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4.7. Consistency of Mark III and VELMOD TF Relations

In constructing the Mark III catalog, Willick et al. (1996, 1997) required that the TF distances for objects common to two or more samples agree in the mean. As noted above, VELMOD yields an independent TF calibration for each sample included in the analysis. As a further consistency check, we can ask whether the VELMOD TF calibrations for the A82 and MAT samples are also mutually consistent.

We compared A82 and MAT TF distances using the VELMOD TF relations for 75 objects common to the two samples, excluding five galaxies with distance modulus differences greater than 0.8 mag (not all of these objects were part of the VELMOD analysis, since some did not meet the criteria outlined in Section 4.1). We found that the VELMOD calibrations yield an average distance modulus difference (in the sense MAT - A82) <Deltaµ> = -0.056 ± 0.046 mag; the Mark III TF calibrations yield <Deltaµ> = 0.018 ± 0.046 mag. The corresponding median distance modulus differences are -0.015 mag (VELMOD) and 0.035 mag (Mark III). Thus, as measured by the criterion of generating mutually consistent TF distances among samples, VELMOD gives the correct result. In Table 2, we list the VELMOD TF parameters and their Mark III counterparts. We see that the A82 zero points, slopes, and scatters derived from the two methods are in almost perfect agreement. The MAT zero points and scatters also agree to well within the errors. The MAT slopes show a somewhat larger discrepancy. However, the two slopes are nearly within their mutual error bars; moreover, the MAT sample used here is only about half as large as that used by Willick et al. (1996) in deriving the MAT TF slope. In any case, this slope difference, even if real, is of no consequence for the determination of betaI, as we now show.

As a final test of VELMOD-Mark III consistency, we ran VELMOD without allowing the TF parameters to vary, instead holding them fixed at their Mark III values. We did so both with and without the quadrupole, while holding sigmav fixed at 150 km s-1 and setting wLG ident 0 (note from Fig. 8 that these latter velocity parameters yield the same betaI as when they are allowed to vary freely). The results of this exercise are shown in Figure 9 and tabulated in Table 2. As can be seen, while there is a large formal likelihood decrease relative to the best solution, using the Mark III TF relations has a negligible effect on the value of betaI obtained from VELMOD. In particular, use of the Mark III TF relations does not bring our VELMOD result appreciably closer to the POTIRAS result, betaI = 0.86, of Sigad et al. (1997). We discuss this issue further in Section 6.1.1. Note that, in contrast with full VELMOD, neglect of the quadrupole now has no effect on the derived betaI, although its inclusion still results in a significant likelihood increase. The indicated formal error bars on betaI should not be taken literally here, because fixing the TF zero points prevents them from compensating for IRAS zero-point errors (cf. Section 3.3).

Figure 9

Figure 9. Same as Fig. 5, except that the A82 and MAT TF parameters have been fixed at their Mark III catalog values. The extremely small formal error bars result from fixing the TF parameters and are unrealistic. The maximum likelihood values of betaI differ negligibly from those obtained when the TF parameters are free.

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