The Galactic Center: Improved Relative Astrometry for Velocities, Accelerations, and Orbits near the Supermassive Black Hole

The central 10'' region of the GC (approximately centered around Sgr A*) has been monitored from the W. M. Keck Observatory with diffraction-limited, near-infrared imaging cameras since 1995. Images used in our multi-epoch astrometric analysis have been obtained in two different modes: speckle imaging from 1995 to 2005 (26 epochs) and laser guide star adaptive optics (LGSAO) imaging from 2005 to 2017 (30 epochs).

All speckle data sets were obtained in the K band (λ₀ = 2.2 μm) using the Near Infrared Camera (NIRC; Matthews & Soifer 1994; Matthews et al. 1996) on the Keck I telescope with a field of view (FOV) ∼5'' × 5'' and a pixel scale of 20 mas. speckle data is taken using very short exposure times (0.1 s) to freeze the atmospheric distortion. Details of the speckle observation in Table 1 can be found in Ghez et al. (1998, 2000, 2005b), Lu et al. (2005), Rafelski et al. (2007), and Z. Chen et al. (2019, in preparation).

Table 1.  Summary of Speckle Imaging Observations

Date		Frames		FWHM	Strehl Ratiob	Nstars	Klimc	${\sigma }_{\mathrm{pos}}$ d	Data Sourcee
(U.T.)	(Decimal)a	Obtained	Used	(mas)	Postprocess		(mag)	(mas)
1995 Jun 9–12	1995.439	15114	5286	46	0.62	380	16.4	1.47	Ref. 1
1996 Jun 26–27	1996.485	9261	2336	47	0.57	246	15.7	3.11	Ref. 1
1997 May 14	1997.367	3811	3486	46	0.65	358	16.4	1.29	Ref. 1
1998 May 14–15	1998.366	16531	7685	47	0.49	251	15.9	0.92	Ref. 2
1998 Jul 3–5	1998.505	9751	2053	42	0.85	226	16.2	0.86	Ref. 2
1998 Aug 4–6	1998.590	20375	11047	46	0.65	293	16.4	0.75	Ref. 2
1998 Oct 9	1998.771	4776	2015	47	0.52	216	16.0	1.32	Ref. 2
1999 May 2–4	1999.333	19512	9427	45	0.78	344	16.7	0.69	Ref. 2
1999 Jul 24-2	1999.559	19307	5776	44	0.77	303	16.8	0.37	Ref. 2
2000 Apr 21	2000.305	805	662	48	0.46	141	15.4	2.48	Ref. 3
2000 May 19–20	2000.381	21492	15591	45	0.62	402	16.9	0.56	Ref. 3
2000 Jul 19–20	2000.548	15124	10678	46	0.61	410	16.7	1.10	Ref. 3
2000 Oct 18	2000.797	2587	2247	47	0.46	209	15.8	1.70	Ref. 3
2001 May 7–9	2001.351	11343	6678	45	0.58	344	16.3	0.95	Ref. 3
2001 Jul 28–29	2001.572	15920	6654	46	0.73	351	16.9	0.57	Ref. 3
2002 Apr 23–24	2002.309	16130	13469	46	0.65	452	16.9	0.90	Ref. 3
2002 May 23–24	2002.391	18338	11860	44	0.74	436	17.1	0.58	Ref. 3
2002 Jul 19–20	2002.547	8878	4192	48	0.52	300	16.5	1.23	Ref. 3
2003 Apr 21–22	2003.303	14475	3715	48	0.53	185	15.5	1.25	Ref. 3
2003 Jul 22–23	2003.554	6948	2914	46	0.65	276	16.2	1.16	Ref. 3
2003 Sep 7–8	2003.682	9799	6324	46	0.67	356	16.6	1.80	Ref. 3
2004 Apr 29–30	2004.327	20140	6212	47	0.66	275	16.1	0.51	Ref. 4
2004 Jul 25–26	2004.564	14440	13085	47	0.61	379	16.9	0.90	Ref. 4
2004 Aug 29	2004.660	3040	2299	49	0.79	289	16.3	0.83	Ref. 4
2005 Apr 24–25	2005.312	15770	9644	47	0.54	282	16.3	0.70	Ref. 5
2005 Jul 26–27	2005.566	14820	5642	50	0.64	332	16.6	1.79	Ref. 5

Notes.

^aDecimal year is defined as the Julian epoch year: 2000.0 + (MJD—51544.5)/365.25. ^bThe Strehl ratio reported here is the postprocess value from the deconvolution method. ^cK_lim is the magnitude at which the cumulative distribution function of the observed K magnitudes reaches 90% of the total sample size. ^dPositional error taken as error on the mean from the three subimages in each epoch and includes stars with K < 15. ^eData originally reported in (1) Ghez et al. (1998), (2) Ghez et al. (2000), (3) Ghez et al. (2005a), (4) Lu et al. (2005), and (5) Rafelski et al. (2007).

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Since 2005, we have used the Keck II LGSAO system (van Dam et al. 2006; Wizinowich et al. 2006) with the near-infrared camera NIRC2 (PI: K.Matthews) in its narrow-field mode, which has an FOV of ∼10'' × 10'' and a plate scale of 9.952 mas/pixel (Yelda et al. 2010). After 2014, the adaptive optics system and NIRC2 camera were realigned, and the plate scale changed to 9.971 mas/pixel (Service et al. 2016). In this paper, we include new data from 2014 to 2017, increasing the time baseline of the LGSAO data set by ∼30% (2005–2017, compared to 2005–2013 in Boehle et al. 2016).⁷ The new LGSAO data sets (Table 2) are obtained in an identical manner and have comparable quality to our previous observations. Additional details about our observational setup are presented in Ghez et al. (2005b, 2008), Lu et al. (2009), Yelda et al. (2012, 2014), and Boehle et al. (2016).

Table 2.  Summary of AO Imaging Observations

Date		Frames		FWHM	Strehl Ratio	Nstars	Klima	${\sigma }_{\mathrm{pos}}$ b	Data Sourcec
(U.T.)	(Decimal)	Obtained	Used	(mas)	Observed		(mag)	(mas)
2005 Jun 30	2005.495	10	10	62	0.29	794	16.2	0.39	LGSAO; Ref. 8
2005 Jul 31	2005.580	59	31	57	0.22	1753	19.0	0.25	LGSAO; Ref. 7
2006 May 3	2006.336	127	107	58	0.32	1951	19.2	0.05	LGSAO; Ref. 7
2006 Jun 20–21	2006.470	289	156	57	0.35	2438	19.5	0.08	LGSAO; Ref. 7
2006 Jul 17	2006.541	70	64	58	0.34	2165	19.3	0.09	LGSAO; Ref. 7
2007 May 17	2007.374	101	76	58	0.35	2492	19.5	0.09	LGSAO; Ref. 7
2007 Aug 10, 12	2007.612	139	78	58	0.30	1877	19.1	0.08	LGSAO; Ref. 7
2008 May 15	2008.371	138	134	54	0.29	2080	19.4	0.06	LGSAO; Ref. 9
2008 Jul 24	2008.562	179	104	58	0.32	2175	19.3	0.04	LGSAO; Ref. 9
2009 May 1, 2, 4	2009.340	311	149	57	0.35	2297	19.4	0.04	LGSAO; Ref. 9
2009 Jul 24	2009.561	146	75	62	0.25	1699	18.9	0.09	LGSAO; Ref. 9
2009 Sep 9	2009.689	55	43	62	0.31	1920	19.1	0.11	LGSAO; Ref. 9
2010 May 4–5	2010.342	219	158	63	0.28	2027	19.2	0.06	LGSAO; Ref. 9
2010 Jul 6	2010.511	136	117	62	0.29	1950	19.1	0.08	LGSAO; Ref. 9
2010 Aug 15	2010.620	143	127	61	0.26	1819	19.1	0.07	LGSAO; Ref. 9
2011 May 27	2011.401	164	114	66	0.25	1557	18.9	0.13	LGSAO; Ref. 9
2011 Jul 18	2011.543	212	167	59	0.26	2017	19.3	0.07	NGSAO; Ref. 9
2011 Aug 23–24	2011.642	218	196	60	0.32	2354	19.5	0.05	LGSAO; Ref. 9
2012 May 15, 18	2012.371	290	201	59	0.30	2256	19.4	0.05	LGSAO; Ref. 10
2012 Jul 24	2012.562	223	162	58	0.33	2317	19.5	0.06	LGSAO; Ref. 11
2013 Apr 26–27	2013.318	267	140	68	0.22	1264	18.4	0.09	LGSAO; Ref. 11
2013 Jul 20	2013.550	238	193	58	0.33	1788	19.1	0.08	LGSAO; Ref. 11
2014 May 19	2014.380	173	147	64	0.28	1468	18.7	0.08	LGSAO; Ref. 12
2014 Aug 6	2014.596	137	127	56	0.33	1760	19.1	0.08	LGSAO; Ref. 12
2015 Aug 9–11	2015.606	288	203	58	0.33	1887	19.1	0.06	LGSAO; Ref. 12
2016 May 3	2016.338	253	166	60	0.31	1655	18.9	0.06	LGSAO; Ref. 12
2016 Jul 13	2016.532	207	144	60	0.26	1378	18.5	0.07	LGSAO; Ref. 12
2017 May 4, 5	2017.343	469	179	59	0.31	1674	19.0	0.06	LGSAO; Ref. 12
2017 Aug 9–11	2017.610	213	111	55	0.32	1476	18.7	0.09	LGSAO; Ref. 12
2017 Aug 23, 24, 26	2017.647	216	112	61	0.29	1216	18.0	0.06	LGSAO; Ref. 12

Notes.

^aK_lim is the magnitude at which the cumulative distribution function of the observed K magnitudes reaches 90% of the total sample size. ^bPositional error taken as error on the mean from the three subimages in each epoch and includes stars with K < 15. ^cData originally reported in (6) Ghez et al. (2005b), (7) Ghez et al. (2008), (8) Lu et al. (2009), (9) Yelda et al. (2014), (10) Meyer et al. (2012), (11) Boehle et al. (2016), and (12) this work.

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Boehle et al. (2016) combined the individual speckle frames using a "holography" method (Schödel et al. 2013) rather than the traditional shift-and-add method (Ghez et al. 2003). We improved this algorithm by using multiple reference stars, removing nearby confusion sources, and subtracting sky background when extracting the PSF. To estimate the positional errors, 100 realizations of each epoch were created using a bootstrap sampling with replacement to combine frames for that epoch. Starfinder is run on each of these realizations, and the standard deviation of the positional measurements of stars are adopted as the uncertainty. We find that this method provides a more robust estimate of astrometric and photometric uncertainties (Z. Chen et al. 2019, in preparation). The new holography method allows us to use more exposures and increases the sensitivity and FOV of the speckle images, resulting in more stars detected at fainter magnitude. On average, 309 stars are detected in speckle data sets down to a 90% limiting magnitude of K_lim = 16.4, with an average position error of 1.1 mas in each direction.

The NIRC2 images were reduced using our standard NIRC2 reduction pipeline, which includes corrections for geometric distortion and differential atmospheric refraction. We used a new photometric calibration described in Gautam et al. (2019) to recalibrate all LGSAO data sets. For observations between 2004 and 2013, we use the distortion solution from Yelda et al. (2010), while for observations obtained in 2014 and later, we use the new static distortion map from Service et al. (2016). This change in optical distortion is a result of changes in the alignment of the AO system (Service et al. 2016). The output of the NIRC2 pipeline is a single combined image for each epoch of data along with three subset images used for error analysis (Ghez et al. 2005a). The positional uncertainties, σ_pos, are the error on the mean of the positions in three subset images for each star. Starlists containing astrometry and photometry were extracted using the PSF-fitting algorithm StarFinder (Diolaiti et al. 2000; Yelda et al. 2014). On average, 1850 stars are detected in AO data sets down to a 90% limiting magnitude of K_lim = 19.0, with average position error of 0.09 mas in each direction. This exquisite precision is a result of the high signal-to-noise ratio (S/N). For a star at K = 15.5 mag, the typical S/N is approximately 3000 and 8000 for speckle and AO separately. Given that the FWHM = 45–65 mas and the lowest possible centroiding error is ${\sigma }_{{\rm{pos}}}\sim {\rm{FWHM}}/({\rm{S}}/{\rm{N}})$, we could potentially reach positional uncertainties as low as 0.01 mas. However, the astrometric precision and accuracy is mostly limited by systematic errors, due to uncertainties in the PSF and noise from the halos of the surrounding stars (Trippe et al. 2010).

3.1.1. Magnitude-dependent Additive Errors

The uncertainty of each star's position comprises three components: (1) ${\sigma }_{\mathrm{pos}}$, the measurement precision on each star's position, (2) ${\sigma }_{\mathrm{aln}}$, the uncertainty in the transformation process for each star, and (3) ${\sigma }_{\mathrm{add}}$, an additive error term to capture additional sources of error. ${\sigma }_{\mathrm{pos}}$ is calculated as the error on the mean of a star's position measured from the three subset images as described in Section 2.2. The alignment error ${\sigma }_{\mathrm{aln}}$ is estimated from a half-sample bootstrap analysis that repeats the alignment process 100 times with random sets of reference stars and is taken as the standard derivation of each star's position over the bootstrap samples. We found that the combination of ${\sigma }_{\mathrm{pos}}$ and ${\sigma }_{\mathrm{aln}}$ alone yields a χ² distribution on the acceleration fits with an unexpected tail of high values that is inconsistent with the standard χ² distribution. This is rectified with ${\sigma }_{\mathrm{add}}$.

Here, χ² is defined as ${\chi }_{{\boldsymbol{p}}}^{2}={\sum }_{i\in \mathrm{epochs}}{({{\boldsymbol{p}}}_{i}-{{\boldsymbol{p}}}_{\mathrm{mod},i})}^{2}/{{\boldsymbol{\sigma }}}_{p,i}^{2}$, where ${{\boldsymbol{p}}}_{i}$ is the position at epoch i, ${{\boldsymbol{p}}}_{{mod},i}$ is the linear/acceleration fit at epoch i, and ${{\boldsymbol{\sigma }}}_{p,i}$ is the positional uncertainty at epoch i (the quadratically summed combination of ${\sigma }_{\mathrm{pos}}$, ${\sigma }_{\mathrm{aln}}$,and ${\sigma }_{\mathrm{add}}$). The linear/acceleration fit is discussed in detail in Section 4. We will use ${\chi }_{\mathrm{acc}}^{2}$ for the χ² of the acceleration fits and ${\chi }_{\mathrm{lin}}^{2}$ for the χ² of the linear fits. The fit in the x and y directions are independent of each other, so there are χ² values in each direction. Ultimately, the χ² distribution of a sample of stars is an important factor for determining the quality of our analysis.

The additive error term, ${\sigma }_{\mathrm{add}}$, was previously determined to be 0.10 mas for AO data (Yelda et al. 2014) and was assumed to be constant with time, position, and brightness. However, the most likely source of additional astrometric error is from inaccurate estimation of the PSF wings, which predominantly impacts the astrometry of neighboring sources. The impact of this effect is largest when the brightness of the neighboring source becomes comparable to the flux in the wings of the many surrounding bright stars. We improved the determination of ${\sigma }_{\mathrm{add}}$ by implementing a magnitude-dependent error term for AO data that is added in quadrature, as described below (see also Fritz et al. 2010; Clarkson et al. 2012).

Defining the Sample of Good Stars—In order to evaluate the ${\chi }_{\mathrm{acc}}^{2}$ distribution and determine the optimal additive error, ${\sigma }_{\mathrm{add}}$, we define a sample of good stars chosen as those that (1) were detected in all 22 epochs of the 06–14 alignment, (2) showed no source confusion in any epoch, as defined in Section 3.4, (3) were located between 08 and 10'' from the SMBH in order to eliminate stars close to the SMBH that show high-order motion beyond acceleration, (4) were not outliers with large acceleration uncertainties, and (5) had ${\chi }_{\mathrm{acc}}^{2}$ smaller than 95 (5 times the dof), where dof is defined as the number of detections minus the number of free parameters. For example, here we have 22 epochs in total, and we use an acceleration fit with three free parameters per axis, thus the dof per axis equals to 22 – 3 = 19. The resulting sample of 370 good stars will be used to evaluate the χ² distribution for different additive errors.

Calculating the Additive Error—To find the optimal ${\sigma }_{\mathrm{add}}$ term, we perform an 06–14 cross-epoch alignment with additive errors from 0.01 to 0.47 mas in steps of 0.01 mas. We then divide the good stars into 9 mag bins with ∼41 stars per bin. The magnitude boundaries are 10, 12.55, 13.61, 14.25, 15.11, 15.39, 15.65, 15.87, 16.52, and 18.72. For each magnitude bin and each trial additive error, the ${\chi }_{\mathrm{acc}}^{2}$ distribution of good stars is calculated (Figure 1). Increasing the additive error causes the ${\chi }_{\mathrm{acc}}^{2}$ distribution to shift to smaller values.

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The optimal additive error should exhibit a ${\chi }_{\mathrm{acc}}^{2}$ distribution that closely matches the standard ${\chi }_{\mathrm{acc}}^{2}$ distribution with dof = 19. We perform a two-sample Kolmogorov–Smirnov test (K-S test) between the observed ${\chi }_{\mathrm{acc}}^{2}$ distribution and a standard ${\chi }_{\mathrm{acc}}^{2}$ distribution for each magnitude bin (Conover 1999). The K-S test statistic (D value), which quantifies the distance between the observed ${\chi }_{\mathrm{acc}}^{2}$ distribution and the cumulative distribution function of the standard ${\chi }_{\mathrm{acc}}^{2}$ distribution, is plotted as a function of additive error in Figure 2. The smaller K-S statistic D value suggests a better agreement with the theoretical predicted ${\chi }_{\mathrm{acc}}^{2}$ distribution. There is a clear trend showing that fainter stars require larger additive errors.

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In Figure 3, we plot the best-fit additive error in each magnitude bin. We derive an analytic function for the additive error as a function of magnitude given as

$$\begin{eqnarray}&&{\sigma }_{\mathrm{add}}(\mathrm{mas})=8\times {10}^{-5}\times {e}^{0.7\times (K^{\prime} -4.9)}+0.1.\end{eqnarray} \tag{ 1 }$$

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With this magnitude-dependent ${\sigma }_{\mathrm{add}}$, we now evaluate the contribution of each source of positional uncertainty to the total. The 2006 May 3 epoch (one of the 06–14 epochs) is used as an example to show how ${\sigma }_{\mathrm{pos}}$, ${\sigma }_{\mathrm{aln}}$, and ${\sigma }_{\mathrm{add}}$ change with magnitude in Figure 4. We can see that both ${\sigma }_{\mathrm{pos}}$ and ${\sigma }_{\mathrm{add}}$ increase with magnitude, almost exponentially, and they are comparable with each other. In contrast, ${\sigma }_{\mathrm{aln}}$ captures the transformation uncertainty and is around 0.1 mas with a weak dependence on magnitude. This is likely due to the fact that ${\sigma }_{\mathrm{aln}}$ is slightly larger in the outskirts of the field, where the faint stars are more easily detected away from the central concentration of bright stars.

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This magnitude-dependent additive error is incorporated for all other AO epochs. The ${\chi }_{\mathrm{acc}}^{2}$ distribution for the 06–14 alignment with this magnitude-dependent additive error is plotted in Figure 5 and shows good agreement with the standard ${\chi }_{\mathrm{acc}}^{2}$ distribution.

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3.1.2. Reference Stars for AO

3.1.3. Artifact Sources on the Edge of AO Epochs

The Keck AO observations deliver near-diffraction-limited spatial resolution; however, the PSF for the AO images is highly structured and varies across the FOV. We found that, near the edges of images from AO epochs, the PSF becomes elongated and structures in the first Airy ring are sometimes identified as separate sources. Figure 6 shows an example.

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Active work is underway to account for PSF variations across the field (e.g., the AIROPA project; Witzel et al. 2016). However, in this paper, we use a simple yet straightforward way to mark and remove those artifact sources. Artifact sources have some common properties. (1) artifact sources are usually within 70 mas of their primary sources. (2) Because the artifact sources are typically in the same relative position compared to their primary sources, the proper motions for the artifact sources and primary sources are similar. (3) The PSF elongation is usually in the same direction as the separation vector between the primary and artifact sources. Combined, these three properties enable us to find potential artifact sources coming from PSF field variability. Figure 7 uses epoch 2011 July 18 as an example to show how this correction works. First, we need to find pairs of sources based on the first two properties: we select every pair of stars that are within 70 mas of each other and have similar proper motions (δμ <3 mas yr⁻¹). Each blue cross is a detected star in epoch 2011 July 18, and the red points are the pairs that pass our selection criteria. The positional offset between each pair is shown by the black and red arrows. The offsets are all in similar directions in the lower right corner, which is strong evidence that they are artifact sources coming from PSF wings, and this agrees with what we see in Figure 6. For those stars that have the same offset direction, we will mark them as PSF artifact sources, which is shown by the red arrows in Figure 7. Among 2729 sources, 88 sources are found to be artifact sources in 26 AO epochs, accounting for about 3.2% of the total number of sources. The artifact sources are excluded from our final sample.

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3.1.4. Update Matching Velocity from 06–14 Alignment

Among the 30 AO epochs, eight were not taken in the standard 06–14 setup including the 2005, 2015, 2016, and 2017 epochs. They have higher order residual distortions from changes in the AO system optics, so the standard second-order polynomial transformation is insufficient to place these images into a common reference frame. Therefore, we need to make local distortion maps for these epochs. Our local distortion maps are calculated based on residuals, which is different from geometric distortions based on the on-sky measurements in Yelda et al. (2010) and Service et al. (2016). The way we calculate the distortion map is described as follows.

First, the 06–14 alignment is used to fit acceleration models for stars within 08 and linear models for stars outside of 08 from Sgr A*. With the acceleration/linear fit, all stars are propagated to each of the nonstandard epochs listed above. The differences between the propagated positions and the observed positions at that epoch are used to estimate the local distortion map. When calculating the distortion map, the following cuts are made to reduce noise in the final distortion map: (1) only stars brighter than 17 mag are used and (2) Outliers are removed by dividing the detected stars into 6 × 6 spatial boxes around the position of Sgr A*, calculating the mean and standard deviation of the distortion in each box, and trimming stars with offsets that are more than 2.5σ from the mean. This gives us the final sample that is used to calculate the local distortion map. On average, 530 stars are used in the final sample.

A high-order Legendre transformation is fit to the residuals to construct the local distortion map. To calculate the uncertainty of the local distortion map, we use the standard deviation among 100 local distortion maps estimated from a full-sample bootstrap with replacement. The distortion and its uncertainty can become very large on the edge of the distortion map as this region is sometimes outside the FOV for a given epoch and the distortion must be extrapolated from very few stars. Therefore, for those "edge" values, whose distortion or distortion uncertainty is larger than 0.3 pixel, we use the nearest "non-edge" value to replace it.

Finally, we need to find the best Legendre transformation order for our distortion map. The residuals will always improve as we increase the transformation order, but the number of free parameters also increases for a higher order transformation. So, we use the F-ratio test to find when an increase in the Legendre transformation order no longer significantly improves the residuals. This is done by finding the point when the (1 – p) value for the F-ratio approaches 0. The F-ratio is calculated as the Legendre transformation order is increased, and the p-value is the probability of obtaining this F-ratio (see Equation (B2) in Lu et al. 2016). Lower p-values indicate more significant benefits to increasing the order of the transformation polynomials. A similar test was done in Lu et al. (2016).

In this way, we derive the final local distortion maps for all non 06–14 AO epochs in arcseconds. The distortion maps are converted into pixel coordinates for each image using the previously derived transformations. The local distortion map for 2016 May 3 is shown in the first two panels of Figure 8 as an example. We apply the local distortion to the stars' original position and add the distortion uncertainty in quadrature. The third panel of Figure 8 shows how the residuals are reduced between the 06–14 alignment propagated starlist and the 2016 May 3 starlist after applying the distortion map. The median residual has been reduced from 0.06 pixels (black arrows) to 0.04 pixels (red arrows).

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Table 3 summarizes the local distortion median value and typical uncertainties for all eight non 06 epochs. To compare with the previously mentioned positional uncertainties in Section 3.1.1, we plot the uncertainties as a function of time in Figure 9. From this plot, we can see AO epochs have reduced ${\sigma }_{\mathrm{pos}}$ and ${\sigma }_{\mathrm{aln}}$ by almost an order of magnitude relative to speckle epochs. For speckle epochs, ${\sigma }_{\mathrm{pos}}$ is slightly larger than ${\sigma }_{\mathrm{aln}}$, and both contribute to the total positional error. For AO epochs, ${\sigma }_{\mathrm{add}}$ and ${\sigma }_{\mathrm{pos}}$ are constant among epochs, but ${\sigma }_{\mathrm{aln}}$ increases with time after 2011, the reason being our reference epoch is 2009, so alignment transformation gets worse when further away from reference epoch. For epochs with a local distortion map, the local distortion map error σ_dist is slightly larger than other uncertainties.

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New holography Data—In the new holography analysis (see Section 2.2), the stars' positional uncertainties are more accurately measured with bootstrapping, which captures unknown uncertainties from confusion or imperfect PSF. So, there is no need to add extra additive error. The typical positional error for bright stars within 2'' from Sgr A* is plotted in Figure 9.

Reference Stars for Speckle—From Section 3.1.2, we have 141 reference stars for AO epochs, but the speckle data have a smaller FOV and shallower detection limit. Therefore, we make a radius cut of stars within 4'' from Sgr A*. This gives us 43 stars out of 141 stars. More importantly, the speckle images were not taken in stationary mode, so the field changed over night. Stars on the edge of speckle images will have less frame coverage compared with stars in the inner region; therefore, they have relatively poor astrometric measurements. To account for this effect, we require stars to be detected in more than 60% of frames in speckle epochs relative to IRS16C, which is one of the brightest and cleanest stars in our FOV. This 60% criterion is lower compared to that of Boehle et al. (2016) because the new holography has more frames. The spatial distribution of AO and speckle reference stars is plotted in Figure 10.

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In conclusion, we have 141 reference stars for AO epochs and 43 reference stars for speckle epochs. This is less than what Boehle et al. (2016) used in her paper, mainly because we pursue the high quality of reference stars over the quantity. On average, we have 21 stars used as reference stars for speckle epochs and 133 stars for AO epochs, while Boehle et al. (2016) had 61 stars for speckle epochs and 230 stars for AO epochs.

Speckle Edge Removal—Since the number of frames that contribute to a given pixel near the edges of the speckle images can be very low because of the field rotation, stars would have poor astrometric measurements on the edge. Therefore, we decided to remove detections without enough frame coverage (fewer than 60% of the frames relative to IRS16C) for speckle epochs. As a result, 3601 detections from 1020 stars were removed, which is almost 44.8% of all stars over all speckle epochs.

Up to this point, 2700 stars have been identified with a total of 56,061 measurements across 56 epochs. However, some measurements are biased or incorrect due to mismatches from stellar crowding and confusion. When matching starlists from different epochs, we require that a new measurement must fall within a radius of 40 mas of the predicted position. However, when two stars get too close to each other (e.g., within the first airy ring), StarFinder cannot easily distinguish them. In this case, only one source will be detected, and the position for this source is biased as it is the flux-weighted average of the two stars. Fortunately, the probability of mismatches for any one star changes with time as stars move past each other in projection; so, the individual stars are not entirely lost if we can remove the biased epochs.

We choose to remove instances of confusion and mismatching using the following method: for each star A, we find nearby stars that are within 100 mas. For every nearby star B, if star B is 5 mag fainter than star A or brighter, then star B is defined as a possible confusion source. Notice that when star B is fainter than star A by more than 5 mag, the confusion from star B will not affect star A's position in a significant way. Both stars A and star B are required to be detected in more than 10 epochs for a reliable proper motion measurement. For star A and its possible confusion sources, if only one source is detected in some epoch where there should be two based on their proper motion prediction, this detection will be removed. Figure 11 shows an example of confusion removal. In total, 5677 detections are removed because of confusion, affecting 751 stars, accounting for 10% of the total detections.

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The sample of 103 stars with significant (>5σ) accelerations is analyzed to determine whether the best-fit accelerations are consistent with the expected acceleration from the SMBH (i.e., negative radial acceleration). The accelerations are projected into the radial (a_r) and tangential (a_t) directions with respect to the SMBH. Significant accelerating sources are defined when one of the following conditions is satisfied: (1) radial acceleration is 5σ larger than its uncertainty, $| {a}_{r}/{\sigma }_{{a}_{r}}| \gt 5$, or (2) tangential acceleration is 5σ larger than its uncertainty, $| {a}_{t}/{\sigma }_{{a}_{t}}| \gt 5$.

Then, we define a sample of significant SMBH-acceleration sources when all of the following conditions are satisfied: (1) significant negative acceleration, ${a}_{r}/{\sigma }_{{a}_{r}}\lt -5$; (2) tangential acceleration consistent with zero, $| {a}_{t}/{\sigma }_{{a}_{t}}| \lt 3$; and (3) negative radial acceleration 3σ smaller than allowed, $({a}_{r,\max }-{a}_{r})/{\sigma }_{{a}_{r}}\,\lt 3$, where ${a}_{r,\max }=-{GM}/{r}^{2}$ is the maximum allowed radial velocity from the SMBH, M is the mass of the SMBH, and r is the 2D projected distance. Because we do not have the line-of-sight distance, r will be the lower limit of the real 3D distance, so ${a}_{r,\max }$ will be the upper limit of the allowed radial acceleration a_r. These criteria result in 27 significant SMBH-acceleration sources.

The remaining 76 accelerating sources are all non-SMBH-acceleration sources, including significant tangential accelerations, significant positive radial accelerations, and too-large negative radial accelerations. The non-SMBH-acceleration sources are likely due to a number of factors, including unrecognized confusion and binarity. A potential binary candidate is analyzed in Section 6.2.

The large number of non-SMBH-acceleration sources suggests that there may be some contaminants in the SMBH-acceleration sample. To determine the degree of contamination, we check all 103 stars by eye, and divide them into three categories: (1) "well-fit"—stars that show no time-coherent residuals from the acceleration model, (2) "confused"—stars that are not fit well by an acceleration model, but show potential confusion from neighboring stars, and (3) "anomalous"—stars that are not fit very well by acceleration and show no potential confusion around them.

Figure 12 gives an example for each category from non-SMBH-acceleration sources. The first two columns are residuals from the acceleration fit, and the third column shows the proper motion of nearby stars. To determine the category an accelerating source belongs to, we first look at the quality of the acceleration fit. In the figure, we can see that S1-24 is fit well by an acceleration model given that the residuals are randomly distributed with time; so, S1-24 is a "well-fit" non-SMBH-acceleration source. If a star is not fit well by an acceleration model, like S1-27 and S2-239, we consider alternative explanations: confusion from nearby stars, bad measurements, or other physical explanations such as astrometric wobble due to binarity or microlensing. To exclude confusion, we check whether there are nearby stars that are not detected in all epochs (note that in Section 3.4 confusion events are removed for cases when both stars are detected in more than 10 epochs, so stars detected in fewer epochs will be missed as potential confusion sources). S1-27 has a nearby star 15star_258, which is only detected in a few epochs, so S1-27 is potentially confused with 15star_258 when 15star_258 is not detected.⁸ In comparison, S2-239 is very isolated. Therefore, S1-27 is classified as a "confused" non-SMBH-acceleration source, and S2-239 is an "anomalous" non-SMBH-acceleration source.

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In summary, Table 4 shows the number of accelerating sources in different categories. Figure 13 plots the spatial distribution of stars from orbit (33), acceleration (103), or linear motion (1048), where acceleration stars are further divided into the categories shown in Table 4. The 24 well-fit SMBH-acceleration sources are particularly interesting and are explained in detail in Section 5.2. The 76 non-SMBH-acceleration sources require further analysis, in which S0-27 is used as an example to explore the potential binarity in Section 6.2.

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Stars are divided into three categories based on the order of the kinematic model needed to fit the on-sky positions: orbit, acceleration, or linear motion (Section 4). To evaluate the aggregated goodness of the linear or acceleration fit, Figure 14 shows the median fitting residual as a function of time. Given the complexity of matching in the dense region around the SMBH, the orbit stars within the central 05 from Sgr A* are not included. Faint stars are more likely to be confused or biased in the measurement, so we only include stars brighter than 16 mag. These two cuts yield a sample of 553 stars that are fit by linear motion or acceleration models based on the categories they belong to. From Figure 14, we can see that the residuals from speckle epochs are eight times larger than AO epochs, which is expected given the larger positional uncertainty in speckle epochs. In fact, speckle and AO epochs both have residuals around 1σ relative to their positional uncertainty.

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To evaluate the quality of our astrometry measurements, we also compare the χ² distribution from linear fits and the distribution of velocity uncertainties, σ_V, with Boehle et al. (2016, referred to as B16 hereafter). To make a fair comparison, we require stars to be detected in more than 20 epochs in the B16 alignment, which gives us a total of 596 stars in the comparison sample. The left panel from Figure 15 clearly shows that the astrometry in this work is more precise (smaller σ_V) and more accurate (smaller χ²). Although the increased time baseline of our data set (up to 2013 versus 2017) partially contributes to the increased measurement precision, the methodology changes in our work increase both the precision and accuracy. From the middle panel, it is clear that almost all stars brighter than 15 mag have a smaller σ_V in our analysis. The median σ_V is reduced by 40% from 0.017 to 0.010 mas yr⁻¹ in the $\alpha \cos \delta$ direction and from 0.019 to 0.010 mas yr⁻¹ in the δ direction for those bright stars. For faint stars, a larger σ_V is expected due to their larger position uncertainties and will give a more robust uncertainty measurement. From the right panel, it is apparent that, for stars of all magnitudes, our analysis typically yields smaller χ² values. In summary, the majority of the stars favors our work relative to B16.

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Figure 16 plots the final reduced χ² distribution for the good stars defined in Section 3.1.1. A total of 328 stars are detected in the final sample among 352 good stars. With the improvements made in Section 3, the final χ² distribution for those 328 stars from 56 years of observation agrees with the predicted χ² distribution.

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From Section 4.1, we have identified 24 well-fit SMBH-acceleration sources outside of r = 05, which are summarized in Table 5. According to Do et al. (2009), 12 (50%) of them are young stars. This is a large fraction considering that only 7% of stars are young stars among our total of 1184 stars. This may suggest that young stars are very centrally concentrated compared to old stars.

Yelda et al. (2014) published six young accelerating sources, four of which are also found accelerating in our analysis: S1-3, IRS16C, S1-14, and IRS16SW. For those four young accelerating sources, the average a_r uncertainty has been reduced by a factor of 2 from 0.39 to $0.16\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{yr}}^{-1}$. The radial acceleration for S1-3, IRS16C, and IRS16SW agrees within 2σ between our analysis and that of Yelda et al. (2014). S1-14 is discrepant by 3σ, but the a_r uncertainties are very large in Yelda et al. (2014). This large discrepancy comes from several aspects, among which the most important reasons are short time baseline and underestimated acceleration uncertainties in Yelda et al. (2014). They only used data until 2011 (6 yr AO observation), while we use data until 2017 (12 yr AO observation), doubling the AO time baseline. Furthermore, the acceleration uncertainties are severely underestimated in Yelda et al. (2014) because they did not use the jackknife method as we do in Section 4. The other two stars that are accelerating in Yelda et al. (2014) are S0-15 and S1-12. S0-15 is also accelerating in our sample, but because of its poor acceleration fit and the presence of nearby stars (Figure 17), we categorize it as a confused source. S1-12 is not accelerating by more than 5σ in our analysis, so it is characterized as a linear moving star. Gillessen et al. (2009b) also reported the following five accelerating sources: S0-70, S0-36, S1-3, S1-2, and S1-13, which are all also in our sample as listed in Table 5.

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We plot our sample of 24 accelerating sources in Figure 18, among which 15 accelerating stars are reported for the first time. Accelerations are detected for sources at 4'', 3 times farther than previously published accelerations in Yelda et al. (2014).

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Since non-SMBH-acceleration sources could exist in all directions, we determine how many potential non-SMBH-acceleration sources exist in our 24 SMBH-acceleration sample. First, we define well-fit SMBH-acceleration sources as those where the star's tangential acceleration agrees with 0 within 3σ, $-3{\sigma }_{{a}_{t}}\lt {a}_{t}\lt 3{\sigma }_{{a}_{t}}$, and the star's radial acceleration is between 0 and ${a}_{r,\max }$ within 3σ, $({a}_{r,\max }-3{\sigma }_{{a}_{r}})\lt {a}_{r}\lt 3{\sigma }_{{a}_{r}}$. In the 2D space of a_t versus a_r, a star in the SMBH-acceleration sample would have a 1σ error ellipse that encloses a_t = 0 and would intersect with a line going from ${a}_{r,\max }$ to a_r = 0. Since ${a}_{r,\max }$ varies between different stars, we divided a_r and a_t by ${a}_{r,\max }$ to normalize them.

Figure 19 shows a_t/${a}_{r,\max }$ versus a_r/${a}_{r,\max }$, where each star is drawn as an ellipse with a semimajor axis of 3${\sigma }_{{a}_{t}}$/${a}_{r,\max }$ and $3{\sigma }_{{a}_{r}}$/${a}_{r,\max }$, respectively. The SMBH-acceleration stars intersect the segment on the horizontal axis from 0 to 1 and are shown in red, and the non-SMBH-acceleration stars are shown in blue. To determine the potential contaminants from non-SMBH-acceleration stars, we randomly draw segments with different orientations originating from (0, 0) and with length of 1. An example is shown with the bold blue ellipses. The average number of intersecting ellipses from non-SMBH-acceleration sources gives the contamination rate. We find that among our 24 SMBH-acceleration sources, 3.5 ± 1.1 could potentially come from non-SMBH-acceleration contaminants.

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To determine if the contaminants are correlated with radius, we perform similar tests for stars within 25 and beyond 25 of Sgr A*. Of the 17 SMBH-acceleration sources within 25, we find that 2.8 ± 1.2 may come from non-SMBH-acceleration contaminants, or 16%. Of the seven SMBH-acceleration sources beyond 25, we find that that 0.6 ± 0.8 may come from non-SMBH-acceleration contaminants, or 9%. Therefore, the contamination rate is not strongly radially dependent. Unfortunately, we cannot identify the specific contaminated sources at this time, and future observations are needed.

One of the goals of the methodology developed in this work is to construct a more stable reference frame for imaging observations of the GC. Currently, systematic uncertainties arising from the construction of the reference frame are assessed using orbital fits of short-period stars by including the astrometric position and velocity of the central SMBH as free parameters in the fit. A fit using S0-2's observations in Boehle et al. (2016) has shown an offset in the position of the SMBH of 2.5 mas and a linear drift of 0.55 mas yr⁻¹. In this section, we assess the improvements induced by our new methodology on S0-2's orbital fit. We use astrometric observations from the cross-epoch alignment presented in Section 3. In addition, we use S0-2's radial velocity measurements obtained using spectroscopic observations reported in Ghez et al. (2003, 2008), Boehle et al. (2016), and Chu et al. (2018), and similar measurements from the VLT reported in Eisenhauer et al. (2003, 2005) and Gillessen et al. (2009a, 2017; a summary of all of S0-2's radial velocity measurements can be found in Chu et al. 2018). The orbital fits are performed using Bayesian inference with the MultiNest sampler (Feroz & Hobson 2008; Feroz et al. 2009). The model used for the fit includes 13 parameters: the mass of the central SMBH, the distance to our GC R₀, the positions (${\alpha }_{0}^{* }$ and δ₀) and velocities (${\mu }_{{\alpha }_{0}^{* }}$, ${\mu }_{{\delta }_{0}}$, and ${v}_{{z}_{0}}$) of the SMBH, and the six orbital parameters of S0-2. Hereafter, we will use α* to represent $\alpha \cos \delta$.

Figure 20 plots the posterior joint probability distributions of ${\alpha }_{0}^{* }$ and δ₀ and of ${\mu }_{{\alpha }_{0}^{* }}$ and ${\mu }_{{\delta }_{0}}$ obtained using these observations. For comparison, we also present the posterior probability distributions obtained using S0-2 astrometry from Boehle et al. (2016). Our new alignment methodology reduces the SMBH offset in both directions by a factor of 2. The linear drift is also severely reduced in the δ-direction by a factor of 4. This analysis shows that the new methodology to align the different astrometric observations presented in this work and in Sakai et al. (2019) improves the quality of the orbital fit of S0-2 significantly.

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The age of the young stars (4–6 Myr) around the SMBH is much smaller than the relaxation timescale (∼1 Gyr) in the GC, so their dynamical structure will greatly help us distinguish different star formation mechanisms (Alexander 2017). Observations have found that around 20% of the young stars move in a well-defined clockwise disk (Levin & Beloborodov 2003; Paumard et al. 2006; Bartko et al. 2009; Lu et al. 2009; Yelda et al. 2014), while the rest are off-disk stars that are more randomly distributed, which may suggest an in situ formation theory. This 20% fraction might only be a lower limit because of stellar binaries (Naoz et al. 2018). An accurate disk membership derivation is required to divide disk and off-disk stars correctly, and later compare useful properties, like the initial mass function, between them.

Disk membership is derived from standard Keplerian orbital elements, including six kinematic variables (${\alpha }^{* },\delta ,z,{\mu }_{{\alpha }^{* }},{\mu }_{\delta },{v}_{z}$). Yelda et al. (2014) assigned disk membership for 116 young stars using data from 1995 to 2011. With a much longer time baseline from 1995 to 2017, which doubles the AO observation time compared to Yelda et al. (2014), and an improved astrometric analysis, our work will give a much more accurate estimate of the Keplerian orbital parameters and thus improve the disk membership assignment.

Among the six kinematic variables, only the line-of-sight velocity v_z comes from the spectroscopic measurements, while the remaining five parameters all come from the astrometric measurement. The projected position (α*, δ) and proper motion (${\mu }_{{\alpha }^{* }}$, μ_δ) can be directly derived from Section 4. For the 48 linear moving stars which are reported both in Yelda et al. (2014) and our final sample, the median velocity uncertainty is reduced from 0.073 to 0.019 mas yr⁻¹, almost a factor of 4. The most difficult part is to measure the line-of-sight distance (z). Fortunately, the absolute value of the line-of-sight distance can be derived using Equation (9) from Yelda et al. (2014), if we can measure significant a_r. Even if we do not have significant measurements of a_r, stars with 3σ acceleration upper limits smaller than ${a}_{r,\max }$ can provide lower limits on the line-of-sight distance. From Section 5.2, we have already detected 12 significant a_r measurements for young stars, two times more than Yelda et al. (2014), and our acceleration uncertainty is reduced by a factor of 2. Therefore, the five kinematic parameters (α*, δ, ${\mu }_{{\alpha }^{* }}$, μ_δ, z) can all be better estimated with our improved astrometry.

In Section 4.1, we identified 76 non-SMBH-acceleration sources with significant acceleration inconsistent with the gravitational force of only the SMBH. These non-SMBH-acceleration sources could be explained by binarity, microlensing, or unrecognized confusion. Here, we use S0-27 as an example to explore the potential for binarity. We use two models to fit its proper motion.

Model A includes only linear motion with four free parameters:

• 1.  
  ${\mu }_{{\alpha }^{* }}$: the proper motion of the star in the X direction;
• 2.  
  μ_δ: the proper motion of the star in the Y direction;
• 3.  
  ${\alpha }_{0}^{* }$: the fiducial position of the star in the X direction; and
• 4.  
  δ₀: the fiducial position of the star in the Y direction.

Then, the astrometric positions α* and δ are given by

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }^{* }(t) & = & {\alpha }_{0}^{* }+{\mu }_{{\alpha }^{* }}(t-{t}_{0})\\ \delta (t) & = & {\delta }_{0}+{\mu }_{\delta }(t-{t}_{0}).\end{array}\end{eqnarray} \tag{ 2 }$$

Here, t₀ is the fiducial time, chosen to be 1990.

Model B includes linear motion plus binarity, which adds six parameters, with 11 free parameters in total: ${\mu }_{{\alpha }^{* }}$, ${\mu }_{\delta }$, ${\alpha }_{0}^{* }$, δ₀, ω, Ω, i, e, t_p, P, a. Here, we follow the models in Koren et al. (2016), which is summarized as follows:

• 1.  
  ${\mu }_{{\alpha }^{* }}$: the proper motion of the system in the X direction;
• 2.  
  μ_δ: the proper motion of the system in the Y direction;
• 3.  
  ${\alpha }_{0}^{* }$: the fiducial position of the system in the X direction;
• 4.  
  δ₀: the fiducial position of the system in the Y direction;
• 5.  
  ω: the argument of periastron of the primary star's orbit in degrees;
• 6.  
  Ω: the longitude of the ascending node of the secondary star's orbit in degrees;
• 7.  
  i: the inclination of the system in degrees;
• 8.  
  e: the eccentricity of the Keplerian in orbit;
• 9.  
  t_p: the time of periastron passage in years;
• 10.  
  P: the period of the Keplerian orbit in years; and
• 11.  
  a: the photometric semimajor axis in milliarcseconds.

Then, the astrometric positions x and y are measured by

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }^{* }(t) & = & {\alpha }_{0}^{* }+{\mu }_{{\alpha }^{* }}(t-{t}_{0})+{BX}(t)+{GY}(t)\\ \delta (t) & = & {\delta }_{0}+{\mu }_{\delta }(t-{t}_{0})+{AX}(t)+{FY}(t),\end{array}\end{eqnarray} \tag{ 3 }$$

where A, B, F, and G are the Thiele–Innes constants (van de Kamp 1967); and X(t) and Y(t) are the elliptical rectangular coordinates.

We use PyMultiNest (Feroz & Hobson 2008; Feroz et al. 2009; Buchner et al. 2014) to explore the parameter space as it efficiently handles degeneracies inherent to binary orbit fitting. The priors and posteriors for the two models are presented in Tables 6 and 7.

Table 6.  Linear Model

Parameters	Prior	Best Fit
${\mu }_{{\alpha }^{* }}$ (mas yr−1)	flat prior in [0,1]	0.56 ± 0.01
μδ (mas yr−1)	flat prior in [2,4]	3.28 ± 0.01
${\alpha }_{0}^{* }$ ('')	flat prior in [0.12, 0.15]	0.1396 ± 0.0002
δ0 ('')	flat prior in [0.45, 0.50]	0.4808 ± 0.0003

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Table 7.  Linear+Binary Model

Parameters	Prior	Best Fit
${\mu }_{{\alpha }^{* }}$ (mas yr−1)	flat prior in [0, 1]	0.61 ± 0.01
μδ (mas yr−1)	flat prior in [2, 4]	3.15 ± 0.02
${\alpha }_{0}^{* }$ ('')	flat prior in [0.12, 0.15]	0.1388 ± 0.0003
δ0 ('')	flat prior in [0.45, 0.50]	0.4834 ± 0.0004
ω (degree)	flat prior in [0, 360]	42 ± 14
		222 ± 16a
Ω (degree)	flat prior in [0, 360]	136 ± 3
		316 ± 3a
i (degree)	flat prior in P(i) = sin(i) in [0, 180]	111 ± 3
e	flat prior in P(e) = e in [0, 1]	0.40 ± 0.08
tp (year)	flat prior in [2000, 2020]	2010.2 ± 0.4
P (year)	flat prior in [0, 30]	12.7 ± 0.6
a (mas)	flat prior in [0, 4]	1.55 ± 0.08

Note.

^aω and Ω have two solutions because they are degenerate with each other.

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The best-fit models for S0-27 are shown in Figure 21, where the left panel is from model A and the middle panel is from model B. Model B, with linear motion plus binarity, is a much better fit as it reduces the Bayesian information criterion (BIC) from 944 to 395, and increases the log of the likelihood from −463 to −172.

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From the linear+binary model, we are able to constrain the semimajor axis a and the period P. If we assume the primary component is a black hole, the photometric semimajor axis a would equal the secondary component's semimajor axis. If we further assume the primary component is much more massive than the secondary component, then we can use a as the total semimajor axis of the binary system. Using Kepler's law, the total mass of this binary system can be calculated using the following equation:

$$\begin{eqnarray}&&\displaystyle \frac{M}{[{M}_{\odot }]}={\left(\displaystyle \frac{a}{[\mathrm{mas}]}\displaystyle \frac{d}{[\mathrm{kpc}]}\displaystyle \frac{1}{| \cos i| }\right)}^{3}{\left(\displaystyle \frac{[\mathrm{yr}]}{P}\right)}^{2}.\end{eqnarray} \tag{ 4 }$$

With the best-fit solutions in Table 7, the total mass of the system is ∼265 ± 40 M_⊙. This is significantly larger than what we expect for a binary system. However, we made several simplistic assumptions that, if broken, would lower the mass. In the future, improved constraints on the total system mass will be derived by combining our astrometric measurements with multi-epoch radial velocity measurements. As astrometric monitoring of the GC continues, more candidate astrometric binaries are likely to be detected, and we will be able to constrain the binary fraction at the GC.