Resonant Orbits of Artificial Satellites and Longitude Terms in the Earth's External Gravitational Potential

Abstract

Summary The zonal harmonics in the Earth's external gravitational potential give rise to long period or secular perturbations of the orbits of close artificial satellites and can be estimated from such perturbations but the tesseral and sectorial harmonics in general produce only short period perturbations which are far more difficult to observe. There is thus little prospect at present of finding the longitude-dependent parts of the potential from the motion of an arbitrary satellite and so it is of interest to see if in special circumstances long period or secular perturbations could arise from these parts. This paper contains a preliminary study of orbits with periods bearing some specific relation to the period of the Earth's rotation. For the purposes of this study, it has been assumed that the eccentricity of the orbit and its inclination to the equator are so small that only the first power of the eccentricity and the square of the inclination need be retained and it is also supposed that, independently of the longitude terms in the potential, the longitudes of the node and perigee change linearly with time. It is found that secular and long period perturbations can arise and that they differ according to the parity of the difference (p-q) for the associated Legendre function P(cos θ) in the spherical harmonic expansion of the potential. One approximate condition for such perturbations is that (q-I)n=qωr where n is the mean motion of the satellite and ωr the Earth's spin angular velocity. The condition is slightly different for odd and even (p-q). When (p-q) is even, there are perturbations of order e-1 (e is the eccentricity of the orbit) in the mean anomaly and the longitude of perigee while when (p-q) is odd there are perturbations of order (sin i)-1 (i is the inclination of the orbit to the equator) in the longitude of the node and of perigee. The simple case of the ellipticity of the equator (P° (cos θ)) is discussed in rather more detail and some preliminary consideration is given to the problems of the realization of resonant orbits. The general expressions for the perturbations, excluding terms proportional to e2 and sin4i or of higher order, are given in the Appendix where the complete set of resonance conditions is listed. The resonances discussed in the body of the paper are generally the most important in that they occur for the smallest orbits but there are resonances with smaller orbits in some special cases.