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

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2. Influence of cutoffs in log VM: schematic treatment

Let us assume that the selection may be described as cutoffs on the wings of the symmetric p distribution. To illustrate, let us consider the simple case when there is only an upper cutoff [FORMULA], so that all [FORMULA] are missed, while all [FORMULA] are measured. The situation is sketched in Fig. 1. The calibrator sample (at a fixed distance) is on the right, while a distant cluster of galaxies is on the left. The [FORMULA]-limit is indicated by a dotted line. The correct regression lines, with the same slopes, are shown. Note that unaware of the p selection, one would derive from the cluster galaxies (or more generally from field galaxies) a too shallow slope. For the moment we simply assume that the correct slope is known. We return to the problem of slope in Sect. 7, and this issue will be discussed elsewhere (Ekholm et al. 1999) in connection with real data.

[FIGURE] Fig. 1. p vs. log of apparent diameter, [FORMULA] diagram for distant cluster and calibrator galaxies. Upper cutoff at p (dotted line) makes the inverse TF distance to the cluster too small when we force the correct slope, followed by both the calibrators and the cluster (dashed lines), through the cluster. The solid vertical line shows the angular diameter limit, the dashed vertical line is the `unbiased plateau' limit for the distant cluster. Cluster members that are not in the observed sample are marked with small symbols.

In inverse TF method (iTF) for deriving distances, one basically shifts one of the regression lines over the other, and the required shift of [FORMULA] gives the (logarithmic) ratio of cluster to calibrator distance.

In practice, one calculates for each cluster galaxy the iTF "distance" [FORMULA] using the calibrating relation [FORMULA], so that [FORMULA]. Averaging over all cluster galaxies gives the unbiased distance. This is equivalent to "shifting the regression lines". The iTF "distance" has a systematic error depending on p, though one does not know how large the error is (c.f. Sect. 5 in T84).

In the depicted situation, even if one knows the correct slope, the attempt to force-fit it on the cluster data will shift the line towards larger [FORMULA], because the upper cutoff has eliminated more galaxies from above the "correct" (unbiased) regression line than from below it. Hence, the inevitable result of an upper cutoff is that the iTF distance to the cluster will be underestimated.

Now, because of the limiting angular size, not all galaxies of the cluster are included. So one should add to the diagram a vertical line, only galaxies to the right of which are large enough to enter the sample. Note that this cutoff does not bias the distance indicator if there is no p-cutoff.

If one moves the cluster closer (to the right), the angular size limit cuts away less of the cluster galaxies. Then the influence of the [FORMULA]-cutoff on the derived distances becomes weaker. In terms of the calculated Hubble constant (actually up to an unknown factor, because [FORMULA] is still unknown), one should see in a [FORMULA] vs. [FORMULA] diagram first at small kinematical distances (proportional to the true ones) a constant "plateau" when the angular size limit keeps out of the cluster. Then there is an increase due to the larger influence of the [FORMULA]-cutoff.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999