Internal Rotation in the Globular Cluster M53

In Boberg et al. (2016), we completed an abundance study of M53 using medium-resolution (R ≈ 18,000) spectra of red giant branch (RGB) stars in the cluster using the Hydra Multi-object spectrograph on the WIYN³ 3.5 m telescope. The RVs of our science targets were also measured to determine cluster membership and remove photometric interlopers from the sample. Multiple RV standards were observed at the beginning of each night when the Hydra spectra were collected. By measuring the RVs of our science targets against multiple standards, we found that the mean standard deviation of the individual measurements is 0.1 $\mathrm{km}\,{{\rm{s}}}^{-1}$, and they have a mean error of 1.1 $\mathrm{km}\,{{\rm{s}}}^{-1}$. The inter-night RV measurements of our sample were very stable with a mean standard deviation of 0.41 $\mathrm{km}\,{{\rm{s}}}^{-1}$. A full description of the methods used to measure the RVs is given in Boberg et al. (2016). In total, the Hydra sample contains 74 stars with a mean velocity of $-63.2\ \mathrm{km}\,{{\rm{s}}}^{-1}$, with a standard error on the mean of $0.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$. This value is consistent with the value found by Kimmig et al. (2015) ($-62.8\pm 0.3\,\mathrm{km}\,{{\rm{s}}}^{-1}$). We will show that the Hydra sample is crucial to measuring the internal rotation of M53 because it covers the innermost regions of the cluster. This sample has 12 stars within two times the core radius (${r}_{c}=0\buildrel{\,\prime}\over{.} 35$) and 49 stars within two times the half-light radius (${r}_{h}=1\buildrel{\,\prime}\over{.} 3$). A summary of the characteristics of the Hydra sample is given in Table 1. The structural parameters are taken from Harris (1996; 2010 Version).

Table 1.  RV Sample Comparisons

Study	Sample Size	Mean RV	Mean RV Error	Median Radius (r)
		($\mathrm{km}\,{{\rm{s}}}^{-1}$)	($\mathrm{km}\,{{\rm{s}}}^{-1}$)	(arcminutes)
(1)	(2)	(3)	(4)	(5)
Boberg et al. (2016)	73	−63.35	1.16	1.73
Kimmig et al. (2015)	91	−63.43	0.33	6.85
Lane et al. (2011)	81	−63.64	3.40	4.49

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The Hydra sample provides good coverage of the innermost regions of M53. In order to expand our sample to larger radii and increase the sample size, we combine our Hydra sample with the data from two previous studies (Lane et al. 2011; Kimmig et al. 2015). The RVs taken from Lane et al. (2011) were determined from low- (R = 3700) and medium-resolution (R = 10000) spectra taken with the AAOmega spectrograph on the Anglo-Australian Telescope. The Kimmig et al. (2015) RVs were determined from relatively high resolution (R ∼ 38,000) spectra taken with the Hectochelle spectrograph on the Multiple Mirror Telescope. There are 11 and 15 stars in common between the Hydra sample and those from Kimmig et al. (2015) and Lane et al. (2011), respectively. In Figure 1 we plot RVs of stars in common between Kimmig et al. (2015) and Lane et al. (2011) versus their measured RVs in the Hydra sample. There is generally good agreement between the three studies, but the Kimmig et al. (2015) and Lane et al. (2011) RVs are systematically higher than those measured in our Hydra study. We find the following mean differences between the Hydra RVs, and those measured by Lane et al. (2011) and Kimmig et al. (2015): $\langle {\mathrm{RV}}_{\mathrm{Hydra}}-{\mathrm{RV}}_{\mathrm{Kimmig}}{\rangle }_{N=11}=-0.72\,\mathrm{km}\,{{\rm{s}}}^{-1}$, $\sigma =0.42\,\mathrm{km}\,{{\rm{s}}}^{-1};$ $\langle {\mathrm{RV}}_{\mathrm{Hydra}}-{\mathrm{RV}}_{\mathrm{Lane}.}{\rangle }_{N=15}=-0.75\,\mathrm{km}\,{{\rm{s}}}^{-1}$, $\sigma =0.60\,\mathrm{km}\,{{\rm{s}}}^{-1}$. These mean offsets are added to the RVs of their respective studies to put them on the same relative scale as the Hydra sample. Stars in common with either study are assigned the RV as found from the Hydra data rather than averaging the different RV measurements from each study. The 44 stars in common between Kimmig et al. (2015) and Lane et al. (2011) were assigned the value as found by Kimmig et al. (2015) due to their typically lower measurement errors. The Kimmig et al. (2015) catalog did not include a cluster membership label. Possible cluster nonmembers were removed by only including stars in the following RV range: $-71\lt \mathrm{RV}\lt -51\,\mathrm{km}\,{{\rm{s}}}^{-1}$. These limits represent the approximate 3σ level of the entire sample and brought the combined RV catalog to 250 stars. We also calculated the absolute value of the difference of each RV with the systemic velocity of the cluster as defined by these 250 stars. If this difference was greater than the 98th percentile of difference distribution, the star was removed from the sample. This was done to remove relatively extreme RVs that might skew the rotation analysis. After this cut, the RV catalog contained 245 stars. In Table 1, we summarize how each sample contributes to the final RV catalog used in this study. The previously mentioned offsets were applied before calculating the statistics shown in Table 1.

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In Figure 2, we plot the cumulative radial distribution of each sample in the cluster. This figure clearly illustrates the benefit and importance of the Hydra sample. The Kimmig et al. (2015) and Lane et al. (2011) samples only have 10 and 17 stars within $2{r}_{h}$, respectively, and neither sample has any stars within $2{r}_{c}$. An integral field unit (IFU) study of the central rotation in GCs by Fabricius et al. (2014) found that all 11 clusters in their sample, including M53, show significant signatures of internal rotation. The IFU used by Fabricius et al. (2014) covered each cluster out to a radius of $\sim 1^{\prime}$, which, in the case of M53, corresponds to $\sim 3\,{r}_{c}$. The Hydra sample allows us to measure and constrain the internal rotation within M53 using individual RVs in a region that bridges the radial gap between the regions covered by Fabricius et al. (2014) and those probed by Kimmig et al. (2015) and Lane et al. (2011). The relatively large sample (N = 48) of Hydra stars within 2r_h provides the essential data needed to measure the central dynamics of the cluster where the signatures of internal rotation may be more pronounced.

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We implement a commonly used method to detect rotation in M53 based on individual line-of-sight RV measurements (Cote et al. 1995; Lane et al. 2009; Bellazzini et al. 2012; Bianchini et al. 2013). The sample of cluster members is divided by a line passing through the center of the cluster. The difference (${\rm{\Delta }}\langle \mathrm{RV}\rangle$) between the mean RV of the stars on either side of the line is calculated and becomes the amplitude of rotation at that position angle (PA). The dividing line is then rotated to a new PA and the difference is recalculated. This process is repeated until the PAs have swept out a full 360^◦. We define $\mathrm{PA}=0^\circ$ to be to the right (east) and $\mathrm{PA}=90^\circ$ to be pointing up (north). The PA values are increased in steps of 15^◦ moving counter clockwise. The resulting curve of ${\rm{\Delta }}\langle \mathrm{RV}\rangle$ as a function of PA is then fit by the sine law shown in Equation (1),

$$\begin{eqnarray}&&{\rm{\Delta }}\langle \mathrm{RV}\rangle ={A}_{\mathrm{rot}}\sin (\mathrm{PA}+\phi ).\end{eqnarray} \tag{ 1 }$$

We determine the amplitude of rotation in the cluster A_rot and the phase (ϕ) of the sine curve by fitting this model to the rotation curve. The axis of rotation of the cluster, PA₀, is related to the phase of the sine model by ${\mathrm{PA}}_{0}=270-\phi$ and is the PA that corresponds to the maximum in the rotation curve. This procedure was also repeated with the PA increasing in steps of $5^\circ ,\ 10^\circ$, and 20^◦. We found that changing the step size in PA did not have a large effect on the fit axis of rotation, which varied by only $\sim 6^\circ$ among the step sizes we tested. As noted in Lane et al. (2010a), this method and fit results in an amplitude (A_rot) that is twice the actual velocity of rotation in the cluster. We note that in a number of studies, A_rot is adopted as a measure of the rotation amplitude instead of ${A}_{\mathrm{rot}}/2$ to account for the effects of the radial variation of the rotational velocity (see, e.g., Bellazzini et al. 2012). The amplitude of rotation is also affected by the inclination (i) of the cluster by a factor of sin(i), which is not measurable in GCs. Therefore, the amplitude of rotation we measure with this method is a lower limit of the actual intrinsic rotation in the cluster.

Following the example of Kacharov et al. (2014), we use the maximum likelihood method outlined in Walker et al. (2006) to calculate mean velocity and velocity dispersions of different stellar samples in our analysis (e.g., the mean velocity on either side of the cluster dividing line). We numerically determine these values by maximizing the log-likelihood formula shown in Equation (2) (Equation (8) in Walker et al. 2006) using the Metropolis–Hastings algorithm (Hastings 1970). In this equation, v_i and ${\sigma }_{i}$ are the individual measured RVs and their errors, respectively. By maximizing this log-likelihood, we can determine the mean RV of a given population $\langle u\rangle$, as well as its intrinsic velocity dispersion ${\sigma }_{p}$. This method allows us to take the measurement error, and RV dispersion, into account when determining these characteristics of a given sample.

$$\begin{eqnarray}\mathrm{ln}(p) & = & -\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{N}\mathrm{ln}({\sigma }_{i}^{2}+{\sigma }_{p}^{2})\\ & & -\displaystyle \frac{1}{2}\displaystyle \sum _{i=1}^{N}\displaystyle \frac{{({v}_{i}-\langle u\rangle )}^{2}}{({\sigma }_{i}^{2}+{\sigma }_{p}^{2})}-\displaystyle \frac{N}{2}\mathrm{ln}(2\pi ).\end{eqnarray} \tag{ 2 }$$

We implemented a bootstrap/Monte Carlo method in order to characterize the variation in the fit model parameters A_rot and PA₀. For each RV measurement in the sample, we added a noise value randomly drawn from a normal distribution centered on zero with a standard deviation equal to the error on that RV measurement. We then ran the noise-added RVs through the rotation-curve procedure outlined in Section 3.1 and redetermined the fit parameters. This process was then repeated 200 times. The procedure also provided us with a measure of the variation of ${\rm{\Delta }}\langle \mathrm{RV}\rangle$ at a given PA. From this analysis we found that the average standard deviation of ${\rm{\Delta }}\langle \mathrm{RV}\rangle$ is $0.3\,\mathrm{km}\,{{\rm{s}}}^{-1}$.

We performed the rotation analysis on subsets of stars that fell within a maximum distance (r_max) from the cluster center. This is done to characterize how the axis and amplitude of rotation vary as a function of radius similar to the analysis in Bianchini et al. (2013). The max radii used ranged from $1.75\mbox{--}18.5{r}_{h}$ in steps of 0.25. This analysis was done on the total combined data set, as well as just the RV sample from Hydra.

In Table 2, we list a subset of the A_rot and PA₀ values resulting from performing the rotation-curve analysis on different samples of stars. Column 1 gives the r_max values used to define each sample. Columns 2–4 give the number of stars, fit amplitude of rotation, and fit axis of rotation, respectively, while columns 5–7 give these same values for the Hydra-only sample. In Figure 3, we plot the rotation curves resulting from ${r}_{\max }=2,3,\mathrm{and}\,5{r}_{h}$. The column of panels on the left are the rotation curves for the total combined sample, and the column of panels on the right are the rotation curves for the Hydra-only sample. The last panel in each column shows the rotation curve for each sample when no radius cut is applied and all stars are used. The rotation curves have a peak amplitude of ${A}_{\mathrm{rot}}=2.76\ \mathrm{km}\,{{\rm{s}}}^{-1}$ and ${A}_{\mathrm{rot}}=2.80\ \mathrm{km}\,{{\rm{s}}}^{-1}$ for the total combined and Hydra-only samples, respectively, for stars within 2r_h. As mentioned earlier, the rotational velocity, V_rot, is equal to ${A}_{\mathrm{rot}}/2$ but this value is a lower limit on the actual rotational velocity due to the effects of the inclination of the cluster. Including the inclination term gives us the following rotational velocities in M53 for stars within 2r_h: ${V}_{\mathrm{rot}}(\mathrm{Total})\cdot \sin (i)=1.38\ \mathrm{km}\,{{\rm{s}}}^{-1}$, ${V}_{\mathrm{rot}}(\mathrm{Hydra})\cdot \sin (i)\,=$ $1.40\ \mathrm{km}\,{{\rm{s}}}^{-1}$. This also shows that the combined and Hydra-only sample produce equivalent rotational velocities, but this could be expected given that the Hydra sample is the dominant contributor of stars in this inner region of the cluster.

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Table 2.  Rotation-curve Fit Parameters

rmax	NTotal	Arot	PA0	NHydra	Arot	PA0
($r/{r}_{h}$)		($\mathrm{km}\,{{\rm{s}}}^{-1}$)	(deg)		$(\mathrm{km}\,{{\rm{s}}}^{-1}$)	(deg)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
2.0	75	2.76	306.15	48	2.80	306.06
2.5	97	2.49	303.58	56	2.64	306.93
3.0	113	2.11	309.81	58	2.38	309.13
3.5	133	1.85	306.32	63	2.00	301.31
4.0	155	1.33	316.11	68	1.55	310.75
4.5	166	1.27	324.95	71	1.22	321.37
5.0	175	1.18	327.21	72	1.21	319.12
5.5	186	1.19	333.11	73	1.22	318.65
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18.5	245	0.60	344.30	⋯	⋯	⋯

Note. A_rot and PA₀ are determined by fitting the model in Equation (1) to the rotation curves. PA₀ is the position angle corresponding to the maximum of the rotation curve, ${\mathrm{PA}}_{0}=270-\phi$. A_rot is the maximum in the rotation curve. The actual peak in the rotational velocity is A_rot/2 (see Section 3.1).

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In Figure 4, we plot the PA₀ distributions resulting from our bootstrap/Monte Carlo analysis for different r_max values. Each of the histograms in these panels represents the 200 test runs of the rotation analysis with the noise-added RV measurements previously described. The panel on the left shows these histograms for the total combined sample for ${r}_{\max }=2,3,5{r}_{{h}}$ and all radii, while the panel on the right shows these histograms for the Hydra-only sample for ${r}_{\max }=2,3,4{r}_{{h}}$ and all radii.

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In Figure 5, we plot the amplitude and axis of rotation as a function of the max radius used to define each sample. The vertical error bars on each point represent the error in the fit parameters. The wide blue error bars on the top panel represent the standard deviation of the PA₀ distributions resulting from the bootstrap/Monte Carlo analysis outlined in Section 3.2 and shown in Figure 4. The length of these error bars was equivalent when using the total combined sample or just the Hydra sample. For $r/{r}_{{h}}\lt 5$, the standard deviations of the bootstrap distributions ranged from $\sim 6^\circ \mbox{--}10^\circ$, and at larger radii they ranged from $\sim 20^\circ \mbox{--}30^\circ$. Similar bootstrap error bars were not included on the A_rot panel because they were equivalent to the error on the A_rot fit ($\sim 0.3\,\mathrm{km}\,{{\rm{s}}}^{-1}$).

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In the bottom panel of Figure 5 we present a measure of the signal-to-noise ratio (S/N) of the strength of the rotation at each r_max value. We calculated this ratio by dividing $\tfrac{1}{2}{A}_{\mathrm{rot}}$ by the mean error on the ${\rm{\Delta }}\langle \mathrm{RV}\rangle$ values used in constructing the rotation curves. The mean error on ${\rm{\Delta }}\langle \mathrm{RV}\rangle$ was calculated as two times the standard deviation in ${\rm{\Delta }}\langle \mathrm{RV}\rangle$ as determined from our bootstrap analysis. At all radii, this value was $0.6\,\mathrm{km}\,{{\rm{s}}}^{-1}$. By calculating this S/N we are able to determine at which r_max values the internal rotation in the cluster is significant (S/N ≥ 1).

Figure 5 illustrates that there is a considerable change in the axis of rotation depending on the sample of stars being used. This characteristic suggests that assigning a global PA₀ value for all stars at all radii in the cluster will not necessarily correspond to the strongest rotation signature. To further illustrate this point, we performed the rotation analysis on two subsets of the RV sample, one with all stars within 2r_h, and another with all stars outside of 2r_h. The rotation curves for each of these samples are shown in Figure 6. The number of stars, rotation amplitude, and axis of rotation for each sample are listed in Table 3. Figure 6 shows clearly that the rotation in the inner and outer regions of the cluster are defined by very different PAs, with the inner sample having ${\mathrm{PA}}_{0}=306^\circ$ and the outer having ${\mathrm{PA}}_{0}=86^\circ$ (note that the direction of the angular momentum vectors of the inner and outer samples are, respectively, 126^◦ and 266^◦; see Figure 7). Interestingly, this analysis also illustrates that there is a signature of counter rotation between the inner and outer regions of M53 (see also Figure 8 below).

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Table 3.  Inner vs. Outer Rotation Parameters

Sample	NTotal	Arot	PA0
		($\mathrm{km}\,{{\rm{s}}}^{-1}$)	(deg)
(1)	(2)	(3)	(4)
Inner sample	75	2.76	306.15
Outer sample	170	0.80	85.9

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We follow convention and use a single value for the PA₀ defined by all stars at all radii in both the total sample and the Hydra-only sample, giving ${\mathrm{PA}}_{0}=344\buildrel{\circ}\over{.} 30$ and ${\mathrm{PA}}_{0}=318\buildrel{\circ}\over{.} 65$, respectively. The implications of the change in PA₀ will be further discussed in Section 5. In Figure 7, we present a polar representation of the positions of stars in our sample, their individual RV measurements, and where the stars fall relative to the axis of rotation.

The rotation-curve analysis clearly showed the signatures of differential rotation in the cluster, as illustrated in Figures 3 and 5 by the decrease in rotation amplitude with increasing radius. By constructing rotation profiles of the cluster, we can further illustrate and quantify the differential rotation in the cluster. We create overlapping bins along the axis that is perpendicular to the axis of rotation determined by fitting the rotation curves. We then calculate the difference between the average velocity in these cylindrical bins with the systemic velocity of the cluster. This results in a measurement of the rotation velocity of the cluster at different radii away from the axis of rotation. In Figure 8, we plot the rotation profiles of each sample using the PA₀ determined by all of the stars in the total sample (31865) and Hydra-only sample ($344\buildrel{\circ}\over{.} 30$), respectively.

These rotation profiles show the strongest signature of rotation toward the cluster center. If we assume that the inner regions of the cluster follow solid-body rotation, we can model this portion of the rotation profile with a simple linear fit (y = mx). We only included the radial bins out to where the rotation profiles begin to turn over. For the total sample this position corresponded to $r=-1\buildrel{\,\prime}\over{.} 0\ \mathrm{to}\ 1\buildrel{\,\prime}\over{.} 5$, and for the Hydra-only sample $r=-2\buildrel{\,\prime}\over{.} 0\ \mathrm{to}\ 1\buildrel{\,\prime}\over{.} 5$. The resulting linear fits are shown by a red dashed line in each panel of Figure 8. The slope of this line is $0.53\ \pm 0.15\ \mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{arcmin}}^{-1}$ for the total sample rotation profile, and $0.87\ \pm 0.06\ \mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{arcmin}}^{-1}$ for the Hydra-only sample. The rotational velocity given by this linear model at the turnover points listed above gives an estimate of the maximum rotational velocity in the cluster. By averaging the values of the linear model at the turnover points for each sample, we find the following rotational velocities: ${V}_{\mathrm{rot}}(\mathrm{Total})\,=0.67\ \pm 0.14\ \mathrm{km}\,{{\rm{s}}}^{-1}$, ${V}_{\mathrm{rot}}(\mathrm{Hydra})$ $\,=\,1.5\ \pm 0.09\ \mathrm{km}\,{{\rm{s}}}^{-1}$. The rotational velocity for the Hydra sample is consistent with the values determined from the rotation-curve analysis presented in Figure 3. It is interesting that both rotation profiles do not converge to 0 $\mathrm{km}\,{{\rm{s}}}^{-1}$ at large perpendicular distances from the axis of rotation. This feature might be another manifestation and consequence of the decoupling between the inner and outer rotation already suggested by the radial variation of the rotation axis. Additional data probing the kinematics of the outermost region would be necessary to further explore the significance of this feature.

We also performed an additional fit of the rotation profile using the following simple analytical model (as in, e.g., Lynden-Bell 1967; Hénault-Brunet et al. 2012; Mackey et al. 2013; Kacharov et al. 2014):

$$\begin{eqnarray}&&{v}_{\mathrm{rot}}=\displaystyle \frac{2{A}_{\mathrm{peak}}}{{r}_{\mathrm{peak}}}\times \displaystyle \frac{r}{1+{(r/{r}_{\mathrm{peak}})}^{2}}.\end{eqnarray} \tag{ 3 }$$

In this equation, v_rot is the rotational velocity at a distance r from the cluster center measured perpendicularly from the axis of rotation, A_peak is the maximum amplitude of the rotational velocity, and r_peak is the distance from the rotation axis where v_rot reaches the maximum value, A_peak.

When performing this fit we only included the bins between $\pm 3^{\prime}$ away from the axis of rotation. This resulted in the following values for the different samples: ${A}_{\mathrm{peak}}(\mathrm{Total})=0.5\pm 0.1\ \mathrm{km}\,{{\rm{s}}}^{-1}$, ${r}_{\mathrm{peak}}(\mathrm{Total})=1\buildrel{\,\prime}\over{.} 2\pm 0\buildrel{\,\prime}\over{.} 5;$ A_peak(Hydra) = $1.1\pm 0.1\ \mathrm{km}\,{{\rm{s}}}^{-1}$, r_peak(Hydra) = $1\buildrel{\,\prime}\over{.} 6\pm 0\buildrel{\,\prime}\over{.} 4$. The peak of the rotation curve, located approximately at the cluster half-light radius, is in general agreement with the simulations presented in Vesperini et al. (2014), Tiongco et al. (2017), and M. A. Tiongco et al. (2017, in preparation).

The total combined sample has extensive radial coverage in both the inner and outer regions of M53, allowing us to construct a velocity dispersion profile of the cluster. To do so, the sample was divided into six radial bins with approximately the same number of stars. The final bin was split in half an additional time because it covered such a large range in radii. In the top panel of Figure 9, we plot the RV of each star as a function of its radius in arcminutes. Along the top of this panel, we also show the radial extent of each sample as wide horizontal bars. The black diamond within each bar marks the median radial distance of each sample. The solid vertical lines mark the inner and outer radius limit of each bin. Within each bin, we use the maximum likelihood method described in Section 3.1 to get an estimate of the mean velocity and velocity dispersion (${\sigma }_{\mathrm{RV}}$). This produces a measurement of the velocity dispersion as a function of radius, as shown in the bottom panel of Figure 9. The radius for each point in this panel marks the mean of the radii in a given bin. We fit a Plummer model (Plummer 1911) to the profile in the bottom panel of Figure 9 in order to get an estimate of the central velocity dispersion in the cluster. This was done using Equation (4) with the central velocity dispersion, ${\sigma }_{0}$, and scale radius, r_s, being the free parameters:

$$\begin{eqnarray}&&{\sigma }^{2}=\displaystyle \frac{{\sigma }_{0}^{2}}{\sqrt{1+\tfrac{{r}^{2}}{{r}_{s}^{2}}}}.\end{eqnarray} \tag{ 4 }$$

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The resulting fit is shown as a solid black line in the bottom panel of Figure 9. From this fit, we find ${\sigma }_{0}=4.0\pm 0.3$ $\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${r}_{s}=4\buildrel{\,\prime}\over{.} 0\pm 1\buildrel{\,\prime}\over{.} 1$ ($20.9\pm 5.7\mathrm{pc}$, assuming R_Sun = 17.9 kpc). These values are consistent with Lane et al. (2009) who found ${\sigma }_{0}=4.4\pm 0.9$ $\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${r}_{s}=17.2\pm 3.8\,\mathrm{pc}$. We performed an additional fit to the velocity dispersion profile using a Plummer model with the scale radius (r_s) fixed to the projected half-light radius (${r}_{h}=1\buildrel{\,\prime}\over{.} 3$). The result of this fit is shown as the dashed line in the bottom panel. This fit was calculated to test how well the Plummer model describes the dispersion profile given that it is expected that the scale radius is equivalent to the projected half-light radius (see Haghi et al. 2009; Lane et al. 2010b; Kacharov et al. 2014). The fit with a fixed scale radius produced a higher central velocity dispersion ${\sigma }_{0}=5.4\pm 0.4$ $\mathrm{km}\,{{\rm{s}}}^{-1}$, although it does not fit the dispersion profile as well as the Plummer model where ${\sigma }_{0}$ and r_s are both free parameters. Although the Plummer model provides a convenient analytical model to determine the central velocity dispersion, this discrepancy clearly suggests that a complete characterization of all the dynamical properties of this cluster will require more sophisticated numerical or analytical models. Some of the peculiar dynamical features found in this study already provide a clear indication of the degree of dynamical complexity of this cluster.