Astron. Astrophys. 319, 435-449 (1997)

## 5. Type dependence in the direct relation

The direct TF-relation is naturally expected also to show a type dependence. However, special care is needed in the analysis, because especially the zero-point shifts may be here deformed by the Malmquist bias. Our tool, as in our several previous studies, is the method of normalized distances. Details may be found in Bottinelli et al. (1995). Because we now know that there are zero-point differences, these must be incorporated in the formula for normalized distances as an additional factor . A few iterations are needed when one calculates from the unbiased "plateau" the regression lines for each type. The procedure is facilitated by our assumption that the slopes of the lines are identical. Note that this is not necessarily true, because intrinsic variations of at a constant logD or M may depend on type.

In order to derive the slope, we made several experiments with different normalized distance limits defining the plateau. It was found that the slope does not change when different reasonable plateau-limits are used. This is as expected from the simulations performed by Ekholm (1996). For the diameter relation, we finally adopted the slope a = 1.04, common for all types. Unfortunately, even with our large sample, the unbiased subsample defined by the plateau is reduced to a few hundreds of objects. Then, only types 3 to 7 still contain a sufficient number of points (see Table 1). Despite the large error bars, we observe again for this type range that the TF slope remains quite stable, and that the zero-points seem to shift progressively as expected from Sect. 3 (see Fig. 6 and top panel of Fig. 7). We emphasize that our models in Sect. 7 are based on the type effect revealed by the inverse relations where it is naturally expected and more easily detected.

 Fig. 6. Direct Tully-Fisher relation in diameter obtained with datas from the unbiased plateau (). Visualisation of the different regression lines corresponding to the different morphological types (full line). The dashed line corresponds to a "forced" slope a=1.04 .
 Fig. 7. Same as Figs. 2 and 4, now for direct diameter relation. A fixed common slope a=1.04 was used in the bottom panel.

Bottom panels of Fig. 7 show the zero-points from the regressions giving the slopes in the top panel, and after fixing the slope to 1.04. The zero-points for the constant slope are given in Table 1. It should be noted that the zero-points in Table 1 are not calibrated in the usual sense, but correspond to the adopted kinematical distance scale.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998