On the Derivation of the Frequently Function of Space Velocities of the Stars from the Observed Radial Velocities

One of the important problems of stellar statistics is the derivation of the frequency function of the space velocities of stars of various spectral types and of different absolute magnitudes. The direct solution of this problem requires the knowledge of the space velocities of a great number of stars. The derivation of the space velocity of a given star is possible only if three different quantities are measured: the radial velocity, the proper motion and the parallax. These quantities are measurable with different relative degrees of accuracy and are exposed to systematic errors of quite different kinds. For some important groups of stars (for example B–type stars) we have very few reliable individual parallaxes. The number of stars with reliable parallaxes is generally small, and among them the radial velocities are known only for a fraction. Therefore several writers have made the attempted to obtain some knowledge about the distribution law of space velocities from the radial velocities alone. However, in every case some more or less arbitrary form of this law was assumed, and the problem was restricted to finding the numerical values of some parameters entering in this form of distribution law. In the majority of cases these constants are the elements of the velocity ellipsoids. Owing to the relative uniformity of the catalogues of the radial velocities, the results of the statistical investigations based on them are almost free from the influence of systematic errors. It seems desirable, then, to try to solve the problem of derivation of the frequency function of space velocities from the distribution of radial velocities without making any hypothesis about the form of this function. So far as it is known to the writer, this problem not only remains yet unsolved, but is not even discussed in any detail. The purpose of the present paper is to derive a general formula for the frequency function of space velocities from the distribution of radial velocities. For the frequency function of the space velocities we will derive and solve an integral equation. In this equation the observed frequency function of the radial velocities for the different parts of the sky enters as the known function.