Visual magnitudes have been determined by Hopmann for the
nuclei of 37 of the nebulae included in the present discussion. These
data, together with types and diameters of the nebulae, are listed
in Table XIII. When the magnitudes are plotted directly against
the logarithms of the diameters, they show little or no correlation.
When, however, the nebulae are reduced to the standard type (by
applying corrections for differences in diameter along the sequence),
a decided correlation is found whose coefficient is 0.76. This is
shown in Figure 9. The simple mean of the two regression curves
is
$m_{n} = 14.45 - 4.94 \log d,$ (6)
where the slope differs by about 1 per cent from that in formula (2).
The list contains 16 elliptical nebulae, 15 normal, and 6 barred
spirals. The nebulae are fairly representative, except that few
late-type spirals are included. This is an effect of selection due to
the fact that nuclei become less and less conspicuous as the sequence
progresses.
The same result can be derived from a study of the differences, m_{n} –
m_{T}, for the individual nebulae. The mean value is 1.55 ± 0.08, and
the average residual is 0.60 mag. Means for the separate types are to
be found in Table XIV.
TABLE XIII
Diameters and Nuclear Magnitudes
=======+============+=========+================+===============
N.G.C. | Type | log d | m_{n} Hopmann | m_{n} Reduced
-------+------------+---------+----------------+---------------
221 | E2 | +0.42 | 9.84 | 11.85
1023 | SBa | .78 | 11.86 | 11.86
2841 | Sb | 0.78 | 12.08 | 11.19
3031 | Sb | 1.20 | 10.94 | 10.05
3115 | E7 | 0.60 | 10.83 | 11.09
3351 | SBb | .48 | 12.31 | 12.15
3368 | Sa | .85 | 11.68 | 11.43
3379 | E0 | .30 | 11.55 | 14.27
3412 | SBa | .40 | 11.59 | 11.59
3489 | Sb | .40 | 11.54 | 10.65
3626 | Sa | .28 | 12.37 | 12.12
3627 | Sb | .90 | 12.03 | 11.14
4125 | E4 | .30 | 11.74 | 13.04
4216 | Sb | .85 | 11.65 | 10.76
4278 | E1 | .0 | 12.02 | 14.38
4374 | E1 | .08 | 11.43 | 13.79
4382 | E4 | .48 | 11.77 | 13.07
4435 | E6 | .11 | 11.65 | 12.26
4438 | Sb | .54 | 11.83 | 10.94
4486 | E0 | .30 | 11.23 | 13.95
4546 | E6 | .18 | 11.75 | 12.36
4552 | E0 | .23 | 11.59 | 14.31
4569 | Sc | .65 | 12.05 | 10.57
4579 | SBc | .45 | 11.48 | 10.07
4621 | E5 | .30 | 11.60 | 12.56
4636 | E1 | .08 | 11.97 | 14.33
4649 | E2 | .30 | 11.57 | 13.58
4697 | E6 | .48 | 10.90 | 11.51
4699 | SBb | .57 | 10.72 | 10.56
4725 | SBb | .70 | 11.97 | 11.81
4736 | Sb | .70 | 10.36 | 9.47
5005 | Sc | .70 | 12.04 | 10.56
5033 | Sc | 0.78 | 12.38 | 10.90
5194 | Sc | 1.08 | 11.38 | 9.90
5322 | E3 | 0.15 | 12.10 | 13.76
5866 | Sa | .48 | 11.76 | 11.51
7331 | Sb | +0.95 | 11.82 | 10.93
Means | | +0.509 | 11.60 | 11.90
-------+------------+---------+----------------+---------------
The low value for Sa-SBa is due to N.G.C. 5866, for which the
magnitude difference of 0.06 is certainly in error, and the high value
for Sc and SBc, to M 51, for which the difference of 3.98 mag. is not
representative. The latter is accounted for in part by the fact that
the m_{T} refers to the combined magnitude of the main spiral and
the outlying mass, N.G.C. 5195. When these two cases are discarded,
the final mean becomes 1.52 ± 0.05, and the average residual, 0.52
mag., is consistent with the probable errors of the magnitude
determinations. The small numbers of objects within each class are
insufficient for reliable conclusions concerning slight variations
along the sequence. From the constancy of m_{n} – m_{T}, the relation
expressed by formula (6) necessarily follows, the small difference in
the constant being accounted for by the different methods of handling
the data.
[Illustration: Fig. 9.—Relation between nuclear magnitudes and
diameters. The nebulae have been reduced to the standard type by
applying corrections to the magnitudes.]
The parallelism of the two curves representing formulae (2) and (6)
indicates that the regular extra-galactic nebulae, when reduced to the
standard type, are similar objects. The mean surface brightness is
constant, and the luminosity of the nucleus, as measured by Hopmann,
is a constant fraction, about one-fourth, of the total luminosity of
the nebulae. If there is a considerable range in absolute magnitude
and hence in actual dimensions, the smaller nebulae must be faithful
miniatures of the larger ones.