Approximating orbits in a rotating gravity field with oblateness and ellipticity perturbations

This paper explores the problem of analytically approximating orbital state for a subset orbits in rotating potential with oblateness and ellipticity perturbations. is done by isolating approximate differential equations orbit radius other elements. The conservation Jacobi integral used to make solvable first-order solutions are characterized as small deviations from an unperturbed circular orbit. approximations developed near-circular initial mean motion $n_{0}$ around body rotation rate $c$. shown be valid values angular ratio $\Gamma = c/n_{0} > 1$, accuracy decreasing \rightarrow singularities at near 1$. Extensions methodology eccentric discussed, commentary on challenges obtaining generally both orbits.