and ϕ is the absolute value of the smallest angle between the Galactic plane and the respective star, thus $0^\circ \leqslant \phi \leqslant 90^\circ$. This definition of ϕ uses the symmetry of the edge-on system in which the dispersion can only vary by R and ϕ. Our bins contain a relatively similar number of stars (Figure 31), between 123 and 325 with an average of 198 stars outside of 12''. Inside of that, we have fewer stars, but we need higher resolution in order to resolve the increased velocity dispersion caused by the SMBH. The innermost bin with $R\lt 0\_\_AMP\_\_farcs;5$ contains 7 stars. Between 12'' and 32'' we choose four azimuthal bins to be able to measure robustly azimuthal variations, for $R\lt 5^{\prime\prime}$ we do not bin the stars azimuthally.

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E.1.1. Radial Velocity Bins

The rotation of the cluster causes a change of the mean radial velocity with ${l}^{*}$ (McGinn et al. 1989; Lindqvist et al. 1992a; Genzel et al. 2000) from negative velocities at negative ${l}^{*}$ to positive values for positive ${l}^{*}$. Our test for the rotation axis (Section 5.2) shows that the radial velocity does not depend much on the sign of ${b}^{*}$ as expected in an edge-on rotating system. Hence, a natural choice for the binning coordinates is (l${}^{*}$, $| {b}^{*}|$). We again choose our bins such that the bins are relatively evenly populated with stars, but the overall number of stars is smaller than for the proper motions. Outside the central 4'', there are on average 51 stars per bin. In the central 4'', we do not use the sign of ${l}^{*}$ since the rotation there is of minor importance compared to the large radial velocity dispersion gradient. We choose quadrangular rings as bins to transit gap free to the rectangular bins further out. The innermost bin includes all of the stars with $| {l}^{*}| \lt 0\_\_AMP\_\_farcs;5$ and $| {b}^{*}| \lt 0\_\_AMP\_\_farcs;5$ (see Figure 32). Further out, in the regime of the maser stars we adopted from the literature, we use more complex bins to achieve comparable star numbers per bin (Figure 32).

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