A Theory of Central Measures for Celestial Mechanics

Abstract

In this paper we create a theory of central measures for celestial mechanics. This theory generalizes central configurations to include continuum mass distributions. Roughly speaking, if $V$ is the Newtonian potential generated by a unit point mass, a central measure $\mu$ for $V$ is a mass distribution in space characterized by the property that gravitational acceleration vector of any $x$ in the support of $\mu$ is a constant multiple of the vector from the mass center of $\mu$ to $x$. This formulation includes central configurations as special cases for which central measures are discrete. We show that concentric spherical shells can be properly arranged so that their mass distributions are central measures for $V$. For any pair of adjacent shells, the ratio of outer and inner radii is between $(\sqrt[3]{2},\infty)$ and this bound is sharp. This provides a bound for outer and inner radii of concentric spheres if the system explodes or collapses homothetically. We also extend our definition of central m...