Chapter III: Part II: Statistical Study of Extra-Galactic Nebulae (1)

THE DATA

The most homogeneous list of nebulae for statistical study is that compiled by Hardcastle[13] 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.

Hopmann[15] 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
==========+===========+========
N.G.C. | m_{T} | log d
----------+-----------+--------

E0 (17)
----------+-----------+--------
404 | 11.1 | +0.11
474 | 12.6 | – .40
1407 | 10.9 | .15
3348 | 11.8 | – .15
3379* | 9.4 | + .30
4283 | 12.2 | – .52
4486* | 9.7 | + .30
4494* | 10.1 | – .15
4552* | 9.9 | + .23
4589 | 11.4 | – .30
4648 | 12.3 | .52
5044 | 11.8 | .30
5216 | 13.3 | .70
5273 | 12.1 | .52
5557 | 12.3 | .40
5812 | 12.0 | – .40
5846 | 10.9 | 0.0
|-----------|---------
Mean | 11.40 | –0.204
----------+-----------+---------

E1 (13)
----------+-----------+---------
467 | 13.0 | –0.70
596 | 11.8 | .22
1400 | 11.1 | .22
2880 | 12.0 | .52
3226 | 12.0 | .10
3962 | 11.8 | – .30
4278* | 10.8 | .0
4374* | 9.9 | + .08
4472 | 8.8 | + .30
4478 | 11.5 | – .10
4636 | 10.9 | + .08
5813 | 12.6 | – .30
7626 | 12.3 | –0.30
----------+-----------+---------
Mean | 11.43 | –0.177
----------+-----------+---------

E2 (14)
----------+-----------+---------
221* | 8.8 | +0.42
1453 | 11.9 | – .10
2672 | 12.8 | – .40
3193 | 12.1 | 0.0
3599 | 12.0 | –0.30
3608 | 11.6 | .22
3640 | 11.1 | – .05
4261 | 11.1 | + .20
4291 | 12.3 | – .52
4377 | 11.9 | – .05
4406* | 10.0 | + .30
4476 | 12.8 | – .30
4649* | 9.5 | + .30
5127 | 13.3 | –0.52
----------+-----------+---------
Mean | 11.52 | –0.088
----------+-----------+---------

E3 (10)
----------+-----------+---------
1052 | 11.8 | –0.15
1600 | 12.7 | + .17
3222 | 13.3 | – .15
4319 | 12.8 | – .52
4365 | 11.4 | + .04
4386 | 12.3 | – .52
5322* | 9.6 | + .15
5982 | 11.4 | .0
7562 | 12.8 | – .22
7619 | 11.8 | – .15
----------+-----------+---------
Mean | 11.99 | –0.133
----------+-----------+---------

E4 (13)
----------+-----------+---------
584 | 10.9 | +0.30
1700 | 12.5 | – .10
2974 | 11.8 | .15
3605 | 12.5 | – .52
3610 | 11.8 | + .15
3894 | 12.8 | – .05
4125* | 10.3 | + .30
4378 | 12.1 | – .15
4382* | 10.0 | + .48
4551 | 12.8 | + .04
4742 | 12.3 | .0
5576 | 12.3 | – .15
7454 | 13.3 | 0.0
----------+-----------+---------
Mean | 11.95 | –0.011
----------+-----------+---------

E5 (6)
----------+-----------+---------
720 | 10.9 | + .11
2693 | 12.3 | – .15
3377 | 10.9 | + .17
4473 | 10.3 | .11
4621* | 10.0 | + .30
4660 | 11.4 | 0.0
----------+-----------+---------
Mean | 10.97 | +0.090
----------+-----------+---------

E6 (7)
----------+-----------+---------
821 | 11.8 | 0.0
2768 | 10.7 | + .18
3613 | 11.8 | .25
4179 | 11.8 | .34
4435* | 10.5 | .11
4546* | 10.3 | .18
4697* | 9.6 | +0.48
----------+-----------+---------
Mean | 10.93 | +0.220
----------+-----------+---------

E7 (5)
----------+-----------+---------
3115* | 9.5 | +0.60
4111 | 10.1 | .54
4270 | 12.1 | .0
4570 | 11.1 | .38
5308 | 12.3 | +0.28
----------+-----------+---------
Mean | 11.02 | +0.360
----------+-----------+---------

Peculiar (8)
----------+-----------+---------
185 | 12.3 | +0.48
205* | 9.3 | .90
524† | 11.9 | .41
3607† | 9.9 | .11
3998† | 12.1 | + .23
4459‡ | 11.3 | – .22
5485‡ | 12.3 | .05
5739 | 13.3 | –0.40
----------+-----------+---------

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
==========+===========+=========
N.G.C. | m_{T} | log d
----------+-----------+---------

SBa (26)
----------+-----------+---------
936 | 11.1 | +0.48
1023* | 10.2 | .78
2732 | 12.3 | .11
2781 | 12.3 | .11
2787 | 11.4 | .36
2859 | 11.1 | .28
2950 | 11.6 | .15
3384* | 10.7 | .48
3412* | 11.2 | + .40
3418 | 13.1 | .0
3458 | 12.8 | – .22
3945 | 11.5 | + .20
4026 | 11.1 | .48
4203 | 11.1 | .36
4346 | 12.0 | .20
4371 | 12.0 | .18
4421 | 12.8 | .17
4442 | 10.9 | .50
4477 | 10.9 | .40
4596 | 12.0 | .25
4643 | 11.1 | .26
4754 | 10.9 | .48
5473 | 12.0 | + .08
5574 | 13.0 | – .05
5689 | 12.0 | + .30
5701 | 12.3 | +0.17
+-----------+---------
Mean | 11.66 | +0.267
----------+-----------+---------

SBb (16)
----------+-----------+---------
1022 | 11.8 | +0.04
2650 | 12.8 | .0
3351* | 11.4 | + .48
3400 | 12.5 | – .10
3414 | 11.5 | + .26
3504 | 11.4 | .30
3718 | 11.8 | +0.48
4102 | 12.0 | +0.36
4245 | 11.1 | .15
4394 | 11.5 | .60
4548 | 11.1 | .60
4699* | 10.0 | .57
4725* | 9.2 | .70
5218 | 12.8 | .25
5566 | 11.1 | .20
7723 | 11.8 | +0.18
+-----------+---------
Mean | 11.48 | +0.317
----------+-----------+---------

SBc (15)
----------+-----------+---------
613 | 10.6 | +0.60
779 | 12.1 | .48
3206 | 13.3 | .45
3344 | 11.4 | .60
3346 | 12.3 | .40
3625 | 13.3 | .0
3686 | 12.0 | .30
3769 | 12.8 | .43
3953 | 11.1 | .74
3992 | 11.5 | .85
4303* | 10.6 | .78
4579* | 9.7 | .45
5383 | 12.6 | .40
5921 | 12.8 | .70
7479 | 12.1 | +0.48
+-----------+---------
Mean | 11.87 | +0.509
----------+-----------+---------

Peculiar (2)
----------+-----------+---------
2782 | 12.3 | +0.26
4314 | 11.1 | +0.34
----------+-----------+---------

TABLE III
Normal Spirals
==========+===========+=========
N.G.C. | m_{T} | log d
----------+-----------+---------

Sa (49)
----------+-----------+---------
488 | 11.8 | +0.48
676 | 13.3 | .30
1332 | 10.9 | .43
2655 | 11.1 | .60
2681 | 10.7 | .48
2775 | 10.9 | .32
2811 | 12.3 | .28
2855 | 12.8 | .11
3169§ | 12.3 | .60
3245 | 11.8 | .30
3301 | 12.4 | .15
3368* | 10.0 | .85
3516 | 12.1 | .20
3619 | 12.3 | .0
3626* | 11.3 | .28
3665 | 12.0 | .0
3682 | 12.1 | .08
3898 | 12.0 | .43
3941 | 10.3 | .30
4036 | 10.9 | .60
4138 | 12.1 | .20
4143 | 11.3 | .11
4150 | 12.0 | .11
4251 | 10.4 | .26
4268 | 12.8 | .0
4274 | 11.1 | +0.54
4281 | 11.5 | +0.18
4429 | 11.5 | .48
4452 | 12.6 | .15
4526 | 11.1 | .70
4550 | 12.1 | .43
4570 | 11.1 | .38
4594 | 9.1 | .85
4665 | 11.8 | + .08
4684 | 12.2 | – .22
4698 | 11.9 | + .43
4710 | 11.8 | .54
4762 | 11.5 | .57
4866 | 12.0 | .50
4958 | 11.4 | .60
5377 | 11.8 | .48
5389 | 12.5 | .25
5422 | 12.1 | + .40
5631 | 12.0 | – .05
5866* | 11.7 | + .48
7013 | 12.8 | .08
7457 | 12.8 | .30
7727 | 11.3 | .43
7814* | 11.4 | +0.48
+-----------+---------
Mean | 11.69 | +0.333
----------+-----------+---------

Sb (70)
----------+-----------+---------
224 | 5.0 | +2.25
672 | 12.8 | 0.54
772 | 11.1 | .70
949 | 13.3 | .0
955 | 12.9 | .40
1068 | 9.1 | .40
1309 | 12.0 | .15
2639 | 12.2 | .0
2715 | 12.5 | .40
2748 | 12.0 | .32
2841* | 9.4 | .78
2985 | 11.4 | 0.48
3031* | 8.3 | +1.20
3182 | 12.9 | –0.22
3190 | 11.9 | + .48
3227 | 12.0 | .48
3277 | 12.6 | .0
3310 | 10.4 | + .18
3380 | 12.1 | – .05
3489* | 11.2 | +0.40
3556 | 11.1 | +0.90
3593 | 11.9 | .60
3623* | 9.9 | .90
3627* | 9.1 | 0.90
3628§ | 11.4 | +1.08
3632 | 13.3 | –0.10
3675 | 11.4 | + .48
3681 | 13.0 | .0
3684 | 13.0 | + .08
3895 | 13.3 | – .05
3900 | 12.1 | + .25
3938 | 12.1 | .65
4020 | 12.3 | .17
4030 | 11.1 | .30
4051* | 11.9 | .60
4085 | 12.5 | .36
4151 | 12.0 | .40
4192 | 10.9 | .90
4216* | 10.8 | 0.85
4244§ | 12.3 | +1.11
4258* | 8.7 | +1.30
4273 | 11.8 | 0.20
4438* | 10.3 | .54
4448 | 11.8 | .48
4450 | 10.6 | + .57
4451 | 12.8 | – .15
4500 | 12.8 | +0.17
4565*§ | 11.0 | 1.17
4736* | 8.4 | 0.70
4750 | 11.8 | .26
4800 | 11.8 | .04
4814 | 12.7 | .56
4826 | 9.0 | .90
5055* | 9.6 | .90
5376 | 12.8 | + .17
5379 | 12.9 | –0.05
5394 | 13.3 | +0.17
5633 | 13.0 | – .10
5713 | 12.3 | + .32
5740 | 12.3 | .48
5746 | 10.4 | .87
5750 | 12.8 | .15
5772 | 12.0 | .25
5806 | 12.3 | .30
5985 | 12.0 | .60
6207 | 11.8 | .30
6643 | 11.9 | .48
7331* | 10.4 | .95
7541 | 12.7 | .41
7606 | 12.0 | +0.78
+-----------+---------
Mean | 11.55 | +0.471
----------+-----------+---------

Sc (115)
----------+-----------+---------
157 | 11.4 | +0.40
253 | 9.3 | 1.34
278 | 12.0 | 0.08
470 | 13.1 | 0.20
598 | 7.0 | 1.78
615 | 12.3 | 0.43
628* | 10.6 | .90
908 | 11.9 | .60
1084 | 11.4 | .34
1087 | 12.1 | .36
1637 | 12.6 | .48
2339 | 13.1 | 0.28
2403* | 8.7 | 1.20
2532 | 13.3 | 0.17
2683 | 9.9 | 1.00
2712 | 12.3 | 0.20
2742 | 11.8 | .40
2776 | 12.3 | 0.34
2903* | 9.1 | 1.04
2964 | 11.6 | .40
2976 | 12.0 | .50
3003§ | 13.3 | .78
3021 | 12.3 | .11
3079§ | 12.0 | .90
3147 | 11.4 | .30
3166 | 12.0 | .0
3184 | 12.7 | .78
3198 | 13.0 | .95
3254 | 12.8 | .60
3294 | 12.0 | .48
3389 | 13.1 | +0.30
3395 | 12.6 | +0.11
3396 | 13.3 | – .10
3430 | 12.6 | + .49
3432 | 12.0 | .79
3437 | 12.4 | .28
3445 | 13.1 | .08
3448 | 12.3 | .26
3486 | 11.8 | .58
3488 | 12.8 | .25
3512 | 12.3 | .0
3521* | 10.1 | .65
3549 | 13.3 | .43
3596 | 13.3 | .60
3631 | 11.8 | .66
3642 | 12.0 | .73
3655 | 11.9 | .04
3666 | 11.8 | .54
3672 | 13.0 | .54
3683 | 12.0 | .15
3780 | 13.0 | .40
3810 | 11.3 | .62
3813 | 12.3 | .32
3877 | 11.8 | .64
3887 | 12.3 | .40
3893 | 11.8 | .61
3949 | 11.8 | .34
3982 | 12.1 | .36
4013 | 13.3 | .60
4041 | 11.4 | .30
4062 | 12.6 | .48
4088 | 11.5 | +0.72
4096 | 12.3 | +0.78
4100 | 12.3 | .60
4145 | 12.3 | .70
4157§ | 12.3 | .77
4212 | 12.3 | .30
4220 | 12.1 | 0.40
4236 | 12.8 | 1.04
4254 | 10.4 | 0.65
4321* | 10.5 | .70
4414 | 10.1 | .48
4419 | 11.8 | .36
4460 | 12.1 | .20
4490* | 10.2 | .60
4501* | 10.5 | .70
4504 | 12.1 | 0.48
4517§ | 12.5 | 1.00
4536§ | 12.3 | 0.85
4559 | 10.7 | .90
4569* | 10.9 | .65
4580 | 12.3 | .15
4605 | 9.9 | 0.48
4631* | 9.5 | 1.08
4632 | 13.1 | 0.50
4666 | 12.0 | .60
4713 | 12.3 | .38
4781 | 11.8 | .48
4793 | 12.4 | .20
4808 | 12.6 | +0.34
4995 | 11.8 | +0.36
5005* | 11.1 | .70
5012 | 11.9 | .43
5033* | 11.8 | 0.78
5194* | 7.4 | 1.08
5204 | 12.8 | 0.59
5236 | 10.4 | 1.00
5247 | 13.3 | 0.70
5248 | 11.5 | .50
5290 | 12.5 | .48
5297 | 12.6 | .60
5364 | 13.3 | .60
5395 | 12.8 | 0.30
5457 | 9.9 | 1.34
5474 | 12.0 | 0.60
5585 | 12.3 | .60
5676 | 11.8 | .48
5678 | 11.8 | .41
5832 | 13.1 | 0.56
5907§ | 11.9 | 1.04
6181 | 12.5 | 0.30
6217 | 12.1 | .25
6503 | 9.9 | .70
7448 | 11.8 | + .30
7671 | 13.3 | – .15
+-----------+---------
Mean | 11.75 | +0.537
----------+-----------+---------

Peculiar Spirals (Unclassified)
----------+-----------+---------
972 | 13.3 | +0.17
2537 | 13.3 | .0
4900 | 11.8 | +0.23
----------+-----------+---------

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

$m_{T} = C - K \log d,$ (1)

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
==========+===========+=========
N.G.C. | m_{T} | log d
----------+-----------+---------
2968 | 12.6 | +0.08
3034* | 9.0 | .85
3077 | 11.4 | .48
3729 | 11.8 | .17
4214* | 11.3 | .90
4449* | 9.5 | .65
4618 | 12.3 | +0.40
4656§ | 11.5 | +1.30
4753 | 11.4 | +0.43
5144 | 12.8 | – .30
5363 | 11.1 | +0.20
----------+-----------+---------
Mean | 11.34 | +0.469
----------+-----------+---------

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

$\Delta \log d = 0.2 (13.0 - C)$

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.

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.

The correlation of the data is very closely represented by the formula

$m_{T} = 13.0 - 5 \log d.$ (2)

This falls between the two regression curves derived from least-square solutions and could be obtained exactly by assigning appropriate weights to the two methods of grouping. The nature of the data is such that a closer agreement can scarcely be expected. No correction to the assumed value of the slope appears to be required. The material extends over a range of 12 mag., and the few cases which have been investigated indicate that the correlation can be extended another 3 mag., to the limit at which nebulae can be classified with certainty on photographs made with the 100-inch reflector. The relation may therefore be considered to hold throughout the entire range of observations.

The residuals without regard to sign average 0.87 mag., and there appears to be no systematic effect due either to type or luminosity. The scatter, however, is much greater for the spirals, especially in the later types, than for the elliptical nebulae. The limiting cases are explained by peculiar structural features. The nebulae which fall well above the line usually have bright stellar nuclei, and those which fall lowest are spirals seen edge-on in which belts of absorption are conspicuous.

TABLE VII
========+============+=========+=========+=====
Type | ¯m¯_{T}$ | ¯log d¯ | C* | d†
--------+------------+---------+---------+-----
E0 | 11.40 | –0.204 | 10.38 | 1.2
1 | 11.43 | .177 | 10.54 | 1.3
2 | 11.52 | .088 | 11.08 | 1.6
3 | 11.99 | .133 | 11.33 | 1.8
4 | 11.95 | – .011 | 11.90 | 2.4
5 | 10.97 | + .090 | 11.42 | 1.9
6 | 10.93 | .220 | 12.03 | 2.5
7 | 11.02 | .360 | 12.82 | 3.7
| | | |
Sa | 11.69 | .333 | 13.35 | 4.7
b | 11.55 | .471 | 13.90 | 6.0
c | 11.74 | .540 | 14.44 | 7.7
| | | |
SBa | 11.66 | .267 | 13.00 | 4.0
b | 11.48 | .317 | 13.16 | 4.3
c | 11.87 | .509 | 14.41 | 7.6
| | | |
Irr | 11.34 | +0.469 | 13.68 | 5.4
------+- ----------+---------+---------+-----

* C = $\overline{m}_{T} + 5 \overline{\log d}.$
† log d = 0.2 (C — m_{T}); m_{T} = 10.0.

EFFECTS OF ORIENTATION

The effect of the orientation is appreciable among the spirals in general. In order to illustrate this feature, they have been divided into three groups consisting of those whose images are round or nearly round, elliptical, and edge-on, or nearly so. The mean values of m_{T} + 5 log d were then computed and compared with the theoretical value, 13.0. The residuals are negative when the nebulae are too bright for their diameters and positive when they are too faint. The results are given in Table IX, where mean residuals are followed by the numbers of nebulae, in parentheses, which are represented by the means.

The numbers of the barred spirals are too limited to inspire confidence in the results, but among the normal spirals there is conclusive evidence that the highly tilted and edge-on nebulae are fainter for a given diameter than those seen in the round. A study of the individual images indicates that the effect is due very largely to dark absorption clouds, which become more conspicuous when the nebulae are highly tilted. These clouds are generally, but not universally, peripheral features. An extensive investigation will be necessary before any residual effect due to absorption by luminous nebulosity can be established with certainty. Even should such exist, it clearly cannot be excessive.

TABLE VIII
=======+=========+==========
Type | C | Δ log d
-------+---------+----------
E0 | 10.30 | +0.54
1 | 10.65 | .47
2 | 11.00 | .40
3 | 11.35 | .33
4 | 11.70 | .26
5 | 12.05 | .19
6 | 12.40 | .12
7 | 12.75 | +0.05
| |
Sa | 13.31 | –0.06
Sb | 13.90 | .18
Sc | 14.45 | .29
| |
SBa | 13.00 | .00
SBb | 13.16 | .03
SBc | 14.41 | .28
| |
Irr | 13.68 | –0.14
------+---------+----------

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 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}.

From the equation of the tangent, PP′,

$y = -x \tan{i} + \sqrt{a^2 \tan^2{i} + b^2}$

$y_{0} = \sqrt{a^2 \tan^2{i} + b^2}$

Since

$b_{1} = y_{0} cos(i)$

$b_{1}^2 = a^2 \sin^2{i} + b^2 \cos^2{i}.$

Let a = 1, then

$\cos^2{i} = \frac{1 - b_{1}^2}{1 - b^2},$

where

$b_{1} = 1 - e_{1},\;b = i - e.$

From these equations, the values of i can be determined for all possible values of e_{1}. 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 e_{1}. 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
--------+---------+--------+--------+--------+------
E7 | 0.055 | 0.111 | 0.114 | 0.116 | 0.121
E6 | .059 | .123 | .126 | .133 | .148
E5 | .067 | .140 | .148 | .166 | .216
E4 | .079 | .169 | .190 | .250 | 0.312
E3 | .100 | .225 | .299 | 0.376 |
E2 | .145 | .378 | 0.477 | |
E1 | 0.300 | 0.700 | | |
E0 | 1.000 | | | |
+---------+--------+--------+--------+------
Total | 1.805 | 1.846 | 1.354 | 1.041 | 0.797
| 0.226 | 0.231 | 0.169 | 0.130 | 0.100
--------+---------+--------+--------+--------+------

========+=====================================
Actual | Apparent
--------+---------+--------+--------+---------
| E5 | E6 | E7 | Total
--------+---------+--------+--------+---------
E7 | 0.132 | 0.164 | 0.187 | 1.000
E6 | .187 | 0.224 | |
E5 | 0.263 | | |
+---------+--------+--------+---------
Total | 0.582 | 0.388 | 0.187 | 8.000
| 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 C_{7} – C_{e} are calculated from the numbers of nebulae in the various columns of Table XI together with the values of C_{7} – C_{e} 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
============+=========================================
| Apparent
Actual +--------+--------+--------+--------+-----
| E0 | E1 | E2 | E3 | E4
------------+--------+--------+--------+--------+-----
E7 | 1.2 | 2.4 | 2.5 | 2.5 | 2.6
E6 | .6 | 1.3 | 1.4 | 1.5 | 1.6
E5 | .8 | 1.7 | 1.8 | 2.0 | 2.7
E4 | .8 | 1.7 | 1.9 | 2.5 | 3.1
E3 | 0.9 | 2.1 | 2.8 | 3.5 |
E2 | 1.1 | 2.9 | 3.6 | |
E1 | 1.7 | 3.9 | | |
E0 | 9.9 | | | |
------------+--------+--------+--------+--------+-----
Total* | 17.0 | 16.0 | 14.0 | 12.0 | 10.0
------------+--------+--------+--------+--------+-----

=========+=========================================
| Apparent
Actual +--------+--------+--------+-------
| E5 | E6 | E7 | Total
---------+--------+--------+--------+-------
E7 | 2.9 | 3.6 | 4.0 | 21.7
E6 | 2.0 | 2.4 | | 10.8
E5 | 3.1 | | | 12.1
E4 | | | | 10.0
E3 | | | | 9.3
E2 | | | | 7.6
E1 | | | | 5.6
E0 | | | | 9.9
---------+--------+--------+--------+-------
Total* | 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
============+===============+===========+======================
| | | Random Orientation
| | +-----------+----------
Class | Pure Classes | Observed | No. as |
| | | Observed | Equal No.
------------+---------------+------------+-----------+----------
C_{7}–C_{7} | 0.00 | 0.00 | 0.00 | 0.00
C_{6} | 0.63 | 0.35* | 0.25 | 0.35
C_{5} | 1.10 | 0.70* | 0.58 | 0.70
C_{4} | 1.51 | 0.85 | 1.11 | 1.01
C_{3} | 1.84 | 1.42 | 0.87 | 1.28
C_{2} | 2.13 | 1.67 | 1.33 | 1.55
C_{1} | 2.39 | 2.01† | 1.54 | 1.83
C_{0} | 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 C_{7} The latter is observed to be 12.75, hence

$m_{T} + 5 log b = 10.13.$ (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.

RELATION BETWEEN NUCLEAR LUMINOSITIES AND DIAMETERS

Visual magnitudes have been determined by Hopmann for the nuclei of 37 of the nebulae included in the present discussion. These data, together with types and diameters of the nebulae, are listed in Table XIII. When the magnitudes are plotted directly against the logarithms of the diameters, they show little or no correlation. When, however, the nebulae are reduced to the standard type (by applying corrections for differences in diameter along the sequence), a decided correlation is found whose coefficient is 0.76. This is shown in Figure 9. The simple mean of the two regression curves is

$m_{n} = 14.45 - 4.94 \log d,$ (6)

where the slope differs by about 1 per cent from that in formula (2). The list contains 16 elliptical nebulae, 15 normal, and 6 barred spirals. The nebulae are fairly representative, except that few late-type spirals are included. This is an effect of selection due to the fact that nuclei become less and less conspicuous as the sequence progresses.

The same result can be derived from a study of the differences, m_{n} – m_{T}, for the individual nebulae. The mean value is 1.55 ± 0.08, and the average residual is 0.60 mag. Means for the separate types are to be found in Table XIV.

TABLE XIII
Diameters and Nuclear Magnitudes
=======+============+=========+================+===============
N.G.C. | Type | log d | m_{n} Hopmann | m_{n} Reduced
-------+------------+---------+----------------+---------------
221 | E2 | +0.42 | 9.84 | 11.85
1023 | SBa | .78 | 11.86 | 11.86
2841 | Sb | 0.78 | 12.08 | 11.19
3031 | Sb | 1.20 | 10.94 | 10.05
3115 | E7 | 0.60 | 10.83 | 11.09
3351 | SBb | .48 | 12.31 | 12.15
3368 | Sa | .85 | 11.68 | 11.43
3379 | E0 | .30 | 11.55 | 14.27
3412 | SBa | .40 | 11.59 | 11.59
3489 | Sb | .40 | 11.54 | 10.65
3626 | Sa | .28 | 12.37 | 12.12
3627 | Sb | .90 | 12.03 | 11.14
4125 | E4 | .30 | 11.74 | 13.04
4216 | Sb | .85 | 11.65 | 10.76
4278 | E1 | .0 | 12.02 | 14.38
4374 | E1 | .08 | 11.43 | 13.79
4382 | E4 | .48 | 11.77 | 13.07
4435 | E6 | .11 | 11.65 | 12.26
4438 | Sb | .54 | 11.83 | 10.94
4486 | E0 | .30 | 11.23 | 13.95
4546 | E6 | .18 | 11.75 | 12.36
4552 | E0 | .23 | 11.59 | 14.31
4569 | Sc | .65 | 12.05 | 10.57
4579 | SBc | .45 | 11.48 | 10.07
4621 | E5 | .30 | 11.60 | 12.56
4636 | E1 | .08 | 11.97 | 14.33
4649 | E2 | .30 | 11.57 | 13.58
4697 | E6 | .48 | 10.90 | 11.51
4699 | SBb | .57 | 10.72 | 10.56
4725 | SBb | .70 | 11.97 | 11.81
4736 | Sb | .70 | 10.36 | 9.47
5005 | Sc | .70 | 12.04 | 10.56
5033 | Sc | 0.78 | 12.38 | 10.90
5194 | Sc | 1.08 | 11.38 | 9.90
5322 | E3 | 0.15 | 12.10 | 13.76
5866 | Sa | .48 | 11.76 | 11.51
7331 | Sb | +0.95 | 11.82 | 10.93
Means | | +0.509 | 11.60 | 11.90
-------+------------+---------+----------------+---------------

The low value for Sa-SBa is due to N.G.C. 5866, for which the magnitude difference of 0.06 is certainly in error, and the high value for Sc and SBc, to M 51, for which the difference of 3.98 mag. is not representative. The latter is accounted for in part by the fact that the m_{T} refers to the combined magnitude of the main spiral and the outlying mass, N.G.C. 5195. When these two cases are discarded, the final mean becomes 1.52 ± 0.05, and the average residual, 0.52 mag., is consistent with the probable errors of the magnitude determinations. The small numbers of objects within each class are insufficient for reliable conclusions concerning slight variations along the sequence. From the constancy of m_{n} – m_{T}, the relation expressed by formula (6) necessarily follows, the small difference in the constant being accounted for by the different methods of handling the data.

The parallelism of the two curves representing formulae (2) and (6) indicates that the regular extra-galactic nebulae, when reduced to the standard type, are similar objects. The mean surface brightness is constant, and the luminosity of the nucleus, as measured by Hopmann, is a constant fraction, about one-fourth, of the total luminosity of the nebulae. If there is a considerable range in absolute magnitude and hence in actual dimensions, the smaller nebulae must be faithful miniatures of the larger ones.

ABSOLUTE MAGNITUDES OF EXTRA-GALACTIC NEBULAE

Reliable values of distances, and hence of absolute magnitudes, are restricted to a very few of the brightest nebulae. These are derived from a study of individual stars involved in the nebulae, among which certain types have been identified whose absolute magnitudes in the galactic system are well known. The method assumes that the stars involved in the nebulae are directly comparable with the stars in our own system, and this is supported by the consistency of the results derived from the several different types which have been identified.

TABLE XIV
===============+==============+=======================================
Type | m_{n} – m_{T}| Number
---------------+--------------+---------------------------------------
E0–E3 | 1.64 | (9)
E4–E7 | 1.43 | (7)
Sa–SBa | 0.97 | (5) 1.27 when N.G.C. 5866 is omitted
Sb–SBb | 1.70 | (11)
Sc–SBc | 1.76 | (5) 1.19 when N.G.C. 5194 is omitted
+--------------+---------------------------------------
Unweighted mean| 1.50 | (37) 1.45 (35)
Weighted mean | 1.55 | (37) 1.52 (35)
---------------+--------------+---------------------------------------

TABLE XV
Absolute Magnitudes of Nebulae
=============+===========+======================
System | M_{T} | M_{S}
-------------+-----------+----------------------
Galaxy | | –5.5
M 31 | –17.1 | 6.5
LMC | 17.0 | 8.0
SMC | 16.0 | 5.5
M 33 | 15.1 | 6.5
N.G.C. 6822 | 13.7 | 5.8
M 101 | 13.5 | –6.3
M 32 | –13.3 |
+-----------+----------------------
| | –6.3
| | –9.0 = M_{S} – M_{T}
+-----------+----------------------
Means | –15.1 | –15.3
Adopted | –15.2 |
-------------+-----------+----------------------

In Table XV are listed absolute magnitudes of the entire system and of the brightest stars involved, for the galaxy and the seven nebulae whose distances are known. The data for the Magellanic Clouds are taken from Shapley’s investigations. The absolute magnitudes of the remaining nebulae were derived from Holetschek’s apparent magnitudes and the distances as determined at Mount Wilson, where the stellar magnitudes were also determined. M 32 is generally assumed to be associated with the great spiral M 31, because the radial velocities are nearly equal and are unique in that they are the only large negative velocities that have been found among the extra-galactic nebulae. M 101 has been added to the list on rather weak evidence. The brightest stars involved are slightly brighter than apparent magnitude 17.0, and several variables have been found with magnitudes at maxima fainter than 19.0. Sufficient observations have not yet been accumulated to determine the light-curves of the variables, but from analogy with the other nebulae they are presumed to be Cepheids. On this assumption, both the star counts and the variables lead to a distance of the order of 1.7 times the distance of M 33. The inclusion of M 101 does not change the mean magnitude of the brightest stars involved, but reduces the mean magnitude of the nebulae by 0.2.