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<p style='text-indent:20px;'>The aim of this work is to continue the analysis, started in [<xref ref-type="bibr" rid="b10">10</xref>], of the dynamics of a point-mass particle <inline-formula><tex-math id="M1">\begin{document}$ P $\end{document}</tex-math></inline-formula> moving in a galaxy with an harmonic biaxial core, in whose center sits a Keplerian attractive center (e.g. a Black Hole). Accordingly, the plane <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> is divided into two complementary domains, depending on whether the gravitational effects of the galaxy's mass distribution or of the Black Hole prevail. Thus, solutions alternate arcs of Keplerian hyperbolæ with harmonic ellipses; at the interface, the trajectory is refracted according to Snell's law. The model was introduced in [<xref ref-type="bibr" rid="b11">11</xref>], in view of applications to astrodynamics.</p><p style='text-indent:20px;'>In this paper we address the general issue of periodic and quasi-periodic orbits and associated caustics when the domain is a perturbation of the circle, taking advantage of KAM and Aubry-Mather theories.</p>

Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. (i) We demonstrate the long-term stability of the semimajor axis within the framework of the $$J_2$$ J 2 problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $${\mathcal {H}}_{J_2}$$ H J 2 . (ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the ‘geolunisolar’ Hamiltonian $${\mathcal {H}}_\mathrm{gls}$$ H gls ), after a suitable reduction of the Hamiltonian to the Laplace plane. (iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $${\mathcal {H}}_{J_2}$$ H J 2 and $${\mathcal {H}}_\mathrm{gls}$$ H gls models, which reflect necessary conditions for the holding of Nekhoroshev’s theorem on the exponential stability of the orbits. We find that the $${\mathcal {H}}_{J_2}$$ H J 2 model is non-convex, but satisfies a ‘three-jet’ condition, while the $${\mathcal {H}}_\mathrm{gls}$$ H gls model restores quasi-convexity by adding lunisolar terms in the Hamiltonian’s integrable part.

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