The Determination of Absolute-Magnitude Dispersion with Application to Giant M Stars.

The problem of finding the distributione according to absolute magnitudes of a group of stars is one of considerable importance. .Except for the.case in which all the stars in the group are at approximately the same distance, the problem can usually be solved only by making certain assumptions as to the nature of the distribution function, either of the absolute magnitudes, or of the accidental and systematic errors in the measured absolute magnitudes. If we may assume that the distribution of the absolute magnitudes in the group is that corresponding to a normal error distribution, or that the errors in the measured absolute magnitudes have this distribution, the problem is definite and can be solved by comparing the distribution of angular velocities with that of the radial velocities. A solution for the case of the second alternative is published in Mount Wilson Contributions, No. 327,1 in connection with the determination of the errors in the spectroscopic absolute magnitudes of giant M stars; but since the modification of the formulas required for the case of the first alternative was only parenthetically referred to, the direct solution of the problem will be given here. It will also be shown that Gyllenberg's solution of the problem can be derived in the same way. We assume that the absolute magnitudes of a group of stars are distributed according to a normal error curve around a mean valueM and with a dispersion q. The individual true absolute magnitudes are denoted by M. The frequency function of the absolute magnitudes is then