Low-altitude, near-polar, near-circular orbits are very desirable as science orbits for missions to planetary satellites, such as the Jupiter's moon Europa. However, past research suggests that a spacecraft into a low-altitude, near-polar orbit around Europa will have an impact in a relatively short time. Frozen orbits around the Moon, natural satellites, asteroids, or are current interest because of various space missions that aim to orbit around such bodies. For this, an approach on stability of these orbits around planetary satellites will be developed to study the orbital motion of artificial satellites around planetary satellites, for example, the Moon and Europa, considering the gravitational perturbation due to the planet and the non-uniform distribution of mass of the planetary satellite. Will be developed semi-analytical approach that will be used in a version of KAM theorem as presented by Kovalev and Saychenko. We present an analytical development to study the stability of equilibrium points of conservative dynamical systems involving the normalization of the Hamiltonian (with two and three degrees of freedom), where is applied the Lie-Hori perturbation method. An application will be made to a problem taking into account the non-uniform distribution of mass of the planetary satellite and the perturbation of the third-body where the equations will be developed in closed form which is valid for a range of applications. A set of parameters will be established (synchronous orbits, inclinations critical, polar orbits, frozen orbits, etc.) that can be used for maintenance of spacecraft in orbit in exploratory missions to planetary satellites. This analysis of stability is of fundamental importance for aerospace research to ensure the success of space missions. An application of the problem considered here is the study of orbital maneuvers necessary for the maintenance of a Keplerian orbit within certain ranges, where the goal is to correct the orbit change was due to perturbations over time. Considering this question, the dynamics studied here is the problem of transferring a spacecraft between two given orbits with minimum fuel consumption possible. In such a transfer there are several other important factors, such as, for example, time spent on the transfer limits on the actuators and/or condition of the vehicle, etc. However, in this work, the fuel consumption is the critical element of our maneuvers, although the time required by the maneuver is also considered.
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