11. COMPARISON OF DISTANCES

This article would be incomplete without some sort of direct comparison of distances from the various methods. However, it is beyond the scope of the review to try to present a compendium of distances, or to try to reconcile the inevitable discrepancies that will arise. We therefore undertook a blind test of the distance methods, where each of the authors provided distances and error estimates for a set of objects, and the results were not compared until all the distances had been collected. Most of the distances are available from the literature, but some unpublished material is also included.

We chose surface brightness fluctuations (SBF) as a basis for comparison because it offers good accuracy, a reasonably large data set, and a substantial distance range for comparison. We worked from a list of 44 groups and clusters, based on the groups defined in Davies et al. (1989), and 15 individual galaxies, for which there was overlap between SBF and other distance methods. Cepheids were not included in the comparison because the overlap was largely limited to the fundamental calibrating galaxies. Similarly, novae were not included because of the small number of comparison galaxies.

As much as possible, the various distances were adjusted by the authors to the assumptions of the SBF calibration: M31 and M32 are assumed to have a true distance modulus of 24.43 and a B band Galactic extinction of 0.31 magnitudes. Extinctions for the comparison galaxies are taken from Burstein and Heiles (1984). We shall describe briefly what we adopted for the zero point, group averages, and group errors for each method's distances. Someday we hope to understand every systematic error in these distance estimates; at this time, however, unknown systematic errors are suspected to be present and various allowances have been included to account for these errors. We have tried to be as explicit as possible when these are introduced, but justification is not possible within the space constraints of this section.

The SBF group distances are an unweighted average of the distances to the galaxies observed in the group. The distance error for each galaxy is the maximum of the formal uncertainty from the analysis and 5%. The distance error for each cluster is the maximum of the average error for the galaxies and the rms scatter among the galaxies observed, divided by N.

Distance errors from the planetary nebula luminosity function (PNLF) method are based on the formal uncertainty in the fit of the luminosity function plus other random uncertainties (e.g., standard stars, filter calibrations). Group distances are based on unweighted averages, and errors are the average error for the individual galaxies, divided by N, with an additional allowance of 5% for any systematic error arising from the definition of the M31 PNLF added in quadrature.

Globular cluster luminosity function (GCLF) distances have errors based on the uncertainty in fitting a luminosity function. The zero point is based on the Milky Way GCLF; the GCLF in M31 has been determined to be 0.2 magnitudes brighter. Thus the GCLF distances do not scale with an assumed distance to M31. Cluster distances and errors are derived from the weighted averages.

D_n- distances and errors were taken directly from Faber et al. (1989). Here the zero point is based on distant galaxies, where an undisturbed Hubble flow is presumed to hold. The Coma cluster is an important calibrator, where the D_n- distance is found to be 7461 ± 273 km s^-1; it has a redshift of 6890 km s^-1 relative to the Local Group (LG) frame as defined by Yahil et al. (1977), or 7200 km s^-1 relative to the cosmic microwave background (CMB) frame. The random error in an individual galaxy's distance is reckoned to be 23% in distance; cluster distance errors are reduced by N.

Distances from Type Ia supernovae (SN Ia) have been based on an absolute B band magnitude of -19.0, and this in turn comes from distances in the Tully Catalog of Nearby Galaxies, which uses H₀ = 75 km s^-1 Mpc^-1 to derive distances. In order to avoid circular reasoning, we have chosen to refer to these distances in terms of velocity, by multiplying the factor of 75 km s^-1 Mpc^-1 back in. The mean distance derived this way for three SN Ia in the Coma cluster is 7160 km s^-1, which is in good agreement with its redshift. We have adopted an error of 0.35 magnitudes in distance modulus for each supernova observed, diminishing this by N for each cluster, since this is roughly the scatter we observe among observations in clusters.

Lastly, distances using the Tully-Fisher (TF) method are founded on Cepheid and RR Lyrae distances to M31, M33, and the Sculptor Group. Photographic and photoelectric (primarily B band), infrared (primarily H band), and CCD (primarily R and I band) observations have been used, with errors assumed of 0.5 mag, 0.4 mag, and 0.3 magnitude per galaxy. Cluster distances arise from averaging together individual galaxy distances and assigning errors which are the maximum of the method error and the scatter in a cluster, divided by N. The three types of observations are combined by adopting weights of 0.5:1:1.