This contribution gives the results of a statistical investigation of
400 extra-galactic nebulae for which Holetschek has determined total
visual magnitudes. The list is complete for the brighter nebulae in the
northern sky and is representative to 12.5 mag. or fainter.
_The classification_ employed is based on the forms of the photographic
images. About 3 per cent are irregular, but the remaining nebulae fall
into a sequence of type forms characterized by rotational symmetry
about dominating nuclei. The sequence is composed of two sections, the
elliptical nebulae and the spirals, which merge into each other.
_Luminosity relations_.—The distribution of magnitudes appears to
be uniform throughout the sequence. For each type or stage in the
sequence, the total magnitudes are related to the logarithms of the
maximum diameters by the formula,
$m_{T} = C - 5 \log d,$
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
$\log D = 4.04 + 0.2 m_{T},$
where m_{T} 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 2.6×10^8 ☉.
_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 10^{17} 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 2.7×10^{10}
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.