  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 $J_2$ problem, by a normal form construction
eliminating the fast angle in the corresponding Hamiltonian and obtaining
$H_{J_2}$ . ii) We demonstrate the stability of the eccentricity and
inclination in a secular Hamiltonian model including lunisolar perturbations
(the 'geolunisolar' Hamiltonian $H_{gls}$), 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 $H_{J_2}$ and $H_{gls}$ models, which reflect necessary conditions for the
holding of Nekhoroshev's theorem on the exponential stability of the orbits. We
find that the $H_{J_2}$ model is non-convex, but satisfies a 'three-jet'
condition, while the $H_{gls}$ model restores quasi-convexity by adding
lunisolar terms in the Hamiltonian's integrable part.
