3. Search for the density law
3.1. Diameter TF distance moduli
We show the procedure in a transparent manner which allows one to see the steps taken and to recognize the impact of possible systematic errors.
First, we have divided the sample into five ranges according to the normalized : 1.5 - 1.7 - 1.9 - 2.1 - 2.3 - 2.5. These intervals cover practically all the sample and the middle one is centered on the median of the distribution (Fig. 2). Inside the extreme ranges the distributions are not symmetric around the mean, due to the systematic effect described in Sect. 3.3.
The distributions of in the intervals are shown in Fig. 3. The bins on should have the width , ( 1,2,3,...) in order to collect the same ranges on the individual distributions as expected from the intervals (0.2) and the slope of the diameter TF relation (1.082). Selecting , we get suitable bin size . For clarity, Fig. 4 shows the distributions artificially shifted on the -axis, revealing their similar forms.
The next step is the normalization of the numbers . For this we use the following procedure:
1) Select the complete part of the distribution. The putative limit is , as expected from previous studies.
2) Find the corresponding part of the first subsample (with smallest ) and count the total number of galaxies below and at : .
3) For the second subsample calculate .
4) Shift curve 2 by .
5) Normalize the remaining curves using the same procedure with increasing by the shift between the subsequent curves. In this manner we utilize progressively deeper complete parts. Then the cumulative correction for the j:th curve is
where . By restricting the steps in this manner, we keep progressively farther from the putative limit , while having an increasing number of galaxies available for normalization.
Fig. 5 shows the resulting normalized composite distribution. One may note two particular things: There is first a rather scattered run of points up to about which can be roughly approximated by a line. After that the common envelope assumes a shallower slope of about 0.46.
Using now the points above for each curve and requiring that each point contains more than 12 galaxies, by which the noisy ranges are removed, we draw a straight line through them. For this line, there are points up to . We use this line, derived by simple least-squares technique, as a reference for further study of the density law using information from the incomplete parts of the µ - distributions. This "envelope line" is shown in Fig. 6 together with points from the incomplete parts where we again use only those which contain more than 12 galaxies The average is written besides each symbol. One can easily discern the sequences corresponding to the same positions in the distributions (c.f. points A and B in Fig. 1). These averages do not vary much along the same sequence. Comparing with the line of slope 0.46 inserted, one sees that the sequences follow rather well this slope, similarly as was found above for the points in the complete part (especially above ).
We tested the completeness limit by assuming limits at 1.3 and 1.4 instead of the above . With these more conservative values we can be more certain of the completeness of the sample, however at the price of reduced number of data points. We got slopes 0.46 () and 0.43 () for limits 1.3 and 1.4 respectively, but the "envelope lines" in both cases were not as well defined as before. For now, we are more satisfied with the completeness limit at giving us the slope 0.46 (see also Sect. 3.4 for another test, using simulated galaxy samples).
Naturally, all this is interestingly close to 0.44 predicted on the basis of fractal dimension 2.2, as obtained e.g in the Di Nella et al. (1996) correlation analysis of LEDA. However, we are well aware that systematic effects may be involved, when we are working rather close to the completeness limit of the KLUN-sample. We have attempted to find any systematic error that could explain the slope shallower than 0.6 and have also looked what happens when one forces a line of slope 0.6 to start at . In Fig. 7 this has been done, and the line which rather well describes the (scattered) run of data below , systematically deviates from the envelopes of the normalized distributions at larger . This behaviour implies that if the deviation from the -law is due to incompleteness problem, the incompleteness in apparent size starts for different ranges at different . This is something that we cannot understand, because the needed effect is quite large. For the largest it means that incompleteness starts around , while for the smallest such a limit would be around 1.2 (in terms of magnitudes this corresponds to a difference of about 2.5 mag). Still another way to state the problem is that if we try to shift the maxima of the distributions to follow the -slope, the curves are everywhere separated, there is no normalization and no common envelope. A clear argument against such a large effect comes also from the method of normalized distances used in Theureau et al. (1997b): in the "unbiased plateau" produced as a part of the method, one should readily recognize such differences in the selection functions of galaxies with small and large . Also, from the manner of how KLUN was created as a diameter limited sample, independent of any considerations of , there is no reason to expect so significant dependence of the selection on . For instance, a recent analysis of the H I line profile detection rates at Nançay radio telescope by Theureau et al. (1997c) does not give any indication that large galaxies are significantly underrepresented in KLUN.
Finally, if erroneous, the present slope 0.46 is just by an accident very close to those obtained by several quite different correlation analyses (e.g. Sylos Labini et al. 1998a).
3.2. Magnitude TF distance moduli
The B -magnitude TF relation has smaller scatter than the diameter relation, which makes it tempting to use also it in this study even though the numbers are then smaller and the selection properties of the resulting sample are complicated by the fact that KLUN has been originally selected on the basis of apparent size. However, if the distribution of the galaxies of different ranges is the same in the - plane, one can use the magnitude relation in a similar manner.
We do not go through the steps in such detail as for the diameter distance moduli. Fig. 8 shows directly the normalized composite diagram. Because of the underlying diameter limit, the incompleteness begins farther from the maxima than in the case of diameters, and the complete part of the envelope is now shorter in . The envelope line constructed from points below the putative magnitude limit , is also shown. It has now the slope 0.40. Because the numbers of galaxies at each point of this diagram are smaller than in Fig. 5, the error of the slope is slightly larger than when using diameters (, compared to with diameters). Inspection of the points in the incomplete part shows that the sequences, analogous to what was discussed for the diameter moduli, have similar slopes. However, now the achievable µ-ranges and numbers are smaller, and we do not show them separately.
3.3. Systematic error caused by finite ranges
One can see a systematic effect in this method where we are forced to use finite ranges instead of ideal infinitesimals. It comes from the fact that galaxies are not quite similarly distributed inside the different ranges. The distribution of has a maximum. Because of the form of the distribution, inside the small - interval one expects an increasing number density of galaxies towards the edge with larger , while for the large - range this trend is reversed (see Fig. 2). In Fig. 6 one sees that the averages within different sequences do not vary very much which suggests that the effect is actually not very important.
In order to check whether decreasing the interval size influences the result obtained above, we made an experiment whereby interval was reduced to 0.1, in the range 1.7 - 2.3, where the numbers of galaxies remain large enough. Now the slope of the envelope line is 0.41, the furthest point of the line being slightly above 100 Mpc (Fig. 9). Again, the diminished number of galaxies make error of the slope larger, , while in Fig. 5.
3.4. Numerical experiment using simulated galaxy sample
In order to check further the reliability of the used method, we have made numerical experiments with simulated galaxy samples. In making these tests, we have kept in mind the following points:
In the experiments we started with large number of galaxies (say ), for which we alloted radial distances using an input value of the radial density gradient in the density law const. For each galaxy we chose a random absolute diameter from a gaussian distribution with and ( in kiloparsecs). Even though the distribution in reality hardly is gaussian, it is not too far from the truth for the KLUN sample. All the galaxies having apparent diameter larger than limit were included in the "observed subsample". To make this observed sample more realistic we also allowed in some galaxies below the diameter limit, percentage of included galaxies being progressively smaller further below the . Each of these observed galaxies were then given a rotational parameter by inverse TF relation () with , and a gaussian dispersion . These values resemble results of the recent KLUN study (Theureau et al. 1997a). Now the - graph (distance moduli from direct TF relation) for the observed subsample could be investigated as above was done for the real KLUN sample.
First we selected and varied the density gradient . For each we chose the initial number of galaxies so that the number of observed galaxies was the same as in KLUN sample. Then we calculated the slope of the "envelope" line in the - graph getting the observed density gradient For each we repeated the simulation 1000 times to get with error bars. For every the resulting was smaller than . However, this tendency was not large enough to explain the deviation of our observed slope (in this section we use as a reference point 0.44, which is a weighed average of the slopes obtained with diameters and magnitudes) from homogeneous galaxy distribution ( 0.6). The simulations showed that for 0.44, 0.47 ( errors). In terms of fractal dimension () we can say that the observed value 2.2 can be affected by our methods so that the true value is 2.35 .
We then tested the effect of the completeness limit by varying , while keeping the other parameters fixed. Fig. 10 shows how when the sample gets more complete. Also, with smaller completeness limits the number of observed points increase, and the errors get smaller. This emphasizes the importance of expanding the database to make it deeper and more complete.
© European Southern Observatory (ESO) 1998
Online publication: May 15, 1998