The numbers of nebulae to different limiting magnitudes can be used to
test the constancy of the density function, or, on the hypothesis
of uniform luminosities, to determine the distribution in space. The
nebulae brighter than about the tenth magnitude are known individually.
Those not included in Holetschek’s list are: the Magellanic Clouds, the
two nebulae N.G.C. 55 and 1097, between 9.0 and 9.5 mag., and the seven
nebulae N.G.C. 134, 289, 1365, 1533, 1559, 1792, and 3726, all between
9.5 and 10.0 mag.
A fair estimate of the number between 10.0 and 11.0 mag. can be derived
from a comparison of Holetschek’s list with that of Hardcastle,
an inspection of images on the Franklin-Adams charts and other
photographs, and a correlation between known total magnitudes and
the descriptions of size and brightness in Dreyer’s catalogues. It
appears that very few of these objects were missed by Holetschek in
the northern sky—not more than six of Hardcastle’s nebulae. For the
southern sky, beyond the region observed by Holetschek, the results are
very uncertain, but probable upper and lower limits were determined as
50 and 20, respectively. The brighter nebulae are known to be scarce in
those regions. A mean value of 35 leads to a total 295 for the entire
sky, and this is at least of the proper order.
The number of nebulae between 11.0 and 12.0 mag. can be estimated on
the assumption that the two lists, Holetschek’s and Hardcastle’s,
are about equally complete within this range. They are known to be
comparable for the brighter nebulae, and, moreover, the total numbers
included in the two lists for the same area of the sky, that north
of declination –10°, are very nearly equal—400 as compared with 408.
The percentages of Holetschek’s nebulae included by Hardcastle were
first determined as a function of magnitude. Within the half-magnitude
interval 11.0 to 11.5, for instance, 60 per cent are in Hardcastle’s
list. If the two lists are equally complete and, taken together, are
exhaustive, the total number in the interval will be 1.4 times the
number of Holetschek’s nebulae. The latter is found to be 50 from
smoothed frequency curves of the magnitudes listed in Tables I–IV.
The total number north of –10° is therefore 70. This can be corrected
to represent the entire sky by applying the factor 1.75, which is the
ratio of the total number of Hardcastle’s nebulae, 700, to the number
north of –10°, 400. In this manner a reasonable estimate of 123 is
obtained for the number of nebulae in the entire sky between 11.0
and 11.5 mag. Similarly, between 11.5 and 12.0, where 50 per cent of
Holetschek’s nebulae are included in Hardcastle’s list, the total
number for the entire sky is found to be 236.
The greatest uncertainty in these figures arises from the assumption
that the two lists together are complete to the twelfth magnitude. The
figures are probably too small, but no standards are available by which
they can be corrected. It is believed, however, that the errors are
certainly less than 50 per cent and probably not more than 25 per cent.
This will not be excessive in view of the possible deviations from
uniform distribution where so limited a number of objects is considered.
Beyond 12.0 mag. the lists quickly lose their aspect of completeness
and cannot be used for the present purpose. There are available,
however, the counts by Fath[24] of nebulae found on plates of Selected
Areas made with the 60-inch reflector at Mount Wilson. The exposures
were uniformly 60 minutes on fast plates and cover the Areas in
the northern sky down to and including the –15° zone. The limiting
photographic magnitudes for stars average about 18.5. The counts have
been carefully revised by Seares[25] and are the basis for his estimate
of 300,000 nebulae in the entire sky down to this limit.
Approximate limiting total magnitudes for the nebulae in two of the
richest fields, S.A. 56 and 80, have been determined from extra-focal
exposures with the 100-inch reflector. The results are 17.7 in each
case, and this, corrected by the normal color-index of such objects,
gives a limiting visual magnitude of about 16.7, which can be used for
comparison with the counts of the brighter nebulae.
The various data are collected in Table XVIII, where the observed
numbers of extra-galactic nebulae to different limits of visual
magnitude are compared with those computed on the assumption of uniform
distribution of objects having a constant absolute luminosity. The
formula used for the computation is
$\log N = 0.6 m_{T} + \textrm{Constant},$
where the constant is the value of log N for m_{T} = 0. The value —4.45
is found to fit the observational data fairly well.
The agreement between the observed and computed log N over a range of
more than 8 mag. is consistent with the double assumption of uniform
luminosity and uniform distribution or, more generally, indicates that
the density function is independent of the distance.
The systematic decrease in the residuals O – C with decreasing
luminosity is probably within the observational errors, but it may also
be explained as due to a clustering of nebulae in the vicinity of the
galactic system. The cluster in Virgo alone accounts for an appreciable
part. This is a second-order effect in the distribution, however, and
will be discussed at length in a later paper.
TABLE XVIII
Numbers of Nebulae to Various Limits
========+===================+===========+========
| LOG N* | |
m_{T} |---------+---------| O – C | LOG D†
| O | C | |
--------+---------+---------+-----------+--------
8.5 | 0.85 | 0.65 | +0.20 | 5.74
9.0 | 1.08 | 0.95 | .13 | 5.84
9.5 | 1.45 | 1.25 | .20 | 5.94
10.0 | 1.73 | 1.55 | .18 | 6.04
10.5 | 1.95 | 1.85 | .10 | 6.14
11.0 | 2.17 | 2.15 | + .02 | 6.24
11.5 | 2.43 | 2.45 | – .02 | 6.34
12.0 | 2.70 | 2.75 | .05 | 6.44
16.7 | 5.48 | 5.57 | –0.09 | 7.38
(18.0) | | (6.35) | | (7.64)
--------+---------+---------+-----------+--------
* Log N = 0.6 m_{T} – 4.45.
† Log D = 0.2 m_{T} + 4.04.
Distances corresponding to the different limiting magnitudes, as
derived from formula (8), are given in the last column of Table
XVIII. The 300,000 nebulae estimated to the limits represented by an
hour’s exposure on fast plates with the 60-inch reflector appear to
be the inhabitants of space out to a distance of the order of 2.4 ×
10^7 parsecs. The 100-inch reflector, with long exposures under good
conditions, will probably reach the total visual magnitude 18.0, and
this, by a slight extrapolation, is estimated to represent a distance
of the order of 4.4×10^7 parsecs or 1.4×10^8 light-years, within
which it is expected that about two million nebulae should be found.
This seems to represent the present boundaries of the observable
region 3 of space.