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Stochastic Gauss Equations

We derive the equations of celestial mechanics governing the variations of the orbital elements under a stochastic perturbation generalizing the classical Gauss equations. Explicit formulas are given for the semi-major axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle and the mean anomaly which are express in term of the angular momentum vector $\textbf{H}$ per unit of mass and the energy $E$ per unit of mass. Together, these formulas are called the \emph{stochastic Gauss equations} and they are illustrated numerically on an example from satellite dynamics.