The Statistics of the Gravitational Field Arising from a Random Distribution of Stars.IV. The Stochastic Variation of the Force Acting on a Star.

Abstract

In this paper the theory of the stochastic variation of the force acting on a star is considered, and the solution to the formal problem is obtained in terms of the average force per unit mass F t acting at time t, given that a force of intensity Fq acted at time / = 0. Various related quantities are also considered, and in particular an explicit formula for the correlation coefficient R{Fq, t) is derived. 1. Introduction.-A basic problem in statistical stellar dynamics is the characterization of the entire stochastic variation of the gravitational force acting on a star. And, as we have already explained in the introductory section to the preceding paper, 1 this requires the specification of the average force per unit mass F t acting on a star at time given that a force of intensity F 0 .acted at time / = 0. In other words, the essential physical quantity which is needed concerns the correlation in the forces acting on a star at two different instants of time. In this paper we propose to present the formal parts of this theory. In a later paper we shall undertake a fuller discussion of the various formu-lae derived and also outline the applications of the theory developed here. 2. The first moments of F t .-As in III, § 2, the solution to the formal problem soon reduces to one of evaluating the characteristic function C(p, cr), associated with the distribution function W(F 0 , governing the probability that forces of intensities F 0 and F t will act, respectively, at times / = 0 and t = t. Similar to HI, equation (7), we now have