Astron. Astrophys. 327, 1004-1016 (1997)

4. Luminosity and mass function

From the CMD we have derived a luminosity function (LF) for the stars of M 55. Fig. 5 shows the LFs in the different annuli defined in the previous section (inner, intermediate and outer).

Fig. 5. Stellar luminosity functions for the inner, , intermediate, , and outer, , annuli of M 55. The color-selected field luminosity function (filled hexagons) is vertically shifted down for clarity.

The three LFs have been normalized to the star counts of the SGB region in the magnitude interval , after subtracting the contribution of the background/foreground stars scaled to the area of each annulus. In the lower part of Fig. 5, we show also the LF of the background/foreground stars estimated from the star counts at vertically shifted for clarity. In order to reduce contamination by those stars, all the LFs have been calculated selecting the stars within (again, is the standard deviation of the mean color) from the fiducial line of the main sequence of the cluster. The LFs do not include the HB and BS stars. The LF of the background stars has a particular shape: it suddenly drops at . This feature has a natural explanation considering the color-magnitude distribution of the field stars around M 55 and the way we selected the stars. The drop in the number of field stars is at the level of the M 55 TO and as can be seen in Fig. 2, or in the lower right panel in Fig. 3, the TO of M 55 is bluer than the TO of the halo stars of the Galaxy, which are the main components of the field stars towards M 55 (Mandushev et al. 1996). Selecting only stars within of the fiducial line of M 55 will naturally cause such a drop.

The completeness correction, as obtained in Appendix C, has been applied to the stellar counts of each field of M 55. As it is possible to see from Table 1, the magnitude limit varies from field to field. We have adopted the same, global, magnitude limit for all the LFs: i.e., that of the fields with the brighter completeness limit (field 16 and 17). This limits all the LFs to , corresponding to a stellar mass , for the adopted distance modulus and a standard 15 Gyr isochrone (see next subsection). The data for the inner annuli come from the central image, which has a limiting magnitude of the corresponding LF fainter than the global value adopted here. This is due to the better seeing of the central image compared to all the other images. We adopted a brighter limiting magnitude in order to avoid problems in comparing the different LFs.

Table 1. For each field of M 55 we list the total number of detected stars in both the V and I frame, the mean airmass of the field, the right ascension and the declination of the field center, the FWHM of the V and I point spread functions of the images, and the V limit magnitude of the observed fields. For each V image the exposure time was of 40 seconds, while for the I image it was of 30 seconds.

Fig. 5 shows clearly different behaviour of the LFs below the TO: they are similar for the stars above the TO, while the LFs become steeper and steeper from the inner to the outer part of the cluster: this is a clear sign of mass segregation. For the inner LF there is also a possible reversal in slope below .

In order to verify that the difference between the three LFs is not due to systematic errors (wrong completeness correction, imperfect combination of data coming from two adjacent fields etc.), we have tested our combining procedure in several ways. In one of our tests we built LFs of two EMMI fields at the same distance from the center of the cluster: i.e., we compared the LF of the field 2 with that of the field 6. After having corrected for the ratio between the covered areas and subtracting the field star contribution, the two LFs were consistent in all the magnitude intervals down to the completeness level of the data (that is lower than the one adopted). Having for field 2 a magnitude limit of 22.2 (see Table 1) and field 6 a limit of 21.5, we also verified that for the latter our star counts are in correct proportion below the completeness level of 50%.

In a second test, we generated two LFs dividing the whole cluster in two octants (dividing along the line that runs from the center of the cluster till the field 19 cf. Fig. 1). For each of the two slices we generated three LFs in the same radial range as in Fig. 5. After comparing all of them we did not find any significant difference. Therefore the differences among the three LFs in Fig. 5 must be real.

Another source of error in the LF construction is represented by the LF of the field stars. As will be shown in Sect.  5, M 55 has a halo of probably unbound cluster stars. The field star LF constructed from the star counts just outside the cluster can be affected by some contamination of the cluster halo. The consequence is that we might over-subtract stars when subtracting the field LF from the cluster LF, modifying in this way the slope of the mass function (the more affected magnitudes are the faintest ones). To test this possibility, we have extracted background LFs in two different anulii outside the cluster (in terms of , and ). Comparing the two background/foreground LFs we found that the number of stars probably belonging to the cluster but outside the tidal radius must be less than of the adopted field stars in the worst case (the faintest bins). The possible over-subtraction is not a problem for the inner and intermediate LFs, where the number of field stars (after rescaling for the covered area) is always less than of the stars counted in each magnitude bin. For the outer LF, the total contribution of the measured field stars is larger, but it is still less than of the cluster stars (the worst case applies to the faintest magnitude bin): this means that the possible M 55 halo star over-subtraction in the field-corrected LF is always less than (), negligible for our purposes.

4.1. Mass function of M 55

In order to build a mass function for the stars of M 55, we needed to adopt a distance modulus and an extinction coefficient. Shade et al. (1988) give , E , while, more recently, Mandushev et al. (1996) give , E . In the absence of an independent measure made by us, we adopted the values published by Mandushev et al. (1996) because they are based on the application, with updated data, of the subdwarfs fitting method. Using the LFs of the previous section we build the corresponding mass functions using the mass-luminosity relation tabulated by VandenBerg & Bell (1985) for an isochrone of and an age of 16 Gyr Alcaino et al. (1992). The MFs for the three radial intervals are presented in Fig. 6. The MFs are vertically shifted in order to make their comparison more clear.

Fig. 6. Mass functions for M 55 for three radial ranges: inner, , intermediate, , and outer, . The effect of mass segregation is clearly visible. The slope x corresponds to the index x of the power law fitted to the data in the range .

The MFs are significantly different: the slopes of the MFs increase moving outwards as expected from the effects of the mass segregation and from the LFs of Fig. 5. Fig. 6 clearly shows that the MF starting from the center out to the outer envelope of the cluster is flat: the index x of the power law, , best fitting the data are: , , and going from the inner to the outer anulii; this means that the slope of the global MF (of all the stars in M 55) should be extremely flat. Indeed, the slope of the global mass function obtained from the corresponding LF of all the stars of M 55 is: This result agrees with the results of Irwin & Trimble (1984), while the results of Pryor et al. (1991) appear in contrast to what we have found here.

Our MF in the outer radial bin can be compared with the high-mass MF of Mandushev et al. (1996), obtained from a field located at  arcmin from the center of M 55. As already reported in Sect. 2, Mandushev et al. (1996) obtained a deep MF for M 55 (down to ) which they describe with two power laws connected at . Their value of for the high-mass end of the mass function () is in good agreement with our value of , obtained in the same mass range for the outer radial bin. The low-mass end of the MF by Mandushev et al. (1996) () has a slope of .

The level of mass segregation of M 55 is comparable to that found in M 71 by Richer & Fahlman (1989). M 71 shares with M 55 similar structural parameters as well as positional parameters inside the Galaxy. The detailed analysis of Richer & Fahlman (1989) of M 71 showed that this cluster should also have a large population of very low mass stars ( ).

By fitting a multi-mass isotropic King model (King 1966; Gunn & Griffin 1979) to the observed star density profile of M 55, we compared the observed mass segregation effects with the one predicted by the models. Here we give a brief description of our assumptions in order to calculate the mass segregation correction from multi-mass King models. A more detailed description can be found in Pryor et al. (1991), from which we have taken the recipe. The main concern in the process of building a multi-mass model is in the adoption of a realistic global MF for the cluster. For M 55 we adopted a global MF divided in three parts:

• a power-law for the low-mass end, , with a fixed slope of (as found by Mandushev et al. 1996);
• a power-law for the high-mass end, , with a variable slope x;
• and a power-law for the mass bins of the dark-remnants where to put all the evolved stars with mass above the TO mass, : essentially white dwarfs. Here we adopted a fixed slope of 1.35, The mass of the WDs were set according to the initial-final mass relation of Weideman (1990).

To build the mass segregation curves we varied the MF slope x (the only variable parameter of the models) of the high-mass end stars in the range , finding for each slope the model best fitting the radial density profile of the cluster. Then we calculated the radial variation of x for the best-fit models in the same mass range of the observed stars: . The radial variations of x are compared with the observed MFs in Fig. 7. The mass function slopes are shown at the right end of each curve. This plot is similar to those presented in Pryor et al. (1986), and allows one to obtain the value of the global mass function of the cluster. The three observed points follow fairly well the theoretical curves. Also the high-mass MF slope value of Mandushev et al. (1996) (open circle in Fig. 7) is in good agreement with the models and our MFs. From these curves, we have that the slope of the high-mass end of the global MF of M 55 is , which is in quite good agreement with the global value of the MF found from the global LF of M 55 (cf. previous section). In Fig. 9 we show the model which best fits the observed radial density profile for a global mass function with a slope .

Fig. 7. Isotropic King model MF slope correction for M 55. The full dots are the slopes of the MFs obtained in this paper, the open circle is the measure of the high-end MF of Manddushev et al., 1996.

Fig. 8. Trivariate relation from Zoccali et al. (1997), between the distance from the Galactic center (), the height above the Galactic plane (), the metallicity of the cluster ([Fe/H]), and the slope of the global stellar mass function () of a globular cluster. The filled square marks the position of M 55.

Fig. 9. Radial density profile of M 55. Small crosses represent the raw stellar counts; filled dots are our star counts after subtracting the background counts contribution; open dots shows the M 55 profile published by Pryor et al. (1991); the continuous line is the single mass King model fitted to our star counts ().

The relatively flat MF of M 55 could be the result of the selective loss of main sequence stars, especially from the outer envelope of the cluster, caused by the strong tidal shocks suffered by M 55 during its many passages through the Galactic disk and near the Galactic bulge (Piotto et al. 1993, for a general discussion of the problem). A flat MF for M 55 agrees well with the results of Capaccioli et al. (1993) who have found that the clusters with a small and/or show a MF significantly flatter than the cluster in the outer Galactic halo or farther from the Galactic plane. Indeed, M 55 is near to the Galactic bulge,  kpc (  kpc), and to the Galactic disk  kpc. Fig. 8 shows that taking into account observing errors, M 55 fairly fits into the relation given by Zoccali et al. (1997), which is a refined version of the one found by Djorgovski et al. (1993). A different conclusion has been reached by Mandushev et al. (1996) using their uncorrected (for mass segregation) value for the MF of M 55. As noted by the referee, M 55 lies further from the average relation defined by the other clusters: of those with a similar abscissa (), M 55 is the one with the lowest value of x. It is not possible to identify the main source of this apparent enhanced mass-loss of M 55 compared to the other clusters; a possible cause can be a orbit of the cluster that deeply penetrate into the bulge of the Galaxy. This cannot be confirmed until is performed a reliable measure of the proper motion of M 55.

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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