SIGNIFICANCE OF THE LUMINOSITY RELATION
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 mT + 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
| (3) |
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
From the equation of the tangent, PP′,
Since
Let a = 1, then
where
From these equations, the values of i can be determined for all possible values of e1. The limits for the observed classes E0 to E7 were chosen midway between the consecutive tenths, E0 ranging from e = 0 to e = 0.05; E1, from e = 0.05 to e = 0.15; E7, from e = 0.65 to e = 0.75. The relative frequencies of the various observed classes are then proportional to the differences in sin i corresponding to the two limiting values of e1. These frequencies must be calculated separately for nebulae of different actual ellipticities.
The results are given in [Table X], where the actual ellipticities, listed in the first column, are followed across the table by the percentages which, on the assumption of random orientation, will be observed as having the various apparent ellipticities. The bottom row will be seen to show the percentages of apparent ellipticities observed in an assembly of nebulae in which the numbers for each actual ellipticity are equal and all are oriented at random.
TABLE X
| Actual | Apparent | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| E0 | E1 | E2 | E3 | E4 | E5 | E6 | E7 | Total | |
| E7 | 0.055 | 0.111 | 0.114 | 0.116 | 0.121 | 0.132 | 0.164 | 0.187 | 1.000 |
| 6 | .059 | .123 | .126 | .133 | .148 | .187 | 0.224 | ||
| 5 | .067 | .140 | .148 | .166 | .216 | 0.263 | |||
| 4 | .079 | .169 | .190 | .250 | 0.312 | ||||
| 3 | .100 | .225 | .299 | 0.376 | |||||
| 2 | .145 | .378 | 0.477 | ||||||
| 1 | 0.300 | 0.700 | |||||||
| 0 | 1.000 | ||||||||
| Total | 1.805 | 1.846 | 1.354 | 1.041 | 0.797 | 0.582 | 0.388 | 0.187 | 8.000 |
| 0.226 | 0.231 | 0.169 | 0.130 | 0.100 | 0.073 | 0.049 | 0.023 | 1.000 | |
From this table and the actual numbers in the observed classes as read from a smoothed curve, the numbers of each actual ellipticity mingled in the observed classes can be determined. For instance, the four nebulae observed as E7 represent 0.187 of the total number of actual E7. The others are distributed among the observed classes E0 to E6 according to the percentages listed in [Table X]. Six nebulae are observed as E6, but 3.6 of these are actually E7. The remaining 2.4 actual E6 nebulae represent 0.224 of the total number of that actual ellipticity, the others, as before, being scattered among the observed classes E0 to E5. [Table XI] gives the complete analysis and is similar to [Table X] except that the percentages in the latter are replaced by the actual numbers indicated by the observations.
Finally, the mean values of C7 – Ce are calculated from the numbers of nebulae in the various columns of [Table XI] together with the values of C7 – Ce for the pure classes as derived from [formula (4)]. The results are listed in the fourth column of [Table XII] following those for the pure and the observed classes. In determining the observed values, N.G.C. 524 and 3998 are included as E0 and E1, although in [Table I] they are listed as peculiar, because they are obviously much flattened nebulae whose minor axes are close to the line of sight.
TABLE XI
| Actual | Apparent | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| E0 | E1 | E2 | E3 | E4 | E5 | E6 | E7 | Total | |
| E7 | 1.2 | 2.4 | 2.5 | 2.5 | 2.6 | 2.9 | 3.6 | 4.0 | 21.7 |
| 6 | 0.6 | 1.3 | 1.4 | 1.5 | 1.6 | 2.0 | 2.4 | 10.8 | |
| 5 | .8 | 1.7 | 1.8 | 2.0 | 2.7 | 3.1 | 12.1 | ||
| 4 | .8 | 1.7 | 1.9 | 2.5 | 3.1 | 10.0 | |||
| 3 | .9 | 2.1 | 2.8 | 3.5 | 9.3 | ||||
| 2 | 1.1 | 2.9 | 3.6 | 7.6 | |||||
| 1 | 1.7 | 3.9 | 5.6 | ||||||
| 0 | 9.9 | 9.9 | |||||||
| Total* | 17.0 | 16.0 | 14.0 | 12.0 | 10.0 | 8.0 | 6.0 | 4.0 | 87.0 |
* The totals represent the numbers in the observed classes as read from a smooth curve.
The observed values in general fall between those for the pure classes and those corresponding to random orientation. They are of the same order as the latter, and the discrepancies are perhaps not unaccountably large in view of the nature and the limited extent of the material. There is a systematic difference, however, averaging about 0.2 mag., in the sense that the observed values are too large, and increasing with decreasing ellipticity. One explanation is that the observed classes are purer than is expected on the assumption of random orientation. This view is supported by the relatively small dispersion in C, as may be seen in [Table I] and [Figure 2], among the nebulae of a given class, but it is difficult to account for any such selective effect in the observations. The discrepancies may be largely eliminated by an arbitrary adjustment of the numbers of nebulae with various degrees of actual ellipticity; for instance, the values in the last column of [Table XII], calculated on the assumption of equal numbers, agree very well with the observed values, although the resulting numbers having the various apparent ellipticities differ slightly from those observed. The observed values, however, can again be accounted for by the inclusion of some flatter nebulae among the classes E6 and E7. Very early Sa or SBa nebulae might easily be mistaken for E nebulae when oriented edge-on, although they would be readily recognized when even slightly tilted. If the numerical results fully represented actual statistical laws, the explanation would be sought in the physical nature of the nebulae. The change from ellipsoidal to lenticular figures, noticeable in the later-type nebulae, would affect the results in the proper direction, as would also a progressive shortening of the polar axis. The discrepancies, however, are second-order effects, and since they may be due to accidental variations from random orientation, a further discussion must await the accumulation of more data.
TABLE XII
Differential Values of C
| Class | Pure Classes | Observed | Random Orientation | |
|---|---|---|---|---|
| No. as Observed | Equal No. | |||
| C7–C7 | 0.00 | 0.00 | 0.00 | 0.00 |
| C6 | 0.63 | 0.35* | 0.25 | 0.35 |
| C5 | 1.10 | 0.70* | 0.58 | 0.70 |
| C4 | 1.51 | 0.85 | 1.11 | 1.01 |
| C3 | 1.84 | 1.42 | 0.87 | 1.28 |
| C2 | 2.13 | 1.67 | 1.33 | 1.55 |
| C1 | 2.39 | 2.01† | 1.54 | 1.83 |
| C0 | 2.62 | 2.17† | 2.15 | 2.25 |
* Read from smooth curve in [Fig. 6]. The small numbers of observed E5 and E6 nebulae justify this procedure. The other values are the means actually observed.
† N.G.C. 524 and 3998 are included as E0 and E1, respectively.
Meanwhile, it is evident that, to a first approximation at least, the polar diameters alone determine the total luminosities of all elliptical nebulae, and the entire series can be represented by the various configurations of an originally globular mass expanding equatorially. A single formula represents the relation, in which the value of C is that corresponding to the pure type E0. From [Table XII], this is found to be 2.62 mag. less than the value of C7 The latter is observed to be 12.75, hence
| (5) |
If this relation held for the spirals as well, the polar diameters could be calculated from the measured magnitudes. Unfortunately, it has not been possible to measure accurately the polar diameters directly, and hence to test the question, but they have been computed for the mean magnitudes of the Sa, Sb, and Sc nebulae as given in [Table III], and the ratios of the axes have been derived by a comparison of these hypothetical values with the means of the measured maximum diameters. The results, 1 to 4.4, 1 to 5.7, and 1 to 7.3, respectively, although of the right order, appear to be somewhat too high. An examination of the photographs indicates values of the order of 1 to 5.5, 1 to 8, and 1 to 10, but the material is meager and may not be representative. The comparison emphasizes, however, the homogeneity and the progressive nature of the entire sequence of nebulae and lends some additional color to the assumption that it represents various aspects of the same fundamental type of system.
From the dynamical point of view, the empirical results are consistent with the general order of events in Jeans’s theory. Thus interpreted, the series is one of expansion, and the scale of types becomes the time scale in the evolutionary history of nebulae. In two respects this scale is not entirely arbitrary. Among the elliptical nebulae the successive types differ by equal increments in the ellipticity or the degree of flattening, and among the spirals the intermediate stage is midway between the two end-stages in the structural features as well as in the luminosity relations.
One other feature of the curves may be discussed from the point of view of Jeans’s theory before returning to the strictly empirical attitude. The close agreement of the diameters for the stages E7 and Sa suggests that the transition from the lenticular nebula to the normal spiral form is not cataclysmic. If the transition were gradual, however, we should expect to observe occasional objects in the very process, but among the thousand or so nebulae whose images have been inspected, not one clear case of a transition form has been detected. The observations jump suddenly from lenticular nebulae with no trace of structure to spirals in which the arms are fully developed.
If the numerical data could be fully trusted, the SBa forms would fill the gap. Among these nebulae, the transition from the lenticular to the spiral with arms is gradual and complete. It is tempting to suppose that the barred spirals do not form an independent series parallel with that of the normal spirals, but that all or most spirals begin life with the bar, although only a few maintain it conspicuously throughout their history. This would also account for the fact that the relative numbers of the SBa nebulae are intermediate to those of the lenticular and of the Sa. The normal spirals become more numerous as the sequence progresses, while the numbers of barred spirals, on the contrary, actually decrease with advancing type.