The range in total magnitudes is sufficiently large in comparison with the dispersion to lend considerable confidence to the conclusion. The total range of four, and the average dispersion of less than 1 mag., are comparable with those in [Table XV] and in [Figure 7], and agree with the former in indicating a constant order of absolute magnitude.
The mean absolute magnitude of the brightest stars in the nebulae listed in [Table XV], combined with the mean difference between nebulae and their brightest stars, furnishes a mean absolute magnitude of –15.3 for the nebulae listed in [Table XVI]. This differs by only 0.2 mag. from the average of the nebulae in [Table XV], and the mean of the two, –15.2, can be used as the absolute magnitude of intermediate- and late-type spirals and irregular nebulae whose apparent magnitudes are brighter than 10.5. The dispersion is small and can safely be neglected in statistical investigations.
This is as far as the positive evidence can be followed. For reasons already given, however, it is presumed that the earlier nebulae, the elliptical and the early-type spirals, are of the same order of absolute magnitude as the later. The one elliptical nebula whose distance is known, M 32, is consistent with this hypothesis.
Conclusions concerning the intrinsic luminosities of the apparently fainter nebulae are in the nature of extrapolations of the results found for the brighter objects. When the nebulae are reduced to a standard type, they are found to be constructed on a single model, with the total luminosities varying directly as the square of the diameters. The most general interpretation of this relation is that the mean surface brightness is constant, but the small range in absolute magnitudes among the brighter nebulae indicates that, among these objects at least, the relation merely expresses the operation of the inverse-square law on comparable objects distributed at different distances. The actual observed range covered by this restricted interpretation is from apparent magnitude 0.5 to 10.5. The homogeneity of the correlation diagrams and the complete absence of evidence to the contrary justify the extrapolation of the restricted interpretation to cover the 2 or 3 mag. beyond the limits of actual observation.
These considerations lead to the hypothesis that the nebulae treated in the present discussion are all of the same order of absolute magnitude; in fact, they lend considerable color to the assumption that extra-galactic nebulae in general are of the same order of absolute magnitude and, within each class, of the same order of actual dimensions. Some support to this assumption is found in the observed absence of individual stars in the apparently fainter late-type nebulae. If the luminosity of the brightest stars involved is independent of the total luminosity of a nebula, as is certainly the case among the brighter objects, then, when no stars brighter than 19.5 are found, the nebulae must in general be brighter than absolute magnitude mT – 25.8 where mT is the total apparent magnitude. On this assumption, the faintest of the Holetschek nebulae are brighter than –12.5 and hence of the same general order as the brighter nebulae.
Once the assumption of a uniform order of luminosity is accepted as a working hypothesis, the apparent magnitudes become, for statistical purposes, a measure of the distances. For a mean absolute magnitude of –15.2, the distance in parsecs is
| (8) |
DIMENSIONS OF EXTRA-GALACTIC NEBULAE
When the distances are known, it is possible to derive actual dimensions and hence to calibrate the curve in [Figure 6], which exhibits the apparent diameters as a function of type, or stage in the nebular sequence, for nebulae of a given apparent magnitude. The mean maximum diameters in parsecs corresponding to the different mean types are given in [Table XVII]. For the elliptical nebulae, values are given both for the statistical mean observed diameters and for the diameter as calculated for the pure types.
Spirals at the last stage in the observed sequence have diameters of the order of 3000 parsecs. Assuming 1:10 as the ratio of the two axes, the corresponding volume is of the order of 1.4×109 cubic parsecs, and the mean luminosity density is of the order of 7.7 absolute magnitudes per cubic parsec as compared with 8.15 for the galactic system in the vicinity of the sun. These results agree with those of Seares who, from a study of surface brightness, concluded that the galactic system must be placed at the end of, if not actually outside, the series of known spirals when arranged according to density.[19]