Continuous Mass Distribution: Clairaut’s Theory

Abstract

In Chapter II of this book we formulated heuristically the closed form of the Roche potential, the gravitational field of which is governed by that of a dipole of two point-masses m₁ and m₂, revolving around the common centre of gravity with an angular velocity ω. The fact that the entire mass of each component was assumed to be confined to a point made it unnecessary for us to concern ourselves about the internal structure of each component—beyond a tacit assumption (underlying the Roche model) that the “mean-free-path” of the infinitesimal mass particle moving in the gravitational field of our rotating dipole is infinite. The aim of the present chapter will, however, be to remove this latter assumption, and to generalize our problem by allowing the mass to be distributed continuously throughout the interior of the respective configuration: i.e., that its internal structure is characterized by a continuous distribution of density ρ, not exhibiting a pole at the centre. Moreover, a restriction to the mean-free-path of infinitesimal mass particles to become finite (even though the latter may be long in comparison with the distance separating the finite masses m₁ and m₂) will give rise to a finite pressure P. In such a case the equations of particle mechanics at the basis of our discussion of Section II-3B should be replaced by Eulerian equations of hydrodynamics, which will constitute the physical generalization of the processes underlying Chapters II and III.