http://arxiv.org/abs/2105.11960
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 $J_{2}$ spherical zonal harmonic. The equatorial track of satellite motion — assuming the co-latitude $\varphi$ fixed at $\pi/2$ — 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 $t = t(c)$ to get the period $T$ of some functions $c(t)$ of mechanical interest and then to 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 $J_{2}$ problem — is benchmarked by applying it to the basic case of keplerian motion, finding again the classic results through our different analytic path.
Keywords: $J_2$ problem, bounded satellite motion, Fourier series, elliptic integrals, Jacobi elliptic functions.
A. Bocci and G. Scarpello
Wed, 26 May 21
41/66
Comments: N/A