THE DATA

The most homogeneous list of nebulae for statistical study is that compiled by Hardcastle13 containing all nebulae found on the Franklin-Adams charts. These are uniform exposures of two hours on fast plates made with a Cooke astrographic lens of 10-inch aperture and 45-inch focal length. The scale is 1 mm = 3′. The entire sky is covered, but since the plates are centered about 15° apart and the definition decreases very appreciably with distance from the optical axis, the material is not strictly homogeneous. Moreover, the published list suffers from the usual errors attendant on routine cataloguing; for instance, four conspicuous Messier nebulae, M 60, M 87, M 94, and M 101, are missing. In general, however, the list is thoroughly representative down to about the thirteenth photographic magnitude and very few conspicuous objects are overlooked. It plays the role of a standard with which other catalogues of the brighter nebulae may be compared for completeness, and numbers in limited areas may be extended to the entire sky.

When known galactic nebulae, clusters, and the objects in the Magellanic Clouds are weeded out, the remaining 700 nebulae may be treated as extra-galactic. Very few can be classified from the Franklin-Adams plates; for this purpose photographs on a much larger scale are required. Until further data on the individual objects are available, Hardcastle’s list can be used only for the study of distribution over the sky. This shows the well-known features—the greater density in the northern galactic hemisphere, the concentration in Virgo, and the restriction of the very large nebulae to the southern galactic hemisphere.

Fortunately, numerical data do exist in the form of total visual magnitudes for many of the nebulae in the northern sky. These determinations were made by Holetschek,14 who attempted to observe all nebulae within reach of his 6-inch refractor. He later restricted his program; but the final list is reasonably complete for the more conspicuous nebulae north of declination –10°, and is representative down to visual magnitude about 12.5. Out of 417 extra-galactic nebulae in Holetschek’s list, 408 are north of –10°, as compared with 400 in Hardcastle’s. The two lists agree very well for the brighter objects, but diverge more and more with decreasing luminosity. At the twelfth magnitude about half of Holetschek’s nebulae are included by Hardcastle. Since the two lists compare favorably in completeness over so large a region of the sky, Holetschek’s may be chosen as the basis for a statistical study and advantage taken of the valuable numerical data on total luminosities.

Hopmann15 has revised the scale of magnitudes by photometric measures of the comparison stars used by Holetschek. New magnitudes were thus obtained for 85 individual nebulae and from these were derived mean correction tables applicable to the entire list. The revised magnitudes are used throughout the following discussion. Hopmann’s corrections extend to about 12.0 mag., and have been extrapolated on the assumption that they are constant for the fainter magnitudes. The errors involved are unimportant in view of selective effects which must be present among the observed objects near the limit of visibility.

The nebulae were classified and their diameters measured from photographs of about 300 of them taken with the 60-inch and 100-inch reflectors at Mount Wilson. Most of the others are included in the great collection of nebular photographs at Mount Hamilton, which have been described by Curtis;16 and, through the courtesy of the Director of the Lick Observatory, it has been possible to confirm the classification inferred from the published description by actual inspection of the original negatives.

Types, diameters, and total visual magnitudes are thus available for some 400 of the nebulae in Holetschek’s list. The few unclassified objects are all fainter than 12.5 mag. The data are listed in Tables I–IV, in which the N.G.C. numbers, the total magnitudes, and the logarithms of the maximum diameters in minutes of arc are given for each type separately. A summary is given in Table V, in which the relative frequencies and the mean magnitudes of the various types will be found.

RELATIVE LUMINOSITIES OF THE VARIOUS TYPES

The frequency distribution of magnitudes for all types together and for the elliptical nebulae and the spirals separately is shown in Table VI and Figure 1. With the exception of the two outstanding spirals, M 31 and M 33, the apparent luminosities are about uniformly distributed among the different types. The relative numbers of the elliptical nebulae as compared with the spirals decrease somewhat with decreasing luminosity, but this is very probably an effect of selection. The elliptical nebulae are more compact than the spirals and become more stellar with decreasing luminosity. For this reason some of the fainter nebulae are missed when small-scale instruments are used, although the same luminosity spread over a larger area would still be easily detected. The effect is very pronounced on photographic plates. It accounts also for the slightly brighter mean magnitude of the elliptical nebulae as compared with the spirals in Table V.

TABLE I
Elliptical Nebulae

The various types are homogeneously distributed over the sky, their spectra are similar, and the radial velocities are of the same general order. These facts, together with the equality of the mean magnitudes and the uniform frequency distribution of magnitudes, are consistent with the hypothesis that the distances and absolute luminosities as well are of the same order for the different types. This is an assumption of considerable importance, but unfortunately it cannot yet be subjected to positive and definite tests. None of the individual similarities necessarily implies the adopted interpretation, but the totality of them, together with the intimate series relations among the types, which will be discussed later, suggests it as the most reasonable working hypothesis, at least until inconsistencies should appear.

TABLE II
Barred Spirals

TABLE III
Normal Spirals

RELATION BETWEEN LUMINOSITIES AND DIAMETERS

Among the nebulae of each separate type are found linear correlations between total magnitudes and logarithms of diameters. These are shown in Figures 2–5 for the beginning, middle, and end of the sequence of types and also for the irregular nebulae. In Figures 2 and 3 adjacent types have been grouped in order to increase the material, and in Figure 5 the Magellanic Clouds have been added to increase the range.

The correlations can be expressed in the form

where K is constant from type to type, but C varies progressively throughout the sequence. The value of K cannot be accurately determined from the scattered data for any particular type, but, within the limits of uncertainty, it approximates the round number 5.0, the value which is represented by the lines in Figures 2–5.

When K is known, the value of C can be computed from the mean magnitude and the logarithm of the diameter for each type. This amounts to reading from the curves the magnitudes corresponding to a diameter of one minute of arc, but avoids the uncertainty of establishing the curves where the data are limited.

TABLE IV
Irregular Nebulae

NOTES TO TABLES I–IV

* Magnitude from Hopmann.

† N.G.C. 524 and 3998 are late elliptical nebulae in which the equatorial planes are perpendicular to the line of sight. They might be included with the E6 or E7 nebulae.

§ Absorption very conspicuous.

‡ N.G.C. 3607, 4459, and 5485 appear to be elliptical nebulae with narrow bands of absorption between the nuclei and the peripheries.

The progressive change in the value of C throughout the sequence may be expressed as a variation either in the magnitude for a given diameter or in the diameter for a given magnitude. Both effects are listed in Table VII and are illustrated in Figure 6, in which magnitudes and diameters thus found are plotted against types. With the exception of the later elliptical nebulae, for which the data are wholly inadequate for reliable determinations, the points fall on smooth curves. In the region of the earlier elliptical nebulae, the curves should be somewhat steeper in order to allow for objects of greater ellipticities which are probably included.

REDUCTION OF NEBULAE TO A STANDARD TYPE

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 normal upper Delta log d equals 0.2 left-parenthesis 13.0 minus upper C right-parenthesis 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

* 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

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.