On the possibility of using a Laplace series for the gravitational potential at the surface of a planet. I

Details On the possibility of using a Laplace series for the gravitational potential at the surface of a planet. I On the possibility of using a Laplace series for the gravitational potential at the surface of a planet. I

Dec 1980

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1980azh....57.1323a&link_type=abstract

(Astronomicheskii Zhurnal, vol. 57, Nov.-Dec. 1980, p. 1323-1330.) Soviet Astronomy, vol. 24, Nov.-Dec. 1980, p. 761-765. Trans

Astronomy and Astrophysics

Astronomy

2

Gravimetry, Gravitational Fields, Planetary Gravitation, Planetary Surfaces, Spherical Harmonics, Laplace Transformation, Legendre Functions, Spherical Coordinates

Scientific paper

Consideration is given to the accuracy of a partial series summation approximation of the spherical harmonics representation of a planetary gravitational potential. If the gravitational potential is expanded in a spherical harmonic series as a function of angular variables with a fixed radial coordinate so that the series converges everywhere, it is shown that the error of the expansion up to order n is proportional to the -3/2 power of n. For a segment of an expansion which converges outside an enveloping sphere, it is found that outside the sphere the approximation behaves as the everywhere-convergent series. Inside the sphere, where the series is, in general, divergent, an expression for the optimum order of the expansion is derived in terms of the ratio of the enveloping sphere radius to the maximum allowable immersion below the sphere, and it is shown that at this optimal order, the error is of the magnitude of the -3/2 power of n.

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