Determination of the Even Harmonics in the Earth's Gravitational Potential

Summary This paper describes a method for using the orbits of artificial Earth satellites to determine the first seven even zonal harmonics in the Earth's gravitational potential, that is, the coefficients J2, J4…J14 of the expansion in terms of Legendre polynomials as given in equation (I). To apply the method, accurate orbital information is required for at least seven satellites with widely differing orbits. Although existing orbital information is inadequate for the full application of the method, the Earth's gravitational potential has been determined as accurately as possible from the orbits of seven satellites covering a wide range of latitude. The potential experienced by these satellites is found to be best represented by the following set of coefficients: 106J2= 1082.86±0.1; 106J4= -1.03±0.2; 106J6= 0.72±0.02; 106J8= 0.34±0.2; 106J10= -0.50±0.2; 106J12= 0.44±0.2. The quoted errors are tentative estimates of the departures from the true physical values. If the potential is to be represented by fewer even harmonics, the best number is three, and the appropriate values are: 106J2= 1082.92±0.1; 106J4= -1.30±0.2; 106J6= 0.85±0.2. To give the complete potential, values of J3, J5, etc., must of course be added.

[1]  I. M. Levitt.  Advances in the astronautical sciences: Vol. 6, edited by Horace Jacobs and Eric Burgess. 898 pages, diagrams, illustrations, 612 × 978 in. New York, The Macmillan Co., 1961. Price, $25.00 , 1961 .

[2]  R. H. Merson,et al.  The orbital elements of Ariel 1 for April–August, 1962 and a comparison of R.A.E. and N.A.S.A ephemerides , 1963 .

[3]  G. E. Cook.  Effect of an oblate rotating atmosphere on the orientation of a satellite orbit , 1961, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[4]  G. E. Cook.  Luni-Solar Perturbations of the Orbit of an Earth Satellite , 1961 .

[5]  H. S. Hopfield,et al.  Odd Harmonics of the Earth's Gravitational Field , 1961, Nature.

[6]  J. O'keefe,et al.  The gravitational field of the earth , 1959 .

[7]  D. Smith,et al.  Determination of the earth's gravitational potential from satellite orbits , 1961 .

[8]  Yoshihide Kozai.  Effect of precession and nutation on the orbital elements of a close earth satellite , 1960 .

[9]  H. Plummer,et al.  An Introductory Treatise on Dynamical Astronomy , 1919 .

[10]  Desmond George King-Hele,et al.  The effect of the earth’s oblateness on the orbit of a near satellite , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[11]  G. E. Cook,et al.  The effect of atmospheric rotation on the orbital plane of a near-earth satellite , 1960, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[12]  Motion of a satellite in the earth’s gravitational field , 1960, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[13]  D. Smith,et al.  A determination of the odd harmonics in the geo-potential function , 1963 .

[14]  R. H. Merson.  The Motion of a Satellite in an Axi-symmetric Gravitational Field , 1937 .

[15]  Yoshihide Kozai,et al.  The motion of a close earth satellite , 1959 .

[16]  G. E. Cook,et al.  Earth's Gravitational Potential: Evaluation of Even Zonal Harmonics from the 2nd to the 12th , 1963, Nature.

[17]  D. King-hele.  The Earth's Gravitational Potential, deduced from the Orbits of Artificial Satellites , 1937 .

[18]  G. E. Cook.  Perturbations of satellite orbits by tesseral harmonics in the earth's gravitational potential , 1963 .