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Astron. Astrophys. 347, 455-472 (1999)

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4. The globular cluster parameters

This section deals with the determination of the metallicity, reddening and distance using the differentially dereddened CMDs. There are two possible ways to achieve the goal. In the first, theoretical models are compared with the CMDs, in the second, empirical relations between parameters and loci in the CMDs are used.

4.1. Isochrone fitting

To derive metallicity, distance and absolute reddening via isochrone fitting, we used the Padova-tracks (Bertelli et al. 1994) with a fixed age of 14.5 Gyr ([FORMULA]). Isochrones with different ages ([FORMULA]) led to identical results. To avoid systematic errors, we used the middle of the broadened structures to fit the isochrones by eye. These loci are easily determined for the ascending part of the RGB, as it runs more or less perpendicular to the reddening vector. Regarding the upper part of the RGB, we take into account that we cannot distinguish between the AGB and the RGB in our diagrams. Hence, the densest regions of the AGB/RGB lie between the model's tracks. We additionally used the HB and the lower part of the RGB, as far as they were accessible. The parameters resulting from the isochrone fit are given in Table 2.


Table 2. Distance modulus [FORMULA], total reddening [FORMULA] and metallicity [FORMULA] of all the sample's clusters derived by isochrone fitting. The errors are eye-estimates of how accurately we could place the isochrones. Note that the isochrones are fitted to the differentially dereddened CMDs.

Figs. 23 to 29 show the differentially dereddened CMDs with the fitted isochrones. For a discussion and comparison of these parameters with the literature, see paragraph 5.3.

[FIGURE] Fig. 23. Isochrone fitting for NGC 5927. [FORMULA] dex, [FORMULA] gyr. The CMD is selected for photometric errors [FORMULA] mag and for radii [FORMULA] pix. The first selection gives preference to brighter stars, i.e. to stars which not necessarily are cluster members. This effect is counterbalanced by the second selection.

[FIGURE] Fig. 24. NGC 6316, selected for photometric errors [FORMULA] mag and with an isochrone [FORMULA], [FORMULA] gyr. The radial selection is due to the correction for differential reddening (see Figs. 9 and 22).

[FIGURE] Fig. 25. Isochrone for NGC 6342 ([FORMULA] dex, [FORMULA] gyr). The CMD is selected for radii [FORMULA] pix.

[FIGURE] Fig. 26. CMD of NGC 6441 with isochrone [FORMULA] dex, [FORMULA] gyr. The CMD is radially selected [FORMULA] pix. The stars to the blue of the HB still are visible.

[FIGURE] Fig. 27. The isochrone for NGC 6760 with [FORMULA] dex and [FORMULA] gyr is slightly too metal-rich, as the AGB/RGB arcs above the model. This result is corroborated by the metallicity- estimates given below. The CMD is radially selected for [FORMULA] pix.

[FIGURE] Fig. 28. The radially selected CMD of NGC 6528 clearly shows the two AGB/RGB's of the cluster's and the background population. The isochrone ([FORMULA] dex, [FORMULA] gyr) is slightly too metal-rich as well.

[FIGURE] Fig. 29. Radially selected CMD ([FORMULA] pix) of NGC 6553 with an isochrone of [FORMULA] dex and [FORMULA] gyr. The strongly differentially reddened field population is now clearly separated from the cluster (to the red of the lower RGB). The CMD is also selected for photometrical errors [FORMULA] mag.

4.2. Metallicity and reddening: relations

4.2.1. Metallicity

The luminosity difference between HB and the turn over of the AGB/RGB in [FORMULA]-CMDs is very sensitive to metallicity in the metal-rich domain (e.g. Ortolani et al. 1997). Moreover, it is a differential metallicity indicator, thus it is independent of absolute colour or luminosity, in contrast to the [FORMULA]-method (see e.g. Sarajedini 1994). We present a preliminary linear calibration of this method,


as there has not been any so far. Because there still are only very few [FORMULA]-CMDs which clearly show both the HB and turn over of the AGB/RGB and which have reliable metallicity determinations, we used the Padova-isochrones and a CMD of NGC 6791 (Garnavich et al. 1994) to set up a calibration. NGC 6791 is one of the richest old open clusters with a good metallicity determination and it is therefore suitable to serve as a zero-point check.

As the form of the RGB depends slightly on age as well (e.g. Stetson et al. 1996), we have to check this dependence before applying our calibration. Fig. 30 shows the linear relation between [FORMULA] and [FORMULA] for four GC-ages. Table 3 contains the respective coefficients. As the metal-poorest isochrones of the Padova-sample ([FORMULA] dex) do not show a maximum of the AGB/RGB, they have not been used. Fig. 30 makes clear that the age has only a minor influence on the resulting metallicity. To be consistent with the isochrone-fit, we used the relation for [FORMULA].

[FIGURE] Fig. 30. [FORMULA]-relation for different ages. The open symbols stand for the isochrone-values. Squares, lines: 9.8 Gyr, diamonds, dotted: 13.2 Gyr, triangles, dashed: 14.5 Gyr, circles, dash-dotted: 15.8 Gyr. The filled square represents the value for NGC 6791 (Garnavich et al. 1994).


Table 3. Calibration coefficients of the [FORMULA]-relation for different cluster ages. The calibration equation is [FORMULA] [FORMULA]

To estimate the metallicities of our clusters, we now only have to measure the relevant luminosities. The results are given in Table 4. The value for NGC 5927 given in column [FORMULA] relates to a single star (François 1991); NGC 6528 and NGC 6553 are from Richtler et al. (1998) and Sagar et al. (1998).


Table 4. Metallicities of all GCs via the differentially dereddened CMDs. Column [FORMULA] contains the values derived by the [FORMULA]-relation. The values of column [FORMULA] have been taken from Harris (1996). Column [FORMULA] gives additional values as discussed in the text. The errors only take account of the uncertainties of the luminosities and the calibration errors of Table 3.

4.2.2. Reddening

It should be remembered, that we used the differentially dereddened CMDs to determine the parameters. Thus, the given reddenings are minimal ones.

As mentioned above, the absolute colour of the RGB at the level of the HB can be used to estimate the metallicity. Conversely (Armandroff 1988), if we know the metallicity, we can determine the absolute colour [FORMULA] and thus the absolute reddening of the cluster.

These relations between colour [FORMULA] and metallicity are well calibrated for the metal-poor to intermediate regime. However, it is difficult to set up a calibration for the metal-rich regime of our clusters. Linear calibrations have been provided by e.g. Sarajedini (1994). A more recent calibration by Caretta & Bragaglia (1998) uses a 2nd order polynomial. To set up a calibration for the metal-rich regime we used again the Padova-tracks together with NGC 6791 to derive the coefficients for a relation of the form


In addition, we used the [FORMULA] and [FORMULA] values for M67 given by Montgomery et al. (1993) to check the zero point. Taking into accound that M67 is even younger than NGC 6791 by 3 to 5 Gyrs, the measured quantities fit reasonably well. Table 5 contains the calibration coefficients, Fig. 31 the graphic relations, again for different ages. As above, we used the relation for [FORMULA].


Table 5. Calibration coefficients for the [FORMULA]-relation (Eq. 4).

[FIGURE] Fig. 31. Calibration of the non-linear [FORMULA]-relation for different ages. The key for the symbols is the same as used in Fig. 30, except for the filled triangle denoting the [FORMULA] pair of M67. In addition, the quadratic relation of Caretta & Bragaglia (1998) is plotted in dash-dotted line. It intersects our relation at low [FORMULA] but shows a difference of [FORMULA] mag in the more metal-rich regime.

Using the metallicities listed in Table 4, column [FORMULA], we get the absolute reddening as given in Table 6. Metallicities as well as reddenings fit very well with the values derived via isochrone-fitting, but are significantly lower than the values given in the literature. This is partly explained by the fact that we take the minimal reddening from the reddening map. Another part of the explanation may be that previous isochrone fits tend to use the red ridge of the RGB and thus overestimate the reddening.


Table 6. Absolute reddening for all GCs. Column [FORMULA] contains values derived via isochrone-fitting, column [FORMULA] values via [FORMULA]-relation. [FORMULA] gives the colour of the RGB at the level of [FORMULA], [FORMULA] the corresponding dereddened colour, calculated via the [FORMULA]-relation. In the last column, we cited values form literature and their sources, which, of course, cannot be more than a selection. Any measurement of colours was done in the differentially dereddened CMDs, which provides the explanation for the difference between our values and that taken from other works.

Sarajedini (1994) proposed a method to simultaneously determine metallicity and reddening. For this, he used the (linear) [FORMULA]-relation and the dependence of metallicity on the luminosity of the RGB at the absolute colour of [FORMULA] mag, in linear form as well. He calibrated both relations for a metallicity range of [FORMULA] dex. We recalibrated these relations in order to use them for our clusters, using NGC 6791 and the Padova-tracks. The graphic results are shown in Figs. 32 and 33; the calibration coefficients are given in Table 7. For a discussion and new calibration of Sarajedini's method see Caretta & Bragaglia (1998). We did not make use of this method, as the extrapolation of Sarajedini's calibration did not seem to be advisable, with reference to Figs. 32 and 33.


Table 7. Coefficients for Sarajedini- and Padova-relations. The equations have the form [FORMULA] and [FORMULA]. The errors are [FORMULA], [FORMULA], [FORMULA] und [FORMULA].

[FIGURE] Fig. 32. Calibration of the linear [FORMULA]-relation according to Sarajedini (filled symbols) in comparison to the recalibration for higher metallicities. The key for the symbols is the same as used in Fig. 30, except for the filled symbols.

[FIGURE] Fig. 33. Calibration of the linear [FORMULA]-relation according to Sarajedini (filled symbols) in comparison to the recalibration for higher metallicities. The key for the symbols is the same as used in Fig. 30, except for the filled symbols.

4.2.3. Distance

The brightness [FORMULA] of the horizontal branch is the best distance indicator for GCs. However, there is a lively discussion on how this brightness depends on the metallicity of the cluster.

We take the LMC distance as the fundamental distance for calibrating the zero point in the relation between metallicity and horizontal branch/RR Lyrae brightness. The third fundamental distance determination beside trigonometric parallaxes and stellar stream parallaxes is the method of Baade-Wesselink parallaxes. It had been applied to the LMC in its modified form known as Barnes-Evans parallaxes. So far, it has been applied to Cepheids in NGC 1866 (Gieren et al. 1994), and the most accurate LMC distance until now stems from the period-luminosity relation of LMC Cepheids by Gieren et al. (1998). We adopt the distance modulus from the latter work, which is [FORMULA] mag, and which is in very good agreement with most other work (e.g. Tanvir 1996).

If we adopt the apparent magnitude of RR Lyrae stars in the LMC from Walker (1992), [FORMULA] mag for a metallicity of [FORMULA] dex, and the metallicity dependence from Caretta et al. (1998), one gets


This zero-point is in excellent agreement with the one derived from HB-brightnesses of old LMC globular clusters, if the above metallicity dependence is used (Olszewski et al. 1991).

With relation 5, with the reddenings (as shown in Table 6, column [FORMULA]) and with the extinction [FORMULA] we can calculate the distance moduli


The values for [FORMULA] and [FORMULA] are listed in Table 4, and the results are given in Table 8. [FORMULA] comes from Table 1.


Table 8. Distances of GCs via [FORMULA]-relation. The given errors only account for the errors in [FORMULA]. The values of column [FORMULA] are taken from Harris (1996). Distances in kpc, brightness in mag.

4.3. Comparison

The distances determined via the [FORMULA]-relation are larger than those determined by the isochrone-fitting (Table 9). However, as the related reddenings do not show any significant differences, this effect is attributed to the [FORMULA]-relation and the isochrone-fitting itself. As described above, the isochrone fitting is lacking the desired accuracy especially because the TOP cannot be resolved for most of the clusters. Moreover, the fact that AGB and RGB cannot be distinguished in our CMDs leads to a systematic error in the isochrone distances in the sense that the isochrones tend to have been fitted with brightnesses which are too large. In the following, we discuss some possible explanations for differences between distances taken from the literature and this work. It should be remembered that the distance errors amount to about 10%.


Table 9. Comparison of the distances taken from Harris (1996), [FORMULA], with the values of this work. [FORMULA] and [FORMULA] contain the distances determined via isochrone-fitting and [FORMULA]-relation. The last column shows recently determined distances for NGC 6528 (Richtler et al. 1998) and for NGC 6553 (Guarnieri et al. 1998).

  1. The distance to NGC 6528 increases by nearly 30% compared to Richtler et al. (1998). Taking into account, that the isochrone (Fig. 28) might have been fitted slightly too low, we still get a distance of about 8.1 kpc. Moreover, Richtler et al. determine the absolute reddening via the differentially reddened CMD, which leads to larger values ([FORMULA]) compared to our [FORMULA]. Thus the distance modulus decreases by about 0.4 mag, as Eq. 6 is corrected more strongly on reddening. Finally, the different slopes of the reddening vector have to be regarded. Richtler et al. assume [FORMULA], our slope, which we determined via the slope of the HB, amounts to [FORMULA]. On the whole, we get a difference between Richtler et al. and this work of 0.7 mag in the distance modulus.

  2. In the CMDs of NGC 5927 and 6760 the differential reddening becomes noticable especially along the steep part of the RGB, as this is running nearly perpendicularly to the reddening vector. Around the turn over of the AGB/RGB and for its redder part, it leads to an elongation, but not to a broadening of the structures. Fitting an isochrone to the broadened RGB, one generally would use the middle of the RGB as an orientation, as one cannot distinguish between reddening effects and photometric errors in the outer regions. However, the red part of the AGB/RGB approximately keeps its unextinguished brightness. Thus the differential reddening might be overestimated, which leads to decreasing distances. A similar point can be made for the determination of [FORMULA] via the [FORMULA]-relation. Measuring the colour [FORMULA] in the differentially reddened diagram is best done at the middle of the broadened RGB again. This leads to an increased reddening, i.e. the distance modulus will be corrected too strongly for extinction. Overestimating the colour by 0.1 mag leads to a decrease in distance of about 10%.

  3. For NGC 6441, Harris (1996) cites a value of [FORMULA] mag. From our CMD we get 17.66 mag. This lower brightness is supported by (V,B-V)-CMDs of Rich et al. (1997), especially as Harris' value comes from a CMD by Hesser & Hartwick (1976), whose lower limiting brightness is around 17.3 mag.

  4. The distances as determined via the [FORMULA]- and the [FORMULA]-relation relate to the differentially dereddened CMDs, i.e to the minimal absolute reddening. However, the papers we obtained the cited values from (Table 6), do not take differential reddening into account (e.g. Armandroff (1988) for NGC 6342 and 6760, Ortolani et al. (1990, 1992) for NGC 6528 and 6553). So their absolute reddenings are systematically larger and the distances smaller. Interestingly, the absolute reddening for NGC 6316 of [FORMULA] mag as determined in this work fits very well the value of [FORMULA] mag given by Davidge et al. (1992); NGC 6316 shows the smallest differential reddening ([FORMULA] mag) of our cluster sample.

  5. Finally, the distances depend on the assumed extinction law. The value varies between [FORMULA] (Savage & Mathis 1979, Grebel & Roberts 1995, see Fig. 21 and discussion). This effect should have the strongest influence on the distances as determined in this work. Taking an absolute reddening of 0.5 mag, the variation between the above cited values results in a difference of 0.25 mag in the distance modulus. This corresponds to about 25% of the distance in kpc.

The distance error mostly depends on the absolute reddening used. The errors in the metallicites have only a minor influence on the distances (see Table 5). They amount to around 3% of the total distance in kpc. In conclusion, the increased distances [FORMULA] (Table 9) are due to the fact that we determine the distance-relevant parameters using the differentially dereddened CMDs.

4.4. Masses

To classify the clusters according to Burkert & Smith (1997), we have to determine the masses from the total absolute brightnesses. Because we could not measure the apparent total brightness, we used the values given by Harris (1996). With the extinctions and distance moduli given above (Table 8), we get the absolute total brightnesses via


We determined the masses using a mass-to-light-ratio of [FORMULA] (Chernoff & Djorgovski 1989). Table 10 shows the results. Thus, NGC 6441 is one of the most massive clusters of the galaxy. [FORMULA] Cen/NGC 5139 has [FORMULA] (Harris 1996).


Table 10. Absolute total brightnesses and masses for all clusters. The apparent brightnesses were taken from Harris (1996).

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Online publication: June 30, 1999