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

Appendix A: image reduction and analysis

The images were reduced using the standard algorithms of bias subtraction, flat fielding and trimming of the overscan, without encountering particular problems. The stellar photometry was done using DAOPHOT II and ALLSTAR (Stetson 1987) The second version of DAOPHOT was particularly useful for the image analysis since we were forced to use a variable point spread function (PSF) through the images. In fact, the stellar images of the EMMI Red Arm together with the F/2.5 field camera presented coma aberration at the edges of the field: the resulting PSF was radially elongated. To better interpolate the PSF we also used an analytic function with 5 free parameters (i.e. the Penny function of DAOPHOT II).

In order to obtain a single CMD for all the stars found in the 18 fields we first obtained the CMD of each field matching the V and I photometry. Then, we combined all the CMDs using the relative zero points determined from the overlapping regions of adjacent fields. All the CMDs were connected to the main CMD, one at a time, following a sequence aimed at maximizing the number of common stars usable for the zero point calculation. The central field CMD was used as the starting point of the combination. For the outer fields we used a minimum of 20 common stars while for the inner fields we had at least 300 stars. The mean error of the zero points was magnitudes, compatible with the errors calculated from the crowding experiments. Since the night was not photometric, we could not directly calibrate our data. We were only able to set an absolute zero point using the unpublished calibrated photometry of the center of M 55 by Piotto (see next section).

In order to perform the photometry of the central field of the cluster, we divided it into 4 subimages of pixels, to minimize the effect of the strong stellar gradients present in this image. We allowed a good overlap to be able to perform the successive combination of the photometry of the stars. In this way we also avoided two problems: we had better control of the PSF calculation and we reduced the number of stars per image to be analyzed. Thanks to the low central concentration of the cluster and the fairly good seeing of the images (even if the crowding was not completely absent), we were able to obtain complete photometry down to .

Appendix B: calibration of the photometry

In principle, the analysis of the radial density profile does not require calibrated photometry. But this operation is necessary if we want to analyze the stellar population of the cluster, together with its stellar luminosity and mass functions. Since we could not use standards taken during the same night, we have performed a relative calibration using existing photometry of M 55. For the V magnitude we linked our data to Piotto's (1994) un published photometry of the central field of M 55 from images taken with the 2.2 m ESO telescope. For the (V-I) we calibrated our data against Alcaino et al. (1992) photometry. They published a CCD BVRI photometry for two different non-overlapping fields outside the center of M 55, named FA and FB, with dimensions of , contained in our central field. Fig. 14 shows our V zero point calculated against Piotto's (1994) while Fig. 13 shows the V zero point of the two fields FA and FB of Alcaino et al. (1992). The mean zero point for the two fields of Alcaino et al. gives which compare well with . The two are in good agreement taking into account the errors. There are no magnitude gradients. The LFs are coming from the photometry in the V-band.

Fig. 13. top: Differences between our instrumental V magnitude () and the calibrated magnitude by Alcaino et al. (1992) (). Bottom: Differences between our instrumental color, , and the calibrated color by Alcaino (1992), .

Fig. 14. Differences between our instrumental V magnitude () and the calibrated magnitude of Piotto (1994) ().

Before the publication of the I-band photometry by Mandushev et al. (1996), the one by Alcaino et al. (1992) was the only photometry in the literature. Unfortunately, the M 55 data set of Mandushev et al. (1996) does not overlap with any of our fields: it is centered just few arcmin south of our field 2. In Fig. 13, we show the difference between our data and those of Alcaino et al. (1992). In this case, the two zero points calculated for Alcaino's fields differ by a significant amount. We do not know the origin of this discrepancy, which we believe is internal to the data of Alcaino et al. (1992). They could not resolve this due to the fact that fields FA and FB do not have stars in common. We believe that the problem is not in our data since both Alcaino's fields are contained in the same subimage of the central field. Lacking other independent (V-I) calibrations, we are forced to adopt as our color zero point the mean of the two values of FA and FB: .

Appendix C: crowding experiments

For each field, we performed a series of Monte Carlo simulations in order to establish the magnitude limit and the degree of completeness of the CMD. The magnitude limit has been defined as the level at which the completeness function reach a value of 0.5, V(50%). This value is reported in Table 1 for each field.

The procedure followed to generate the artificial stars for the crowding experiments is the standard one Piotto et al. (1990b). The completeness function used to correct our data is the combination of the results of the experiments in both the V and I images for each field. In the outer fields the stars were added at random positions in a magnitude range starting from (just 0.5 mag below the main sequence TO). In the I band experiments, we used the same star positions of the V experiments, with the I magnitudes set according to the corresponding main sequence color. For each outer field, we performed 10 experiments with 100 stars. For the inner fields (fields number 2, 3 and 6) the experiments were 10 with 100 stars in an interval of only 1 magnitude for 5 different magnitudes (a total of 50 experiments). Moreover, in these fields the stars were added taking into account the radial density profile of the cluster. For the 4 subimages of the central field, we performed independent crowding experiments. For each subimage, we ran 10 experiments in 0.5 mag. steps in the range , with the stars radially distributed as the density profile of the cluster. In this way, we were able to better evaluate the level of the local completeness of the photometry.

The completeness function has been calculated for each field taking into account the results of the two different experiments in V and I. As an example in Fig. 15 we show the completeness functions for field number 3 (top) and 19 (bottom). The results of the experiments were fitted using the error function:

is the magnitude at which the completeness level is 50%, V(50%); gives the rapidity of the decrease of the incompleteness function and is connected with the read out noise and the crowding of the image. For the star counts correction we used the interpolation with the previous equation instead of using directly the noisy results of the experiments (these were too few to lower the small number statistical noise of the results). In this way we avoid the adding of noise to the star counts. In every case we verified that the fitting function is an acceptable interpolation that gives very low residuals compared to the error distribution function.

Fig. 15. Top: Completeness function for the field number 3. Bottom: Completeness function for the field number 19. Both functions have been obtained combining the crowding experiments on in V and I.

© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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