### 6. THE COMA CLUSTER OF NEBULAE

According to Hubble and Humason, (10) the Coma cluster of nebulae "consists of about 800 nebulae scattered over an area roughly 1°.7 in diameter .... Photographic magnitudes range from about 14.1 to 19.5, with 17.0 as the most frequent value." The distance of the cluster is about 13.8 million parsecs.

In order to get some preliminary data on the distribution of nebulae in the Coma cluster, photographs of this cluster were taken on Mount Palomar with the new 18-inch Schmidt-type telescope of the California Institute of Technology. The faintest nebulae which on limiting exposures (30-60 min) with this telescope can still be clearly distinguished from stars have an apparent magnitude close to 16.5. In Figure 3 dots represent nebulae which can be distinguished on 30-minute exposures on panchromatic films. To avoid crowding of points near the center of the cluster, not all the nebulae in this region are marked which can be seen on the original photographs. The counts given in Table 1 at different distances r from the center of the Coma cluster include, however, all the nebulae which I have been able to identify on half-hour exposures.

The nebula NGC 4874 ( 12h 56m, 28° 20', 1930), which lies at (0°, 0°) in Figure 3, was taken to be the approximate center of the cluster, no effort being made to determine a mathematically accurate central point. Concentric circles were then drawn around the adopted center, with radii r = nr0, where n is a whole number running from n = 1 to n = 32, and r0 5 minutes of arc. The unit of area to which all counts are reduced is s = r20 or about 1/46.4 sq. deg. The numbers of nebulae per unit area, nr, are averages for the ringlike areas which lie between r = nr0 and r = (n + 1)r0. The first four figures in Table 1, however, are averages for the full circles the radii of which are r = r0/5, r0/3, r0/2, and r0, respectively. The corresponding numbers nr of nebulae per square degree are nr = 46.4nr.

 r/r0 nr log10nr 1/5 50 1.699 1/3 63 1.799 1/2 52 1.716 1 31 1.491 2 14 1.146 3 10.8 1.033 4 6.33 0.801 5 4.22 0.625 6 4.10 0.613 7 1.85 0.267 8 2.73 0.436 9 1.76 0.245 10 2.16 0.334 11 1.76 0.245 12 1.61 0.207 13 1.20 .079 14 1.44 .158 15 0.66 -.181 16 0.48 -.319 17 0.60 -.222 18 0.48 -.319 19 0.43 -.367 20 0.64 -.194 22 0.26 -.585 24 0.36 -.444 26 0.27 -.569 28 0.16 -.796 30 0.25 -.602 32 0.15 -0.824

In Figure 4 values of log10 nr are plotted against values of r. The full curve is drawn only approximately and does not correspond to any definite mathematical function. From the general character of this curve it is seen immediately that the Coma cluster extends to much greater distances than was originally assumed by Hubble and Humason. 10 At the edge of a circle the diameter of which is 4°.5 instead of only 1°.7, the average number of nebulae per unit area is still higher than the corresponding number in the surrounding general field. Since our counts include only the brighter nebulae, it is to be expected that counts made with more powerful telescopes will enable us to follow the extensions of the Coma cluster still farther into the general field.

 Figure 4. Counts of nebulae in the Coma cluster.

The high central condensation, the very gradual decrease of the number of nebulae per unit volume at great distances from the center of a cluster, and the great extension of this cluster become here apparent for the first time. It is quite as we should expect from the considerations of section v. According to these considerations, a cluster of nebulae analogous to an isothermal gravitational gas sphere may in some cases be expected to extend indefinitely far into space, until its extension is stopped through the formation of independent clusters in the regions surrounding it.

The actual shape of the distribution curve in Figure 4 is also of great interest. We notice at once the great similarity of this curve to the luminosity-curves of elliptical nebulae derived by Hubble. (11) According to him, the distribution of the intrinsic luminosity in globular nebulae corresponds very closely to the distribution of mass density in isothermal gas spheres as computed by R. Emden. 3 The same is approximately true for the distribution of the brighter nebulae in the Coma cluster. This result also checks the general conclusions drawn from the basic principles which, according to the discussion in section v, determine the stationary configurations of clusters of nebulae.

The total number of nebulae in the Coma cluster that can be identified on photographs taken with the 18-inch Schmidt telescope is obtained as follows: The curve in Figure 4 apparently has a horizontal asymptote which corresponds to a value of nr not higher than n = 0.159. Thus, the corresponding number n of nebulae in the general field should be n = 46.4 × n = 7.38. The total number of nebulae counted to the distance r = 2°40' from the center is 834. From this we must subtract 22.3 × 7.38 = 165 nebulae, since the area in question covers 22.3 sq. deg. The Coma cluster therefore comprises a number of nebulae the brightnesses of which are greater than m = 16.5:

 (51)

Since the most frequent apparent magnitude is about 17, we conclude that 16.5 is less than half the total number of nebulae incorporated in the Coma cluster. This number must be at least equal to = 1500, and it may be even greater.

Finally, a word with respect to the limiting magnitude m = 16.5 which we have used in the preceding discussion. According to Hubble, 2 the number nm of nebulae per square degree which are brighter than m near the galactic pole is given by

 (52)

Since the Coma cluster lies within a few degrees of the galactic pole, we may use relation (52) directly without introducing any correction for partial obscuration. Our counts of nebulae lead to an average number of 7.38 nebulae in the general field surrounding the Coma cluster. Inserting nm = 7.38 into the equation (52), we therefore find that m = 16.6 represents approximately the limiting magnitude at which it is still possible to distinguish images of average nebulae from those of stars on photographs taken with the 18-inch Schmidt telescope.

10 E. Hubble and M. L. Humason, Ap. J., 74, 131, 1931; see also the descriptions of the Coma cluster by M. Wolf, Heidelberg Pub., 1, 125, 1902, and H. Shapley, Harvard Bull., No. 896, 1934. Back.

11 E. Hubble, Ap. J., 71, 131, 1930. Back.