The most serious difficulty encountered in studies of satellites is the problem of separating the physical companions from the optical ones, that is, from the background-foreground field. A decrease in the limiting magnitude by one magnitude class will increase the number of optical companions by a factor of 4, as will also a doubling of the radius of the survey area. In the analysis reported in the preceding section the difficulty was not of a serious nature on account of the rather bright limiting magnitude. The problem will be of another order of size when we try to pick out all dwarf satellites on the Palomar Sky Atlas. On account of the tremendous number of galaxies down to the limit of the Atlas, it is quite impossible to use the same type of survey area, with a radius that is considerably larger than the maximum separation of the satellites. In order to reduce the ratio of optical to physical companions, we have to be satisfied with rather small fields, including only those satellites that are comparatively close to the central galaxy. In the present work the practical limit has, as will be demonstrated later, been found to be survey areas with a radius of 50 kpc According to the preceding section, areas of this size will include about 30% of the total number of satellites. Still, the number of optical companions is in some cases uncomfortably large.
In order, to determine empirically the number of optical companions, it is necessary with the above procedure to introduce comparison areas located at suitable distances from the survey area. With two fields, one on each side of the survey area, possible gradients in the distribution of galaxies are largely eliminated. If the three areas are of the same size, and if the galaxies are registered in exactly the same way, we ought to get a reliable measure, at least in a statistical sense, of the size of the background-foreground population.
It may be remarked that the successful separation of physical from optical companions ultimately depends on the difference in space density between the two groups. The final analysis of 160 satellite systems leads to a mean space density of satellites (in the volumes corresponding to survey areas with a radius of 50 kpc) of about 100 per Mpc3, whereas the smoothed-out density in the general field amounts to only 0.17 per Mpc3 (cf. sect. 14 - 15; for practical reasons, the densities refer to galaxies brighter than M = - 15.0). There is thus a difference in density by a factor of about 600. For comparison, it may be noted that the Virgo cluster has a space density of about 20 per Mpc3 (cf. sect. 13).
Another problem to be discussed is whether it is possible to identify on the Sky Atlas prints distant dwarf galaxies of low surface luminosity, objects that are comparable to some of the fainter members of the Local Group. The present survey is supposed to include dwarf satellites down to an absolute pg magnitude of about -10.6 (abs. disc. mod. 30.0) or down to -11.8 (mod. 31.2), which with a galactic absorption of 0.3 magn. would mean a limiting apparent magnitude 19.7. At the largest distances, the smallest dwarfs are found to have major diameters of about 0.'2, as measured on the blue prints, or about 0'.3, as reduced to the photometric diameter scale of Holmberg (1958). Practical experience shows that it is indeed feasible to pick up dwarf galaxies down to these limits: The reasons are (a) that the reproduction of the original Sky Atlas plates was aimed at. preserving the faintest details at the limit of the plates (cf. Minkowski and Abell 1963), and (b) that the surface brightness of a galaxy is independent of the distance. The surface magnitude of the faintest dwarfs would in fact be more or less comparable to that of the Leo I system in the Local Group (cf. Table 1), which is a fairly outstanding object on the Sky Atlas prints. Even at a great distance, a system of this type would be readily recognizable if looked for in a very small survey area and studied through an eyepiece of adequate magnification.
It is interesting to find that it is possible to get valuable information on the dwarf satellites around distant spiral galaxies but very difficult, if at all possible, to make up a list of dwarf systems of the same absolute luminosity in the Local Group. The nearby dwarfs ought to be clearly visible on the Sky Atlas prints, but their distances are difficult to estimate (cf. van den Bergh 1966); the local systems cannot be separated from more distant galaxies. The position of the observer inside the Local Group is in this case a disadvantage.
A final question is, what kind of information can be acquired about the individual galaxies in the survey areas. Magnitudes would be of great value but cannot be measured for the faintest dwarf systems with present observational technique. Attempts to estimate magnitudes from the Sky Atlas prints would involve a great deal of work and would undoubtedly yield results of low accuracy. The diameters of the galaxies can, on the other hand, be readily measured down to about 0'.2. The morphological types can usually be estimated for the medium-sized and large galaxies. It will be shown that even information that is limited to diameters and types may be sufficient to obtain interesting results.