where C varies progressively from type to type, indicating a variation in diameter for a given magnitude or vice versa. By applying corrections to C, the nebulae can be reduced to a standard type and then a single formula expresses the relation for all nebulae from the Magellanic Clouds to the faintest that can be classified. When the minor diameter is used, the value of C is approximately constant throughout the entire sequence. The coefficient of log d corresponds with the inverse-square law, which suggests that the nebulae are all of the same order of absolute luminosity and that apparent magnitudes are measures of distance. This hypothesis is supported by similar results for the nuclear magnitudes and the magnitudes of the brightest stars involved, and by the small range in luminosities among nebulae whose distances are already known.
Distances and absolute dimensions.—The mean absolute visual magnitude, as derived from the nebulae whose distances are known, is –15.2. The statistical expression for the distance in parsecs is then
where mT is the total apparent magnitude. This leads to mean values for absolute dimensions at various stages in the sequence of types. Masses appear to be of the order of 2.6×108 ☉.
Distribution and density of space.—To apparent magnitude about 16.7, corresponding to an exposure of one hour on fast plates with the 60-inch reflector, the numbers of nebulae to various limits of total magnitude vary directly with the volumes of space represented by the limits. This indicates an approximately uniform density of space, of the order of one nebula per 1017 cubic parsecs or 1.5×10–31 in C.G.S. units. The corresponding radius of curvature of the finite universe of general relativity is of the order of 2.7×1010 parsecs, or about 600 times the distance at which normal nebulae can be detected with the 100-inch reflector.
Recent studies have emphasized the fundamental nature of the division between galactic and extra-galactic nebulae. The relationship is not generic; it is rather that of the part to the whole. Galactic nebulae are clouds of dust and gas mingled with the stars of a particular stellar system; extra-galactic nebulae, at least the most conspicuous of them, are now recognized as systems complete in themselves, and often incorporate clouds of galactic nebulosity as component parts of their organization. Definite evidence as to distances and dimensions is restricted to six systems, including the Magellanic Clouds. The similar nature of the countless fainter nebulae has been inferred from the general principle of the uniformity of nature.
The extra-galactic nebulae form a homogeneous group in which numbers increase rapidly with diminishing apparent size and luminosity. Four are visible to the naked eye;[2] 41 are found on the Harvard “Sky Map”;[3] 700 are on the Franklin-Adams plates;[4] 300,000 are estimated to be within the limits of an hour’s exposure with the 60-inch reflector.[5] These data indicate a wide range in distance or in absolute dimensions. The present paper, to which is prefaced a general classification of nebulae, discusses such observational material as we now possess in an attempt to determine the relative importance of these two factors, distance and absolute dimensions, in their bearing on the appearance of extra-galactic nebulae.
The classification of these nebulae is based on structure, the individual members of a class differing only in apparent size and luminosity. It is found that for the nebulae in each class these characteristics are related in a manner which closely approximates the operation of the inverse-square law on comparable objects. The presumption is that dispersion in absolute dimensions is relatively unimportant, and hence that in a statistical sense the apparent dimensions represent relative distances. The relative distances can be reduced to absolute values with the aid of the nebulae whose distances are already known.