5 Measuring proper motions and velocity dispersions

The shifts obtained (in pixels) for the individual WFs were transformed to a common grid, i.e. the Galactic coordinates, as well as to arcsec per century ('' cent^-1). In Fig. 2 these proper motions are plotted together with the histograms for the proper motions in l and b. The full data set for stars brighter than V $_{\rm 555}=19$ (proper motions, positions, and magnitudes) is given in the Tables in Appendix A.

Figure 2: Stellar proper motions and the resulting histograms for $\mu _{l}$ and $\mu _{b}$. The proper motions are measured in arcsec per century.

In an ideal scenario, where the globular cluster has an appreciable motion in relation to the bulge stellar population, the bulge and cluster stars will form two distinct distributions in the proper motion diagram. See for example the recent results by Zoccali et al. (2001) for NGC 6553, King et al. (1998) for NGC 6739 and Bedin et al. (2001) for M 4, or those for NGC 6712 by Cudworth (1988). The bulge stars have a larger velocity dispersion than the cluster stars, and hence a larger scatter in the proper motion diagram.

Figure 3: Histograms for the l and b proper motions ($\mu _{l}$, $\mu _{b}$) for different magnitude ranges (as indicated). For each histogram Poissonian errorbars are drawn as well as the fitted Gaussians. The dotted lines show the two Gaussians needed to fit the data and the full line the final combined distribution. The fitted parameters are given in Table 4. Proper motions are measured in arcsec per century.

In our case Fig. 2 shows that the motion of the globular cluster in relation to the Galactic bulge is very small. This is as expected since the heliocentric radial velocity relative to the local standard of rest for NGC 6528 is high ( $184.9\pm3.8$ km s^-1, Harris 1996; $\simeq$210 km s^-1, Carretta et al. 2001), which suggests that NGC 6528 is on a mostly radial orbit away from us. However, we note that the histograms for the velocities in the l and b coordinates have broad wings. In fact when we tried to fit our histograms with Gaussian distributions it became clear that a single Gaussian distribution could not fit the observed distributions and two Gaussians were needed.

To separate the bulge and cluster stars using the measured proper motions, we divide the stars into different magnitude ranges and find the best fitting Gaussians, as shown in Fig. 3. We found that two Gaussians were required to fit the data well, indicating that, as expected, we have two stellar populations with different velocity dispersions. Based on previous measurements of bulge and cluster velocity dispersions, we associate the narrow Gaussian with NGC 6528 and the broad Gaussian with the bulge stars.

 

 

Table 4: Gaussian fits to proper motion distributions in Fig. 3. An error of 0. indicates that that particular parameter was kept fixed during the fitting procedure. We give the amplitude, the center ($<\mu >$), and the $\sigma$ for two Gaussians (shown with dotted lines in Fig. 3). Both are in arcsec per century. The first fours rows give the results for the l-coordinate and the last for for the b-coordinate.

Galactic l	Field			Cluster
mag. range	Amp1	$<\mu_1>$	$\sigma_1$	Amp2	$<\mu_2>$	$\sigma_2$
		(arcsec century-1)			(arcsec century -1)
V<18	$32.783\pm5.990$	- $0.097\pm0.027$	$0.305\pm0.025$	$95.539\pm 11.934$	- $0.005\pm0.010$	$0.088\pm0.011$
V<19	$63.362\pm7.231$	- $0.084\pm0.017$	$0.306\pm0.012$	$142.831\pm14.992$	- $0.018\pm0.009$	$0.088\pm0.009$
$19\leq V<20$	$114.083\pm9.793$	- $0.072\pm0.013$	$0.305\pm0.009$	$105.318\pm16.496$	$0.017\pm0.014$	$0.087\pm0.015$
$20\leq V<21$	$74.461\pm4.957$	$0.004\pm0.008$	$0.309\pm0.007$	$155.211\pm 7.769$	$0.010\pm0.005$	$0.099\pm0.006$
Galactic b	Field			Cluster
V<18	$43.627\pm5.509$	- $0.052\pm0.018$	$0.214\pm0.014$	$95.539\pm0.$	- $0.014\pm0.012$	$0.088\pm0.$
V<19	$86.681\pm7.088$	- $0.051\pm0.013$	$0.221\pm0.009$	$142.831\pm0.$	- $0.015\pm0.010$	$0.088\pm0.$
$19\le V<20$	$126.907\pm6.433$	- $0.050\pm0.011$	$0.273\pm0.007$	$105.318\pm0.$	- $0.009\pm0.015$	$0.088\pm0.$
$20\le V<21$	$87.990\pm3.073$	- $0.044\pm0.007$	$0.265\pm0.004$	$155.211\pm0.$	- $0.010\pm0.005$	$0.099\pm0.$

We found that the $\mu _{l}$-diagram was easily fit by two Gaussians by our routine, however $\mu _{b}$ proved more difficult. In fact the fitting routine found one badly fitting broad Gaussian for the data in Fig. 3. However we expect the cluster stars to have the same velocity dispersion and amplitude in l and b and so we fixed the parameters for the narrower Gaussian, which represents the cluster stars, from the fitting of the $\mu _{l}$-distribution. In particular we fixed the height and the width but left the position free to be fitted. The second Gaussian had all three parameters (width, height and position) free for fitting. The results for all the fitted Gaussians are given in Table 4.

We find that all four $\mu _{l}$-distributions are fit by two Gaussians, one narrow and with a $\sigma_{l} \approx 0.08$ arcsec per century, and one broader with $\sigma_{l} \approx 0.30$ arcsec per century. The centers of these Gaussians vary with magnitude range, see Table 4. For the broad Gaussian in $\mu _{l}$ the three brightest magnitude ranges agree very well within the calculated errors while for the last bin the center has moved from $\sim$-0.07  to 0 arcsec per century. For the narrow Gaussian the centers for the two brightest magnitudes agree within the errors, as do those for the two faintest magnitudes.

Figure 4: Histograms for the b and l proper motions ($\mu _{l}$, $\mu _{b}$) for blue, V-I<1.6, and red, V-I>1.6, stars all with $V_{\rm 555}\leq 19$. For each histogram Possonian errorbars are drawn as well as the fitted Gaussians. For the blue samples only one Gaussian is fitted, while for the red two Gaussians are necessary to fit the data (see text for discussion). The fitted parameters are given in Table 5. Proper motions are measured in arcsec per century.

Figure 5: Histograms for the b and l proper motions ($\mu _{l}$, $\mu _{b}$) for blue, V-I<1.6, and red, V-I>1.6, stars all with $19 < V_{\rm 555}<20$. For each histogram Possonian errorbars are drawn as well as the fitted Gaussians. For the blue samples only one Gaussian is fitted, while for the red two Gaussians are necessary to fit the data (see text for discussion). The fitted parameters are given in Table 5. Proper motions are measured in arcsec per century.

For $\mu _{b}$ we fixed the fwhm and height for the narrow Gaussian before fitting (see discussion above). Thus by definition $\sigma_ b=\sigma_ l$ and ${\rm amplitude}_b = {\rm amplitude}_l$ for each magnitude range for the narrow Gaussian. For the broad Gaussian we find a $\sigma_b$ around 0.21. Unlike the $\mu _{l}$ distribution, the fitted centres for the broad and narrow Gaussians in $\mu _{b}$ remain consistent within the errors for all the magnitude bins. The centers for the two Gaussians are found to be $-0.05 \pm 0.01$ and $-0.012\pm 0.012$ arcsec per century.

   

Table 5: Gaussian fits to proper motion distribution in Figs. 4 and 5. An error of 0. indicates that that particular parameter was kept fixed during the fitting procedure. We give the amplitude, the center ($<\mu >$), and the $\sigma$ for two Gaussians (shown with dotted lines in the figures). Both are in arcsec per century.

Galactic l		Field			Cluster
Mag. range	Colour	Amp1	$<\mu_1>$	$\sigma_1$	Amp2	$<\mu_2>$	$\sigma_2$
			(arcsec century-1)			(arcsec century -1)
V<19	V-I<1.6	$34.332\pm3.485$	- $0.193\pm0.028$	$0.324\pm0.020$
V<19	V-I>1.6	$17.912\pm3.496$	- $0.030\pm0.026$	$0.327\pm0.027$	$98.707\pm8.215$	- $0.024\pm0.007$	$0.089\pm0.007$
19< V<20	V-I<1.6	$51.985\pm6.185$	- $0.090\pm0.018$	$0.314\pm0.014$	$36.609\pm9.312$	$0.014\pm0.024$	$0.092\pm0.028$
19< V<20	V-I>1.6	$28.649\pm5.302$	- $0.059\pm0.025$	$0.302\pm0.025$	$65.822\pm9.429$	$0.029\pm0.012$	$0.085\pm0.013$
Galactic b		Field			Cluster
V<19	V-I<1.6	$55.828\pm5.611$	- $0.048\pm0.017$	$0.202\pm0.011$
V<19	V-I>1.6	$23.495\pm3.135$	- $0.063\pm0.021$	$0.254\pm0.017$	$98.707\pm 0.$	- $0.019\pm0.008$	$0.089\pm0.$
19< V<20	V-I<1.6	$56.149\pm3.678$	- $0.051\pm0.015$	$0.291\pm0.010$	$36.609\pm 0.$	$0.007\pm0.025$	$0.092\pm0$.
19< V<20	V-I>1.6	$37.056\pm3.878$	- $0.066\pm0.018$	$0.241\pm0.013$	$65.822\pm 0.$	$0.001\pm0.013$	$0.085\pm0.$

The different behaviour of the centers of the $\mu _{l}$ and $\mu _{b}$histograms with magnitude suggests the presence of a stellar component in addition to the bulge and cluster stars that is adding an additional proper motion component to $\mu _{l}$. One possibility for this component is disk stars, which may be the main component of the blue plume seen in the colour-magnitude diagram in Fig. 1. We therefore investigate below whether we see any difference in Gaussians fit separately to blue and red samples.

In Figs. 4 and 5 we divide the two bright magnitude samples, $V_{\rm 555}\leq 19$ and $19 < V_{\rm 555}<20$ into red and blue stars and fit Gaussians to them in the same way as before. The fitted parameters are given in Table 5.

For the blue stars ( $V_{\rm 555}-I_{\rm 814}<1.6$) in the brightest sample, i.e. $V_{\rm 555}<19$, it proved impossible to fit two Gaussians both for $\mu _{l}$ and $\mu _{b}$. Since there are few stars in these two histograms we have carefully checked that the chosen binning did not effect the final result. The results are given in Table 5.

The red samples, on the other hand, show a very strong central peak and broad wings which means that two Gaussians are needed to achieve a good fit. As before we first fitted the $\mu _{l}$distribution and then fixed the $\sigma$ and amplitude for the narrow Gaussian when fitting the $\mu _{b}$ distribution. It is interesting to find that indeed the narrow Gaussian has a $\sigma=0.08$arcsec per century, exactly the same as found when the full colour-range was investigated.

We also investigate the distributions for the magnitude range 19 to 20, Fig. 5. Here, again, we see a rather broad dominating dispersion in the blue while the red is dominated by the narrow distribution, even if not as prominently as in the case of the brightest stars.

The main change between the fit parameters for the sample split by colour and for the whole sample, is that in l the center for the blue sample is significantly different from the red sample, and both the blue and red centers are different to that found when the whole sample is fitted. We suggest that this is due to some of the bright blue stars being from a third stellar population, namely the Galactic disk, which has an additional velocity component in l. We note that the stars identified with the Galactic disk have a mean proper motion in l that is negative relative to that of the Galactic bulge. Disk dwarf stars at the magnitudes observed here should be within a kpc or less from us (see e.g. Sadler et al. 1996). If the rotation curve of the disk (which is not well sampled for these type of objects) differs somewhat from pure differential rotation then this can explain our measured proper motions for stars at these distances.