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].
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Original Russian Text © E.D. Kuznetsov, K.V. Kholshevnikov, V.Sh. Shaidulin, 2016, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2016, No. 3, pp. 490–499.
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Kuznetsov, E.D., Kholshevnikov, K.V. & Shaidulin, V.S. On the representation of the gravitational potential of several model bodies. Vestnik St.Petersb. Univ.Math. 49, 290–298 (2016). https://doi.org/10.3103/S1063454116030079
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DOI: https://doi.org/10.3103/S1063454116030079