Astron. Astrophys. 322, 730-746 (1997)

3. The method of normalized distances

In BGPT86 the method of normalized distances, based on a discussion of the Malmquist bias in TF distance moduli (Teerikorpi 1984), was first applied to a sample of field galaxies. The method is based on the expectation that the mean Hubble ratio averaged over all galaxies at the same (kinematical) distance and with the same value of , starts to increase at some limiting kinematical distance. Here the magnitude limit begins to cut away the faint part of the luminosity function . For different the limiting distance will be different by the factor from the TF relation . Hence, if one multiplies by , the behaviour of will be the same along the thus derived normalized distance axis. Finally, one inspects the vs. diagram for the unbiased plateau seen at small normalized distances as a horizontal part, now for all together.

The luminosity function is supposed to be nearly gaussian, with a constant dispersion over all values of p. These assumptions are realistic in view of the distribution of absolute magnitudes for different ranges. In some recent studies (Giovanelli 1996) it has been argued that is larger at smaller . However, we did not find such a tendency in our plateau sample (cf. Fig. 1). The TF relation itself also shows a regular distribution of the points along the regression line, allowing us to consider equal to the intrinsic TF scatter, except perhaps for small galaxies where measurement errors in diameters become important. The method of normalized distances was discussed and verified in BGPT86, Bottinelli et al. (1988), Bottinelli et al. (1995), and also in Ekholm (1996) by using synthetic data sets.

Fig. 1. Diameter (left) and magnitude (right) TF relations and their dispersions at different .

3.1. Analytical description

In its original form, the formula for the normalized kinematical distance was essentially given as

(magnitude relation, BGPT86).

This expression is for the ideal case where the sample is complete up to a given apparent magnitude limit or down to a given apparent diameter limit. But we may desire to use data from several different catalogues, each being complete up to a different apparent limit. It is then necessary to introduce an additional term for the definition of the normalized distance:

where may be arbitrarily chosen and depends on the catalogue. An example may be found in Bottinelli et al. (1988), where the study was done with different samples selected according to different ranges.

The influence of the magnitude limit on the bias properties has been well illustrated also by Sandage (1994a) using the device of the Spaenhauer diagram. The importance of a well defined limit was further discussed by Federspiel et al. (1994) in their analysis of the Mathewson-Ford-Buchhorn sample. Consequently, using the type dependent TF relation, we have to pay attention to the fact that, the sample being diameter selected, the magnitude limit will be different from one type to another. Thus, the factor has to be used.

It is also important to note that galaxies of the same type and , but of different inclinations form separate classes as far as the Malmquist bias is concerned. Highly inclined galaxies are much fainter in the sky than similar face-on galaxies at similar distance. Hence, they fall below the (magnitude) detection limit at a smaller distance than the face-on galaxies. In the same way, one must take into account the galactic extinction : in regions behind an enhanced extinction, galaxies appear smaller and fainter than elsewhere, hence, their normalized distance is larger. These effects were taken into account in Bottinelli et al. (1995), where the formula was written:

for magnitude relation, or

for diameter relation.

Here the factor or the constant C takes care of the influence of the inclination angle on the observed magnitude or diameter. However, this term practically vanishes when one uses the diameter relation: (Bottinelli et al. 1995).

Following the same idea, one must also take into account the fact that the zero-point of the TF relation depends strongly on morphological type T (Theureau et al. 1997). Using , all galaxies are shifted to type , which was chosen as a reference because it is one of the better populated and shows the nicest TF regression.

3.2. On the Spaenhauer diagram method

One can but note the similarities that exist between our normalized distance method and the method based on the use of the Spaenhauer diagram (Sandage, 1988a, 1994a, 1994b, Federspiel et al., 1994). The philosophical basis and hypotheses are indeed approximately the same, even if the methods are formally different.

Both methods assert the importance of the Malmquist effect on the estimation of the Hubble constant when using a flux-limited sample. Both assert that the bias considered here depends on the distance (Teerikorpi 1975, 1984, Sandage 1994a) and that the selection of an unbiased subsample is the only way to derive the good TF slope and an unbiased value of from field galaxies. It is also clear that the bias depends on (BGPT86, Bottinelli et al. 1988, Sandage 1994b), and consequently that the limit of the unbiased regime is a function of . Moreover, it is shown that attempts to get the volume-limited (unbiased) subsample or to correct for the bias, require the strict completeness of the sample either in magnitude or in diameter, depending on the TF relation used (BGPT86, Federspiel et al. 1994). Hence, redshift, , and magnitude limit, constituting the elements of Sandage's "triple-entry-correction", are also included in the method of normalized distance, in fact, in formulae (4) and (5). Note also that, both methods assume the symmetry of the luminosity function of a given class of objects (fixed , given type) and its invariability with distance.

However, the two methods have also differences. Sandage's method is, in a certain way, more empirical: it consists of dividing the sample in discrete boxes of to identify for each range separately, the corresponding unbiased region. Our more analytical method allows to treat the values continuously, all the objects being used in the same diagram to extract an optimal "plateau", statistically better defined due to the larger number of points. In addition, the normalized distance method permits one to take into account also more subtle effects due to inclination (i.e. opacity), galactic absorption, and morphological type or mean surface brightness. In addition, the normalized distance scale, the unbiased plateau, and the derived TF slope, are obtained by iteration and therefore optimized. Finally, the Spaenhauer method requires as assumption the uniformity of density to determine the unbiased subsample by fitting the estimated envelopes of the sample to the data plot. Our method does not need such a strong hypothesis to fix confidently the limit of the plateau.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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