CHAPTER XVIII

THE PROGRESSION IN THE NUMBER OF STARS AS THE BRIGHTNESS DIMINISHES

WE

Hither, as to their fountain, other stars

Repairing, in their golden urns draw light.-MILTON.

E mentioned in an earlier chapter that, when we compare the number of stars of each successive order of magnitude with the number of the order next lower, we find it to be, in a general way, between three and four times as great. The ratio in question is so important that a special name must be devised for it. For want of a better term, we shall call it the star-ratio. It may easily be shown that there must be some limit of magnitude at which the ratio falls off. For a remarkable conclusion from the observed ratio for the stars of the lower order of magnitude is that the totality of light received from each successive order goes on increasing. Photometric measures show, as we have seen, that a star of magnitude m gives very nearly 2.5 times as much light as one of magnitude m+1. The number of stars of magnitude m+1 being, approximately from 3 to 3.75 times as great as those of magnitude m, it follows that the total amount of light which they give

us is some 40 or 50 per cent. greater than that received from magnitude m. Using only rough approximations, the amount of light will be about doubled by a change of two units of magnitude; thus the totality of stars of the sixth magnitude gives twice as much light as that of the fourth; that of the eighth twice as much light as that of the sixth; that of the tenth twice as much again as of the eighth, and so on as far as accurate observations and counts have been made.

To give numerical precision to this result, let us take as unity the total amount of light received from the stars of the first magnitude. The sum-total for this and the other magnitudes, up to the tenth, will then be :

That is, from all the stars to the tenth magnitude combined, we have more than seventy times as much. light as from those of the first magnitude.

There must, evidently, be an end to this series, for, were this not the case, the result would be that which we have shown to follow if the universe were

infinite; the whole heaven would shine with a blaze of light like the sun. At what point does the rate of increase begin to fall off?

We are as yet unable to answer this question, because we have nothing like an accurate count of stars above the ninth, or at most, the tenth magnitude. All we can do is to examine the data which we have and see what evidence can be found from them of a diminution of the ratio.

It must be pointed out, at the outset, that the ratio must be greater in the galactic region than it is in other regions. This follows from the fact that the proportion of small stars increases at a more rapid rate in the galaxy than elsewhere. This is shown by the comparisons we have already made of the Herschelian gauges with the counts of the brighter stars. While the galactic region is less than twice as dense as the remaining regions for the brighter stars, it seems to be ten times as dense for the Herschelian stars. If we knew the limiting magnitude of the latter, we could at once draw some numerical conclusion. But unfortunately this is quite unknown. All we know is that they were the smallest stars that Herschel could see with his telescope.

The ratio in various regions of the heavens has been very exhaustively investigated by Seeliger, in the work already quoted. The bases of his investigations are the counts of stars in the Durchmusterung. Instead of taking the ratio for stars differing by units of magnitude, as we have done, Seeliger divides them according to half-magnitudes. The

reproduction of his numbers in detail would take more space than we can here devote to the subject and would not be of special interest to our readers. I have, therefore, derived their general mean results for different parts of the sky with reference to the Milky Way and for stars of the various orders of magnitude. The following table shows the conclusions:

In the first column we have the designation of the zone or region of the sky, as already given.

In the second and third columns we have the mean ratio of increase for whole magnitudes as derived from the Durchmusterung and the Southern Durchmusterung, respectively. It will be recalled that region I., around the north galactic pole, is entirely wanting in the S. D., while the adjoining regions, II. and III., are only partially found, and that, in like manner, the D. M. includes none of region IX. around the south galactic pole, and but little of the adjoining region.

It will be seen that there is a very remarkable systematic difference between the two lists, the ratio of the number of faint to that of bright stars being much greater in the S. D. This difference is shown

in the fourth column. I have assumed that the two systems are equally good, and so diminished all the ratios of the S. D. by 0.25, and increased those of the D. M. by the same amount. The mean of the two corrected results was then taken, giving the principal weight to the one or the other, according to the number of stars on which they depend.

It will be seen that the increase of the ratio from either galactic pole to the Milky Way itself is as well marked as the increase of the richness of the respective regions in stars in general. We may condense the results in this way:

It will be recalled that zone V. is a central belt 20° broad, including the Milky Way in its limits. But the latter, as seen by the eye, especially its brightest portions, does not fill this zone. These portions, as we know, comprise the irregular collection of cloudlike masses described in the last chapter. Seeliger has investigated the ratio within these masses, and compared it with the stellar density, or the number of stars per square degree. The mean results are:

In that portion of the galaxy extending from Cassiopeia to the equator near 6h. of R. A., ratio=4.02.