MONITORING STELLAR ORBITS AROUND THE MASSIVE BLACK HOLE IN THE GALACTIC CENTER

Only star images that are not visually distorted (e.g., due to a confusion event) were used from the SHARP data.

• 1.  
  Reference stars. We obtained the reference stars' positions from the four single-pointing maps from each epoch. Due to the limited field of view in each frame, only a subset of the reference stars is present.
• 2.  
  Brighter S-stars. For the brighter S-stars (e.g., S2, S1, S8, S10, S30, S35) typically all four different pointing positions could be used. The astrometric position of each star was determined from the corresponding four pixel positions using the astrometric average position (see Section 3.4.3).
• 3.  
  Fainter S-stars. In order to detect faint S-stars, we used the fifth co-added map which can be trusted astrometrically only for the innermost arcsecond. The limiting magnitude for a nonconfused source was typically m_K ≈ 15.8. We determined the pixel positions of the weaker S-stars as well as the ones of the brighter S-stars. The latter served as a reference for relating the fainter stars to the astrometric coordinate system (see Section 3.4.3).

Since we had two different deconvolutions at hand, we extracted pixel positions from both sets of images. Thus, up to eight (= two deconvolutions × four pointings) pixel positions were obtained per star and epoch.

For the NACO data, we used both the 27 mas pixel⁻¹ data and the 13 mas pixel⁻¹ data.

• 1.  
  SiO maser stars. Positions for the SiO maser stars were obtained by Gaussian fits to the stars' images in the 27 mas pixel⁻¹ mosaics. The SiO maser stars were unconfused in all mosaics.
• 2.  
  Reference stars. The positions of the reference stars were measured both on the 27mas pixel⁻¹ mosaics and on the 13 mas pixel⁻¹ maps (Table 1), since they serve as cross-calibration between the two sets. They were selected to be unconfused; thus, essentially it was possible to use all reference stars visible on any given frame.
• 3.  
  S-stars. For isolated S-stars, the positions were obtained from a simple Gaussian fit to the manually identified stars in the maps. Due to the higher sampling rate with NACO, confusion events can be tracked much better in the AO data than in the SHARP data. Therefore, it was reasonable to also measure positions when stars are partly overlapping. In such a case, a simultaneous, multiple Gaussian fit to the individual peaks was used, resulting of course in larger statistical uncertainties of the obtained positions.

Using the results from Reid et al. (2007), we calculated the expected radio maser positions for the given observation epochs. The different maser images contained between seven and nine SiO masers of which we used six to eight, since we excluded IRS7 due to its brightness of m_K ≈ 6.5. By allowing for a linear transformation of type $\vec{x} = \vec{x_0}+M.\vec{p}$ between the astrometric positions $\vec{x}$ and the pixel positions $\vec{p}$ in the respective image, we determined a transformation by which any detector position can be converted into astrometric coordinates. Note that the use of a linear transformation is justified since the IR images were distortion-corrected mosaics. The rms of the one-dimensional residuals of the SiO masers (thus applying the transformation to the SiO masers' pixel positions and comparing the result with the expected radio positions) was 2.28 mas. Correspondingly, we expect that from our 11 images a coordinate system can be defined to at most an one-dimensional accuracy of $2.28/\sqrt{11}$ mas ≈0.7 mas if the measurement errors from the 11 images are uncorrelated. The transformation was applied to the sample of reference stars in each image. We then fitted the resulting astrometric positions of the reference stars as a function of time with linear functions. From these linear fits, we obtained residuals, allowing obvious outliers to be identified and rejecting them. We excluded stars that had a residual different from the median residual by twice that value, if the deviation was larger than 2 mas. That excluded between 0 and at most 10 of the 91 reference stars for the various mosaics, the reason for these outliers being confusion that in some mosaics affects the fainter reference stars due to the varying image quality. After this moderate data cleaning, we repeated the fitting. The mean rms of the one-dimensional residuals per image had then a value of 1.45 mas.

The next step of refinement was to compare all measured positions in one mosaic with the positions expected from the fits, effectively checking how well a given image fits to the other 10 images. A visual inspection of maps of residual vectors showed that the residuals are not randomly distributed, but unveiled some systematic shift and rotation for each image. Since each image is compared with 10 other images, any systematic problem in the given image is most likely to come from that image and not from a combined effect of the others. Indeed, the interpretation of the observed systematic effect is straightforward, it means that each individual mosaic is not registered perfectly with respect to the sample average, i.e., the transformation for the respective image is slightly wrong. This systematic error is naturally explained by measurement errors of the positions of the SiO maser stars in the respective image. Such an error translates into an error of the parameters of the linear transformation used to tie the astrometric frame to the pixel positions in the mosaic and shows up as a systematic effect in the residuals of the independent set of reference stars. Thus, we were able to determine better transformation parameters by adding to the original linear transformation, the linear transformation that minimizes the residuals of the reference star sample, yielding a corrected linear transformation. We applied it to the data and obtained the final linear motion models for the reference stars. The rms of the one-dimensional residuals now was 0.55 mas. This step changed the position of the origin by (Δα, Δδ) = (−0.01,  0.05) mas and the velocity of the system by less than 4 μas yr⁻¹; these quantities being the mean differences of the respective quantities for the reference stars before and after the refinement. Hence, the refinement effectively did not change the coordinate system calibration. We call the coordinate system so defined the "maser system" in the following.

The position of the origin of the maser system and its velocity are uncertain due to two effects: (1) the nonzero errors of the SiO maser stars' radio positions and velocities and (2) the IR positions of the SiO maser stars show some residuals to the best-fitting linear motion, indicative of residual image distortions and of measurement errors in the pixel positions in the IR images. The propagation of the statistical errors into the definition of the coordinate system was addressed using a Monte Carlo technique. We varied the input to the transformations according to the measured errors and residuals. We created 10⁵ realizations of transformations, assuming a Gaussian distribution of the simulated values around the original values. The standard deviation of the positions obtained for Sgr A* estimates the positional uncertainty of the maser system under the assumption of uncorrelated measurement errors. We obtained (Δα, Δδ) = (0.46, 0.77) mas. Similarly, the standard deviation of the velocities obtained for Sgr A* estimates the uncertainty of the maser system's velocity under the same assumption. We obtained (Δv_α, Δv_δ) = (0.29, 0.55) mas yr⁻¹.

However, in our data the assumption that the errors from the 11 maser images are uncorrelated is not fulfilled. We rather observe a typical residual per SiO maser star for all epochs when comparing the transformed, measured positions with the predicted radio positions. Possible reasons are as follows. First, the linear motion models obtained for the SiO masers could be inaccurate due to some unknown some unknown systematic problem of the radio positions. Second, the radio positions could not be applicable to the IR positions, for instance if the maser emission would originate from far away of the stellar surface. Third, the correlation could arise due to some unaccounted systematics in the IR frames, such as uncorrected distortion. We fitted the residuals of each star with linear functions and obtained in that way estimates for the mean-position and mean-velocity uncertainties for each star. Then we calculated the mean deviation (over the SiO maser stars which are <15'' away from Sgr A*) of these linear motion model parameters as estimates for the positional and velocity uncertainty of the maser system given the correlations in our data. With our initial transformation, we obtained (Δα, Δδ) = (0.92 ± 0.42,  2.22 ± 0.43) mas and (Δv_α, Δv_δ) = (0.41 ± 0.24,  0.29 ± 0.24) mas yr⁻¹. After the refinement, we got (Δα, Δδ) = (0.95 ± 0.73,  2.35 ± 0.58) mas and (Δv_α, Δv_δ) = (0.38 ± 0.41,  0.28 ± 0.33) mas yr⁻¹. Finally, we conservatively adopt for the uncertainties of the maser system (Δα, Δδ) = (1.0,  2.5) mas and Δv_α = Δv_δ = 0.5 mas yr⁻¹. The positional uncertainty is considerably larger than what one would have obtained for uncorrelated residuals.

While the maser system is a direct cross-calibration of maser and reference stars, the resulting velocities of the reference stars are directly sensitive to errors in the velocities of the SiO maser stars, both in the radio data and the NIR 27 mas pixel⁻¹ mosaics.

The sensitivity of the reference star velocities to errors in the SiO maser velocities can be avoided by an additional step of cross-calibration. We can measure the positions of the reference stars in all maser images with respect to a much larger sample of stars in these images. This cluster of stars is assumed to be nonrotating and not moving with respect to Sgr A*. The cluster is tied to the astrometric frame for just one epoch, as given by the radio positions of the SiO masers, which can be calculated for the chosen epoch from Reid et al. (2007). For all other epochs it is assumed that the mean cluster is stationary in time. Hence, the velocity calibration relies on the statistical argument that for a sufficiently large sample of cluster stars the mean velocity of the cluster is expected to become very small. For a typical velocity of v for a cluster star and N stars, the error of this mean should be of order $v/\sqrt{N}$.

Effectively, the few maser stars are only used once in this scheme. Any error in their radio positions, radio velocities, or NIR detector positions will therefore translate into a positional offset, but not into a systematic velocity of the coordinate system. The latter is instead connected to the validity of the assumption that the cluster mean is stationary. In order to ensure the best estimate of the velocity calibration, we adopted the fol- lowing procedure.

• 1.  
  We selected the maser mosaic from 2005 May 12, which was chosen since it is of good quality and roughly corresponds to the middle of the range in time covered with NACO. Building upon the work done by Trippe et al. (2008), we selected an ensemble of stars in that mosaic of which the positions can be measured with high reliability. Take all stars that have a peak flux of more than 25 counts which at the given noise level of 1.9 counts selects high-significance stars. In a second step, many stars get excluded again. All stars with more than 700 counts (they could be saturated in other frames with longer single-detector integration times) and all stars that have a potential source (peak with five counts) within 10 pixels. Furthermore a Gaussian fit was required to yield a FWHM<0.05 pixel and the fitted position must coincide with the position obtained using DAOPHOT FIND. This yielded a sample of 433 stars. We determined the astrometric positions at the given epoch of the 433 stars by means of a linear transformation that was determined from the eight maser stars.
• 2.  
  We then determined preliminary astrometric positions for a much larger sample of 6037 stars in all 11 mosaics, by tying their pixel positions at all 11 epochs to the astrometric positions of the 433 stars at the reference epoch with a linear transformation. Note that not taking into account proper motions in that step makes the velocity calibration independent from the radio measurements. The error due to the omission of the proper motions is minimized by using 433 stars instead of few masers for the cross-calibration.
• 3.  
  We fitted linear motion models to all 6037 stars. After the fit, we determined the residuals of all star positions in all mosaics. By inspecting the residuals of any mosaic as function of position, we were able to map the residual image distortions in the given mosaic. These residual image distortions can arise due to imperfect registration of the individual exposures to the respective mosaic, or due to an error in the distortion correction applied to the individual frames.
• 4.  
  For each star in each mosaic, we determined an estimate of the residual image distortion by calculating the mean of the residuals of the stars in the vicinity (r < 2'') of the chosen star. The radius was chosen such that a suitable number of stars were present in the area from which the correction was determined and the area was sufficiently local. A value of r < 2'' was a good compromise, typically yielding 30–50 stars. The estimate for the residual image distortion was then subtracted from the given star; the typical values applied were Δx = 0.66 ± 0.20 mas and Δy = 0.62 ± 0.24 mas.
• 5.  
  In a second fit, we used the corrected astrometric positions in order to obtain updated linear motions models for the 6037 stars.
• 6.  
  Then we defined the final cluster. It consists of all stars which were present in all 11 mosaics, with radii between 2'' and 15'' and which are not known early-type stars. These criteria yielded a cluster sample size of 2147 stars.
• 7.  
  We determined the cluster mean velocity, yielding (−0.04,  0.00) mas yr⁻¹, and subtracted that value from all velocities of the 6037 stars. This then is the final, velocity-calibrated list of linear motions from which the reference star sample is extracted. The mean radius of the 2147 stars of the cluster sample is 989, the root mean square (rms) speed of the stars in the sample is 157 km s⁻¹≈4.15 mas yr⁻¹ (for R₀ = 8 kpc).

We call the coordinate system defined in this way the "cluster system." Since we expect the mean of the cluster to show a net motion of order $157/\sqrt{2147}$ km s⁻¹ = 3.4 km s⁻¹= 0.09 mas yr⁻¹, we estimate the uncertainty of the velocity calibration to be of the same size. We checked this number more thoroughly by means of a Monte Carlo simulation. We divided the cluster into nine radial bins with boundaries [2, 5, 7, 8.5, 10, 11, 12, 13, 14, 15] mas (selected such that in each bin roughly the same number of stars is present). For each bin, we have determined the rms velocity yielding 3.7, 3.3, 3.0, 2.9, 2.8, 2.7, 2.6, 2.6, 2.6 mas yr⁻¹ respectively. We then simulated clusters in proper motion space, for each bin the Gaussian width of the velocity distribution was set to the respective rms velocity and the number of stars was matched to the real numbers in each bin. For each simulated cluster, we were able to obtain in that way a mean velocity; simulating 10,000 clusters allowed us then to estimate the uncertainty of the mean cluster velocity. We obtained 0.06 mas yr⁻¹; even a bit better than the simple estimate. Hence, if the assumption of isotropy is correct, the cluster system should allow for a better calibration of the reference star velocities than with the maser system. The assumption could be wrong, for example if a net streaming motion were present in the GC cluster.

The statistical positional uncertainty of the origin of the cluster system was estimated by the same means as for the maser system. We obtained (Δα, Δδ) = (0.85, 1.51) mas. In addition to these uncertainties, the residuals of the SiO masers also need to be considered, for the epoch at hand the mean deviation is (Δα, Δδ) = (1.87, 3.12) mas. The uncertainties here are greater than the respective numbers for the maser system due to the fact that the position of Sgr A* in the cluster system is effectively measured only on one frame, while in the maser system it is measured in several, and the residuals are not fully correlated.

The maser system has a smaller systematic error in its position calibration, while the cluster system is superior with respect to the velocity calibration. Hence, by combining the two, we were able to construct a system that combines both advantages. The idea simply is to correct either the velocity calibration of the maser system such that it agrees with the one from the cluster system, or to correct the origin of the cluster system such that it coincides with the origin of the maser system (taking into account that the systems refer to two different epochs). Note that this implicitly uses the fact that the second, refining transformation of the maser system did not change its calibration properties.

We used the sample of reference stars to compare the two systems. The mean positional offset between the two lists of positions for the epoch of the cluster system was

$$\begin{eqnarray} \vec{p}_\mathrm{CSys} - \vec{p}_\mathrm{MSys} & =& \left(\begin{array}{@{}c@{}}-1.87\\ +1.87\end{array}\right) \pm \left(\begin{array}{@{}c@{}}0.04\\ 0.04\end{array}\right)\,\mathrm{mas}. \end{eqnarray} \tag{ 1 }$$

Here, "Csys" denotes the cluster system, "MSys" the maser system. The errors are the standard deviation of the sample of differences. We also calculated the differences of the reference star velocities, as given by the two linear motion models obtained for each reference star. We obtained

$$\begin{eqnarray} \vec{v}_{CSys} - \vec{v}_{MSys} &=& \left(\begin{array}{@{}c@{}}-0.60\\ +0.56\end{array}\right) \pm \left(\begin{array}{@{}c@{}}0.08\\ 0.06\end{array}\right)\,\frac{\mathrm{mas}}{\mathrm{yr}}, \end{eqnarray} \tag{ 2 }$$

where again the errors are the standard deviation of the sample of differences.

This means that the two coordinate systems differ significantly in position and velocity calibration in a systematic way. It should be noted that only the difference between the two coordinate systems is that well defined; for the question how well each of the coordinate systems relates to Sgr A*, the larger, systematic errors of Sections 3.2.1 and 3.2.2 need to be considered. It is exactly the fact the difference between the coordinate systems is well defined that allowed us to combine the two coordinate systems and to gain accuracy in the combined system that way. Also note that the size of the offsets occurring here are consistent with the combined uncertainties of the two coordinate system; much larger offsets would have meant that the coordinate systems would be inconsistent with each other.

Finally, we chose the method which corrects the cluster system by a positional offset. The positional difference from Equation (1) was subtracted from all positions of the cluster stars (and thus also from the reference stars that are a subset of the cluster). This combined coordinate system has the same prior as the cluster system, namely that the cluster is at rest with respect to Sgr A*. The linear motion models so obtained were then used for the further analysis.

The accuracy in two-dimensional position (Δx, Δy) and two-dimensional velocity (Δv_x, Δv_y) of the combined coordinate system is given by the numbers in Sections 3.2.1 and 3.2.2. In the third dimension, we don't use any priors for Δz, since we wish to determine R₀ from our data.

For Δv_z, we use the prior that Sgr A* is not moving radially, based both on theoretical arguments and on radio and NIR measurements. Even if Sgr A* is dynamically relaxed in the stellar cluster surrounding it, some random Brownian motion due to the interaction with the surrounding stars is expected. Merritt et al. (2007) calculated this number and concluded that the motion should be ≈0.2 km s⁻¹. This is consistent with the findings of Reid & Brunthaler (2004) who show that Sgr A* has a proper motion of v_l = 18 ± 7 km s⁻¹ in galactic longitude and v_b= −0.4 ±  0.9 km s⁻¹ in galactic latitude (assuming R₀ = 8 kpc). The significance of the fact that v_l ≠ 0 is disputed, and furthermore it is not clear, whether it is truly due to a peculiar motion of Sgr A*, or due to a difference between the global and local measures of the angular rotation rate of the Milky Way (Reid & Brunthaler 2004). Clearly, the motion of Sgr A* perpendicular to the galactic plane is very small as expected. In the third dimension, the velocity of Sgr A* can only be determined indirectly by radial velocity measurements of the stellar cluster surrounding it. Using a sample of 85 late-ytpe stars Figer et al. (2003) found that the mean radial velocity of the cluster is consistent with 0: v_z= −10 ± 11 km s⁻¹. Trippe et al. (2008) used a larger sample of 664 late-type stars and found consistently v_z = 4.6 ± 4.0 km s⁻¹. Compared to that, the uncertainty ΔU ≈ 0.5 km s⁻¹ in the definition of the local standard of rest is much smaller (Dehnen & Binney 1998). We conclude that all measurements are consistent with Sgr A* being at rest at the dynamical center of the Milky Way, and we assume a prior of v_z = 0 ± 5 km s⁻¹ for our coordinate system.

Summarizing, our combined coordinate system should be accurate to the numbers listed here, of which finally used the conservatively rounded values:

$$\begin{eqnarray} \Delta x &=& 0.95 \,\mathrm{mas} \approx 1.0 \,\mathrm{mas} \nonumber \\ \Delta y &=& 2.35 \,\mathrm{mas} \approx 2.5\,\mathrm{mas} \nonumber \\ \Delta v_x &=&0.06\, \rm mas\; yr^{-1} \approx 0.1 \,\rm mas\; yr^{-1} \nonumber \\ \Delta v_y &=&0.06\,\rm mas\; yr^{-1} \approx 0.1 \,\rm mas\;yr^{-1} \nonumber \\ \Delta v_z &=&5\,\rm km\; s^{-1}. \end{eqnarray} \tag{ 4 }$$

Potentially, there are two more degrees of freedom, which could affect the reliability of the chosen coordinate system, namely rotation and pumping. An artificial rotation can be introduced if the selected stars by chance preferentially move on tangential tracks with a preferred sense of rotation. Similar, artificial pumping can occur: suppose that by chance all selected stars move on perfect radial trajectories and that stars further out move faster than stars closer to Sgr A*. Such a pattern, which would be somewhat similar to the Hubble flow of galaxies, would yield under the set of transformations a time-dependent plate scale and otherwise stationary stars. Both effects can affect the selection of the reference star sample and (less important) the selection of cluster stars.

The chosen coordinate system relies on the assumption that the cluster does not show any net motion (see Section 3.2.2), net rotation, or net pumping. The selection of a finite number of cluster stars, however, limits the accuracy with which these conditions can be satisfied. Given 2147 stars with a rms velocity of ≈157 km s⁻¹ and a typical distance of 10'', we expect that any selection leads to a pumping, or rotation effect of the order of 9 μas yr⁻¹ arcsec⁻¹.

Due to the errors in the SiO maser positions, the maser system can show artificial pumping, or rotation. Similar to what was done in Sections 3.2.1 and 3.2.2 we simulated in a Monte Carlo fashion the error propagation. From 10⁵ realizations of the transformations, assuming the observed errors of the SiO maser positions in the NIR and radio, we created perturbed sets of reference stars. The standard deviation of the pumping and rotation motion (v_r/r and v_t/r respectively) over these sets then estimate the stability of the maser system. We obtained

$$\begin{eqnarray} v_r/r|_\mathrm{MSys} &=& 37\, \mu \mathrm{as\,yr}^{-1} \,{\rm arcsec}^{-1},\nonumber \\ v_t/r|_\mathrm{MSys} &=& 33 \,\mu \mathrm{as\,yr}^{-1} \,{\rm arcsec}^{-1}. \end{eqnarray} \tag{ 5 }$$

The cluster system (and therefore also the combined system) can be checked against the maser system. By calculating the difference in velocity for each reference star and subtracting from those the difference of the two coordinate system velocities, we obtained a vector field of residual velocities, which is well described by

$$\begin{eqnarray} v_r/r|_\mathrm{CSy}-v_r/r|_\mathrm{MSy} &=& (32 \pm 2|_\mathrm{stat} \pm 9|_\mathrm{sys}) \,\mu \mathrm{as\,yr}^{-1}\,{\rm arcsec}^{-1}\nonumber \\ v_t/r|_\mathrm{CSy}-v_t/r|_\mathrm{MSy}& = &(6 \pm 2|_\mathrm{stat} \pm 9|_\mathrm{sys})\, \mu \mathrm{as\,yr}^{-1}\,{\rm arcsec}^{-1}\nonumber \\ \end{eqnarray} \tag{ 6 }$$

The combined size of the effects from Equations (5) and (6) estimate the error made when using the assumption that the combined coordinate system is nonrotating and nonpumping. At 1'', these effects can sum up over 15 years to at most 0.7 mas, while for the maximum projected distance of S2 (≈0.2'') the resulting positional errors are even a factor 5 smaller. We therefore neglected these effects in the following.

This paragraph deals with the uncertainties of the stellar positions on a given image; the unit of this error term as measured is therefore pixels. The error which is most easily accessible is the formal fit error of the Gaussian fit to a source. However, in deconvolved and beam-restored images it might be a bad estimator for the positional uncertainties. Therefore, we compared additionally different deconvolutions of the same image for each epoch in order to get a more robust estimate.

For the SHARP data, we used up to eight (= two deconvolutions × four pointings) pixel positions. The standard deviation of the astrometric positions was included in the error estimate for the statistical position error. For stars which were present only in one frame, the typical error of the epoch was used instead.

For NACO, we split up each data set into two parts and deconvolved both co-added images with the same PSF as the co-added image of the complete data set (see Section 2.2). We determined the pixel positions of the reference and S-stars in the two deconvolved frames and applied a pure shift between the two lists of pixel positions such that the average pixel position is the same for both. The remaining difference between respective positions of one star estimates the statistical uncertainty for that star. The error estimates obtained this way were a strong function of the stellar brightness. Therefore, we described the error estimates as a function of flux for each epoch (see Figure 3) using a simple empirical model of the form ax⁻ⁿ + b. The mean floor $\bar{b}$ over all data sets is 99 μas, while for lower fluxes the error increases up to 2 mas. We used the empirical description of each image to assign an error to all stellar positions obtained from that frame. Finally, we checked whether the formal fit error of the positions was greater than the estimate from the empirical error model. In such a case, we used the formal fit error instead. Figure 4 shows the final distribution of statistical errors for the NACO data. It is effectively the mean error model folded with the brightness distribution of the S-stars.

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For the SHARP data, we obtained a broad distribution of the statistical pixel position errors with no clear maximum and a tail to 2 mas. The median error is 360 μas, the mean error 760 μas in the SHARP data.

A main source of error at the sub-milliarsecond level is image distortions. We estimated this error term by comparing distances of stars in different pointing positions with a dither offset of 7'' (see Figure 5). If we had used only the raw positions and linear transformations, the resulting mean one-dimensional position error would be as large as 1 mas for the 13 mas pixel⁻¹ NACO data. By applying a distortion model (see Section 2.2) plus a linear transformation this error can be reduced to 600 μas. Allowing for a cubic transformation onto a common grid yields an error of 240 μas only. This justifies our choice to use a high order transformation rather than to dedistort the 13 mas pixel⁻¹ NACO images. The numbers obtained in this way are actually the combined error of the statistical and transformation uncertainties with the residual image distortions. Subtracting the former, we conclude that residual image distortions have a contribution of 210 μas to the error budget of each individual astrometric data point. We thus added this value in squares to all other error terms, effectively acting as a lower bound for the astrometric errors.

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We applied the same analysis to the 27 mas pixel⁻¹ NACO data which had a dither offset of 14'' (see Figure 6). The raw differences showed a skewed distribution, indicating the presence of image distortions. The rms of this distribution is 2.1 mas. After applying the distortion model the typical residual error is reduced to 1.3 mas and the distribution is a nice Gaussian. Interestingly, mapping the positions with a cubic transformation onto each other does less well here. The distribution becomes less skewed; however, it is still non-Gaussian, and the rms is 1.6 mas. This justifies a-posteriori the use of the distortion correction for the 27 mas pixel⁻¹ NACO data when determining the motion models for the reference stars. For the SHARP data, we obtained a characteristic error due to residual image distortions of 0.8 mas and a median of 1.2 mas.

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It is important to notice that any error in deriving the motions for the reference stars only translates into a global uncertainty of the coordinate system (which could show up as an offset of the center of mass from 0/0, or a net motion of the coordinate system). It will, however, not affect the accuracy of individual data points in this system. Only the selection of reference stars and transformation errors contribute to the errors of the individual data points. We estimated them by performing all coordinate transformations not only once but also with subsets of the available reference stars. The standard deviation of the sample of obtained astrometric positions was then included in the astrometric error estimate. The typical uncertainty introduced by the transformations was quite small, namely 23 μas for the NACO data. This is consistent with the fact that ≈100 stars have been used of which each can be determined with an accuracy of ≈200 μas. For the SHARP data, we found a value of 100 μas, again consistent with the characteristic single position error of ≈1 mas.

At the sub-mas level, there is a multitude of differential effects over the field of view that can influence astrometric positions. The most prominent ones are relativistic light deflection in the gravitational field of the sun, light aberration due to Earth's motion, or refraction in the atmosphere. Since our analysis is based on relative astrometry, the absolute magnitudes of the effects do not matter. Only the differential effects over the field of view can contribute to the positional uncertainties.

• 1.  
  Over a field of view of 20'', the differential effects of aberration can be described by a global change of image scale (Lindegren & Bastian 2006). Since we fit the image scale for each epoch separately, the differential aberration is absorbed into the linear terms of the transformation and thus is not affecting the astrometry. The size of the effect for a small field of view with a diameter f amounts to f × v/c × cos Ψ where Ψ is the angle between the observation direction and the apex point. For f ≈ 10'' and v ≈ 30 km s⁻¹ this yields ≈1 mas at most.
• 2.  
  The light deflection can be approximated by $4\,\mathrm{mas}\times \cot \Psi /2$ where Ψ is the angle between observation direction and Sun (Lindegren & Bastian 2006). The differential effect over 20'' will not exceed 100 μas as long as Ψ>36, which is guaranteed for all our data.
• 3.  
  From the usual refraction formula R = 44'' tan z (for a standard pressure of 740 mbar at Paranal), we find a differential effect of 4–8 mas over 20'', or 2–4 mas over the field in which we selected the reference stars. The effect will be a change in one direction (toward zenith) of the image scale. Since the effect is at most quadratic over the field, it will be absorbed completely into the first- and second-order terms of the transformations. Note that it is crucial to allow also for skew terms, i.e., it is not sufficient to use a shift, rotation and scale factor only in the linear terms, but the off-diagonal terms in the transformation matrix are also required.

One important contribution to the position errors is the fact that stars can be confused and that sometimes the confusion is not recognized. This problem is more severe for the SHARP data than for the NACO data due to the lower resolution. Of course, we excluded positions for which we know that they are confused. However, unrecognized confusion cannot be dealt with by principle. We therefore simply accept that these events happen. This means, in turn, that we expect to find a reduced χ² > 1 when trying to describe the motions with smooth functions. In addition, we note that for a sufficiently large amount of data points unrecognized confusion events should only lower the precision but not the accuracy, since no global bias is expected. Still, if a confusion event happens during an unfortunate part of the orbit (for example at an end point, or during pericenter passage) a bias in the results of an orbit fit can be introduced.

Gravitational lensing might affect the measured positions. A quantitative analysis shows that the effects are very small except in unusual, exceptional geometric configurations. For a star at a distance z ≪ R₀ sufficiently far behind Sgr A* the angle of deflection as measured from Earth is

$$\begin{equation} \theta =\frac{z}{R_0} \frac{4 G M_\mathrm{MBH}}{c^2\,b}, \end{equation} \tag{ 7 }$$

where b is the impact parameter. For the GC, this evaluates to θ ≈ 20 μas × z/b, indicative of a very small astrometric effect unless z/b ≫ 1 This rough estimate is consistent with the rigorous treatment of the problem from Nusser & Broadhurst (2004), who show that in order to achieve a displacement of 1 mas, a star at z ≈ 1000 AU needs to have b ≈ 2 mas≈16 AU. In our data set, none of the stars gets close to the regime that gravitational lensing actually becomes important. Therefore, we neglected the effect.

We were able to check how well our error estimates agree with the intrinsic noise of the data. For this purpose, we fitted all measured positions of the reference stars with simple quadratic functions. After exclusion of 3σ outliers, we have calculated the reduced χ² for each reference star. The mean reduced χ² for the NACO data is 2.0 ± 0.7, while for the SHARP data, we obtained values between 0.5 and 2.0 with a mean of 1.0.

Since our data set consists of two subsets (SHARP and NACO), each covering roughly the same amount of time, the relative weight of the two subsets matters. Given that we seem to underestimate the errors for NACO a little, while the SHARP errors seem consistent with the noise in the data, we decided to apply a global rescaling factor of r = 1.42 to all NACO data points. This procedure adjusts the relative weight between the two subsets. Still we expect a reduced χ² > 1 when performing orbit fits due to unrecognized confusion events.

In Figure 7, we show the final error distributions (after rescaling all NACO errors with the global factor) for the S-stars and reference stars, both for the NACO and SHARP data. The characteristic error for a reference star in the NACO data is 360 μas, in the SHARP data it amounts to 760 μas. For the S-stars, the histogram of the NACO errors has a peak also around 325 μas and a tail toward larger errors, essentially telling us that for bright S-stars the astrometry is as good as one could hope for (since it is equally good as for the reference stars). The tail is due to the fact that many of the S-stars are faint (hence the statistical error is severe) and probably also unrecognized confusion events affect the statistical error since confusion can alter the shapes of the images of faint stars. In the SHARP data, the typical S-stars error is 2 mas and the lower end of the distribution at ≈1 mas is consistent with what could be expected from the reference stars.

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First, we used the S2 data only to determine a Keplerian gravitational potential (see Figure 13). Using the priors as obtained in Equation (4), the fits yield the numbers in the first and second rows of Table 4. The two values for R₀ differ by more than what the errors suggest, indicating that the 2002 data influence R₀. This confirms the presumption from Section 3.5. We exploited this further in Figure 14. Assigning the 2002 data higher weights (smaller errors) pushes the distance estimate up, smaller weights lower it.

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Mass and distance are strongly correlated parameters, see Figure 15. The scaling of mass with R₀ in our data set is a power law with M_MBH ∼ R²₀. For a purely astrometric data set, one would have an exponent of 3 and a complete degeneracy; the fact that the exponent is <3 and that the degeneracy is not complete is due to the influence of the radial velocity information in our data set and due to the use of priors. The degeneracy can be understood qualitatively. Changing R₀, effectively changes the conversion from measured angles (in mas) to physical lengths (in pc), i.e., changing R₀ changes the semimajor axis. Since the orbital period is well determined in our data, the mass has to change in order to fulfill Kepler's third law.

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The strong dependency means that the uncertainties for mass and distance are coupled. Fixing the distance yields a very small fractional error on the mass of ΔM_MBH ≈ 0.02M_MBH. This shows that the error of the fitted mass is completely dominated by the uncertainty in the distance. Once the distance is known, the mass immediately follows from the scaling relation

$$\begin{eqnarray} M_\mathrm{MBH} &=& (3.99 \pm 0.07|_\mathrm{stat} \pm 0.32|_\mathrm{R_0})\nonumber \\ && \times 10^6\,M_\odot \left(\frac{R_0}{8\,\mathrm{kpc}} \right)^{2.02}\;(\mathrm{incl. 2002}),\nonumber \\ M_\mathrm{MBH} &=& (4.08 \pm 0.09|_\mathrm{stat} \pm 0.39|_\mathrm{R_0})\nonumber \\ &&\times 10^6\,M_\odot \left(\frac{R_0}{8\,\mathrm{kpc}} \right)^{1.62}\;(\mathrm{excl. 2002}), \end{eqnarray} \tag{ 8 }$$

where the error due to R₀ corresponds to the fit error reported in Table 4.

By construction the position of the radio source Sgr A* in our coordinate system is located at the origin. Since it is clear that Sgr A* is the MBH candidate, we used this fact when applying the priors of Equation (4). However, our data actually allow us to test this hypothesis. By leaving the position and proper motion of the mass completely free, we can check how well the position of the mass coincides with Sgr A*. Using the S2 data only, no two-dimensional priors but the prior in v_z from Equation (4), we obtained the numbers presented in the third and fourth rows of Table 4. We note that the mass is located within ≈2 mas at the expected position. The current accuracy by which this statement holds is an improvement of a factor ≈2 over the work from Schödel et al. (2003).

We also report the S2-only fits when not using any coordinate system priors at all (Rows 5 and 6 in Table 4). This enlarges the errors on R₀ and M_MBH substantially, the fit values, however, are not significantly different from the respective fits in which the v_z-prior was applied. Not applying the v_z-prior also shows a large uncertainty on v_z of ≈50 km s⁻¹; this parameter also is degenerate with R₀.

From the numbers, it seems that the fit excluding the 2002 data agrees better with the expectations for the coordinate system (Equation (4)) than the fit including it. The latter is marginally consistent with the priors, while the former is fully consistent. This means that the 2002 data not only affect R₀ (which we want to measure and thus cannot use to judge the result), but also the position and the velocity of the mass, for which we have an independent measurement via the coordinate system definition. This argument points toward rejecting the 2002 data.

At 22 epochs, we have identified a source in the NACO data between 2003 and 2008 that might be associated with Sgr A*. In some cases, e.g. when a bright flare occurred, the identification seems unproblematic. In other cases, one cannot be sure that the emission is not due to an unrecognized star at or very close to the position of Sgr A*; an example is Figure 1. Due to this probably very frequent confusion, we expect that the measured positions are very noisy, and we decided not to include them into the orbital fits. However, we checked whether the measured positions are compatible with the orbital fits. Fitting a linear motion model to the Sgr A* data, we obtained

$$\begin{eqnarray} \hspace{-12pt}\alpha [\mathrm{mas}]&=& (1.2 \pm 0.8) + (0.15 \pm 0.46) \times (t[\mathrm{yr}] - 2005.91)\quad \nonumber \\ \hspace{-12pt}\delta [\mathrm{mas}]&=& (2.7 \pm 0.7) - (0.73 \pm 0.39) \times (t[\mathrm{yr}] - 2005.91)\quad \end{eqnarray} \tag{ 9 }$$

The errors here are rescaled for a reduced χ² of 1. The velocity errors are approximately a factor 5 larger than the priors from Equation (4), justifying our choice not to incorporate these data into the orbital fits. Given the uncertainties, the position of the IR counterpart of Sgr A* is consistent with the position of the central point mass. Interestingly, these data seem to prefer a position of Sgr A*, marginally North of the expected position, which is also the case for the orbit fits that include the 2002 data of S2. This weakens again the conclusion from Section 6.2.2 that the 2002 data should be rejected.

Given the large uncertainties due to the 2002 data of S2, we decided to obtain more information about the potential by using a combined orbit fit and the coordinate system priors. For comparison, we also excluded S2 completely. We used the stars S1, S2, S8, S12, S13, S14. We selected these stars from the sample that can contribute to the potential (Section 6.1) since for them a large fraction of the respective orbit is covered. We did not select S4 and S17 as they suffered confusion in the SHARP data. S19 was omitted because its time base is quite short still (the star was not detected before 2003). Since S24 would only contribute marginally to the potential, it was left out, too. Finally, we did not select S31, since the nearby sources S59 and S60 were confused with S31 in the earlier NACO data. Not surprising, the final sample contains the same stars as Eisenhauer et al. (2005) had reported orbits for.

In order to balance the relative weights of the stars used, we had fitted the five additional stars first alone, leaving also the potential free (but applying the priors). While the such obtained fits were not of interest per se, they still provided a smooth, unbiased model for each star. Hence, we used the resulting reduced χ² values to rescale the astrometric and radial velocity errors such that all stars yielded a value of 1. The scaling factors applied ranged from 1.20 to 2.33, the latter value being extreme and occurring for S13, which perhaps suffered from confusion in the SHARP data and of which the data in 2006/2007 were affected by confusion with S2. Our procedure guaranteed that such a star with a high astrometric noise would not contribute overly much to the combined χ². We obtained the results given in rows 7, 8, and 9 of Table 4. These numbers agree with each other within the uncertainties. The combined fit including the S2 2002 data also agree with the corresponding S2-only fit. This is not true for the combined fit excluding the S2 2002 data, which is hardly compatible with the respective S2-only fit. A possible reason is that the S2 data before 2002 are only relying on the SHARP measurements, which not only have larger formal errors but also are more affected by unrecognized confusion events than the NACO data.

By fitting the combined data at various, fixed values of R₀, we obtain again the scaling of mass and distance:

$$\begin{eqnarray} M_\mathrm{MBH} &=& (3.95 \pm 0.06|_\mathrm{stat} \pm 0.18|_\mathrm{R_0})\nonumber \\ &&\times 10^6\,M_\odot \left(\frac{R_0}{8\,\mathrm{kpc}} \right)^{2.19}\;(\mathrm{incl. 2002})\,,\nonumber \\ M_\mathrm{MBH} &=& (4.01 \pm 0.07|_\mathrm{stat} \pm 0.18|_\mathrm{R_0})\nonumber \\ &&\times 10^6\,M_\odot \left(\frac{R_0}{8\,\mathrm{kpc}} \right)^{2.07}\;(\mathrm{excl. 2002})\,,\nonumber \\ M_\mathrm{MBH} &=& (3.88 \pm 0.10|_\mathrm{stat} \pm 0.41|_\mathrm{R_0})\nonumber \\ && \times 10^6\,M_\odot \left(\frac{R_0}{8\,\mathrm{kpc}} \right)^{3.07}\;(\mathrm{excl. S2})\,, \end{eqnarray} \tag{ 10 }$$

Beyond what was considered before, the physical model for the potential is another source of uncertainty. For example using a relativistic model instead of a Keplerian orbit model increased the distance by ΔR₀ = 0.18 kpc (0.09 kpc) when including (excluding) the 2002 data. This is consistent with the formal error on R₀. Since we do not detect explicitly relativistic effects, we stay with Keplerian orbits and consider the shift of the value as an uncertainty for R₀. Fitting a Plummer model (as in Section 6.3) instead of a point mass potential increases the distance by a similar value: 0.14 kpc (0.03 kpc) when including (excluding) the 2002 data. The additional degree of freedom in this fit increased the formal uncertainty by 0.11 kpc added in squares. Finally, we adopted for the uncertainties of the potential an error of ΔR₀ = 0.25 kpc.

An additional, systematic error is whether the use of priors (Equation (4)) is correct. In order to address this, we repeated the combined orbit fits without the two-dimensional priors. We obtained the numbers in rows 10 and 11 of Table 4. The influence of the priors on the value of R₀ is relatively small (compare rows 7 and 8 with 10 and 11 in Table 4). We adopt for this source of uncertainty ΔR₀ = 0.10 kpc.

Furthermore, rows 7, 8 and 9 of Table 4 show that the uncertainty of the weights of the 2002 data from S2 in a combined fit alters R₀ by ΔR₀ = 0.13 kpc. Deselecting S2 from the fits changes the result by ΔR₀ = 0.07 kpc. Finally, we assign ΔR₀ = 0.15 kpc for the uncertainties related to the selection of data.

Adding up the uncertainties yields that the uncertainty of the distance to GC is still rather large with Δ_totalR₀ = 0.35 kpc. Table 5 summarizes the error terms for R₀.

We finally adopt the potential from the combined fit, including the S2 2002 data, the difference to the one excluding that data are negligible, given the formal fit errors (Section 6.2.4). This potential will be used in Section 6.4 to determine the orbits of the other stars, for which we expect to find an orbital solution. Hence, we find

$$\begin{equation} R_0 = 8.33 \pm 0.17|_\mathrm{stat} \pm 0.31|_\mathrm{sys}\,\mathrm{kpc}. \end{equation} \tag{ 11 }$$

It should be noted that this value is consistent within the errors with values published earlier (Eisenhauer et al. 2003, 2005). The improvement of our current work is the more rigorous treatment of the systematic errors. Also it is worth noting that adding more stars did not change the distance much over the equivalent S2-only fit. For the mass, we adopt

$$\begin{eqnarray} M_\mathrm{MBH} &=& (3.95 \pm 0.06|_\mathrm{stat} \pm 0.18|_\mathrm{R_0,\;stat} \pm 0.31|_\mathrm{R_0,\;sys})\nonumber \\ &&\times 10^6\,M_\odot \left(\frac{R_0}{8\,\mathrm{kpc}} \right)^{2.19}\nonumber \\ &=&(4.31 \pm 0.38) \times 10^6\,M_\odot \;\mathrm{for}\; R_0=8.33\,\mathrm{kpc}. \end{eqnarray} \tag{ 12 }$$

Figure 19 illustrates the orientations of the orbital planes for all stars from Table 7. Paumard et al. (2006) suggested that the six stars S66, S67, S83, S87, S96 and S97 (E17, E15 (S1-3), E16 (S0-15), E21, E20 (IRS16C) and E23 (IRS16SW) in their notation) are members of the clockwise disk. Our findings explicitly confirm this. All six stars have an angular distance to the disk between 9° and 21° with a mean and standard deviation of 15° ± 4°. This is somewhat (a factor of 2) more than the disk thickness of 14° ± 4° found by Paumard et al. (2006). However, statistically the difference is not very significant and only the inner edge of the disk is sampled here. All six disk stars have a semi major axis of a ≈ 1'' and a small eccentricity (e ≈ 0.2 − −0.4) in agreement with the estimates from Paumard et al. (2006). The orbital plane of S5 is also consistent with the disk given its distance of 18°. However, the lower brightness (m_K = 15.2) of the star and the higher eccentricity (e > 0.8) of the orbit make it unlikely that S5 is a true disk member. The next closest star to the disk beyond the six disk stars and S5 is S31 with an angular distance of 27°. We also note that the orbital solutions for S96 and S97 derived from marginal accelerations are consistent with the disk hypothesis. Therefore, we are confident in these orbits, too.

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We used a Rayleigh test (Wilkie 1983) to check whether the distribution of orbital angular momenta for the 22 other stars for which we found orbits is compatible with a random distribution. We found a probability of randomness of p = 0.74; meaning that the nondisk stars do not show a preferred orbit orientation. Using the projection method from Cuesta-Albertos et al. (2007), we obtained p = 1.0. The same statement also holds when testing for randomness of the subset of early-type stars.

Figure 20 shows the cumulative probability distribution function (pdf) for the semimajor axes of stars which have semi major axis smaller than 05, thus excluding the stars that are identified to be members of the clockwise disk. The statistic is limited still (15 stars make up this sample), but nevertheless the distribution allows us to estimate the functional behavior of the pdf n(a). Due to the small number of data points, we did not bin the data but used a log-likelihood fit for n(a). We found n(a) ∼ a^0.9±0.3. This can be converted to a number density profile as a function of radius (Alexander 2005). We obtain n(r) ∼ r^−1.1±0.3, consistent with the mass profile in Genzel et al. (2003b) who found ρ(r) ∼ r^−1.4 and with the newer work in Schödel et al. (2007) who found ρ(r) ∼ r^−1.2 for the innermost region of the cusp.

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The distribution of eccentricities allows us to estimate the velocity distribution. Figure 21 shows the cumulative pdf for the eccentricities of those young (early-type) stars which are not associated with the clockwise stellar disk. Using again a log-likelihood fit, we find n(e) ∼ e^2.6±0.9. The profile still is barely consistent with n(e) ∼ e, corresponding to an isotropic, thermal velocity distribution (Schödel et al. 2003; Alexander 2005). This would be the expectation for a relaxed stellar system. However, given that the maximal lifespan for B stars (≲10⁸ yr) is much shorter than the local two-body relaxation (TBR) time (≈10⁹ yr, Alexander 2005) one does not expect a thermal distribution. In this light, it is interesting to notice that the distribution appears to be a bit steeper (i.e., peaked toward higher eccentricities) than a thermal distribution. This might be a first hint toward the formation scenario for the S-stars. For example, it is exactly what one expects in the binary capture scenario (Perets et al. 2007), in which the S-stars are initially captured on very eccentric orbits (e ≳ 0.98), and then subsequent relaxation gradually smears out the distribution of eccentricities toward a thermal distribution. From the timescales involved, one expects that the latter is not reached completely, so a high eccentricity bias remains, which in turn might fit nicely together with our indication for a steeper than thermal eccentricity distribution.

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This lays out a very interesting perspective for the continuation of the orbital monitoring. Increasing the statistics of the eccentricity distribution by determining more stellar orbits will allow us to test explicitly whether it truly deviates from a thermal distribution and thus provides us with a quantitative test for formation scenarios of the S-stars.

Genzel et al. (2003b) and Schödel et al. (2007) have inferred a stellar density profile for the GC from completeness-corrected stellar number counts. Assuming that the luminous objects trace the total mass, the number density profile, which is determined reliably on the >0.01 pc scale, can be extrapolated to the S2 orbit. We obtain

$$\begin{equation} \eta = 3.7 \times 10^{-4}\times \bigg(\frac{m_\star }{10} \bigg). \end{equation} \tag{ 18 }$$

This extrapolation is quite uncertain, since mass segregation predicts that the SBHs should have a much steeper slope than the less-massive luminous stars in the central 0.01 pc, where the SBHs dominate the total mass (Hopman & Alexander 2006; Alexander 2007). Therefore, both the mass-to-number ratio and the slope of the density profile are expected to have a significant radial dependence.

The drain limit is a conservative theoretical upper limit of the number of compact objects that can exist in a steady state around an MBH. It is given by the condition that the number of SBHs that can be packed inside any given radius in the steady state has to be smaller than the number of SBHs scattered into the MBH over the age of the Galaxy (Alexander & Livio 2004). This can be translated into a theoretical limit for η. Close to m_⋆ = 10, and using M_MBH = 4 × 10⁶M_☉ and t = 10 Gyr the relation can be approximated by

$$\begin{equation} \eta \lesssim 0.0011 \times \bigg(\frac{m_\star }{10} \bigg)^{-0.7}. \end{equation} \tag{ 19 }$$

The drain limit could be violated for a non-steady-state situation. Indeed, the existence of the young star disks with a relatively well-defined age of 6 Myr suggests that star formation in the GC is episodic. However, the amount of mass from SBHs would hardly exceed 10³ M_☉ even assuming an optimistic, top-heavy initial mass function, given that the total amount of mass in the disks is ≈10⁴ M_☉.

The expected degree of central concentration of SBHs around the MBH can be estimated by modeling the dynamical evolution of a system with a present-day mass function similar to that of the GC (Alexander 2005). Monte Carlo simulations of the GC using the Hénon method and including also stellar collisions and tidal disruptions (M. Freitag 2008, private communication; see also Freitag et al. 2006), but neglecting star formation yield a rather flat mass density profile of 10⁸(M_☉pc⁻³)(r/0.01 pc)^−0.5, which translates to η ∼ 10⁻⁴. Due to the statistical nature of this method the density profile at the very center is not well determined. An alternative analytic solution for steady-state distribution using a much more idealized formulation of the mass segregation problem (Hopman & Alexander 2006) yields a similar result of η ∼ 5 × 10⁻⁴. However, in this method the fixed boundary conditions far from the MBH may artificially maintain a high density in the center by preventing the expansion of the system. Nevertheless, the fact that these two different methods yield similar results also consistent with the drain limit lends some credence to this estimate.

A cluster of compact objects will accrete the surrounding gas and thus lead to X-ray emission, which for current X-ray satellites (≈1'') would be barely resolved. Indeed, the X-ray source at the position of Sgr A* is slightly extended (Baganoff et al. 2003). We fit the radial profile of Sgr A* as reported by Baganoff et al. (2003) by the superposition of a point source with a Gaussian width of σ = 0375 (Baganoff et al. 2003) and an extended component with a free width σ_ext. We obtain as empirical description for the surface brightness profile of Baganoff et al. (2003), Figure 6:

$$\begin{equation} B(r)\,[\mathrm{cts/arcsec}^{2}] = 73.5 \, e^{-r^{2}/2 \sigma _{}^2} + 40.3 \,e^{-r^2/2 \sigma _\mathrm{ext}^2}, \end{equation} \tag{ 20 }$$

with σ_ext = 105. Thus, we obtain for the extended luminosity (assuming the same spectral index of point like and extended component) L_X,ext = 1.95 × 10³³ erg s⁻¹, accounting for ≈80% of the total X-ray luminosity.

The expected X-ray luminosity of a single compact object is given by the mass accretion rate and the radiation efficiency. A simple estimate is given by assuming Bondi accretion (Bondi 1952):

$$\begin{equation} \dot{M}_B = 4\pi \lambda (G M_\star)^2 n_e \mu \,m_p\, c_s^{-3}\, \approx 10^9\,\rm g\; s^{-1}, \end{equation} \tag{ 21 }$$

where λ = 1/4, n_e = 26 cm⁻³ the electron number density, μ = 0.7 the mean atomic weight, m_p the proton mass, $c_s=\sqrt{5\, k\, T_e/3\,\mu \,m_p}$ the speed of sound and T_e = 1.3 keV (Baganoff et al. 2003). Pessah & Melia (2003) estimate the accretion rate by

$$\begin{equation} \dot{M}_P = \pi r_\mathrm{acc}^2 \,\rho \, v\approx 10^9\,\rm g\; s^{-1}, \end{equation} \tag{ 22 }$$

with the accretion radius r_acc = 2GM_⋆/v²_eff ≈ 3 × 10¹¹ cm. Using the density from above and the Keplerian velocity at r = 1'' one obtains consistently $\dot{M_P}\approx 10^9\,{\rm g}\; {\rm s}^{-1}\,{\approx}\, \dot{M}_B$.

The radiation efficiency depends on the type of object considered (Haller et al. 1996). For neutron stars 10% is assumed (Pessah & Melia 2003), since the accreted material will fall onto a hard surface and the energy released can be radiated away, resulting in a luminosity of L_NS ≈ 10²⁹ erg s⁻¹. For SBHs due to the absence of a surface most of the emission will be thermal bremsstrahlung yielding only L_⋆ ≈ 2 × 10²⁰ erg s⁻¹ (Haller et al. 1996). This shows that L_X,ext cannot be due to SBHs, since one would need 10¹³ objects to explain the observed luminosity. In the case of neutron stars, one would need ≈20, 000 objects within r ≲ 1'' in order to account for the observed luminosity, corresponding to η ≈ 0.07. However, this number exceeds the estimate of the segregated cusp model of Hopman & Alexander (2006), who predict only ≈100 neutron stars there.

Muno et al. (2005) report an overabundance of X-ray transients in the inner parsec of the GC compared to the overall distribution of X-ray sources. The sources are classified as X-ray binaries (XRBs). These authors suggest a dynamical origin of the XRBs, namely an exchange of type Binary +  SBH →  XRB +  Star. The rate density for this reaction is

$$\begin{equation} \gamma _+=n_\star \,n_\mathrm{b}\, \Sigma \, \sigma _1, \end{equation} \tag{ 23 }$$

where n_b is the density of binaries,

$$\begin{equation} \Sigma = \pi a^2 + 2\pi a G (M_\mathrm{b}+M_\star)/\sigma _1^2 \end{equation} \tag{ 24 }$$

the exchange cross-section and σ₁ = (GM_MBH/3r)^1/2 the one-dimensional velocity dispersion. According to Muno et al. (2005) the number of XRBs is limited by dynamical friction, which yields a characteristic life time of τ = 10 Gyr(M_⋆/M_☉)⁻¹(r/)^1/2. Refining this argument, we also take into account the back reaction XRB +  Star → Binary + SBH and assume for simplicity equal exchange cross-sections for forward and backward reaction. Both effects together yield a rate density of

$$\begin{equation} \gamma _-=\case{1}{2} n\, n_\mathrm{XRB}\,\Sigma \, \sigma _1 + \frac{ n_\mathrm{XRB}}{\tau }, \end{equation} \tag{ 25 }$$

where n is the number density of stars, and n_XRB is the number density of XRBs.¹¹ In equilibrium one has γ₊ = γ₋, which allows one to solve for n_XRB. After integrating n_XRB over volume out to 1 pc and assuming a = 0.1 AU, n = 10⁵⁻³(r/)⁻², n_b = 0.1n, M_b = 3M_☉, M_⋆ = 10M_☉ and n_⋆ = f_⋆n one obtains the number of XRBs in the central parsec as N_XRB = 7 × 10⁴f_⋆, where f_⋆ is the relative number of SBHs to ordinary stars. A certain fraction f_X of those will shine up as X-ray transients: f_XN_XRB = N_X. Calculating η from this yields

$$\begin{eqnarray} &&\eta = f_\star \frac{M_\star }{M_\mathrm{MBH}} \int _\mathrm{peri}^\mathrm{apo} 4\pi r^2 n(r) dr \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} &&= 3.8\times 10^{-7}\, \frac{N_\mathrm{X}(<1)}{f_\mathrm{X}}, \end{eqnarray} \tag{ 27 }$$

relating the number of X-ray transients in the central parsec with η. Using the values f_X ≲ 0.01 and N_X = 4 (Muno et al. 2005), we obtain η ≈ 1.5 × 10⁻⁴.

A more detailed investigation by Deegan & Nayakshin (2007) shows that within r < 0.7 pc a cusp of SBHs with ≈20, 000 members is consistent with the number of discrete X-ray sources in the GC. Converting this number for the assumed profile of n(r) ∼ r^−7/4 into η yields η ≈ 2.1 × 10⁻⁴, which is very similar to our estimate in the previous paragraph.

There are at least three other aspects of an extended mass component in the GC which are worth mentioning but beyond the scope of this paper.

• 1.  
  Star formation in the presence of a dark cluster. The process of star formation in the GC might be altered significantly by the presence of a substantial dark component. The additional perturbative gravitational forces due to the SBHs might assist star formation since they increase the inhomogeneities in a star forming gas cloud. On the other hand, close encounters between individual clumps and SBHs might result in disruption of the clumps.
• 2.  
  Interaction of the spin of the MBH with the dark cluster. The spin of the MBH is subject to evolution by several processes. While gas accretion and major mergers can increase the spin, the accretion of SBHs tends to decrease the spin (Hughes & Blandford 2003; Gammie et al. 2004), assuming many random infall events of isotropically distributed SBHs. Furthermore there is the general relativistic spin-orbit-coupling between a SBH and the MBH spin, leading to a change of the spin direction of the MBH but not to a change of its modulus (Lodato & Pringle 2006).
• 3.  
  Dark matter. Dark matter, which is widely accepted in cosmology, might also show up in dynamic measurements in the GC. However, Gendin & Primack (2004) show that the density of the dark matter at 0.01 pc is ρ_DM ≈ 3 × 10⁵ M_☉/³, which is negligible compared to the theoretically predicted stellar density there (Hopman & Alexander 2006). See also Vasiliev & Zelnikov (2008).

The various estimates for η all consistently point toward an expected value of ≈10⁻³–10⁻⁴, approximately two orders of magnitude smaller than what we can measure with orbital dynamics today. Nevertheless, some astrophysical insights are possible.

Among the most important scientific questions in the GC is the origin of the S-stars, being a population of apparently young stars close to the MBH (Ghez et al. 2003; Martins et al. 2008). One possible origin is that these stars have reached their current orbits by TBR. Then the S-stars would have an isotropic, thermal velocity distribution, naturally explaining the observed random distribution of angular momentum vectors (Figure 19). The number of stars visible is by far too low to make TBR efficient enough to account for the present population of S-stars. A hypothetical cluster of SBHs could accelerate the process. The Chandrasekhar TBR timescale (Binney & Tremaine 1987) is given by

$$\begin{equation} t_r \approx \frac{0.34\, \sigma ^3}{G^2 \langle M_\star \rangle ^2 n_\star \ln \Lambda }. \end{equation} \tag{ 28 }$$

For a power-law cusp around an MBH, the velocity dispersion and the density are related to each other. Assuming ln Λ ≈ 10, a power-law index of −3/2 (which is approximately what is observed) and a population of stars with a single mass one obtains a relaxation time independent of radius

$$\begin{equation} t_r \approx 1.8 \times 10^{5} \,\mathrm{yr}\, \,\eta ^{-1} \left(\frac{m_\star }{10}\right)^{-1}. \end{equation} \tag{ 29 }$$

Thus, if the S-stars formed at the same epoch as the stellar disks 6 × 10⁶ yr ago (Paumard et al. 2006) and reached their present-day orbits by TBR, one needs η ≳ 0.033 for m_⋆ = 10 (Timmes et al. 1996). This exceeds the expectations by at least two orders of magnitudes. If the S-stars were not born in the presently observed disks, but in older, now-dispersed disks, one can use Equation (29) with the typical age of B stars (≈5 × 10⁷ yr). For m_⋆ = 10 this yields η ≳ 3.5 × 10⁻³, which could be marginally compatible with the other estimates for η.

In order to assess the expected progress, we simulated future observations with existing instrumentation and similar sampling. Continuing the orbital monitoring for two more years will lower the statistical error to Δη ≈ 0.01, corresponding to t_r ≈ 2 × 10⁷ yr. This means we will soon be able to test the hypothesis that the S-stars formed in the disks and reached their current orbits by TBR. Furthermore there is a chance to rule out any TBR origin of the S-stars observationally in the near future, namely when η ≲ 3.5 × 10⁻³ is reached.