THE ACS SURVEY OF GALACTIC GLOBULAR CLUSTERS. X. NEW DETERMINATIONS OF CENTERS FOR 65 CLUSTERS

We began with the catalog as constructed above and converted the apparent F814W magnitudes into an absolute magnitude scale using the distance moduli for each cluster from the Harris Catalog (1996). All stars with an absolute F814W magnitude fainter than eight were immediately excluded. This has the effect of smoothing out the distribution, as stars fainter than this are not likely to be found near much brighter stars, and they are very incomplete in the cores of denser clusters. When stars of all magnitudes are included in the density distribution, many underdense regions around bright stars appear. These holes do not reflect the true underlying density distribution, and so an arbitrary cutoff was implemented to limit these effects (Figure 1). For denser clusters, stricter magnitude cutoffs were used. The densest clusters required that only stars brighter than an absolute F814W magnitude 2 be used, in order to remove the appearance of these holes. These regions of incompleteness begin to significantly overlap as you approach the center of the cluster. The number density of very bright stars in the cores of clusters is so large that very faint stars are almost impossible to reliably detect. As a result, fainter stars are almost entirely incomplete in the inner region of the cluster. Because density is a function of radius, and the ability to reliably detect a star is dependent on the density of the bright stars in the region surrounding that star, the incompleteness of fainter stars is also a function of radius. So, although incompleteness is also present to a lesser extent in stars above the implemented cutoff, the radial dependence of the incompleteness means that there will be no directional bias introduced by the incompleteness, as evidenced in Figure 2, and center determinations should not be affected.

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The coordinate system used for the determination is oriented to right ascension and declination, but with the necessary cosine term to generate a projection onto the plane tangent to the sky at the center of the field. The density of the cluster was then constructed over a grid of points. This grid was centered on the R.A. and decl. from the Harris Catalog (1996). The grid extends 100'' in each direction, sampling every 2''. As mentioned in Anderson & van der Marel (2010), significant overbinning is required to generate a smooth distribution. Circular bins were used with a radius of 25'' in most cases. However, clusters with very flat distributions required larger bins (up to a radius of 40'') to smooth the distribution. To remove the effects of the edge of the field on the density distribution, the density values for points on the sample grid that were within a bin radius of the edge of the field were set to 0, so as not to be included in any of the contours. That is to say, if the circular bin at that point included area outside of the field, that point was excluded from the ellipse fitting portion of this method.

After the distribution was generated and the edge effects were accounted for, the distribution was broken up into eight contours. The contours were spaced evenly between the minimum non-zero density and the maximum density. The outer three contours were not used due to the fact that they were often quite azimuthally incomplete, and this could potentially bias the ellipse fits. The innermost contour was also ignored as it is actually a solid two-dimensional region of points and could not be fit well with an ellipse. Each of the four remaining contours was then fit with an ellipse. This step is outlined in Figure 3. For most of the denser clusters, it is possible to fit many more than four ellipses (in some cases up to 20). However, larger numbers of smooth contours cannot be generated for the clusters with low numbers and/or flat distributions. In order to maintain consistency, eight contours and four ellipses were adopted as the maximum numbers that could be used for every cluster. The final center value was determined as the average center of these four ellipses. The uncertainty was estimated as the standard deviation of the ellipse centers. A plot of the density contours as well as the ellipses fit to these contours and their centers for the cluster NGC 1261 is shown in Figure 4. Plots such as this one are available for all 65 clusters in Table 1 as part of the supplementary material.

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View figure set (65 panels)

The consistency of the method discussed above as well as the error estimates for these measurements were analyzed through bootstrapping. Synthetic cluster distributions were generated and analyzed with the density contours method. The number of stars in the synthetic distributions ranged from 25,000 to 250,000 in increments of 25,000. These values cover roughly the range of the 65 clusters presented in this paper after the initial faint magnitude exclusion. For each increment between 25,000 and 250,000, one hundred distributions were generated, and their centers were measured through the density contours method. The standard deviation of the centers used in each determination, as well as the actual errors from the true center of the generated distributions were averaged over these 100 samples. The estimated uncertainty and actual error are plotted against the number of stars in Figure 5. This figure suggests that the standard deviation of the ellipse centers provides a very good estimate of the actual uncertainty in the determination.

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A distribution of the differences in X and Y from the catalog centers is shown in Figure 6. X and Y are oriented to R.A. and decl. and represent seconds of arc on the sky. The center coordinates were also compared to those presented in Noyola & Gebhardt (2006). There appears to be no systematic difference between the centers determined in this paper and those determined by Noyola & Gebhardt. The means of the differences in R.A. and decl. are consistent within the standard errors with a distribution about the origin. With respect to the centers in the Harris Catalog, the distribution of differences in X is consistent with a random normal distribution centered around zero, suggesting that there are no systematics biasing the centers in R.A. However, the average difference in the Y-direction is greater than the standard error of the differences in that direction, indicating that the distribution is just barely inconsistent with a center of zero. This suggests a possible systematic difference of about 17 in the decl. direction between the centers provided in Table 1 and the centers given in the Harris Catalog (1996). The centers presented by Noyola & Gebhardt also exhibit this systematic difference with a distribution of differences from the Harris Catalog centered approximately 1'' south of the origin.

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To begin, a grid of sample points was constructed around the center value from the Harris Catalog. The grid extends 40'' in each direction, sampling every 2''. At each point on the grid, the stars within a radius of 12 were divided into eight different pie slices corresponding to the cardinal and semi-cardinal directions in R.A. and decl. (see Figure 7). A cumulative radial distribution was then generated for each of the eight pie slices. The four pairs of opposing distributions were then compared. Figure 8 shows two pie slices and the corresponding cumulative distributions of the stars within these regions. To compare the distributions at a single point, the integrated area between the curves was used. Four pairs of distributions were compared, giving four values for each point on the grid. These values were then added together to yield a measure of the symmetry of the cluster at that point on the sample grid. When determined in this way, higher values correspond to less symmetry, as this indicates a larger difference in the opposing distributions at that point. This was done for every point on the sample grid. The result is a two-dimensional array of values that describe the symmetry of the cluster as a function of position in projected space.

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As discussed in the previous section and shown in Figure 1, bright giant stars create large holes in the apparent stellar distribution. These holes are often asymmetrically distributed due to their small numbers, and as a result large differences appear in opposing radial distributions at locations that are otherwise quite symmetric. Because of this effect, many of the denser clusters required that an incompleteness mask be applied before using the pie-slice method. In this case, simply imposing a strict magnitude cutoff does not work, as this lowers the number of stars in each pie slice to the point where the sample sizes are no longer useful. This is discussed in more detail in Anderson & van der Marel (2010, see Figure 12 of that paper).

It is important to note that a minimum of two pairs of orthogonal pie slices must be used to get a center determination with this method, as each pair of pie slices is only sensitive to changes along one axis. This is demonstrated in Figure 8. Moving a pair of pie slices perpendicular to their orientation keeps them in a position that is still symmetric along their axis. For this reason, a second pair of pie slices must be used to sample the symmetry in the direction orthogonal to the first. For our sample of clusters, it was found that eight pie slices worked better than four; however, switching to 16 pie slices resulted in sample sizes that were too small to be useful.

Determining the size of pie slices to use also takes some consideration. The radius of the pie slices must be determined based on the size of the field and the size of the sample grid. It is important to constrain the radius so that no point on the grid is affected by the edge of the field. It is also important that the radii of the pie slices remain constant across the grid, so as not to bias the determination. This creates a problem, as larger radii create larger samples and thus better distributions, however the sample grid must then be smaller so that the pie slices at the outer edges of the grid do not run out of the field. For this same reason, sampling over a larger area requires the radii of the pie slices to be smaller.

After the symmetry value is calculated at each point in the sample grid, contours can be created and a center determined by finding the most symmetric point on the grid. Due to the orientations of the pie slices, these contours will be eight-sided figures (or 2N-sided figures for N pairs of pie slices), and so it is not appropriate to fit them with ellipses as discussed in the density contours method. This method can only show the most symmetric point on the sample grid, and so the uncertainty of this center is limited by the spacing of the grid points. Even for centrally concentrated clusters with good numbers across the field, the symmetry did not appear to change significantly on scales smaller than 2'', and so a grid spacing of less than this was not used. This makes the centers determined by this method considerably less precise than those determined by the density contours method.