The distance of N.G.C. 4594 is unknown, but the assumption that it is a normal nebula with an absolute magnitude of –15.2 places it at 700,000 parsecs. The orders of the mass by the two methods are then
MASS OF N.G.C. 4594
| Spectrographic rotation | 2.0×109 ☉ |
| Öpik’s method | 2.6×108 |
Here again the resulting masses are of the same order. They can be made to agree as well as those for M 31 by the not unreasonable assumption that the absolute luminosity of the nebula is 2 mag. or so brighter than normal.
Öpik’s method leads to values that are reasonable and fairly consistent with those obtained by the independent spectrographic method. Therefore, in the absence of other resources, its use for deriving the mass of the normal nebula appears to be permissible. The result, 2.6 × 108 ☉, corresponding to an absolute magnitude of –15.2, is probably of the right order. The two test cases suggest that this value may be slightly low, but the data are not sufficient to warrant any empirical corrections.
NUMBERS OF NEBULAE TO DIFFERENT LIMITING MAGNITUDES
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.