The slope, K, in the formula relating magnitudes with diameters,
appears to be closely similar for the various types, but accurate
determinations are restricted by the limited and scattered nature
of the data for each type separately. With a knowledge of the parameter
C, however, it is possible to reduce all the material to a standard
type and hence to determine the value of K from the totality of the
data. The mean of E7, SBa, and Sa was chosen for the purpose, as
representing a hypothetical transition-point between the elliptical
nebulae and the spirals, and was designated by the symbol “S0.”
The corresponding value of C, in round numbers, is 13.0. Corrections
were applied to the logarithms of the diameters of the nebulae of each
observed class, amounting to
$\Delta \log d = 0.2 (13.0 - C)$
where C is the observed value for a particular class.[17] When
the values of C are read from the smooth curve in Figure 6, these
corrections are as shown in Table VIII.
TABLE V
Frequency Distribution of Types
=========+==========+==============+===========
Type | Number | Percentage | Mean Mag.
---------+----------+--------------+-----------
Elliptical Nebulae
---------+----------+--------------+-----------
E0 | 17 | 18 | 11.40
1 | 13 | 14 | 11.43
2 | 14 | 15 | 11.52
3 | 10 | 11 | 11.99
4 | 13 | 14 | 11.95
5 | 6 | 6 | 10.97
6 | 7 | 8 | 10.93
7 | 5 | 5 | 11.02
Pec | 8 | 9 | 11.55
+----------+--------------+-----------
Total | 93 | 23* | 11.53
---------+----------+--------------+-----------
Normal Spirals
---------+----------+--------------+-----------
Sa | 49 | 21 | 11.69
b | 70 | 29 | 11.55
c | 115 | 49 | 11.75
Pec | 3 | 1 | 12.80
+----------+--------------+-----------
Total | 237 | 59* | 11.68
---------+----------+--------------+-----------
Barred Spirals
---------+----------+--------------+-----------
SBa | 26 | 44 | 11.66
b | 16 | 27 | 11.48
c | 15 | 26 | 11.87
Pec. | 2 | 3 | 11.70
+----------+--------------+-----------
Total | 59 | 15* | 11.66
---------+----------+--------------+-----------
Irregular Nebulae
---------+----------+--------------+-----------
| 11 | 3* | 11.34
---------+----------+--------------+-----------
Totals
---------+----------+--------------+-----------
All types 400 100 11.63
---------+----------+--------------+-----------
* Percentages of 400, the total number of nebulae investigated. The
percentages of the subtypes refer to the number of nebulae in the
particular type.
TABLE VI
Frequency Distribution of Magnitudes
====================+=========+================
| Numbers of Nebulae
Magnitude Interval +---------+----------+-----
| E | S | All
--------------------+---------+----------+-----
8.1– 8.5 | 0 | 2 | 2
8.6– 9.0 | 2 | 4 | 7
9.1– 9.5 | 4 | 6 | 11
9.6–10.0 | 7 | 7 | 19
10.1–10.5 | 7 | 13 | 20
10.6–11.0 | 8 | 14 | 32
11.1–11.5 | 9 | 24 | 49
11.6–12.0 | 21 | 57 | 88
12.1–12.5 | 20 | 52 | 86
12.6–13.0 | 10 | 33 | 51
--------------------+---------+----------+-----
The corrected values of log d were then plotted against the observed
magnitudes. This amounts to shifting the approximately parallel
correlation curves for the separate types along the axis of log d
until they coincide. Since the mean magnitudes of the various types
are nearly constant, the relative shifts will very nearly equal the
differences in the mean observed log d, and hence the effect of errors
in the first approximation to the values of K will be negligible.
[Illustration: Fig. 1.—Frequency distribution of apparent magnitudes
among nebulae in Holetschek’s list.]
[Illustration:Fig. 2.—Relation between luminosity and diameter among
nebulae at the beginning of the sequence of types—E0 and E1 nebulae.
The plot is shown in Figure 7, in which the two Magellanic Clouds
have been included in order to strengthen the bright end of the curve
which would otherwise be unduly influenced by the single object, M
31. The magnitudes +0.5 and +1.5, which were assigned to the Clouds,
are estimates based upon published descriptions.
[Illustration: Fig. 3.—Relation between luminosity and diameter among
nebulae at the middle of the sequence of types—E7, Sa, and SBa nebulae.]
[Illustration: Fig. 4.—Relation between luminosity and diameter among
nebulae at the end of the sequence of types—Sc and SBc nebulae.]