$m_{T} = 13.0 - 5 \log d.$ (2)
This falls between the two regression curves derived from least-square
solutions and could be obtained exactly by assigning appropriate
weights to the two methods of grouping. The nature of the data is such
that a closer agreement can scarcely be expected. No correction to the
assumed value of the slope appears to be required. The material extends
over a range of 12 mag., and the few cases which have been investigated
indicate that the correlation can be extended another 3 mag., to the
limit at which nebulae can be classified with certainty on photographs
made with the 100-inch reflector. The relation may therefore be
considered to hold throughout the entire range of observations.
[Illustration:Fig. 5.—Relation between luminosity and diameter among
the irregular nebulae. The Magellanic Clouds are included. N.G.C. 4656
is an exceptional case in that it shows a narrow, greatly elongated
image in which absorption effects are very conspicuous; hence the
maximum diameter is exceptionally large for its apparent luminosity.
The residuals without regard to sign average 0.87 mag., and there
appears to be no systematic effect due either to type or luminosity.
The scatter, however, is much greater for the spirals, especially in
the later types, than for the elliptical nebulae. The limiting cases
are explained by peculiar structural features. The nebulae which fall
well above the line usually have bright stellar nuclei, and those which
fall lowest are spirals seen edge-on in which belts of absorption are
conspicuous.
TABLE VII
========+============+=========+=========+=====
Type | ¯m¯_{T}$ | ¯log d¯ | C* | d†
--------+------------+---------+---------+-----
E0 | 11.40 | –0.204 | 10.38 | 1.2
1 | 11.43 | .177 | 10.54 | 1.3
2 | 11.52 | .088 | 11.08 | 1.6
3 | 11.99 | .133 | 11.33 | 1.8
4 | 11.95 | – .011 | 11.90 | 2.4
5 | 10.97 | + .090 | 11.42 | 1.9
6 | 10.93 | .220 | 12.03 | 2.5
7 | 11.02 | .360 | 12.82 | 3.7
| | | |
Sa | 11.69 | .333 | 13.35 | 4.7
b | 11.55 | .471 | 13.90 | 6.0
c | 11.74 | .540 | 14.44 | 7.7
| | | |
SBa | 11.66 | .267 | 13.00 | 4.0
b | 11.48 | .317 | 13.16 | 4.3
c | 11.87 | .509 | 14.41 | 7.6
| | | |
Irr | 11.34 | +0.469 | 13.68 | 5.4
------+- ----------+---------+---------+-----
* C = $\overline{m}_{T} + 5 \overline{\log d}.$
† log d = 0.2 (C — m_{T}); m_{T} = 10.0.