4. APPLICATION TO MARK III CATALOG DATA

4.1. Sample Selection

To apply VELMOD to the real TF data, we first needed to identify a suitable subsample of the Mark III catalog. As discussed in Section 2.2.2, we elected to restrict the TF sample to cz_LG 3000 km s^-1, where, here and throughout, we correct heliocentric redshifts to the Local Group frame following Yahil, Tammann, & Sandage (1977). We thus use the A82 and MAT TF samples, which are rich in local galaxies, restricted to this redshift interval. The two cluster samples in the Mark III catalog, HMCL and W91CL, contain only clusters at greater redshifts and thus are not used here. Finally, only a small fraction of the W91PP and CF samples is found at cz_LG 3000 km s^-1 (~ 2% of W91PP, ~ 15% of CF). This small number of additional galaxies was not worth the additional six free parameters that would be required for the likelihood maximization procedure (Section 2).

We made several further cuts on the data, as follows:

1. An RC3 B-magnitude limit of m_B = 14.0 mag was adopted for A82. As discussed by Willick et al. (1996), A82 galaxies within 3000 km s^-1, and subject to this magnitude limit, are well described by the "one-catalog" selection function of Willick (1994) that enters into the likelihood equations (Section 2).

2. An ESO B-band diameter limit of d_ESO = 1.6 arcminute was adopted for MAT. As discussed by Willick et al. (1996), this allows the MAT subsample to be described by the one-catalog selection function of Willick (1994).

3. Only galaxies with axial ratios log (a / b) 0.1 were included. This cut, corresponding to an inclination limit of i 38° (Willick et al. 1997), reduces TF scatter due to velocity width errors.

4. Galaxies with < -0.45 (rotation velocities less than about 55 km s^-1) were excluded. In practice, this criterion applied to the MAT sample only, which contains numerous, very low line width galaxies. The need for excluding such objects was discussed by Willick et al. (1996).

5. Two objects within the Local Group, defined as having raw forward TF distances less than 100 km s^-1, were excluded. However, no lower bound was placed on the redshifts of sample objects.

This left a sample of 856 A82 and MAT galaxies. As discussed by Willick et al. (1996, 1997), real samples exhibit a mainly Gaussian distribution of TF residuals, but with an admixture of a few percent of non-Gaussian outliers. We excluded 18 additional galaxies (four in A82, 14 in MAT), or ~ 2% of our sample, because of their extremely large residuals from the TF relation. Finally, then, 838 galaxies, 300 in A82 and 538 in MAT, were used in the VELMOD analysis. Of these, 53 of them are objects found in both samples (although with different raw data) and thus are used twice in the analysis.