Theory of the Orbit of an Artificial Satellite with Use of Spheroidal Coordinates.

Abstract

view Abstract Citations (12) References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Theory of the Orbit of an Artificial Satellite with Use of Spheroidal Coordinates. Vinti, John P. Abstract The author has developed the orbital theory for an artificial satellite, using a gravitational potential, (Vinti, 1959, J. Research Nat. Bur. Standards 63B 105) in spheroidal coordinates, p, ~, 4), which leads to quadrature solutions. This procedure accounts for all of the second harmonic (of coefficient J2) and for most of the fourth, but omits entirely the local anomalies, drag, lunar-solar perturbations, and the odd harmonics when the origin is chosen at the earth's center of mass. A previous report (Vinti, 1960, Bull. Am. Phys. Soc. 5, 8) outlined a solution for the secular motions of CL, and ~. The complete orbital solution now requires evaluation of the integrals in the kinetic equations and their subsequent solution for p and ,i; all other quantities follow readily from these. Only the p integrals cause trouble; if .1... P4 are the zeros of a certain quartic F(p) that they all involve, one must first find Pi+P2, P3+P4, PiP2, and p~p~, where p' and P2 are those zeros of F(p) which would reduce to the perigee and apogee radii if J2 vanished. With this fact as a start, a method of successive approximations leads to power series in J2 for these sums and products, the present treatment stopping with J22 Expressions of order J22 follow readily for the p integrals and the ~ integrals, the former involving two angles closely related to v and E, the latter involving two other angles closely related to CL, + ~ and 4) - ~. Here v and E are the true and eccentric anomalies, CL, is the argument of perigee, and 4) and ~ are the right ascensions of the satellite and the node. The constants that appear are all expressible in terms of initial conditions. A method of successive approximations, started by neglecting terms involving J2, then leads to the solution for p and ~ as functions of time, through terms of order J22 Calculation of the polar coordinates is then easy. This work was supported by the U. S. Air Force, through the Office of Scientific Research of the Air Research and Development Command. Publication: The Astronomical Journal Pub Date: August 1960 DOI: 10.1086/108267 Bibcode: 1960AJ.....65..353V full text sources ADS |