Frozen orbits with inner planar perturbing body up to triakontadipole level of approximation.

In the exploration of celestial bodies, including planets, moons, and asteroids, it is useful to identify stable orbits in which the orbital elements remain, on average, constant (referred to as frozen orbits). This paper aims to investigate the feasibility of frozen orbits for a small body, such as a probe, orbiting a main body under the perturbation due to the gravitational attraction of an inner third-body, on a circular (or slightly eccentric) orbit coplanar with the main body equator, by means of an approach based on mean element theory. The disturbing potential has been developed in Legendre polynomials up to order l = 5 (triakontadipole level of approximation). Thus, the double-averaged potential function and the associated constants of motion have been determined up to the same order. Then, by applying the Lagrange Planetary Equations, the secular and long-term orbital element variations have been obtained. Therefore, frozen orbital solutions have been evaluated, and the effects on them of the addition of main body oblateness perturbation have been considered. Furthermore, the procedure has been repeated, and the frozen solutions are re-evaluated in the case of a disturbing body with small eccentricity up to order l = 4 (hexadecapole level of approximation). These solutions offer significant advantages in exploring bodies subject to a notable perturbation from an inner third-body, such as binary asteroid systems. • Frozen orbits for the inner third-body problem. • Inner circular planar third-body: Double-averaged potential up to triakontadipole. • Inner slightly eccentric planar third-body up to hexadecapole. • Orbital solutions for binary asteroid observation. [ABSTRACT FROM AUTHOR]