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5.1. Calibration of SBF

It may appear from the brief description above that SBF is a purely ``geometrical method,'' like parallax. If this were true, it would free the method from the nagging questions that plague other DIs: are they really universal, or do they depend on galaxy type, age, environment, and so forth? In reality, however, SBF dependent on the stellar populations in the galaxies to which it is applied. Not only does this mean that we need to be cautious with regard to its univerality, but, also, that it is difficult to derive an absolute calibration of SBF from first principles. Like the other DIs considered in this Chapter, absolute distances obtained from the SBF technique are tied to the Cepheid distance scale. If the latter were to change, so would the SBF distances. In particular, estimates of H0 derived from SBF studies (see below) may well require revision as the HST Key Project (Section 2) yields new results.

The stellar population dependence of SBF arises because the stars which contribute most strongly to the fluctuations are those that lie at the tip of the giant branch. Tonry and coworkers parameterize this effect in terms of ``effective fluctuation magnitudes'' Mbar (absolute) and mbar (apparent). The quantity Mbar may be thought of as the absolute magnitude of the giant branch stars which dominate the fluctuations; mbar is an apparent magnitude obtained from the observed fluctuations. If all galaxies had identical stellar populations, they would all have the same value of Mbar, and their distance moduli would be given simply by mbar - Mbar.

Because galaxies do not have identical stellar populations, it is necessary to determine an empirical correction to Mbar as a function of a distance-independent galaxian property. Tonry and coworkers use (V - I) color for this purpose. Their most recent calibration is

Equation 10 (10)

which is valid for 1.0 leq (V-I) leq 1.3 (Tonry et al. 1997). We discuss the zero point of this relation in Section 5.2. The color dependence indicated by equation 10 is readily seen in the mbar versus (V - I) diagrams of several tight groups and clusters, as shown in the upper panel of Figure 5. A line of slope 4.5 - the solid lines drawn throught the data points - fits the fluctuation magnitude-color data in each group well. (The solid points, as well as the small open squares, are thought to be non-members of the groups.) The different intercepts of the solid lines reflect the different distances to the groups.

Figure 5a
Figure 5b
Figure 5. Top panel: fluctuation apparent magnitudes mbar versus (V - I) color for several nearby groups and clusters. The solid lines drawn through the data points all have slope 4.5. Bottom panel: a plot of the theoretical Mbar versus (V - I) relation, from the stellar population synthesis models of Worthey (1994). The different point types indicate different metallicities relative to the Milky Way, as coded in the inset. For each point type, there are several distinct points, corresponding to different stellar population ages, as indicated by the arrow. The solid line is a fit to the theoretical models with slope fixed at 4.5. The dashed line is the empirical relation, equation 10. Adapted from Tonry et al. (1997).

The correlation of fluctuation magnitude with color is quite strong. Accurate colors are therefore required in order to minimize systematic effects. The possibility that the slope or zero point of this correlation may not be universal, but instead depend on some as yet undetermined galaxy properties as suggested by Tammann (1992), merits further attention. However, Tonry et al. (1997) show that the most likely manifestation of such a problem, a trend with metallicity of residuals from the Mbar - (V - I) relation, does not exist. It is reassuring, moreover, that theoretical stellar population synthesis models predict a trend of fluctuation magnitude with color that is very similar to the empirical one. This is shown in the lower panel of Figure 5, in which the population synthesis models of Worthey (1994) are plotted in the Mbar - (V - I) plane. The points represent models of various metallicities relative to the Milky Way (indicated by point type as coded in the inset of the figure), and of various ages (the trend with age, at a given metallicity, is indicated by the arrow). The solid line, which has slope 4.5 and an intercept determined by fitting to the theoretical models, is seen to be a reasonable fit. The dashed line is the emprical relation, equation 10. The zero point of the theoretical relation differs from that of the empirical one by only 0.07 mag.

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