Absolute and Relative Motion Satellite Theories for Zonal and Tesseral Gravitational Harmonics

Abstract

In 1959, Dirk Brouwer pioneered the use of the Hamiltonian perturbation methods for constructing artificial satellite theories with effects due to nonspherical gravitational perturbations included. His solution specifically accounted for the effects of the first few zonal spherical harmonics. However, the development of a closed-form (in the eccentricity) satellite theory that accounts for any arbitrary spherical harmonic perturbation remains a challenge to this day. In the present work, the author has obtained novel solutions for the absolute and relative motion of artificial satellites (absolute motion in this work refers to the motion relative to the central gravitational body) for an arbitrary zonal or tesseral spherical harmonic by using Hamiltonian perturbation methods, without resorting to expansions in either the eccentricity or the small ratio of the satellite’s mean motion and the angular velocity of the central body. First, generalized closed-form expressions for the secular, long-period, and short-period variations of the equinoctial orbital elements due to an arbitrary zonal harmonic are derived, along with the explicit expressions for the first six zonal harmonics. Next, similar closed-form expressions are obtained for the sectorial and tesseral (collectively referred to as tesserals henceforth) harmonics by using a new approach for the exact Delaunay normalization of the perturbed Keplerian Hamiltonian. This approach reduces the solution for the tesseral periodic perturbations to quadratures. It is shown that the existing approximate approaches for the normalization of the tesseral problem, such as the method of relegation, can be derived from the proposed exact solution. Moreover, the exact solution for the periodic variations due to the tesseral harmonics produces a unified artificial satellite theory for the sub-synchronous and super-synchronous orbit regimes without any singularities for the resonant orbits. The closedform theories developed for the absolute motion are then used to develop analytic solutions in the form of state transition matrices for the satellite relative motion near a perturbed elliptic reference orbit. The expressions for differential equinoctial orbital elements for establishing a general circular orbit type satellite formation are also derived to avoid singularities for the equatorial and circular reference orbits. In order to negate the along-track drifts in satellite formations, an ana- lytic expression for the differential semimajor axis is derived by taking into account the secular effects due to all the zonal harmonics. The potential applications of the proposed satellite theories range from fuel-efficient guidance and control algorithms, formation design, faster trade and parametric studies to catalog maintenance, conjunction analysis, and covariance propagation for space situational awareness. Two specific applications, one for solving a perturbed multiple revolution Lambert’s problem and the other for rapid nonlinear propagation of orbit uncertainties using point clouds, are also given. The theories presented in this work are implemented for computer simulations in a software tool. The simulation results validated the accuracy of these theories and demonstrated their effectiveness for various space situational awareness applications.