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

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5. Calculation of the average bias

In order to have an idea of how large an influence the p-cutoffs have on the value of [FORMULA] calculated from the whole sample using a (correct) inverse relation, we derive the average bias for a special case of radial space density behaviour:

[EQUATION]

where r is the radial distance from our Galaxy and [FORMULA] corresponds to homogeneity. It is easy to understand that the average bias depends on how the observed galaxies are distributed along the [FORMULA]-axis, which must depend on how galaxies are distributed in space.

Rather than trying to start with averaging the derived [FORMULA] over [FORMULA] (Eq. 6), we use another route, based on the fact that the bivariate distribution [FORMULA] is known for an angular diameter limited sample, when space density behaves as Eq. (8). We assume for simplicity that the cosmic distribution of p (as well as of [FORMULA]) is normal, which is a rough approximation for fixed Hubble types, see e.g. Fig. 1 in Theureau et al. (1997a).

As illustrated by Fig. 1 in T84 for the case of magnitude TF relation, in magnitude and volume limited samples the direct regression lines are parallel and separated by [FORMULA] (when [FORMULA]). The separation happens so that the volume-limited line (carrying with it the bivariate distribution) glides along the unchanged inverse line which goes through the average M of both the volume ([FORMULA]) and magnitude limited ([FORMULA]) bivariate distributions. Fig. 3 gives a similar diagram for the diameter TF relation, showing the three kinds of regression lines for a synthetic sample of field galaxies.

[FIGURE] Fig. 3. A volume limited subsample of a synthetic data-set is shown as crosses, and the diameter limited subsample of the same parent data-set is marked with circles. Direct TF relations of these subsamples, [FORMULA] and [FORMULA], are parallel and separated vertically by [FORMULA] (Eq. 11). The inverse relation [FORMULA] (Eq. 12), that under ideal conditions is the same for both subsamples, intersects the direct regression lines at [FORMULA] and [FORMULA].

In the present calculation we need to know how the diameter direct relation is constructed from the parameters defining the iTF (c.f. T84, Teerikorpi 1993, where this was given for the magnitude TF relation):

[EQUATION]

where

[EQUATION]

The Malmquist shifted (angular size limited) direct relation is

[EQUATION]

and the inverse relation is

[EQUATION]

One can express the average bias at a fixed p using the inverse and (Malmquist-shifted) direct relations:

[EQUATION]

where

[EQUATION]

so that

[EQUATION]

This simple expression is then easy to integrate over all p in the sample, when one notes that the distribution of p is

[EQUATION]

where [FORMULA] gives the Malmquist-shifted average value of p (shifted by [FORMULA] times the shift of [FORMULA]):

[EQUATION]

Averaging [FORMULA] over all p from [FORMULA] to [FORMULA] gives:

[EQUATION]

where [FORMULA] and [FORMULA] are

[EQUATION]

Note that the information on the space density (via [FORMULA] and Eq. 16) is contained in [FORMULA] and [FORMULA], which show how far away from the Malmquist-shifted average [FORMULA] the cutoffs are, in terms of the dispersion [FORMULA] of the cosmic (Gaussian) distribution function of p. The factor before [FORMULA] can also be written in terms of [FORMULA] and [FORMULA]. Then

[EQUATION]

Such an analytic calculation is possible only in the given special cases of space density. Generally, the distribution of points in the p vs. [FORMULA] diagram will differ from the Gaussian bivariate case: at each p, the shift of [FORMULA] will be due to different Malmquist biases of the 1st kind. However, even in this case the inverse relation is not affected and the calculation and prediction of Sect. 3 is valid.

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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