The luminosity functions and CM diagrams of the off-centre field of M 32 are seen in Figs. 1 (click here) and 2 (click here). With the artificial-star experiments as our basis we will now discuss the uncertainty associated with the identified stars.
Figure 1: Luminosity functions for objects identified in I and V, respectively, in the HST PC-frame at from the centre of M 32. A total of 19461 stars were found to match in I and V. No correction for M 31 has been made, see text. The slope of the I-band luminosity function is found to be 0.42. (This value is not used elsewhere in this paper)
Figure 2: Colour-magnitude diagram of the off-centre field in M 32. This figure was produced by matching the objects in Fig. 1 (click here). The 212 star signs show the magnitude and colour of the stars in our artificial-star experiments
Figure 3: Results of our I-band artificial-star experiments. For each magnitude group of added stars (vertical line) the histogram of recovered stars is shown. Note, the number-axis is arbitrary (Sect. 3). To clarify the implications of image crowding we plot several error indicators, in addition to the mean absolute deviation (MAD), of the magnitude of the recovered star . This demonstrates that the distribution of recovered stars is skewed towards brighter magnitudes, that is, the faintest objects are found too bright. This "bin jumping" was discussed by Drukier et al. (1988)
Figure 4: Same as Fig. 3 (click here) but now for the V-band
In Figs. 3 (click here) and 4 (click here) the magnitude of each group of added stars is indicated by a vertical line. The magnitudes of the recovered stars are binned into histograms of bins, one histogram for each magnitude group of added stars. Each of the histograms has been normalised so that they all cover the same number of stars (implying that the number-axis is arbitrary). This illustrates that the brightest stars are easily identified, whereas fainter stars are "spread out" from their original magnitude (the vertical line) to surrounding magnitudes. Firstly, note that for the faintest bins the probability of a star being found in a bin other than the one to which it was added is nonvanishing. If we want to correct the luminosity function for incompleteness, this "bin jumping" has to be taken into account (Drukier et al.\ 1988). Secondly, the "bin jumping" is clearly asymmetric with an extended tail of bright measurements. As the measurement errors are clearly not Gaussian we shall replace the usual, but now inadequate, standard deviation by the mean absolute deviation (MAD). (The MAD is related to the median as the standard deviation is related to the mean, but the MAD does not require that the distribution of the errors is Gaussian and it is a more robust estimator than is the mean, Press et al. 1992). In addition, we supplement with the lower and upper quartile, and several percentiles, of the magnitude of the recovered star . These numbers are indicated in Figs. 3 (click here) and 4 (click here) and illustrate that the faintest stars in the CM diagram are very likely identified too bright by several tens of a magnitude as compared to their true, unaffected magnitude.
Note, the straight lines represented by diamond symbols in Figs. 3 (click here), 4 (click here), and 9 (click here) indicate the magnitude of the added stars and the expected magnitude of the recovered stars if they are not affected by "binjumping", i.e., . In order to claim that we are able to carry out reliable photometry, must be a unique function of , and we must be certain that we are measuring the added star and not a statistical lump of unresolved stars.
The distribution of the colour of recovered stars is, however, fairly symmetric and much less wide than that of the two passbands individually (Fig. 5 (click here)). That is, the added stars which undergo "bin jumping" tend to keep their original colour, mainly because they are most likely identified on top of a star with a colour similar to their own - the majority of stars in the CM diagram are found along the giant branch with . We may therefore expect that the colours of the stars in the CM diagram are correct.
Figure 5: Colour deviation of recovered stars for the 9 artificial I-band magnitudes (error indicators are as in Figs. 3 (click here) and 4 (click here)). This shows that stars are very likely to be found with the correct colour. The explanation is that the majority of stars in the CM diagram (Fig. 2 (click here)) have more or less identical colours
Figure 6 (click here) illustrates that the asymmetric distribution in the histogram of recovered stars is not a result of plotting the number of stars as a function of magnitude. We generated a Gaussian distribution of fluxes according to , where N0 = 106, f0 = 1.0, and . The corresponding magnitudes were binned into bins. The increase in the number of stars fainter than f0 (m > 0) exceeds a small increase just below m = 0 and is clearly opposite the asymmetry seen in Figs. 3 (click here) and 4 (click here). Note that Secker & Harris (1993) adopt a Gaussian function to represent the measurement error as a function of magnitude.
Figure 6: Asymmetry that arises due to the effect of plotting a histogram of stars as a function of magnitude , where the distribution in flux f is Gaussian (arbitrary number-axis). The asymmetry is opposite that of the histograms in Figs. 3 (click here) and 4 (click here)
The limiting magnitudes at half probability are V1/2 = 26.2 and I1/2 = 25.1 for the M 32 off-centre field. The limiting magnitudes are based on all the stars plotted in Figs. 3 (click here) and 4 (click here), respectively, i.e., regardless of the associated error estimate. Note, the standard deviation , as calculated by ALLSTAR, amounts to only and at the limiting magnitudes for I and V, respectively, whereas the MAD is higher by a factor of 3.8 and 2.8 as compared to the standard deviation (for a Gaussian distribution ). This merely indicates that in the present investigation the standard deviation, as given by ALLSTAR, may be considered too optimistic.