Analytic Inversion of Closed form Solutions of the Satellite’s J2 Problem

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.

The Dynamics of One Way Coupling in a System of Nonlinear Mathieu Equations

Background: This paper extends earlier research on the dynamics of two coupled Mathieu equations by introducing nonlinear terms and focusing on the effect of one-way coupling. The studied system of n equations models the motion of a train of n particle bunches in a circular particle accelerator. Objective: The goal is to determine (a) the system parameters which correspond to bounded motion, and (b) the resulting amplitudes of vibration for parameters in (a). Method: We start the investigation by examining two coupled equations and then generalize the results to any number of coupled equations. We use a perturbation method to obtain a slow flow and calculate its nontrivial fixed points to determine steady state oscillations. Results: The perturbation method reveals the existence of an upper bound on the amplitude of steady state oscillations. Conclusion: The model predicts how many bunches may be included in a train before instability occurs.