On the representation of the gravitational potential of several model bodies

Abstract

A Laplace series of spherical harmonics Y _n (θ, λ) is the most common representation of the gravitational potential for a compact body T in outer space in spherical coordinates r, θ, λ. The Chebyshev norm estimate (the maximum modulus of the function on the sphere) is known for bodies of an irregular structure:〈Y _n 〉 ≤ Cn ^–5/2, C = const, n ≥ 1. In this paper, an explicit expression of Y _n (θ, λ) for several model bodies is obtained. In all cases (except for one), the estimate 〈Y _n 〉 holds under the exact exponent 5/2. In one case, where the body T touches the sphere that envelops it,〈Y _n 〉 decreases much faster, viz.,〈Y _n 〉 ≤ Cn ^–5/2 p ⁿ, C = const, n ≥ 1. The quantity p < 1 equals the distance from the origin of coordinates to the edge of the surface T expressed in enveloping sphere radii. In the general case, the exactness of the exponent 5/2 is confirmed by examples of bodies that more or less resemble real celestial bodies [16, Fig. 6].