Mutual gravitational energy of spheroidal bodies

The problem of mutual gravitational energy Wmut for a system of two homogeneous prolated spheroids whose symmetry axes are on the same line is set and solved. The method of equigravitating elements is applied, where the external potentials of three-dimensional spheroids are represented by the potentials of one-dimensional inhomogeneous focal rods. The solution of the problem is reduced to the integration of the potential of one rod over the segment of the second rod. As a result, the expression Wmut for two prolated spheroids can be obtained in a finite analytic form through elementary functions. The force of attraction between the spheroids is found. The function Wmut is also represented by a series of degrees of eccentricity of the spheroids. Possible applications of the obtained results are discussed.