where b is the minor diameter in minutes of arc and e is the ellipticity. The term mT + 5 log b is observed to be constant for a given type. If it were constant for all elliptical nebulae, then the term Ce + 5 log (1 – e) would be constant also. On this assumption,
where C0 is the value of C for the pure class E0. Hence
| (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 Ce – C0 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 C7 – Ce, 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 x0 and y0 be the intercepts of the tangent on the X- and Y-axis, respectively. The apparent ellipticity is determined by bx, 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 b1, and hence of the apparent ellipticity, e1 may fall within certain designated limits. This requires that the angle i be expressed in terms of b1.
Fig. 8