Non-Gaussian error propagation in orbital mechanics
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abstract
The problem of propagating orbit initial condition uncertainty is examined. The dominant Earth oblateness (J2) and atmospheric drag perturbations are included in the equations of motion. The covariance due to uncertainty in position and velocity is propagated forward in time in the conventional rectangular coordinates, polar coordinates and the orbit element space. The orbit elements are considered as a candidate set because all but one of them are "slow-varying" varying functions of time. The orbit averaged effects of the perturbations are derived to establish variational expressions valid over long time periods. Certain measures of "nonlinearity" are developed to evaluate the validity of the linearization approximations. After linearization in polar coordinates, the nonlinear transformation from polar to rectangular coordinates analytically maps Gaussian statistics in polar coordinates into highly non-Gaussian statistics in rectangular coordinates. These results are in close qualitative agreement with Monte-Carlo simulations. Results due to linearization in orbit elements appear most promising for long-term uncertainty propagation.
