On the representation of the gravitational potential of several model bodies

Abstract

Laplace series with respect to spherical functions Yn(θ, λ) represents now a most popular form of the gravitational potential representation for a compact body T in the outer space in spherical coordinates r, θ, λ. There exists a well-known estimate Yn Cn−5/2, C = const, n 1 of the Chebyshevian norm (maximum of the modulus) for bodies of irregular structure. In the present paper an explicit representation of Yn(θ, λ) for several model bodies is obtained. The indicated estimate Yn is valid under the exact exponent 5/2 for all cases except one. If the segment touches the enveloping sphere, then Yn decreases much faster. Namely, Yn Cn−5/2pn, C = const, n 1. The quantity p < 1 equals to the distance from the origin of coordinates to the edge of the segment expressed in radii of the enveloping sphere. The exactness of exponent 5/2 is valid in common case by examples of bodies which more or less reminiscent of real celestial bodies. Refs 16. Figs 6.