Probability distribution of distance in a uniform ellipsoid: Theory and applications to physics

A number of authors have previously found the probability Pn(r) that two points uniformly distributed in an n-dimensional sphere are separated by a distance r. This result greatly facilitates the calculation of self-energies of spherically symmetric matter distributions interacting by means of an arbitrary radially symmetric two-body potential. We present here the analogous results for P2(r;ε) and P3(r;ε) which respectively describe an ellipse and an ellipsoid whose major and minor axes are 2a and 2b. It is shown that for ε=(1−b2/a2)1/2⩽1, P2(r;ε) and P3(r;ε) can be obtained as an expansion in powers of ε, and our results are valid through order ε4. As an application of these results we calculate the Coulomb energy of an ellipsoidal nucleus, and compare our result to an earlier result quoted in the literature.