Analytical Description of Frozen Orbits Under Zonal, Sectorial, and Tesseral Gravitational Harmonics

Frozen orbits are equilibrium solutions to the nonlinear set of equations that describe the motion of a satellite around a primary with a nonuniform mass distribution. In particular, frozen orbits are selected to exhibit no secular change in one or more orbital elements over a long period of time. This provides them with favorable characteristics, such as their periodicity, which is applicable to many missions, especially satellite constellations, formation f light, telecommunications and observation missions. As such, these orbits are extensively used in industry. Analytical techniques have been devised to find approximate solutions for frozen orbits and have been successful at doing so accounting for zonal perturbations. However, many of these solutions tend to struggle to handle sectorial and tesseral effects due to their dependence on time, which makes the set of differential equations that describes the system more difficult to integrate. To overcome this challenge, numerical methods have been implemented based on natural parameter continuation to correct zonally perturbed frozen orbits for a gravity f ield that includes the additional sectorial and tesseral effects. Unfortunately, these numerical techniques are computationally expensive due to their iterative nature and do not provide a lot of insight about the system. As such it is preferable to find an analytical method that can handle all three types of gravitational perturbation. In this thesis, a second-order solution is introduced for low eccentric frozen orbits that accounts for zonal, sectorial, and tesseral effects up to degree and order three. This frozen orbit solution is found by producing an approximate solution for the evolution of the orbital elements under the same perturbations and determining, under what conditions, the secular variation of these orbital element equations is equal to zero. These results are applied to the gravity fields of Earth, the Moon, and the asteroid Eros to demonstrate the accuracy of the solution for bodies with varying mass distributions. The approximate solution for the evo lution of the orbital elements is also compared to other published solutions to quantitatively evaluate the quality of the approximation against another modern approaches.