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.
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