3.2. The Problem of False Pairs
For convenience, we will refer to the sample of model pairs as M-pairs and to real objects in the catalogue as K-pairs. Analysis of the sample of M-pairs reveals very many cases of galaxies lying close to one another projected on the sky but at great distances from one another in space. Among these, it is not rare to encounter cases with rather small differences in radial velocity. The number of such objects in the computer realisations appeared substantially higher than intuitive expectations. In addition to these chance optical systems, a substantial number of the M-pairs consisted of group and cluster members. These are not dynamically distinguished from the other members of these systems but two galaxies within the system may satisfy the isolation criterion in projection on the sky. Obviously, at some degree of isolation one will select the two brightest members of a cluster of galaxies. Because the peculiar velocities of galaxies in groups and clusters are not very high, say 200 to 500 km/s, such pseudo-pairs are not easy to distinguish from real physical pairs.
In modelling the apparent distribution of galaxies, we adopted the proposition that every second galaxy in a unit volume belongs to a double system (Table 3). This value may be shown to be unrealistically high. Nevertheless, in a sample of modelled galaxies brighter than magnitude 15.7 the number of objects being components of a pair is M* = 2 × 301/9433 = 0.064. This exceeds, by about one and a half times, the observed number of pairs of double galaxies in the catalogue K* = 2 × 585/27841 = 0.042. This numerical deficit of the catalogue in comparison to the model has two causes: a) for the above galaxy luminosity function, the probability of finding pairs in the sample brighter than magnitude 15.7 incorporating both galaxy components is smaller (photometric selection), b) many physical pairs do not satisfy the basic isolation criteria because of projection with regard to galaxies in the foreground or background (isolation criterion). On the other hand, increasing the catalogue selection occurs at the expense of including false, non-isolated systems.
Among 301 M-pairs, 34 (11%) appear to result from chance projections along the line of sight of galaxies which are not in fact close to one another (optical pairs). Besides this, the isolation criterion is satisfied by 96 M-pairs, the galaxies of which are in fact the brightest members of triplets, groups or clusters. The presence of pseudo-pairs occurring as a sort of `apex' of systems of galaxies can falsify the sample analysis, distorting the main characteristic distributions in, for example, apparent separation and relative velocity.
The occurrence of optical pairs in the samples may be decreased by remembering that, in addition to the basic weak selection criterion, the catalogue also includes a strict criterion and two intermediate criteria, respectively (+ +) and (+ - , - +). The effectiveness of each criterion may be judged from its results when applied to the model distribution of galaxies. Table 4 shows the number of optical pairs and members of systems which satisfy one or another of these criteria. In the final column we present another important characteristic: the probability that a physical pair with components brighter than magnitude 15.7 will fail the criterion and will not appear among the number of isolated pairs. As shown from these data, going from the weak criterion to the strict one results in a marked decrease in the number of optical pairs in the sample, by a factor three, while cutting in half the number of pseudo-pairs.
Upon application of the strict criterion, more than half of the physical pairs failed the requirement to have no galaxies seen nearby in projection. Such strong selection effects make it more difficult to infer the characteristics of the spatial distribution of the sample of pairs from their measured catalogue distributions.
The estimates of the effectiveness of the criteria presented in Table 4 are not, of course, absolute. They follow from the choice of parameters for systems of galaxies which were adopted in the modelling. For example, the probability of finding a physical pair depends sensitively on the value of the characteristic separation between components of physical pairs (the quantity r = 8 kpc in Table 3). Neglecting any of several arbitrarily chosen parameters and ignoring their effect on selection criteria of the first and second kind can have an important effect on one's estimate of the effectiveness of those criteria. We emphasise that from an analytical point of view, the approach from a Poisson distribution of galaxies makes a definition of the extent of contamination of the sample by members of systems (pseudo-pairs) quite impossible.
The important role of projected, non-isolated `pairs' in the selection of catalogues was remarked by Karachentsev and Fesenko (1979), and also by White et al. (1983) and van Moorsel (1983). Starting from first principles van Moorsel (1983) showed that the fraction of optical pairs in our catalogue was about 11% but that the number of pairs composed of members of systems of galaxies is 34%. These estimates are practically identical with the data given in the first line of Table 4. We will return several times in the following chapter to the problem of non-isolated optical pairs.