Some transformation properties for the Coefficients in a spherical harmonics expansion of the earth’s external gravitational potential

Abstract

The earth's external gravitational potential may be mathematically expressed as an infinite series of spherical harmonics. The coefficients in such series depend numerically on the location and orientation of the adopted system of reference with respect to the body of the earth. The present subject is to derive general formulae for the numerical changes of the harmonic coefficients due to both a re-orientation and a re-location of the reference frame. Such formulae could be of value in critical studies of the impact of artificial satellites on dynamical geodesy, especially when combining the results with those from terrestrial geodesy. Use has been made of a transformation property of spherical harmonics which seems to be not a part of standard mathematics. Some simple numerical examples are given. ( n -S ) ! 1 ( n + s)! Ma" IM Ji=cE,--__ r" cossl . Pi(sinp)dm.Here GM stands for the product of the universal constant of gravitation G and the total mass M of the earth; the mass of the atmosphere will be neglected. a is an arbitrary scaling factor which conveniently can be taken as to coincide with the equatorial radius of the earth. r, 1 and ( =latitude) are the conventional spherical coordinates and the notation of the associated Legendre functions of the first kind could follow Hobson (1931): 1 if s = O 2 if s>O. (?a + a)! Tellus XXI (1969), 4