Contribution to the Theory of Critical Inclination of Close Earth Satellites.

Recently many authors have shown interest in the critical inclination problem of close satellites. Almost all have treated only the leading terms, which are of the order of (J2) 2 in the equations of motion. But, as first shown by Izsak, continuation to higher approximations from the ordinary treatment of libration, including only leading terms, could break down; and a peculiar kind of libration would then occur, especially in the case of small eccentricity. Izsak discussed the results only 71 from the viewpoint of the form of the Hamiltonian. He did not try to solve the equations of motion explicitly. This paper shows the development of a higher- order theory of the motion of close satellites in the vicinity of critical inclination, provided that the earth potential is plane symmetrical. The equations of secular motion, which I have solved here, are very similar to Izsak's. The solution includes not only the case of small eccentricity, but also the case of moderate eccentricity, with the terms corresponding up to the order of (J2) 3, counting the square of the eccentricity as the order of J2 in the former case, in the original equations of motion. For the case of moderate eccentricity, the expression of the solution involves not only the elliptic functions of Jacobi, but also the elliptic integral of the second kind. On the other hand, for the case of small eccentricity, the expression of the solutions involves only the elliptic functions. It is also shown that in order to get higher approximation beyond the third order, especially for the case of small eccentricity, the solutions must involve the elliptic integral of the third kind. This work was performed under a National Academy of Sciences Postdoctoral Resident Research Associateship.