Analytic Inversion of Closed form Solutions of the Satellite’s J2 Problem
Alessio Bocci *
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, Italy.
Giovanni Mingari Scarpello
Ordine Degli Ingegneri di Milano, Italy.
*Author to whom correspondence should be addressed.
Abstract
This report provides some closed form solutions -and their inversion- to a satellite’s bounded motion on the equatorial plane of a spheroidal attractor (planet) considering the J2 spherical zonal harmonic. The equatorial track of satellite motion- assuming the co-latitude φ fixed at π/2- is investigated: the relevant time laws and trajectories are evaluated as combinations of elliptic integrals of first, second, third kind and Jacobi elliptic functions. The new feature of this report is: from the inverse t = t(c) we get the period T of some functions c(t) of mechanical interest and then we construct the relevant c(t) expansion in Fourier series, in such a way performing the inversion. Such approach-which led to new formulations for time laws of a J2 problem- is benchmarked by applying it to the basic case of keplerian motion, finding again the classic results through our different analytic path.
Keywords: J2 problem, bounded satellite motion, Fourier series, elliptic integrals, Jacobi elliptic functions