Astron. Astrophys. 343, 713-719 (1999)

## 3. "Unbiased plateau" for the inverse relation

The plateau in the vs. diagram is not an unbiased plateau; it still gives too large a value of , because of a residual effect of the -limit. The bias can be reduced by applying an upper cutoff to the cluster sample in , as illustrated by the dashed vertical line in Fig. 1. The galaxies between the upper and lower cutoffs now provide the best iTF distance estimate.

The position of the critical righthand side vertical line in Fig. 1, which cuts away the "bad" interval, depends on the distance of the cluster (compare the cluster and the calibrators). However, its position is practically constant on the linear size scale , independently of the cluster distance. This is clearly seen if one shifts in Fig. 1 the cluster galaxies to the right, i.e. decreases their distance until it coincides with the distance of the calibrator cluster. Because the TF relations are essentially the same, the critical vertical lines will almost coincide. Hence, in the general case of a field galaxy sample, one may shift all galaxies independent of their distances, to one diagram of vs. , where the abscissa is up to an unknown constant (as far as is unknown) the same as linear diameter, . If one now accepts galaxies left of the (common) critical line only, an unbiased average distance is derived.

One might term (or ) the normalized distance, analogously to the case of the direct TF normalized distance introduced in T84. Then in the vs. diagram, using the correct slope , there is at small normalized distances an unbiased plateau, after which starts to grow.

The resulting unbiased plateau should be horizontal, if the slope is correct. ET97 pointed out a simple device of deciding which is the correct inverse slope: in the vs. diagram the run of points should be horizontal. This was, however, made under the premise of no p-selection. Now we understand that instead of one should use on the abscissa, in order to identify the cutoff . Only galaxies below this cutoff are expected to show a horizontal run on the vs. diagram, if is correct. Fig. 2 shows how in the more general case, the lower cutoff causes too small below the unbiased plateau. Ekholm et al. (1999) will discuss the evidence for upper and lower cutoffs in in the KLUN sample and how the described method can in practice be implemented.

 Fig. 2. Effect of the p-cutoffs on the value of at different . The normalized distance is defined as .

Though in this manner one may detect sharp cutoffs in and derive an unbiased value of without detailed modelling, it is important to know how large an influence the cutoffs would have on , if their presence were ignored. We discuss this in Sects. 4-6.

© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999