## 2. The sample and the method## 2.1. KLUNThe 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 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
## 2.2. The method of TF distance moduliWe 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 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 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
.
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 aspectsActually, 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. © European Southern Observatory (ESO) 1998 Online publication: May 15, 1998 |