4.5 Uncertainties

We now summarize the error budget in the GCLF method as it actually works. Consider an application in which the GCLFs for 2 or 3 giant ellipticals in the same cluster of galaxies have been measured. Thus if the uncertainty e (m₀) averages ± 0.3 mag per galaxy, the individual values can be combined and the mean < m₀ > for the entire group obtained to ± 0.2 mag (see Harris et al. 1991). The resulting distance modulus will then have an internal uncertainty due to the measurement process made up of:

(i) ± 0.2 mag for e (<m₀>) (note that for the Virgo system which represents the current best case, e (<m₀>) is 0.1 mag);

(ii) ± 0.2 mag for the internal uncertainty in M₀, as established from the calibrating galaxies;

(iii) ± 0.05 mag (typically) in the photometric scale zeropoint;

(iv) ± 0.05 mag for foreground absorption uncertainty (in most cases of interest, A_V is quite small since the program galaxies are at high latitude; for Virgo, the uncertainty would be about half this value).

Since these uncertainties are uncorrelated, they should be added in quadrature and the net internal uncertainty e (m - M)₀ for the system will be typically ± 0.3 mag.

There are also several potential sources of external error having to do with properties of the galaxies themselves: as usual, these are harder to quantify, but the known items include:

(1) The uncertainty in the fundamental RR Lyrae luminosity scale. Although M_V(RR) is a continuing matter of debate, recent discussions both observational and theoretical (e.g. Hesser et al. 1987; Jones et al. 1988; Fernley et al. 1990; Sandage and Cacciari 1990; Lee et al. 1990; Walker 1992) indicate e (M₀) ± 0.15 mag from this source.

(2) The systematic mean difference M₀ (turnover) between galaxies of different types, and most importantly between the Local Group spirals and the more distant large ellipticals. Calibrating this term thoroughly is the most urgent observational need at the present time for developing the GCLF technique. With additional measurements of M₀ in several more key galaxies (see below), it should be possible to reduce the error due to M₀ to a level less than ~ 0.1 mag.

(3) The intrinsic scatter of M₀ about the mean relation of M₀ vs. galaxy type. Again, this is not well known at present, but the existing comparisons within the Local Group and Virgo (Figure 8) suggest e (M₀) ± 0.2 mag from this source.

(4) Scatter due to metallicity differences between globular cluster systems in different galaxies. The clusters in giant ellipticals have mean metallicities consistently near [Fe/H] = -1.0 (Mould et al. 1987; Mould et al. 1990b; Geisler and Forte 1990; Couture et al. 1990, 1991), whereas in the Milky Way and M31 the average extends down to [Fe/H] -1.6 depending on the region of the halo (Zinn 1985; Elson and Walterbos 1988). For the usual broadband indices, such metallicity differences translate into ~ 0.1-mag shifts in mean color, or in luminosity if the observations are done in bands such as B which are metallicity-sensitive (Wagner et al. 1991; Bridges et al. 1991; Couture et al. 1990).

(5) Systematic errors in fitting the fiducial GCLF curve to the observations, leading to an incorrect estimate of m₀. This error can arise in cases where the photometric limit of the observations is near or at the turnover, so that the curve fitting must be done after having corrected the data for image detection incompleteness, internal photometric uncertainty, and background contamination, all of which are strong functions of magnitude near the photometric limit. Effects of increasing photometric error with magnitude tend to bias m₀ faintward, while the other effects may bias the result either positively or negatively. Biases in either direction may also occur (see above) if the wrong dispersion is used for the fitted curve. Although these errors seem to be no larger than the other items on this list, they have not yet been completely modeled in most real cases to date; for other related comments, see Hanes and Whittaker (1987), van den Bergh et al. (1985), and Harris et al. (1991). More powerful maximum-likelihood methods for matching the data to a model including all these effects simultaneously are being developed (Secker and Harris 1992) and will enable more nearly unbiased fits to be made.

The total budget for the external errors, under the most pessimistic assumption that they are uncorrelated, is 0.27 mag. Combining this with the internal errors yields a total uncertainty for the method of 0.4 mag per galaxy.

As imaging capabilities continue to improve, other effects which are currently too small to be important may become noticeable. For example, over the past decade, technical improvements have pushed image quality to the point (< 0."4) where globular clusters in galaxies as distant as Virgo can actually be resolved as nonstellar images. While this should not seriously affect the photometry when performing either point-spread function fitting or aperture photometry, it may also provide a means for better rejection of both field stars and background galaxies from the samples.