The determination of space density is based on the redshifts listed in the Reference Catalogue by de Vaucouleurs et al. (1964), including a later addition by the same authors (1967). In order to have a material as homogeneous as possible, only the northern galactic hemisphere above B^{II} = +30° has been included; below this latitude it is difficult to estimate the degree of completeness of the material, as well as the systematic selection effects. To avoid all disturbances from the Virgo cluster, a rather large region ( = 12^{h} -13^{h}, = -10° - +20°) has been excluded; the remaining area amounts to 23.9% of the entire sky. The apparent pg magnitudes have, in order of preference, been taken from Holmberg (1958), Humason et al. (1956), and (in a few cases) the Reference Catalogue. All magnitudes have been corrected for galactic absorption by ^{-}m.25 csc B^{II}; since the maximum redshift (corr. for solar motion) is 4000 km/sec, the magnitudes are free from any appreciable redshift effects.
Table 6 presents the distributions of the corrected magnitudes and logarithmic redshifts for all the galaxies (in the redshift lists) located in the above area. Each square of the table gives the number of objects, as well as the mean absolute pg magnitude corresponding to a Hubble parameter H = 80; the absolute magnitude ranges from -16.5 to -21.5. The table makes possible a check of the marginal distributions of apparent magnitude and redshift: As regards the magnitude, it seems quite clear that the interval m = 11 - 12 is under-represented. For m < 11 the magnitude distribution can be described by a logarithmic expression of the same type as that of eq. (3 a); if the distribution is reduced to one square degree, the constant term will be about the same or -8.68, which indicates that the redshift material is essentially complete down to m = 11. In order to agree with this distribution, the frequencies of the interval m = 11 - 12 have to be multiplied by the factor 2.15; the corrected frequencies are also listed in the table. It may be pointed out here that it would be impossible to pursue the investigation to fainter magnitudes; in the interval m = 12 - 13 the correction factor is increased to about 10.
8^{m} -9^{m} | 9^{m} - I10^{m} | 10^{m} -11^{m} | 11^{m} -12^{m} | |
2.4-2.6 | 3 | 3 | 5 | 1 2 |
(-19.5) | (-18.5) | (-17.5) | (-16.5) | |
2.6-2.8 | 2 | 7 | 6 | 7 15 |
(-20.5) | (-19.5) | (-18.5) | (-17.5) | |
2.8-3.0 | 1 | 5 | 21 | 23 49 |
(-21.5) | (-20.5) | (-19.5) | (-18.5) | |
3.0-3.2 | - | 1 | 27 | 36 77 |
(-21.5) | (-20.5) | (-19.5) | ||
3.2-3.4 | - | - | 2 | 38 82 |
(-21.5) | (-20.5) | |||
3.4-3.6 | - | - | - | 10 22 |
(-21.5) | ||||
In a renewed analysis, the procedure is repeated for galaxies belonging to the type group E-So-Ir, their total number down to m = 12 being 65; with the same correction factor in the interval m = 11 - 12 the number is increased to 114. The morphological types have been taken from the original redshift lists, attention being paid to the information given in the Reference Catalogue.
Summations along the diagonals of the distribution tables give the total numbers referring to the different classes of absolute magnitude; divisions by the corresponding volumes lead to the space densities. As appears from Table 6, reliable results can be obtained only for absolute luminosities higher than M = -17.0. The densities, as referred to 1 Mpc^{3}, have been plotted in Fig. 10. The results are in perfect agreement with the luminosity curves derived from the satellite groups. It should be noted that the results have not been corrected for possible disturbances by the peculiar motions of the galaxies. A detailed analysis of the redshift distributions corresponding to the different classes of apparent magnitude shows that the theoretical corrections would be small, and that they would partly be of opposite sign, as compared to the corrections indicated for the luminosity curve derived from the satellite groups. Since the fluctuations in the individual class frequencies of the magnitude-redshift tables are considerable, and since there may remain systematic selection effects in the material (especially in the class m = 11 - 12), it did not seem possible to derive any corrections that would be statistically significant.
By means of an extrapolation based on the luminosity curve of Fig. 10, the above analysis leads to a total space density of galaxies brighter than M = -15.0 of 0.18 per Mpc^{3}. The integrated result is practically independent of any disturbance that may be caused by the peculiar motions of the galaxies. The result is in good agreement with those obtained from the magnitudes and the diameters.
The total space density derived here agrees with the result obtained by Kiang (1961) in a similar analysis of galaxian redshifts. On the other hand, the luminosity functions do not agree, due to the fact that Kiang's curve is based on the assumption that the redshift material represents a selection according to the apparent magnitude, and to the fact that in the fainter magnitude classes the material has been increased by the incorporation of data of questionable quality. In the writer's opinion, the redshift material available for the present does not permit the drawing of any reliable conclusions about the luminosity function below M = -17.