Random Gravitational Forces in a Star Field

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Abstract
The statistical properties of the gravitational field due to a random distribution of stars in space can be expressed in terms of the probability-density of the star field. In this paper we establish that the mean value of the gravitational force at any point is equal to the force produced at that point by the gravitational attraction of a gas whose actual density is equal to the probability-density of the stars. The dispersion of the gravitational force is finite if the point at which it is measured lies within an empty region. It is then proportional to the square of the mass density and inversely proportional to the number of stars per unit volume. We also consider the joint probability distribution of force components at two different points. It is found that, for separations of greater order than a stellar diameter, the correlation coefficient varies inversely as the cube of the separation, and we conclude that the force field fluctuates rapidly as we pass from one point to another.

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