Astron. Astrophys. 348, 418-436 (1999)

3. Structural parameters

3.1. Core radii

Fig. 11 shows the distribution of King core radii for the NGC 5128 GC candidates and for the Milky Way GCs. The Milky Way GCs have been shifted to the distance of NGC 5128 and their core radii have been converted to arcseconds ( pc for Mpc). Galactic GCs with a central brightness cusp (which is believed to be a signature of a collapsed core) have been omitted from the sample. The faintest GC candidate in our sample has an absolute total magnitude of . In order to ensure that we are comparing similar objects we have excluded all Milky Way GCs fainter than .

Fig. 11. This figure compares the distribution of core radii for GC candidates in NGC 5128 with the distribution of core radii for selected GCs (see Sect. 3.1) in the Milky Way. The vertical axis is the fraction of the total number of GCs and the error bars show the Poisson uncertainties in each bin.

In order to facilitate a comparison between the two GC systems we plotted the fraction, , of the total number of GCs in each bin, , where n is the total number of GCs in each data set. The uncertainty in was computed from the Poisson uncertainties in the number of GCs in each bin, and in the total number of GCs using

The mean core radius for the 21 NGC 5128 GC candidates is (standard error) while the mean for the 73 selected Milky Way GCs is (se). A Kolmogorov-Smirnov (KS) test shows that we can reject the hypothesis that the two samples are drawn from the same distribution at the 46% confidence level. Therefore, there is no evidence that the core radii of the GC candidates in NGC 5128 are distributed differently from the core radii of the Milky Way GCs.

The most noticeable difference between the core radii of the NGC 5128 GC candidates and the core radii of the Milky Way GCs in Fig. 11 is the lack of a tail extending to large core radii in the NGC 5128 data. This may be an artifact of the small number (21) of NGC 5128 GC candidates in our sample. The mean of a distribution is sensitive to the presence of tails and outliers, but the median is much more robust against outliers. Therefore we computed the median core radius for each data set. Both the NGC 5128 GC candidates and the Milky Way GCs have median core radii of ( pc, or pixels on the WF CCDs and 1.4 pixels on the PC CCD). The similarity in the median core radii suggests that the "typical" core radius of a GC candidate in NGC 5128 is similar to that of the Milky Way GCs.

There may be systematic biases in the core radii that we have derived for the GC candidates in NGC 5128. Fitting Michie-King models to objects with core radii that are similar to the pixel scales of the images requires that the centers of the objects be accurately known. Small errors in determining the center of a candidate GC, and small systematic errors introduced by integrating Michie-King model profiles over the area of a pixel, may be sufficient to bias the fitted core radii towards smaller values. In addition to pixelation effects, the similarity between the core radii and the FWHMs of the PSFs may also be biasing our fits toward smaller core radii.

3.2. Tidal radii

The tidal radius of a GC is affected by the gravitational potential of its parent galaxy (e.g., Innanen et al. 1983; Heggie & Ramamani 1995). In order to compare the tidal radii of GC candidates in NGC 5128 with those of GCs in the Milky Way it is necessary to correct for the tidal fields of both galaxies. The first step is to normalize the tidal radius of each GC candidate by its mass, , to get . If we assume that the gravitational potentials of the Milky Way and NGC 5128 can be approximated by a spherical logarithmic potential of the form

where R is the galactocentric distance, is a scale length, and C is a constant, then we can compare the mean value of the normalized tidal radii of the GC candidates using

where is the amplitude of the flat part of the rotation curve of the galaxy, G is the Newtonian gravitational constant, and is a slowly varying function of the orbital eccentricity of the GC that has values of for circular orbits. The mean value of the normalized tidal radius of the GC candidates in NGC 5128, , can then be related to the mean normalized tidal radius of the Milky Way GCs by

where the subscript MW denotes the value for the Milky Way.

Eq. 6 assumes that the shape of the galactic potential is the same in both galaxies, but allows the total mass, as parameterized by , of each galaxy to vary. It also requires a knowledge of the distribution of GC orbits in each galaxy, as parameterized by and . We assumed a rotation velocity of for the Milky Way and for NGC 5128 (Hui et al. 1995). The only information available on the distribution of orbits for GC candidates in NGC 5128 is the projected radial distances of the GC candidates from the center of NGC 5128, so we have assumed that the NGC 5128 GC system is dynamically similar to the Milky Way GC system. This involves two assumptions about the nature of the GC orbits. First, we assume that the mean eccentricity of the NGC 5128 GC orbits is the same as that for the Milky Way GCs (, Odenkirchen et al. 1997). Second, we assume that each GC is near the apogalacticon of its orbit, , so . This is a reasonable assumption since a GC having spends % of its time in the outer half of its orbit. An additional complication is that the observed galactocentric distance for a NGC 5128 GC candidate is the projection of the true galactocentric distance onto the plane of the sky. If we assume that the orbits are oriented randomly then for a particular object's orbit is most likely to be observed to lie at a projected distance of . Therefore, we estimated the perigalactic distances for each of the NGC 5128 GC candidates from the observed distance from the center of NGC 5128 using

Fig. 12 shows the distribution of the normalized tidal radii for the NGC 5128 GC candidates and the Milky Way GCs. The cluster masses were computed from their total V-band luminosities assuming a mass-to-light ratio of two.

Fig. 12. This figure compares the distribution of normalized tidal radii (see Sect. 3.2) for GC candidates in NGC 5128 with the distribution of normalized tidal radii (after correcting for the difference in mass between the two galaxies) for GCs in the Milky Way. The vertical axis is the fraction of the total number of GCs and the error bars show the Poisson uncertainties in each bin.

Our sample of GC candidates has (se) (). The mean normalized tidal radius for 73 selected Milky Way GCs is (se) . The multiplicative factor in Eq. 6 is 0.703, which yields a corrected of (se) pc/ . A KS test says that we can reject the hypothesis that the two samples are drawn from the same distribution at the 74% confidence level. Therefore, there is insufficient evidence to state that the distribution of the tidal radii of the NGC 5128 GC candidates differs from that of the Galactic GCs if the difference in the masses of the two galaxies is taken into account . However, we wish to stress that this calculation assumes that the distribution of GC orbits are statistically similar for both galaxies.

3.3. Half-mass radii

The distribution of half-mass radii for the NGC 5128 GC candidates is shown in Fig. 13 along with the same distribution for our subsample of 73 Milky Way GCs. The half-mass radii for the Milky Way GCs were determined by computing a Michie-King model (with concentrations and core radii taken from W. Harris 1996) for each Milky Way GC. This allowed us to make a direct comparison between the half-mass radii of the best-fitting single-mass Michie-King models for the NGC 5128 GC candidates and the half-mass radii of the best-fitting single-mass Michie-King models for the Milky Way GCs.

Fig. 13. This figure compares the distribution of half-mass radii for GC candidates in NGC 5128 with the distribution of half-mass radii for a subset of GCs in the Milky Way. The vertical axis is the fraction of the total number of GCs and the error bars show the Poisson uncertainties in each bin.

The mean half-mass radius for the 21 NGC 5128 GC candidates is (se) while the mean for 73 selected Milky Way GCs is (se). However, a KS test says that we can reject the hypothesis that the two samples are drawn from the same distribution at only the 74% confidence level, so there is no evidence that the distribution of half-mass radii in our sample of NGC 5128 GC candidates is different from that of the GCs in the Milky Way.

3.4. Ellipticities

The distribution of ellipticities for the NGC 5128 GC candidates is shown in Fig. 14. The 21 NGC 5128 objects have (se) while the White & Shawl (1987) sample of Milky Way GCs yields (se) for the 73 Milky Way GCs. A KS test says that we can reject the hypothesis that the two samples are drawn from the same distribution at the 99.7% confidence level. Therefore, we conclude that the NGC 5128 GC candidates may have a different distribution of ellipticities from the Milky Way GCs.

Fig. 14. The upper panel shows the distribution of ellipticities for the GC candidates in NGC 5128. The lower panel shows the distribution of ellipticities for the Milky Way's GCs (from White & Shawl 1987).

The NGC 5128 GC candidates appear to be systematically more elliptical than the Milky Way GCs. There appears to be a lack of objects with low ellipticities and an excess of GC candidates with . The lack of GC candidates with is probably due to the elliptical PSF not being fully removed from the data. Another possible source of ellipticity is the stochastic distribution of bright stars near the center of the object. Geisler & Hodge (1980) found that the random placement of stars with respect to the adopted center of a GC can introduce a systematic error in the observed ellipticity of for GCs which are intrinsically spherical. This effect acts to make nearly spherical GCs appear to be more elliptical than they actually are. They also found that this systematic error decreases as the intrinsic ellipticity of the GCs increases. This would explain the lack of nearly circular GC candidates in NGC 5128, relative to the Milky Way.

AM97 identified 125 GC candidates in the inner of NGC 5128. Table 6 lists the ellipticities from those GC candidates in common between the two studies. The mean difference between our ellipticities and the AM97 values is .

Table 6. A comparison of our ellipticities with those of AM97.

3.5. Correlations of cluster properties with distance

GCs live within the tidal field of their parent galaxy, and the properties of a GC may depend on its position within the protogalactic cloud at the time the GC formed (e.g. Murray & Lin 1992). In Fig. 15 we plot , (both radii in arcseconds), c, and as functions of , where D is the observed distance from the center of NGC 5128 in arcminutes. None of the properties show strong correlations with the distance from the center of NGC 5128, but the trends are qualitatively similar to the trends found by Djorgovski & Meylan (1994) for the same properties of the Milky Way's GCs (see their Fig. 8). The correlations for and c for the NGC 5128 GC candidates are in the same sense as those for the Milky Way GCs, although Spearman rank correlation coefficients suggest that the correlations are weaker for the NGC 5128 GC candidates. This is to be expected because we are using the observed distance from the center of NGC 5128 (i.e. the distance projected onto the plane of the sky), while Djorgovski & Meylan (1994) used the true Galactocentric distance.

Fig. 15a-d. This figure shows the relationships between four properties of the NGC 5128 GC candidates and their observed distance, D, from the center of NGC 5128. The Spearman rank correlation coefficients, and the confidence levels for rejecting the hypothesis that the observed correlations are due to chance, are shown at the top of each panel. The lines show the best-fitting straight lines to the data. Panel a shows the core radius (in arcseconds), panel b shows the central concentration, panel c shows the half-mass radius (in arcseconds) of the best-fitting single-mass Michie-King model, and panel d shows the ellipticity.

We find a weak trend for the half-mass radius of a GC candidate to decrease as the distance from the center of NGC 5128 increases, while the opposite trend is seen in the Milky Way. The Spearman rank correlation coefficient for our data is -0.356, which corresponds to a significance of 0.887, while for the Milky Way GCs the correlation coefficient is . Therefore, there is insufficient evidence to for us to conclude that the observed trend for to decrease as the galactocentric distance increases is real.

Djorgovski & Meylan (1994) found no correlation between the ellipticity of a GC and its Galactocentric distance. We find a correlation coefficient of -0.257, corresponding to a significance of 0.739. An examination of Fig. 15, however, suggests that there is no significant correlation between ellipticity and galactocentric distance for the NGC 5128 GC candidates.

© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999
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