The true spatial distribution of the SVs of galaxies we have assumed to be isotropical. Then, due to projection effects, i can be distributed , B can be distributed , the variables L and P can be distributed randomly, and formulae (1) and (2) can be used to calculate the corresponding values of and . Our results are based on calculations including virtual galaxies.
3.1. Selection effects and the isotropic polar angle distribution
In Fig. 1 we demonstrate that selection effects either concerning inclination and position will change the expected isotropic polar angle distribution dramatically. Consider first what happens with this distribution when selecting different inclinations. When all possible inclinations are included in the measurements, the isotropic distribution of is cosine. It is this curve which was already used previously as the isotropic distribution for the polar angle by different authors. However, all the other curves in Fig. 1 show selection effects at work. When face-on galaxies are missing, this curve is deformed significantly. However, this deformation is dependent on the supergalactic latitude of the galaxy sample (this effect was already noted by Godlowski 1993). When this sample is located at high B, a `hump' develops at small values of and a long range `dip' at large angles when neglecting face-on galaxies more and more (see Fig. 1b). These effects work the other way round when the region in question is located at low supergalactic latitudes (Fig. 1d): a `hump' develops at large values of and a `dip' at small values. It should be noted here, that we call here and in the following `dip' and `hump' (under apostrophes) deviations relative to a reference curve (here cosine).
To study these effects in more detail, Fig. 1a and c gives the constituents of these cumulative effects. It is shown how the isotropic polar angle distribution changes and deviates from cosine for a sequence of galaxies from nearly face-on to nearly edge-on. Let us begin the discussion for low values of B (Fig. 1c,d) for nearly face-on galaxies () a `hump' at small angles and a `dip' at large angles is found. As the number of face-on galaxies is reduced from the sample, the reverse pattern is achieved: a `dip' at small angles and a `hump' at large angles for nearly edge-on galaxies. When going to higher supergalactic latidudes, all these individual isotropic distribution curves will become narrower and shifted. Note, that distribution curves which peaked at high values of for low B are shifted to low values of when increasing B, and vice versa.
Fig. 1 makes clear that the expected isotropic -distributions of edge-on galaxies and pole-on galaxies are very different due to selection effects. Maybe this effect can play a role, at least in part, in results already published. In this way the correct treatment of face-on galaxies is a crucial point. Artificial structures will play an important role because PAs for nearly face-on galaxies cannot be measured with proper accurancy (in particular when measured visually).
3.2. Selection effects and the isotropic azimuthal angle distribution
To study these effects we have first devided the whole sky into 12 parts having with no selections on B and i. Fig. 2a shows in which way the isotropic azimuthal angle distribution changes due to selection effects concerning L. It can be seen that the position of galaxies (i.e. the region investigated) has marked influence upon the expected isotropic distribution of . We found a wavelike structure: a `dip' (`hump') at and a `hump' (`dip') at for galaxies with (, resp.). These artificial structures are more pronounced when L is approaching (, resp.). So the nature of the isotropic distribution curve for the azimuthal angle is different for northern and southern hemisphere galaxies. However, this effect could be removed by using the (analytical) method of Flin & Godlowski 1986 (see Flin & Godlowski 1990), because there is no selection on i.
Fig. 2b,c,d demonstrate how selection on the inclination will change the isotropic distribution for constant. As an example, in Fig. 2b and c we show the case for . It can be seen that for edge-on galaxies there is a `dip' at negative regions of and a `hump' at the other side. When going to face-on galaxies the maxima of the distribution curves is shifted to negative values of . For face-on galaxies this trend is opposite.
Moreover, the influence of this selection is dependent also on the choice of L, as is shown in Fig. 2d, where we have put our results for . The isotropic distribution now is different from that of the above choice of L and more complex.
In addition, selection effects of B will make the shape of the expected isotropic distribution in Fig. 2b,c and d even more complex.
Up to now, no authors have used homogeneous data covering both the northern and southern hemisphere for their investigations on SVs of galaxies. Flin & Godlowski (1986) mainly used UGC galaxies. Later Godlowski (1993, 1994) added galaxies taken from the ESO B Atlas. Kashikawa & Okamura (1992), Yuan et al. (1997), and Hu et al. (1998) used the Photometric Atlas of Northern Bright Galaxies (Kodaira et al. 1990, hereafter PANBG) and investigated galaxies with .
To show that these artefacts from selection effects can play an important role in interpreting data we give some examples from the literature. Hu et al. (1998) and Yuan et al. (1997) have used the PANBG as data base for their studies. Because they have used only northern galaxies, we can expect isotropic azimuthal distributions similar to those in Fig. 2. Indeed, in their histograms of azimuthal angle distributions, these authors found a remarkable dip at to and a hump at to which could be mainly due to the positions of their galaxy samples. In addition, if we make a closer look on the histograms of azimuthal angle distribution of Kashikawa & Okamura (1992), Hu et al. (1995, 1997), a similar trend in all histograms of distribution for their subsamples can be seen. This trend can be due to the inhomogenous distribution of their galaxies.
From Fig. 2 (subset 3b S(VI)) of Hu et al. (1995) we have taken data points and error bars directly from the published figure. To calculate the expected isotropic polar angle distribution for the database which was used by these authors we have taken the region and for the Virgo cluster region and distributed the inclination angles within the intervall .
The result is given in Fig. 3a. The authors noticed a dip at small due to missing face-on galaxies and corrected it roughly by distributing galaxies randomly in this range. However, after this correction an anisotropic feature at still remains, according to the authors. However, accepting our expected isotropic distribution, also this feature is just an artefact due to selection effects.
A further example is shown in Fig. 3b. These data were taken from Fig. 7 of Hu et al. (1998). Similar data can be found in Yuan et al. (1997). To calculate the expected isotropic distribution of we have used exactly those regions which were treated in that article. No selection were made on inclination, because the PAs were measured by fitting a 25 mag arcsec-2 isophote level. This example clearly demonstrates the need of considering selection effects in investigations of SVs of galaxies. The global shape of the measurements can be fitted with the expected isotropic distribution.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001