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

## 3. Type dependence as revealed by the inverse diameter relation

As noted in the Introduction, we study primarily the type dependence using the inverse TF-relation (), i.e. presenting the data and calculating the regression lines as against logD where D is the linear diameter calculated from the kinematical distance and the angular size . The panels of Fig. 1 show the inverse relations for the types T = 1 to 8. It is seen even by eye that there are systematic differences along the Hubble sequence. Note also the large scatter for Sa galaxies and the occasional outliers in every type. It may be of some significance that in the early types there is a slight curving down at small diameters, possibly due to the influence of the bulge profile on (Sect. 6). In Fig. 2 we give the slopes and the zero-points, respectively, for each type, together with the error bars. The mean value of the slopes is 0.503 . We checked this value by combining all the types in one diagram and shifting them to have the same . Such a normalization is necessary in order to overcome the bias due to the particular locations of the clouds of data points for different types. This resulted in the slope = 0.504 0.01 . From the small scatter of slopes in Fig. 2 we conclude that one may use the same slope for all types, and adopt = 0.5 . This slope is shown as the dashed lines in Fig. 1. The middle panel of Fig. 2 gives the progression of the zero-points according to type T. There is a clear decrease towards the later types. The fluctuations around the general trend are clearly due to the fluctuations in the slopes of the regression lines. When we fix the slope to = 0.5, we get the final zero-points in the bottom panel of Fig. 2. The numerical values are listed in Table 1.

 Fig. 1. Inverse Tully-Fisher relation for diameters. Visualisation of the different regression lines for the different morphological types (full line). The dashed line corresponds to a "forced" slope a' = 0.5.
 Fig. 2. From top to bottom: the slopes, zero-points, and zero-points when using a fixed common slope a'=0.5, of inverse diameter relation, plotted against the morphological Hubble type of galaxies.

Table 1. slope and zero-points for the direct (a,b) and the inverse (a',b') diameter TF-relations

Table 2. slope and zero-points for inverse magnitude TF-relation

In the panels of Fig. 3 we give average surface brightness vs. residual from the iTF relation. Each type reveals a significant dependence between these quantities: positive residuals go together with a brighter surface of the galaxy. This indicates that the scatter in the diameter iTF relation is not due to errors in logD (nor in ), but is mostly intrinsic. Additional support for the small errors in logD (coming e.g. from errors in kinematical distances) is obtained from our study of the M vs. logD relations which for late types give the slope 5 as expected from the pure disc model, but significantly different slopes for the early types. That the errors in logD do not dominate, gives further significance to the slope 0.5 derived above.

 Fig. 3. Correlation between mean surface brightness and iTF residuals in , for the different types.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998