The correlations thus far derived are between total luminosities and
maximum diameters. In the most general sense, therefore, they
express laws of mean surface brightness. The value, K = 5.0, in formula
(1) indicates that the surface brightness is constant for each separate
type. The variations in C indicate a progressive diminution in the
surface brightness from class to class throughout the entire sequence.
The consistency of the results amply justifies the sequence as a basis
of classification, since a progression in physical dimensions
is indicated, which accompanies the progression in structural form.
Although the correlations do not necessarily establish any generic
relation among the observed classes, they support in a very evident
manner the hypothesis that the various stages in the sequence represent
different phases of a single fundamental type of astronomical body.
Moreover, the quantitative variation in C is consistent with this
interpretation, as is apparent from the following considerations.
TABLE IX
Residuals in m_{T} + 5 log d as a Function of Orientation
===============+=============+===============+============
Type | Round | Elliptical | Edge-On
---------------+-------------+---------------+------------
Sa | –0.02 (13) | –0.27 (13) | +0.57 (23)
Sb | .77 (24) | .0 (35) | 1.71 (11)
Sc | –0.08 (35) | –0.13 (57) | +0.66 (22)
All S | –0.26 (72) | –0.11 (105) | +0.83 (56)
SBa | 0.0 (10) | –0.30 (7) | +0.31 (8)
SBb | – .16 (10) | + .07 (6) |
SBc | +0.19 (9) | –0.50 (4) | +0.32 (2)
All SB | +0.01 (29) | –0.21 (17) | +0.31 (10)
All spirals | –0.22 (101) | –0.13 (122) | +0.73 (66)
---------------+-------------+---------------+------------
Among the elliptical nebulae it is observed that the nuclei are sharp
and distinct and that the color distribution is uniform over the
images. This indicates that there is no appreciable absorption, either
general or selective, and hence that the luminosity of the projected
image represents the total luminosity of the nebula, regardless of
the orientation. If the observed classes were pure, that is, if the
apparent ellipticities were the actual ellipticities, formula (1) could
be written
$C_{e} = m_{T} + 5 \log b - 5 \log (1 - e),$ (3)
where b is the minor diameter in minutes of arc and e is the
ellipticity. The term m_{T} + 5 log b is observed to be constant for a
given type. If it were constant for all elliptical nebulae, then the
term C_{e} + 5 log (1 – e) would be constant also. On this assumption,
$C_{e} + 5 \log (1 - e) = C_{0},$
where C_{0} is the value of C for the pure class E0. Hence
$C_{e} - C_{0} = -5 \log (1 - e),$ (4)
a relation which can be tested by the observations. An analysis of the
material indicates that this is actually the case, and hence that among
the elliptical nebulae in general, the minor diameter determines the
total luminosity, at least to a first approximation.[18]
The observed values of C vary with the class, as is seen in Table VII
and Figure 6, but, excepting that for E7, they are too large because of
the mixture of later types of nebulae among those of a given observed
class. It is possible, however, to calculate the values of C_{e} –
C_{0} for the pure classes and then to make approximate corrections
for the observed mixtures on the assumption that the nebulae of any
given actual ellipticity are oriented at random. In this manner, mean
theoretical values can be compared with the observed values. The
comparisons are shown in Table XII in the form C_{7} – C_{e}, because
E7 is the only observed class that can be considered as pure. The
significance of the table will be discussed later.
The following method has been used to determine the relative
frequencies with which nebulae of a given actual ellipticity, oriented
at random, will be observed as having various apparent ellipticities.
In Figure 8, let the co-ordinate axes OX and OY coincide with the
major and minor axes, a and b, of a meridian section of an ellipsoid
of revolution. Let OO′ be the line of sight to the observer, making an
angle i with OX, and let OR be perpendicular to OO′. Let PP′ be a
tangent to the ellipse, parallel to and at a distance from OO′. Let
x_{0} and y_{0} be the intercepts of the tangent on the X- and Y-axis,
respectively. The apparent ellipticity is determined by b_{x}, which,
for various values of the angle i, ranges from b to a. The problem is
to determine the relative areas on the surface of a sphere whose center
is O, within which the radius OY must pass in order that the values of
b_{1}, and hence of the apparent ellipticity, e_{1} may fall within
certain designated limits. This requires that the angle i be expressed
in terms of b_{1}.
[Illustration: Fig. 8]
From the equation of the tangent, PP′,
$y = -x \tan{i} + \sqrt{a^2 \tan^2{i} + b^2}$
$y_{0} = \sqrt{a^2 \tan^2{i} + b^2}$