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10.2.2 Correction For Biases

In principle, once the above corrections have been applied, the Dn-sigma method can be used to estimate relative distances. However there are two selection biases, which potentially afflict any sample of galaxies chosen in a diameter or magnitude limited way, that require correction. The first, a ``Malmquist-like'' bias, affects principally those galaxies which are not in rich clusters. As there are more galaxies at greater distances in any sample limited by diameter or apparent brightness, there will be more large/bright galaxies erroneously included in the sample from larger distances than there are galaxies omitted from the sample at smaller distances. Thus, a diameter limited sample is contaminated by galaxies that are measured to be large for their dispersion. This generates the illusion that galaxies are nearer than their true distance and would generate spurious positive peculiar velocities. Unless corrected, the effect of this would be to introduce a scale change as if to overestimate the Hubble constant. LFBDDTW show that the correction for this depends on the size of the dispersion in the distance estimator (i.e., ln Dn), Delta, and on the distribution of galaxies. If the latter is parameterized as a power law in number density such that the number of field galaxies in an annulus between r to r + dr is n(r)dr / r, so that:

Equation 23 (23)

(for a uniform distribution of galaxies alpha = 3), then the correction for the bias is given by equation 2.11 of LFBDDTW:

Equation 24 (24)

where de is the raw estimate of the distance and d is the corrected value. The correction is smaller for aggregates of galaxies, falling as 1/N when there are N galaxies with independently estimated distances de.

The second correction is applied only to galaxies in clusters where the selection by diameter causes the cluster to have an over-representation of large galaxies and an under-representation of small ones. LFBDDTW estimated the size of the correction for this effect by combining the galaxies with sigma > 100 km s-1 from clusters containing three or more galaxies using the preliminary distance estimates and then by truncating the sample at a range of diameters. The correction factors are tabulated in their appendix B and can amount to a 10% correction for aggregates at 6000 km s-1.

The steps outlined above allow relative distances of aggregates of galaxies to be determined self-consistently and accurately from Dn-sigma data for individual and cluster elliptical galaxies. This is one of the 2 methods of Aaronson et al. (1982): distances are calculated from the distance estimator and the parameters of the flow model are derived by maximizing the likelihood in the velocity field residuals. The second method of Aaronson et al. (1982) (discussed in the context of the Dn-sigma method by LFBDDTW and by Faber and Burstein 1989), involves determining the distance to galaxies on the basis of the observed velocity and a model of the velocity field and using this to predict the velocity dispersion for a given diameter. The model parameters are determined by maximizing the likelihood in the Dn-sigma relation. The two methods each have advantages depending on the exact application being pursued; Faber and Burstein (1989) give an extensive discussion of the pros and cons of each method and develop a hybrid approach.

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