On the integral equation governing the distribution of the true and the apparent rotational velocities of stars

Abstract

In this paper the integral equation governing the distribution of the true (ν) and the apparent (νsin i) rotational velocities of stars is reconsidered, with the object of suggesting the most suitable methods for analyzing observed frequency functions affected by a random orientation factor, sin i, in the manner of the rotational velocities of stars. First, it is shown that there is a simple relation the moments of the true and the observed frequency functions which enables us to pass from the moments of the one to the moments of the other. The mean and the mean square of the true rotational velocities of stars can therefore be determined directly from the corresponding means of the observed distribution of ν Sin i. When it is felt that something more should be said about the true distribution of ν than its mean and mean square, it is suggested that a comparison be made between the observed distribution and those derived from certain assumed forms of the true frequency function. Reasons are given for preferring this method to an inversion of the integral equation by a numerical procedure. The form f(x)=1/√π{e^-(x-x₁2)+e^-(x+x₁2)} suggested by Elsa van Dien has therefore been considered in some detail, and a one-parameter family of frequency functions for x sin i has been derived. The methods of analysis suggested in this paper have been applied to a rediscussion of the rotational velocities of stars.