Читать книгу Foundations of Space Dynamics - Ashish Tewari - Страница 37

A spherical body of constant radius with a radially symmetrical mass distribution has the density , varying only with the radius, . A body consisting of concentric spherical shells, each of which has a different (either constant or radially varying) density, has a radially symmetrical mass distribution. Such a distribution results in the vanishing of all the integrals of and in Eqs. (2.109)–(2.111), which produces , thereby implying a potential which is independent of the co‐latitude and longitude. Hence the gravitational potential of a body with a spherically symmetric mass distribution of radius, , is given by

(2.124)

Clearly, a spherical body with a radially symmetric mass distribution behaves exactly as if all its mass were concentrated at its centre. The acceleration due to gravity is thus given by

(2.125)