11.2 The Comparison
We performed the comparison between methods by determining the mean and the scatter in the natural logarithm of the ratio of a given method's distances and those of SBF. PNLF and GCLF offer sufficient accuracy in a single measurement to warrant comparison on individual galaxies, so these results are reported as well as the comparison of cluster averages.
Table 3 presents the results. Column (1) names the method compared with SBF; column (2) gives the number of objects compared; columns (3) and (4) are the weighted average ratio between the method's distance and SBF and ^{2} for the resulting scatter around this mean; columns (5) and (6) list the unweighted average ratio and the rms scatter in the natural logarithm around this unweighted average; columns (7) and (8) give the median number of galaxies in a group used in computing the average distance from the method and the median error estimate among these galaxies; and columns (9) and (10) list the same quantities for the SBF measurements. Please note in columns (3) and (5) that the ratio between D_{n}- and SBF and SN Ia and SBF has units of km s^{-1} Mpc^{-1}.
(unwgt) | Method | SBF | ||||||||
Method | N | < ratio > | ^{2} | < ratio > | rms | <n> | <d/d> | <n> | <d/d> | Notes |
PNLF GCLF PNLF GCLF SN Ia D_{n}- D_{n}- D_{n}- TF TF TF | 14 9 7 6 6 34 32 12 34 24 14 | 1.027 1.156 1.072 1.117 85.4 86.5 89.3 91.1 0.921 0.960 0.986 | 23.6 3.7 6.4 3.0 20.0 70.3 32.8 11.8 100.8 67.6 15.2 | 1.054 1.157 1.094 1.127 92.4 92.5 92.8 92.5 0.909 0.974 0.970 | 0.088 0.140 0.078 0.152 0.248 0.227 0.165 0.123 0.275 0.237 0.127 | 1 1 1 1 2 2 2 4 4 5 9 | 0.044 0.120 0.056 0.160 0.133 0.149 0.149 0.105 0.157 0.122 0.083 | 1 1 1 2 2 2 2 2 1 1 1 | 0.052 0.050 0.057 0.052 0.050 0.052 0.045 0.052 0.052 0.050 0.055 | Individual galaxies Individual galaxies Cluster averages Cluster averages All clusters 2.5 clip applied 3 in cluster All clusters Has H or CCD data 5 in cluster |
There are three entries for TF because photographic and photoelectric TF measurements are sometimes considered less reliable (see Sec. 7.1.1). Hence, we fitted the distribution after those clusters with only these data were removed. Although H and red CCD data do improve the results, the distances are still highly inconsistent with SBF. Thirdly, we fitted the distribution of only those clusters in which at least 5 spirals had been observed. (The well-mixed groups of Dorado, Telescopium, and Leo are also included in this sample.) In these groups, we believe there is sufficient overlap, or co-mingling between the spirals and early-type galaxies to be confident that both types are at the same distance. In this case, the TF results are consistent with those of the SBF.
Figure 25. SBF is compared with PNLF abd GCLF for individual galaxies. The left-hand panels are the PNLF distance plotted against the SBF distance, and the ratio of the two plotted against SBF distance. The right-hand panles give the same for the GCLF distances. The dashed lines are drawn according to the weighted mean ratio given in Table 3. |
The comparisons between the various distances are shown graphically in Figures 25-29. Figure 25 shows the comparison between SBF and PNLF and SBF and GCLF for individual galaxies, and Figure 26 presents the comparison for PNLF and GCLF using cluster averages.
Figure 26. SBF is compared with PNLF and GCLF using cluster averages. The left-hand panels show the PNLF distance plotted against SBF distance, and the histogram of the ratios. The right-hand panels are the same for GCLF. The dashed lines in the distance plots are the weighted mean and the rms scatter from Table 3. Note that these are Gaussians in the natural logarithm, and since the abscissa is the ratio itself, the Gaussians are skewed and the peak does not correspond exactly to the mean value. |
It appears from the GCLF comparison that the GCLF method is more accurate than its error estimates, perhaps by 40%. There is again an offset in zero point relative to SBF of about 13% in the same sense as with PNLF which is significant at the 2 level. Either SBF distances are too small by 13%, or the GC's in the early type galaxies studied here are as much fainter than the GC's in the Milky Way, as the Milky Way is fainter than M31.
Figure 27 plots the comparison for D_{n}- and SN Ia. In these plots the distances have units of km s^{-1}, and the ratios have units of km s^{-1} Mpc^{-1}, and thus represent Hubble ratios in which the zero-points have been set using the SBF distances.
Figure 27. SBF is compared with D_{n}- and SN Ia. The left-hand panels are the same for SN Ia distances. The dashed lines in the distance plots are the weighted mean ratios from Table 3, and the smooth curves on the histogram plots are the Gaussians derived from Table 3. The D_{n}- curves come from the fits which have the two outliers removed. In these plots the distances have units of km s^{-1} Mp^{-1}. |
Figure 28 shows how the TF distances compare with SBF. The very poor agreement between TF and SBF distances is a surprise. ^{2} indicates that the errors in TF distances to groups with ellipticals are significantly worse than the error estimates. The factor is the same for H band and red CCD data as for photographic and photoelectric data, so the latter are not at fault. We plotted the ratio of D_{n}- and SBF against the ratio of TF and SBF, and found no correlation at all, so it is unlikely that the scatter comes from the SBF distances.
Figure 28. SBF is compared with TF distances. The left-hand panels show all of the TF data, as in Fig. 26, and the right-hand panels show the TF data which has been pruned of all the clusters for which only the B-band data are available. |
In order to explore the hypothesis that this inconsistency arises from the TF distances to spirals which may not be associated with elliptical galaxies with SBF distances, we plot in the upper left panel of Figure 29 the velocity difference versus the distance difference for the TF clusters with H or red CCD data and the corresponding SBF clusters. There is a poor but significant correlation present, suggesting that some of the clusters are separated in space and have a velocity difference arising from an undisturbed Hubble flow. That is, it appears to be more difficult to assign cluster membership to the spirals than to the ellipticals. In the lower left we reproduce the panel from the lower right of Figure 28, with lines showing SBF distance plus and minus 8 Mpc. This delineates the envelope of the points pretty accurately, and plausibly reflects the km s^{-1} Mpc^{-1} velocity selection criterion used to decide whether a spiral group was coincident with an elliptical group.
Figure 29. The upper left-hand panel shows the velocity difference vs the distance difference for the TF clusters with H or red CCD data and the corresponding SBF clusters. In the lower left-hand side we reproduce the panel from the lower right-hand side of Fig. 28, with lines showing SBF distance ± 8 Mpc. The right-hand panels show the comparison of SBF and TF when the sample is restricted to populous groups of spirals which we judged are likely to be co-mingled with their elliptical counterparts. |
The right panels of Figure 29 show the distribution of TF distances relative to SBF distances when the TF sample is restricted to those which are most likely to be closely associated with the adjacent cluster of elliptical galaxies. In this case the differences between TF and SBF become consistent with the errors. The TF-SBF comparison, in general, deals with disparate galaxies (spirals versus ellipticals) in which few, if any, galaxies are common to both methods. Consequently, we see differences that are larger than the formal expectations unless the samples are carefully chosen.
Thus, we believe that the TF error estimates probably are accurate, but that spiral and elliptical groups are often substantially disjoint, more so than is widely appreciated. At the risk of overinterpreting Figure 29, about half of the outlying spiral groups appear to follow the Hubble flow relative to the elliptical group, and about half have the same velocity at different distances, equally divided between near and far. The implications that this has for cluster virialization and large scale flows are beyond the mandate of this review.
The ratio of the restricted TF sample and SBF is not significantly different from unity (~ 0.98 ± 0.03) and the level of agreement is comparable to that of the PNLF-SBF agreement. The larger uncertainties in the TF method are statistically reduced by a larger sample.