Orbital dynamics in a stochastic atmosphere

Abstract

An algorithm is developed to analyze the orbital dynamics of a near-Earth satellite subjected to the random perturbations of a atmosphere. The orbital equations include the perturbation caused by an oblate geopotential and a drag model where the atmospheric density varies exponentially with height and temporally with the solar-activity and semiannual cycles. A random process representative of short-term density fluctuations modulates this deterministic density function. The solution begins with the analytical averaging of both the gravity and the drag perturbations, following the principle of the Method of Averaging. The drag averages are expanded in a series of newly defined functions called the two-dimensional hyperbolic Bessel functions/' Their nice analytical properties greatly simplify the next part of the solution where the equations of motion are further transformed into a stochastic Taylor expansion.'' Unlike the standard linear approach used in the treatment of systems, this technique preserves the full nonlinearity of the equations of deterministic motion and, therefore, avoids the usual restriction to small excursions from an initial or reference state. The analytical nature of the drag averages provides insight into the orbital effects of random density fluctuations and results in an efficient algorithm that prescribes the evolution of the mean and variance of the orbital state. Based on this algorithm, a computer program has been developed to generate confidence intervals that bound, in a probabilistic sense, the trajectory of a satellite. Typical numerical results assuming a 300-km-high near-circular orbit are illustrated. Although orbital error analysis is the main application of the paper, its analytical results can also be used in the development of an orbit determination and density estimation algorithm.