IMPROVING GALACTIC CENTER ASTROMETRY BY REDUCING THE EFFECTS OF GEOMETRIC DISTORTION

To characterize the optical distortion in the NIRC2 camera, it is ideal to compare the measured set of stellar positions to those in a distortion-free reference frame. As this idealized reference frame does not exist, we choose observations of M92 (NGC 6341; α = 17 17 07.27, δ = +43 08 11.5) made with the well-characterized Advanced Camera for Surveys Wide Field Channel (ACS/WFC) on the Hubble Space Telescope (HST), which has a plate scale ∼49.9933 ± 0.0005 mas pixel⁻¹ and position angle (P.A.) offset = −00006 ± 00023 (van der Marel et al. 2007), as our reference frame. The static distortion in this camera has been corrected down to the ∼0.01 pixel (∼0.5 mas) level (Anderson 2005, 2007; Anderson & King 2006) and is therefore a useful reference for our purposes given the level of distortion in the NIRC2 camera. While several clusters were considered during the planning phase of this project, M92 was chosen because it had been extensively observed with ACS/WFC, was observable in the northern hemisphere during the summer, was sufficiently crowded, and had an isolated natural guide star (NGS) available. The HST observations of M92 used for this analysis were made on 2006 April 11 with both the F814W (I) and F606W (V) filters as part of the ACS Survey of Globular Clusters (GO-10775, PI: A. Sarajedini). The details of the observations and data reduction can be found in Anderson et al. (2008), and the catalog of positions was provided in advance of publication by J. Anderson. We used the Anderson (2007) correction for the linear skew in ACS to ensure that our reference frame was free of skew.

Observations of M92 were made from 2007 June to 2009 May using the AO system on the W. M. Keck II 10 m telescope with the facility near-infrared camera NIRC2 (PI: K. Matthews). Aside from the 2007 July data set, these observations were obtained upon completion of our primary science program for the night (GC astrometry), or when conditions were not optimal for the primary science program (e.g., clouds were present or seeing was relatively poor). All images were taken with the narrow field camera, which maps the 1024 × 1024 pixel array into ∼10'' × 10'' field of view, and through the K' (λ₀ = 2.12 μm, Δλ = 0.35 μm) bandpass filter. While the natural guide star adaptive optics (NGSAO) system was used to obtain the majority of the data, the laser guide star (LGS) AO system was used for one run in 2008 June. The NGSAO atmospheric corrections and the LGSAO low-order, tip-tilt corrections were made using visible observations of USNO-B1.0 1331-0325486 (R = 8.5 mag). The resulting image point-spread functions (PSFs) had Strehl ratios of ∼0.55 and FWHM of ∼50 mas, on average.

M92 was observed at 79 different combinations of P.A.s) and offsets (see Figure 1), with three identical exposures taken at each pointing. This allowed for a given star to fall on several different parts of the detector over the course of the observations. We note for clarity that the reported P.A. value is the angle (eastward) of the camera's columns with respect to north. The field of view of NIRC2's narrow camera contained the NGS in each pointing, and in most cases two other nearby stars, which are circled in Figure 2; this facilitated the process of combining the positional information from all of the different pointings. Table 1 provides the details of the NIRC2 M92 observations.

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The M92 images are calibrated and stellar positions are measured from these images using standard techniques. Specifically, the images are first dark- and sky-subtracted, flat-fielded, and bad-pixel and cosmic-ray-corrected. The images are then run through the PSF fitting program StarFinder (Diolaiti et al. 2000), which is optimized for AO observations of crowded stellar fields to identify and characterize stars in the field of view. StarFinder iteratively constructs a PSF from a set of bright stars in the field, which have been pre-selected by the user. For M92, a total of 16 stars spread out across the detector are used to obtain a PSF that is representative of the entire field. The resulting PSF is then cross-correlated with the image and detections with a correlation peak of at least 0.7 are considered candidate stars. Relative astrometry and photometry are extracted by fitting the PSF to each candidate star. This results in a star list, which contains the NIRC2 pixel coordinates for the detected stars, for each of the 237 images.

Final star lists for each pointing are produced by combining each set of three star lists from images with the same observational setup. Positions are taken from the first of the three images and the centroiding uncertainties are estimated empirically using the three images at each pointing and computing the rms error of each star's position.⁷ The median centroiding uncertainty is ∼0.035 pixel (∼0.35 mas; see Figure 3). Two initial criteria are used to trim out false or problematic source detections. First, only stars detected in all three images are kept. Second, we remove the two brightest stars (the NGS and a comparably bright star ∼51 to the east that appears in the images of 147 out of 237 pointings) and any other source identified within a 60 pixel (∼06) radius of these stars (see Figure 2). These two sources are ∼1 mag brighter than any other detected star and are often detected at levels that saturate the detector. Saturation leads to poor PSF matching with the empirical PSF estimate, and consequently poor positional estimates for these two stars, as well as ∼20–50 false detections in their halos. With these selection criteria, the 79 final star lists contain a combined total of 3846 stellar position measurements of more than 150 independent stars.

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Two types of GC observations were obtained with the NIRC2 narrow camera and the LGSAO system at Keck. First, two sets of deep GC observations centered roughly on Sgr A* offer a test of the new distortion model because something is changed in each. In a data set from 2007 May, previously reported in Ghez et al. (2008), 103 frames were obtained with a camera orientation of P.A. = 0° (May 17) and another 20 frames were collected at P.A. = 200° (May 20). In a new data set collected on 2008 May 15, the first 22 images were obtained with the LGSAO system and the remaining 112 images were taken with the NGSAO system; the Strehl ratio was 31% and 22% for the LGSAO and NGSAO data that evening, respectively. Second, three new epochs of observations, designed to measure the relative positions of seven IR-bright SiO masers, were carried out in 2008 May, 2009 June, and 2010 May, bringing the total number of such observations to six. These observations are primarily to generate an astrometric reference frame in which Sgr A* is at rest, but are also used as an additional test of the new distortion solution. The 2008 May data set is identical to those reported in detail in Ghez et al. (2008, Appendix C) and consists of a total of 27 images, which were obtained in a widely dithered (6'' × 6'') nine-point box pattern with three images at each of the nine pointing positions. The 2009 June data set differed in that we repeated the box pattern three times, resulting in a deeper image by ∼0.5 mag, and the 2010 May data set was similar in total exposure time to the initial mode, but made a trade of less co-adds for more recorded images in an attempt to compensate for the shorter atmospheric coherence times that evening. We note that these observations were generally carried out under poorer seeing conditions than our other GC observations (see below), since the quality of the seeing is not the limiting factor in measuring the positions of the SiO masers. These observations are summarized in Table 2. USNO 0600-28577051, which is offset by 94 E and 169 N from Sgr A*, served as the tip-tilt star for all of the LGSAO observations and as the NGS for the NGSAO observation.

All the GC images are calibrated in a similar manner to what was carried out with the M92 data sets, with a few exceptions. First, all images are corrected for DAR (see Section 3.1) and the geometric optical distortion using the new model derived in Section 3.1. Second, all the images from each epoch (and each configuration in the case of the deep central 10'' × 10'' observations) are combined into a final average map as described in Ghez et al. (2008). In addition, three subset images, each containing 1/3 of the data, are created for estimating centroiding uncertainties.

Astrometry was extracted using StarFinder (Diolaiti et al. 2000) in a similar manner as in Ghez et al. (2008), with a few minor modifications. Images from the individual pointings were analyzed with a correlation threshold of 0.9 in order to minimize spurious detections. The maser mosaics and the deep central 10'' × 10'' average maps were run at a correlation threshold of 0.8, but used an improved algorithm to minimize spurious detections, which is described in Appendix A.

To find the best-fit model for NIRC2's geometric optical distortion from the M92 observations, one must account for the fact that the ACS/WFC data do not suffer from DAR, while the NIRC2 data come from ground-based observations and therefore will be affected by the Earth's atmosphere (see Figure 4).

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DAR will compress an image along the zenith direction, causing the apparent separation between a pair of stars to be smaller than their true separation. Since the stellar positions are first geometrically distorted by the atmosphere and then the telescope/instrument, it is best to "undo" these effects in the reverse direction. Assuming that one has a distortion solution for NIRC2 at hand, to convert observed positions to their counterparts in rectilinear space, the distortion solution should first be applied to the observed positions and then corrected for DAR. When attempting to find the distortion solution, however, the positions observed at NIRC2 cannot be corrected for DAR to compare with HST because we do not know the distortion-corrected positions this process require. Instead, DAR is applied to the HST positions to produce a set of reference positions that should correspond to distortion-free positions as observed through the atmosphere. Because the effects of DAR depend on the elevation and, to a much lesser extent, the atmospheric conditions of the observations, it is necessary to create a separate DAR-transformed ACS/WFC star list for each NIRC2 image and associated star list. To account for DAR, we follow the prescription for DAR given in Gubler & Tytler (1998). The stellar positions are only corrected for achromatic DAR, as the error from chromatic DAR is negligible (<0.2 mas) relative to the residual distortion in ACS/WFC (∼0.5 mas). Neglecting chromatic effects, the DAR term (ΔR) depends on (1) the observed zenith angle of star 1, (2) the wavelength of the observations, (3) the observed zenith separation of star 1 and star 2, (4) the temperature at the observatory, (5) the pressure at the observatory, and (6) the relative humidity at the observatory. The atmospheric parameters of interest are downloaded from an archive maintained at the Canada–France–Hawaii Telescope (CFHT)⁸ for the night of each observation. These values are recorded every five minutes, allowing us to find the appropriate atmospheric conditions on Mauna Kea within three minutes of the observation (Lu 2008). As shown in Figure 4, the magnitude of the achromatic effect over the range of elevations for the M92 observations is expected to be ∼2–4 mas across NIRC2's 10'' field of view, along the elevation axis.

Each of the NIRC2 star lists described in Section 2.1 is then used as a reference coordinate system into which the ACS/WFC star list of positions is transformed. In this process, the ACS/WFC star list is transformed by minimizing the error-weighted (NIRC2 positional errors) net displacement for all the stars, allowing for translation, rotation, and a global plate scale (i.e., a four-parameter transformation model). This process is described in greater detail in Ghez et al. (2008) and Lu et al. (2009). Only sources that are cross identified in both the NIRC2 and ACS star lists are used in the remaining analysis. From the 79 separate alignments, a total of 2743 matches in stellar positions are obtained for a total of 150 independent stars. The differences in the matched positions, or deltas, are a result of the optical distortion in NIRC2.

The mapping of ACS positions to NIRC2 positions shows clear spatial structure across the detector, as expected from optical distortion (see Figure 5). However, some deltas are inconsistent with those in their immediate surroundings. These outliers are found by examining the vector deviations in 205 × 205 pixel bins and determining the average and standard deviation. Any 3σ outliers in either the X or Y direction are removed. A total of 75 deltas are removed based on this criterion. An additional cut (>3σ) in each of these bins is made on NIRC2 positional uncertainties, as they may vary with respect to detector position. This cut removes 73 data points, four of which were also eliminated by the first cut. These bins are examined a second time for vector outliers, as they often show a rather wide distribution. The average and standard deviation in each bin are recalculated and the vector outliers (>3σ) are removed once again. This resulted in an additional loss of 26 deltas.

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Many of the eliminated measurements come from common stars or images. We therefore remove all measurements of the 9 out of 150 stars and of the 8 out of 79 images that were eliminated more than 20% of the time by the sigma-clipping process. Many of these problematic stars have close neighbors (<02) that are not resolved or not well measured in the lower-resolution ACS observations (FWHM ∼ 70 mas for the F814W observations). The majority of the rejected frames have exposure times less than 10 s, while the remaining frames are at least 30 s. This results in significantly higher centroiding uncertainties, residual atmospheric effects, and fewer stars detected. We note that our trimming criteria mentioned above also result in the exclusion of all data from the 2008 June epoch (t_int = 9 s), which coincidentally was the only M92 data set taken in LGSAO mode. Although the 2008 April data set had relatively long exposure times (t_int = 48 s), the observations were heavily impacted by clouds and the AO system was often unable to remain locked on the NGS. These rejected frames show a value of −1.0 in the last column of Table 1. Our final data set consists of 2398 positional deviations between ACS and NIRC2, with median centroiding uncertainty for the NIRC2 images of 0.035 pixel (∼0.35 mas). The vector plot for this cleaned sample is shown in the bottom panel of Figure 5.

A bivariate B-spline is fit to the distortion map (Figure 5) using the SciPy package interpolate, and a look-up table sampled at each of the 1024 × 1024 NIRC2 pixels is subsequently produced. The effect of the smoothing factor (f; which is related to the number of nearest-neighbor measurements used to calculate the smoothing) used in the interpolation routine was investigated extensively in order to find a good compromise between the closeness of fit and the smoothness of fit. The residuals between the original distortion vectors in the bottom panel of Figure 5 and the computed shift at the nearest pixel (from the smoothed look-up table) were measured. The median deviation is found to increase until f ∼ 150, where it plateaus at a value of ∼0.27 pixel. We choose for our interpolation the smoothing factor that gave nearly the lowest median deviation, f = 135. Although the deviations were lower for distortion solutions created with smaller smoothing factors, the edge effects were prominent in the look-up tables and the distribution of deviations was much larger (for details on surface fitting and the choice of smoothing factors, see Dierckx 1995). The resulting look-up tables for shifts in X and Y are shown in Figure 6 and are produced in the form of FITS files that may be fed into the IRAF routine, Drizzle (Fruchter & Hook 2002), to correct for the optical distortion. Figure 7 (left) shows a histogram of these values.

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Statistical uncertainties in the distortion solution were computed by running a bootstrap analysis with 1000 trials. In each trial, we generated a random set of data pulled from the observed data, allowing for replacement after each data point was sampled, and then derived a distortion model from this resampled data set. The rms error with respect to the distortion solution (i.e., the actual distortion solution was taken as the average) was calculated at each pixel and the results are shown in the bottom of Figure 6. The average errors in X and Y are (σ_X, σ_Y) = (0.05, 0.04) pixel ∼ (0.5, 0.4 mas), respectively. We can see that the uncertainties are highest near the edge of the detector, where the spline algorithm is less robust. The uncertainties are also shown in the form of a histogram in Figure 7 along with a histogram of the distortion solution itself.

To solve for the global plate scale and orientation that results from this new solution, we re-reduce the raw NIRC2 observations of M92 from all epochs and apply corrections for distortion and DAR to these images. The distortion correction and DAR correction are applied to each image at the same time in the form of look-up tables using the Drizzle algorithm as implemented in IRAF (Fruchter & Hook 2002). The look-up tables are specified in Drizzle using the xgeoim and ygeoim keywords and are FITS files of the same dimensions as the science image. Because DAR depends on the zenith angle and atmospheric conditions, both of which vary in time, the look-up tables are created by first including the distortion solution and then applying the necessary DAR correction. Two FITS files, one for shifts in X and one for shifts in Y, are created for each NIRC2 observation and contain the shifts to be applied to each pixel in the image. From these distortion- and DAR-corrected NIRC2 images, star lists were generated and aligned with the original ACS star list (without DAR) as described above. The weighted average of the plate scale is 〈s〉 = 9.950 ± 0.003_stat ± 0.001_abs mas pixel⁻¹. The difference between the orientation given in the header of the NIRC2 images⁹ and the measured orientation is on average (weighted) 0254 ± 0014_stat ± 0002_abs. We use the rms errors of the average values from each epoch as the statistical uncertainties, and the absolute errors are the rms errors in the ACS/WFC plate scale and orientation angle (van der Marel et al. 2007). The results from each epoch of M92 data are shown in Table 4.

The new distortion solution and its errors are made public and may be obtained in the form of FITS files.¹⁰

There are two parts to the error in the distortion model when applied to a real data set: static distortion error (hereafter "residual distortion," discussed here) and time-varying effective distortion (discussed in Section 3.3). The residual distortion map on the detector is unknown, but is likely to be highly spatially correlated. The spurious position shift due to residual distortion when comparing two measurements is a function both of the size of the residual distortion itself, and the difference ΔR in location on the detector between the two measurements.

To estimate the size of the residual distortion from our model, we consider two cases. In the first, sets of images are taken at two very different P.A.s so that the distance ΔR between two measurements of the same object (and therefore the degree to which the residual distortion varies between measurements) is a strong function of position on the detector. In the second, images are taken at the same P.A. so that ΔR is constant over the image, but are widely dithered (60% of the detector side-length) so that the residual distortion is sampled at widely separated detector locations for all objects.

In both cases, we compare our new distortion model with two previous solutions, which we refer to as the "pre-ship" and the "P. Brian Cameron (PBC)" solutions. The pre-ship solution,¹¹ which is known to ∼4 mas, was found using a pinhole mask and is in the form of a third-order polynomial. The more recent solution by P. B. Cameron, also from a pinhole mask, is a fourth-order polynomial and improves upon the former solution mainly along the X axis.¹²

First, we use the two high precision data sets taken of the central 10'' × 10'' on 2007 May 17 and May 20 at two different P.A.s (0° and 200°) with roughly the same central position (Section 2.2). The typical NIRC2 positional uncertainties (the standard deviation, δ_pos) for the P.A. = 0° and P.A. = 200° images are ∼0.013 pixel and ∼0.018 pixel, respectively. The P.A. = 200° image was transformed into the P.A. = 0° image's coordinate system, again allowing for translation, rotation, and global plate scale. The differences in the aligned positions of stars with K < 14.5 are shown in Figure 8. This analysis gives an average residual distortion by comparing the positions of a star at two distinct locations on the detector. Our new solution shows significantly less residual structure than the previous solutions. To estimate the magnitude of the residual distortion (σ), we compute the rms error of the offsets (Δ) between the positions in the two images in the X and Y directions separately¹³ and correct for the positional measurement error from both images (δ):

$$\begin{equation} \sigma _{x} = \sqrt{\frac{1}{2}\displaystyle \sum _{i}^{N_{\rm stars}}\frac{(\Delta _{x,i} - \langle \Delta _x\rangle)^2}{(N_{\rm stars} - 1)} - \frac{1}{2}\big(\delta ^2_{{\rm pos},0^{{\circ }}} + \delta ^2_{{\rm pos},200^{\circ }}\big),} \end{equation} \tag{ 1 }$$

where N_stars is the number of stars matched across the two images, and $\delta _{{\rm pos},0^{\circ }}$ and $\delta _{{\rm pos},200^{\circ }}$ are the positional uncertainties (quoted above) for stars brighter than K = 14.5 in the P.A. = 0° and P.A. = 200° images, respectively. Average positional uncertainties are subtracted, as opposed to each star's individual uncertainty since most stars brighter than K = 14.5 have similar centroiding uncertainties. We compute σ_y similarly. We note that the division by 2 is necessary to determine the residual distortion incurred per NIRC2 image. This results in estimates of the residual distortion of (σ_x,0, σ_y,0) = (0.12, 0.11) pixel, (0.17, 0.28) pixel, and (0.27, 0.22) pixel for the new, PBC, and pre-ship solutions, respectively. Thus, the new solution results in smaller residuals by a factor of ∼2–2.5 over both of the previous solutions. The uncertainty in the distortion models has not been removed from these values, as the uncertainty in the PBC and pre-ship solutions is unknown. We can, however, remove the average uncertainty in our new model in quadrature (Figure 6, bottom) to obtain a final measure of the residual distortion: (σ_x,0, σ_y,0) = (0.11, 0.10) pixel.

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As an additional check on the distortion solution, we use the images from widely dithered (6'') 2008 May data set taken at P.A. = 0° (Section 2.2), which, unlike the test just described, maintain the independence of the X and Y axes as they are shifted relative to one another by only a translation. Largely dithered data sets are essential in testing the distortion solution because stars are placed on very different locations on the detector and therefore provide a sensitive test of residual distortion. These data have an average rms error on the positions of 0.05 pixel. Only four overlapping fields, each of which was imaged three times and whose centers are the corners of a 6'' × 6'' box, were examined from this data set (Figure 9). Only stars detected in at least 6 of the 12 images (and therefore at least two of the four overlapping fields) were kept in the analysis. The stars had to also be detected in all three exposures at each dither position, and the average of the positions in these three exposures was taken as the position at the corresponding dither position. The error (σ_x,i, σ_y,i) from residual optical distortion of each star's offsets (Δx, Δy) from IRS 16SW-E (which was in each of the four fields) was computed as

$$\begin{eqnarray*} &&\hspace{-190pt}\sigma _{x,i} =\nonumber \end{eqnarray*}$$

$${\selectfont\fontsize{8}{9}{\begin{eqnarray} &&\sqrt{\frac{1}{2} \displaystyle \sum ^{N_{\rm fields}}_j\frac{({\Delta }x_j - \langle {\Delta }x_j\rangle)^2}{N_{\rm fields} - 1} - \frac{1}{2}\frac{1}{N_{\rm fields}} \displaystyle \sum ^{N_{\rm fields}}_j\bigg (\frac{\delta _{{\rm pos,IRS\,16SW\mbox{-}E}}^2}{N_{\rm exp} - 1} + \frac{\delta _{{\rm pos},i}^2}{N_{\rm exp} - 1}\bigg)}\nonumber\\ \end{eqnarray}}} \tag{ 2 }$$

and likewise for σ_y,i, where we divide by the number of overlapping fields in which a star was detected (N_fields), and we correct for the NIRC2 positional measurement error (the standard deviation, δ_pos) per exposure (N_exp) for both IRS 16SW–E and star i. The factor of 2 in the denominator accounts for the fact that the distortion affects both stars, IRS 16SW–E and star i. These errors from the residual optical distortion are shown in Figure 10 for all three solutions. The median values (σ_x, σ_y) are (0.05, 0.06) pixel, (0.07, 0.15) pixel, and (0.18, 0.17) pixel, for the new, PBC, and pre-ship solutions, respectively. The new solution was found to significantly improve positional measurements overall as compared to both of the previous solutions and in particular, it is a factor of 3 better in the Y direction over the more recent PBC solution. As mentioned above, the uncertainties in the distortion models have not been removed from these values, as they are unknown for the two previous solutions. For the new distortion solution, however, the rms offsets are consistent with the average uncertainty in the distortion model itself.

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The errors computed using the pairwise analysis are approximately half the size of those reported using the GC observations at two P.A.s. We note that the pairwise analysis uses measurements that have uncertainties that are a factor of 3 larger than our other test and is therefore more sensitive to the removal of the measurement bias term. We take as the final residual error term for the new NIRC2 distortion solution the value from the first test: (σ_x,0, σ_y,0) = (0.11, 0.10) pixel.

While the new distortion solution represents a significant step forward in our astrometric capabilities, it still leaves ∼0.1 pixel or ∼1 mas residual distortion in LGSAO images that are widely dithered or taken at different P.A.s. The residual distortion is twice as large as the estimated uncertainties in the distortion solution (∼0.05 pixel; Figure 7) and must come from sources of uncertainty that are not accounted for in our analysis. Below we consider two possibilities, time-variable distortion, and the difference between NGSAO and LGSAO observations.

To test the stability of the camera's distortion, we created a distortion solution with data points from 2007, the year with the most data (N = 1711). A smoothing factor of f = 120 was used for the spline fitting and was determined in the same manner as our new distortion solution. As the number of data points from each of the years 2008 (N = 253) and 2009 (N = 489) was not sufficient to make separate distortion solutions for these years, we take the differences between the 2007-only distortion solution and the actual measured data from each of the individual years (see Table 1). We find no significant differences (0.05 ± 0.30 pixel and 0.09 ± 0.29 pixel for 2008 and 2009, respectively), suggesting that the distortion solution is stable within our measurement uncertainties. As a check, comparing the data from 2007 to the 2007-only solution gives an average difference of 0.01 ± 0.22 pixel, which has a smaller rms error since it was the data set used to create the single-year model. Based on this analysis, we conclude that there is no evidence for time-dependent changes.

While we tested our distortion solution on LGSAO data, the model itself was computed using only NGS data, as the six LGS frames from 2008 June were thrown out based on the cuts mentioned in Section 3.1. To test the possibility that the NGS and LGS AO systems have different distortion solutions, we compare GC data taken in both LGS and NGS modes, but otherwise the same setup and in the same night in 2008 May. The data were reduced using the usual data reduction steps (see Ghez et al. 2008), and final LGS- and NGS-only images of the GC were produced. The astrometric precision for each of these images was 0.007 pixel (NGS) and 0.012 pixel (LGS) for stars with K < 15. The NGS image was transformed into the LGS image's coordinate system allowing only for translation between the two frames. Differences between the transformed positions would indicate a possible difference in the distortion between the LGS and NGS observing modes. The rms difference in the aligned positions, corrected for measurement error bias, is only (Δ_x, Δ_y) = (0.06, 0.05) pixel (1σ) and is therefore comparable to the error in the distortion model (∼0.05 pixel, Figure 7). Thus, given the uncertainties in the distortion solution, we do not see a difference in the astrometry from images taken in NGS or LGS mode and conclude that this is a negligible contribution to the residual distortion.

While we have not identified the source of residual distortion, we can functionally include it by adding a constant term of 0.1 pixel (1 mas) in quadrature with the error map of the distortion (see Section 3.1) when analyzing astrometric data. We note that while these error terms are important for our localization of Sgr A*-radio in our infrared reference frame (Section 4.1), they do not come into consideration for relative proper motion measurements based on data using similar observational setups.

Measurements of seven SiO masers that are detectable in the radio and infrared wavelengths are used to transform the IR maser mosaics into an Sgr A*-radio rest frame. At radio wavelengths, each of these masers has well-measured positions and velocities with respect to Sgr A* (see, e.g., Reid et al. 2003, 2007). In this analysis, we use radio positions and proper motions from M. Reid (2010, private communication), who has improved the values compared to what is published in Reid et al. (2007) by adding one more epoch of observations and by applying a correction for the effects of differential nutation. The radio maser positions were propagated using the radio proper motion measurements to create a star list at the epoch of each IR mosaic. Each of the six infrared mosaics was aligned with a four-parameter model (two-dimensional translation, rotation, and a single pixel scale) to the radio maser star list by minimizing the error-weighted net displacements, D, for the masers, where the infrared positional errors include the positional rms errors (from the three subset images; see Section 2.2), as well as errors from the distortion model (see Appendix B). The net displacement and the weighting scheme used are described in Appendix A of Ghez et al. (2008). Errors in the transformation to the Sgr A*-radio rest frame in each epoch were determined using a jack-knife sampling technique, in which one maser at a time is excluded from the alignment. The various sources of error in our astrometry are broken down in Table 3. Distortion errors generally dominate the individual IR positional uncertainties of these masers, with the exception of IRS 12N and IRS 28, each of which is in only one pointing and is furthest from the tip-tilt star, where the AO corrections are the poorest. All the transformed IR positions agree with the radio positions to within ∼1σ of each other, suggesting that our uncertainties are well characterized and that we are not missing large systematic error sources.

The NIRC2 pixel scale and orientation values obtained from the SiO maser alignment are similar to those obtained from the M92 study (Table 4). We note, however, that the rms scatter shows a larger variation between the epochs than the uncertainty inferred from the jack-knife analysis of each epoch. We therefore take the rms values as our estimates of the uncertainties for our average pixel scale and orientation angle given in Table 4. The weighted average NIRC2 plate scale and angle offset from the IR to radio alignments are 9.953 ± 0.002 mas pixel⁻¹ and 0249 ± 0012, respectively. We average the results from the two methods (SiO masers and M92) to obtain our final values for the NIRC2 pixel scale and orientation angle, 9.952 ± 0.002 mas pixel⁻¹ and 0252 ± 0009, respectively. Thus, 0252 must be added to the presumed P.A. (ROTPOSN–INSTANGL) in order to get the true P.A. of an NIRC2 image (i.e., the NIRC2 columns must be rotated eastward of north by 0252).

Our transformed IR positions from the mosaic images provide a calibrated astrometric reference frame in which Sgr A* is at rest at the origin. Comparison of the SiO masers as measured in the IR and radio provide estimates of how well we can localize the position and velocity of Sgr A*-radio within this reference frame. For each maser, a linear motion model is obtained by fitting a line to the star's transformed infrared positions as a function of time. In this initial step, the positional uncertainties include only the centroiding and alignment errors; the distortion uncertainty (see Appendix B) is omitted here, since it is correlated across all epochs. Furthermore, the model fit is calculated with respect to T_0,IR, which is the average time of the IR positional measurements, weighted by the average of the X and Y positional uncertainties. Figures 11 and 12 show the resulting fits, which have an average reduced χ² value of 0.62 (see Table 5), and the uncertainties in the resulting fit parameters were determined from the covariance matrix. In these figures, and in all other cases, X and Y increase to the east and north, respectively. While the reduced χ² values suggest that the uncertainties may be overestimated (possibly due to IRS 7; see Appendix C), we err on the conservative side and do not re-scale our positional measurements. The velocities measured in the infrared are statistically consistent with the radio proper motion values.¹⁴ Based on the weighted average of the velocity differences between the infrared and radio reference frames, we conclude that Sgr A* is at rest to within ∼0.09 mas yr⁻¹ (compared to ∼0.03 mas yr⁻¹ in the radio reference frame).

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Table 5. Astrometry of SiO Masers

Maser	K	$\tilde{\chi }^{2}$a	IR T0	IR + Radio T0b	(IR−Radio) X Positionc	(IR−Radio) Y Positionc	(IR−Radio) X Velocity	(IR−Radio) Y Velocity
	(mag)		(year)	(year)	(mas)	(mas)	(mas yr−1)	(mas yr−1)
IRS 9	9.063	0.29	2007.7	2007.2	−1.32 ± 1.11 ± 0.34	0.35 ± 1.17 ± 0.57	0.18 ± 0.18 ± 0.08	−0.23 ± 0.26 ± 0.14
IRS 7	7.658	0.68	2007.7	2007.2	1.46 ± 1.11 ± 5.00	−6.24 ± 1.22 ± 5.00	−0.25 ± 0.09 ± 0.22	−0.28 ± 0.31 ± 0.30
IRS 12N	9.538	0.60	2006.4	2006.4	−1.88 ± 1.26 ± 0.42	−2.74 ± 1.59 ± 0.47	−0.28 ± 0.49 ± 0.05	1.24 ± 1.03 ± 0.06
IRS 28	9.328	1.19	2007.6	2007.6	−2.12 ± 1.34 ± 0.62	3.93 ± 1.41 ± 0.52	0.16 ± 0.47 ± 0.22	−0.95 ± 0.49 ± 0.24
IRS 10EE	11.270	0.64	2008.1	2007.7	0.62 ± 1.14 ± 0.29	1.75 ± 1.13 ± 0.32	0.17 ± 0.12 ± 0.04	−0.01 ± 0.24 ± 0.05
IRS 15NE	10.198	0.06	2007.4	2007.1	2.31 ± 1.16 ± 0.36	3.90 ± 1.31 ± 0.47	−0.33 ± 0.20 ± 0.05	0.68 ± 0.39 ± 0.06
IRS 17	8.910	0.90	2007.7	2005.1	−0.68 ± 1.22 ± 3.72	0.80 ± 1.08 ± 1.91	−0.23 ± 0.24 ± 0.99	−0.33 ± 0.15 ± 0.54
Weighted averaged		0.62		2006.9	−0.31 ± 0.55	1.44 ± 0.59	0.02 ± 0.09	−0.06 ± 0.14

Notes. Infrared (first) and radio (second) formal uncertainties are reported for each maser's position and velocity. Average distortion errors (σ∼1 mas) for each maser are added in quadrature to the infrared formal uncertainties. X and Y increase to the east and north, respectively. ^a$\tilde{\chi }^2$ is the average of the X and Y χ² per degree of freedom. ^bAverage T₀ from both IR and radio measurements weighted by velocity errors (see Equation (3)). ^cPositional offsets computed for the common epoch of 2006.9. ^dWeighted average and error in the weighted average are reported for all columns except the $\tilde{\chi}^2$ and T₀ columns, where we report the average.

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While increasing the time baseline and depth of the infrared measurements will improve upon this precision, we note that 4 out of 14 one-dimensional relative velocity measurements are already limited by the radio measurements. Further improvements in the radio will ultimately be required to create a reference frame that is more stable than 0.03 mas yr⁻¹ (the current limits from the radio measurements alone).

In order to compare the IR and radio positional measurements, two additional steps are required. First, we determine the time, T_0,(IR+radio), at which the positional difference is expected to have the smallest uncertainty for each maser,

$$\begin{equation} T_{0,({\rm IR+radio})} = \frac{\sigma ^2_{v,{\rm IR}}T_{0,{\rm IR}} + \sigma ^2_{v,{\rm radio}}T_{0,{\rm radio}}}{\sigma ^2_{v,{\rm IR}} + \sigma ^2_{v,{\rm radio}}}, \end{equation} \tag{ 3 }$$

where σ_v,IR and σ_v,radio are the velocity errors in the IR and radio, respectively. We take the average of these seven times, 2006.9, and find the IR and radio positions and uncertainties at this common epoch. Then the correlated distortion error (including both the uncertainty in the model and the residual distortion, ∼0.1 pixel) for each maser is added in quadrature to the formal uncertainty from the infrared fit. Comparison with the radio positions indicates that the position of Sgr A*-radio is known to within ∼0.57 mas in year 2006.9 in our infrared reference frame. We note that the localization of Sgr A*-radio in the IR reference frame is time dependent (Figure 13). Decreasing the impact of uncertainties in the IR distortion model, with either more highly dithered measurements or a better distortion model, would improve this precision. Overall, our current measurement uncertainties for both the position and velocity of Sgr A* are a factor of 3–4 better than earlier measurements—either those of Ghez et al. (2008), when treated in the same manner,¹⁵ or those reported by Gillessen et al. (2009b) using their "maser system" method, which is comparable to the method used here.¹⁶,¹⁷

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Sgr A* is detected in three of the IR maser mosaics and its position is consistent with Sgr A*-radio in our IR reference frame (Figure 13). In 2008 May, Sgr A* was as bright as K = 14.8 mag, which is one of the brightest IR detections of Sgr A* (see, e.g., Do et al. 2009; Sabha et al. 2010). In 2009 June and 2010 May, it was detected with K = 16.4 and K = 15.3, respectively. The magnitude of Sgr A* in 2009 is more in line with the faint end of what is observed for this highly variable source. The later detections were possible because the mosaics were deeper than the previous mosaics. No other mosaics show Sgr A* since these observations are composed of very short exposures to avoid saturation on the infrared-bright SiO masers. Figure 13 shows that all three detections are consistent within 3σ in X and Y with the position of Sgr A*-radio in the IR reference frame. Furthermore, the IR position is more consistent with the radio position (within 1σ) when Sgr A*-IR is in a relatively bright state and less prone to astrometric biases from underlying sources (e.g., Ghez et al. 2008; Gillessen et al. 2009b). This independent comparison confirms that we have a well-constructed reference frame.

By carrying out the same analysis in polar coordinates (as opposed to Cartesian), we find that the infrared and radio Sgr A*-rest coordinate systems show no net relative expansion (V_r) nor rotation (V_t/R) to within 0.12 mas yr⁻¹ and 0.11 mas yr⁻¹ arcsec⁻¹ (1σ), respectively (Table 6).

The Sgr A*-rest reference frame generated in Section 4.1 is more stable than the cluster-rest reference frame that has been used as the principle coordinate system for most of the previous proper motion studies. Here we are quantifying frame stability as the uncertainty in the velocity of the object that is defined to be at rest. Since the cluster-rest frame previously used is defined by assuming that a set of reference stars has no net motion, the translational stability of this reference frame is limited to be $\sigma / \sqrt{N}$, where σ is the intrinsic dispersion of the stars and N is the number of reference stars used. With a dispersion in the plane of the sky of roughly 3 mas yr⁻¹ (Trippe et al. 2008; Schödel et al. 2009; see also Section 4.3), ∼1100 reference stars would be needed in order to match the stability of our current Sgr A*-rest frame. This is a factor of ∼2–10 more than have been used in earlier studies (e.g., Trippe et al. 2008; Ghez et al. 2008; Lu et al. 2009; Schödel et al. 2009) and comparable to that of Gillessen et al. (2009b).

The stability of reference frames for observations made with fields of view that are too small to tie into the masers directly can be significantly improved by using secondary astrometric standards generated by the proper motion measurements for infrared stars other than the masers from the measurements presented in Section 4.1. To create these secondary standards, we use the same linear motion modeling done for the masers and estimate the positions and proper motions of stars that are detected in at least four of the six maser mosaics (N = 1445). These stars have K' magnitudes that are brighter than 16 mag and a χ² distribution that is consistent with their positional uncertainties and degrees of freedom (Figure 14). From these, we select the 1279 stars that have velocities less than 10 mas yr⁻¹ (to exclude mismatches) and velocity errors less than 1.5 mas yr⁻¹ in both the X and Y directions (Figure 15). The positions and proper motions of these stars are reported in Table 7, and the left panel of Figure 16 shows the cumulative distribution as a function of radius for the entire sample, as well as for those known to be old and young. Reference frames defined based on these secondary astrometric standards (which can be young or old), as opposed to one defined on the premise that the old stars have no net motion, are significantly more stable, and the exact advantage depends on the field coverage; for the Keck speckle and 10'' × 10'' AO data (see, e.g., Ghez et al. 2008; Lu et al. 2009), the translational stability is expected to be a factor of 17.5 and 12.5 times better, or 0.03 and 0.02 mas yr⁻¹, respectively (see right panel of Figure 16). Similarly the rotational stability, quantified as the uncertainty in the average rotational velocity, is expected to be a factor of 20 and 9 times better, for the Keck speckle and AO data, or roughly 0.03 and 0.02 mas yr⁻¹ arcsec⁻¹, respectively. This improvement has been seen and will be presented in a separate forthcoming paper (see also Yelda et al. 2010, conference proceedings showing these results).

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The proper motions for the secondary astrometric standards listed in Table 7 also offer the first opportunity to study the kinematic properties of the central stellar cluster directly in an Sgr A*-rest frame. Since all previous proper motions have been made in the cluster-rest reference frame, any net rotation of the cluster in the plane of the sky is removed from these earlier measurements. Rotationally, we find no motion in the plane of the sky in the tangential velocities (−0.09 ± 0.14 mas yr⁻¹) nor in the angular velocities (0.26 ± 0.36 mas yr⁻¹ arcsec⁻¹) about Sgr A* (Figure 17). However, we do detect rotation in the plane of the Galaxy, as has been previously reported by both Trippe et al. (2008) and Schödel et al. (2009) and is shown in Figure 18, which shows that there is a preferred angle for the proper motion vectors of 254 ± 163, consistent with the angle of the Galactic plane (314; Reid & Brunthaler 2004). This rotation in the Galactic plane is also seen in the flattening of the distribution of velocities in the direction parallel to the Galactic plane as compared to the velocities in the perpendicular direction (Figure 19). Translationally, we find that the weighted average velocity of all the stars in the sample that are not known to be young (N = 1202) is 0.21 ± 0.13 mas yr⁻¹ (∼8.0 ± 4.9 km s⁻¹ at 8 kpc) and 0.13 ± 0.14 mas yr⁻¹ (∼4.9 ± 5.3 km s⁻¹) in the X and Y directions (where X and Y increase to the east and north), respectively. Figure 20 compares the mean translational motion of the nuclear stellar cluster to the motion of Sgr A* (as determined in Section 4.1) and shows that there is no relative motion between the cluster and the black hole. The uncertainties quoted here are the rms errors of the weighted average velocities from a bootstrap analysis with 10⁵ trials, where in each trial a random set of data was sampled (with replacement) from the observed cluster velocity distribution.

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