Analytical Model for Orbital Motion Under J₂

Abstract

As the number of operational satellites and debris objects in Earth orbit continues to accelerate, the ability to predict orbital trajectories with both accuracy and efficiency has become an indispensable capability. Numerical integration of the full Cartesian equations of motion offers generality but at high computational cost, while traditional analytical theories are efficient but often restricted by singularities in the classical orbital element set. Analytical formulations expressed in nonsingular elements can combine efficiency with global validity, and provide physical insight into the structure of orbital perturbations.

This thesis develops a globally valid analytical model for orbital motion under the Earth's second zonal harmonic (J₂) in the modified equinoctial element (MEE) framework. The MEE set eliminates the singularities present in circular and equatorial orbits, allowing uniform treatment across all regimes. Two principal contributions are made. First, explicit first-order mean equations of motion are derived using a generalized averaging method applied to the J₂ disturbing function. The resulting system reduces to two planar rotations of the eccentricity and inclination vectors with constant rates, together with a secular drift in the true longitude. These equations reproduce Brouwer's classical secular results when mapped back to Keplerian elements, while retaining the nonsingular advantages of the MEE formulation. Second, closed-form mean--osculating transformations are obtained, enabling consistent recovery of short-period variations from the mean solution. These transformations allow a dual representation: efficient mean propagation combined with reconstruction of instantaneous orbital states.

The analytical model is validated against high-fidelity Cartesian propagation across a set of representative orbit classes, including LEO, GEO, GTO, and Molniya orbits. In all cases, the mean element evolution predicted by the MEE-based theory shows close agreement with numerical integration. Over week-long propagation intervals, relative position errors remain small, while computational cost is substantially reduced compared to Cowell integration. These results establish the MEE-based analytical framework as both theoretically rigorous and practically effective, providing a foundation for accurate, efficient, and globally valid orbit prediction.