Astron. Astrophys. 334, 395-403 (1998)

2. The sample and the method

2.1. KLUN

The KLUN galaxy sample and the quality of the measured quantities of its members have been recently described by Theureau et al. (1997b). Originally intended to be complete down to the isophotal diameter , it now consists of 5171 galaxies covering the type range Sa - Sdm, having redshifts and the measured 21cm-line widths for the Tully-Fisher relation. 4577 galaxies have also the magnitude.

For calculation of distance moduli µ, we use the direct TF relations established by Theureau et al. (1997b) and the same corrections for type effect, internal extinction, and galactic absorption. For example, the diameter TF relation gives the distance modulus µ as

where . In this formula, is the angular diameter corrected for galactic absorption and internal extinction, is the maximum rotational velocity measured from the 21cm-line width of the galaxy, is the TF zero point for galaxy type T, and a is the slope of the TF relation.

2.2. The method of TF distance moduli

We apply here a method outlined in Sect. 3 of Teerikorpi (1993), where the TF distance moduli can be used, in principle, up to the limits of the sample, in order to derive the form of the density law. The method utilizes galaxies in narrow ranges to construct several differential distributions of µ, one for each range. These distributions are displaced relative to each other in µ, because of different (absolute magnitude) and the common diameter limit (or more exactly, the common incompleteness law). They have also generally different amplitudes, because different ranges are populated by different numbers of galaxies, though their forms are similar if inside the ranges the galaxies are evenly distributed.

One expects that above some , the sample is complete, hence below certain limits each distribution reflects the true average space distribution. We illustrate this with Fig. 1 (see also Fig. 1 of Teerikorpi 1993), which helps one to understand why division of the sample into -ranges allows one to separate the effect of true space density distribution from the effect of incompleteness in the distribution of µ. After having identified this µ-range for the subsample with smallest , hence smallest , one can normalize the other subsamples to this one, shifting their vs. µ distributions vertically, so that up to all have the same average normalized number counts. This is allowed because for the subsamples with larger , the complete µ-ranges extend beyond the corresponding to smaller .

Fig. 1. Schematic illustration of how the distribution of TF distance moduli differ for two narrow ranges separated from each other by . Here it is assumed that the true space distribution produces a constant slope in the vs. diagram (corresponding to density law ). Small- subsample reveals the correct density law up to where the incompleteness starting at begins to affect. Large- subsample shows the same behaviour, though shifted towards larger by . In the present method such distributions are in addition shifted vertically so that the complete parts come together forming a common envelope which follows the true space distribution law. Points A and B correspond to the same angular size in the incomplete region. After normalization they also define the correct density gradient.

The common envelope of thus normalized distributions should follow the space distribution law up to the for the subsample with the largest . An interesting feature of the method is that one may construct the space distribution still farther, as is shown by Fig. 1b where the points A and B corresponding to the same value of () in the incomplete region are connected by the line having the correct space density slope. The method is also quite simple in essence and allows one quickly to "see" the run of the density law and the factors influencing its derivation. Of course, working with real data is more problematic than the ideal theoretical case. For instance, one cannot use indefinitely small ranges.

2.3. Special aspects

Actually, one must use the normalized when making the described division of the sample:

The concept of normalization has been introduced and discussed in a whole series of studies connected with the so-called method of normalized distances (see e.g. Theureau et al. 1997b). In this manner, galaxies of different types and with different inclination and galactic absorption corrections, but with equal normalized , have the same effective limiting diameters. The -distributions become thus better defined, and more closely follow the ideal distribution determined by the space density and incompleteness curves. In other words, for a given , the scale reflects faithfully the scale.

Finally, we note that the method is in principle insensitive to the "patchy incompleteness" (Guzzo 1997) that may be present in such somewhat non-uniform collections of data as LEDA (the host database of KLUN which, however, was intended to be a homogeneous part of LEDA): In different regions of the sky the limiting diameters (magnitudes) may be different. Let us consider two regions, 1 and 2, where incompleteness starts at and , respectively, and suppose we cannot identify these regions from the data. The summed relation, vs. µ, for a constant still follows the correct slope at small distances (, where the constant c depends only on the ratios of limiting angular sizes and region areas) while now the incomplete part of the curve is deformed. However, for different the form of the curve is similar and the situation is analogous to what is depicted by Fig. 1.

© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998

helpdesk.link@springer.de