The number of nebulae of known distance is too small to serve as a
basis for estimates of the range in absolute magnitude among nebulae in
general. Further information, however, can be derived from a comparison
of total apparent magnitudes with apparent magnitudes of the brightest
stars involved, on the reasonable assumption, supported by such
evidence as is available, that the brightest stars in isolated systems
are of about the same intrinsic luminosity.
The most convenient procedure is to test the constancy of the
differences in apparent magnitude between the brightest stars involved
and the nebulae themselves, over as wide a range as possible in the
latter quantities.
An examination of the photographs in the Mount Wilson collection has
revealed no stars in the very faint objects or in the bright elliptical
nebulae and early-type spirals. This was to be expected from the
conclusions previously derived. Observations were therefore confined to
intermediate- and late-type spirals and the irregular nebulae to
the limiting visual magnitude 10.5. The Magellanic Clouds and N.G.C.
6822 were added to the nebulae in Holetschek’s list. Altogether,
data were available for 32 objects, or about 60 per cent of the total
number in the sky to the adopted limit. For this reason it is believed
that the results are thoroughly representative.
TABLE XVI
Difference in Magnitude between Nebulae and
Their Brightest Stars
==========+=============+============+===============
N.G.C. | m_{s} | m_{T} | m_{s} – m_{T}
----------+-------------+------------+---------------
Sb
----------+-------------+------------+---------------
224 | 15.5 | 5.0 | 10.5
1068 | 17.5 | 9.1 | 8.4
2841 | >19.5 | 9.4 | >10.1
3031 | 18.5 | 8.3 | 10.2
3310 | >19.0 | 10.4 | > 8.6
3623 | >20.0 | 9.9 | >10.1
3627 | 18.5 | 9.1 | 9.4
4438 | >19.0 | 10.3 | > 8.7
4450 | 19.5 | 10.0 | 9.5
4736 | 17.3 | 8.4 | 8.9
4826 | >19.5 | 9.2 | >10.3
5055 | >19.0 | 9.6 | > 9.4
5746 | >19.5 | 10.4 | > 9.1
7331 | 19.0 | 10.4 | 8.6
----------+-------------+------------+---------------
SBb
----------+-------------+------------+---------------
4699 | >19.5 | 10.0 | > 9.5
----------+-------------+------------+---------------
Sc
----------+-------------+------------+---------------
253 | 18.3 | 9.3 | 9.0
598 | 15.6 | 7.0 | 8.6
2403 | 17.3 | 8.7 | 8.6
2683 | >20.0 | 9.9 | >10.1
2903 | 19.0 | 9.1 | 9.9
4254 | 18.5 | 10.4 | 8.1
4321 | 18.8 | 10.5 | 8.3
4414 | >19.5 | 10.1 | > 9.4
4490 | 18.8 | 10.2 | 8.6
5194 | 17.3 | 7.4 | 9.9
5236 | 18.6 | 10.4 | 8.2
5457 | 17.0 | 9.9 | 7.1
----------+-------------+------------+---------------
Irr.
----------+-------------+------------+---------------
LMC | 9.5 | 0.5 | 9.0
SMC | 12.0 | 1.5 | 10.5
3034 | >19.5 | 9.0 | >10.5
4449 | 17.8 | 9.5 | 8.3
6822 | 15.8 | 8.5 | 7.3
----------+-------------+------------+---------------
The data are listed in Table XVI and are shown graphically in Figure
10. The luminosities of the brightest stars are given in photographic
magnitudes. For the Magellanic Clouds, M 33, and N.G.C. 6822, these
were obtained from published star counts. For M 31, 51, 63, 81, 94,
and N.G.C. 2403, they depend upon unpublished counts, for which the
magnitudes were determined by comparisons with Selected Areas. For the
remaining nebulae, the magnitudes of stars were estimated with varying
degrees of precision, but are probably less than 0.5 mag. in error.
[Illustration: Fig. 10.—Relation between total magnitudes of
extra-galactic nebulae and magnitudes of the brightest stars involved.
Differences between total visual magnitudes of nebulae and the
photographic magnitudes of the brightest stars are plotted against
the total magnitudes. The dots represent cases in which the stars
could actually be detected; the incomplete crosses represent cases in
which stars could not be detected, and hence give lower limits for the
magnitude differences. The diagonal line indicates the approximate
limits of observation, fixed by the circumstance that, in general,
stars fainter than 19.5 probably would not be detected on the nebulous
background.]
The sloping line to the right in Figure 10 represents the limits
of the observations, for, from a study of the plates themselves,
it appeared improbable that stars fainter than about 19.5 could be
detected with certainty on a nebulous background. Points representing
nebulae in which individual stars could not be found should lie in this
excluded region above the line, and their scatter is presumably
comparable with that of the points actually determined below the line.
When allowance is made for this inaccessible region, the data can be
interpreted as showing a moderate dispersion around the mean ordinate
$m_{s} - m_{T} = 9.0.$ (7)
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 m_{T} – 25.8 where m_{T} 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
$\log D = 0.2 m_{T} + 4.04.$ (8)