Orbit determination using analytic partial derivatives of perturbed motion

Abstract

State transition matrices for the motion of space vehicles have been calculated both numerically and analytically. The former method, for example, integrates the equations. It yields accurate results when the same force model is used in the variational equations as in the equations of motion. The main advantage of the latter method, i.e., using analytic partial derivatives, is faster computation, but, if the transition is made by Keplerian formulas to a time many revolutions from epoch, the accuracy of the matrix is severely degraded. It is shown in this paper that the state transition matrix of a satellite orbit may be calculated analytically including the effects of perturbations. This is accomplished by generating each perturbation in the Gaussian form (using three mutually perpendicular components of the perturbing acceleration) to find the transition matrix from epoch state to the state at any other time. This analytic method is considerably faster than the numerical integration of the variational equations and it is shown that the methods are in good agreement. The state vector considered includes orbit parameters, which are not singular for any elliptic orbits except retrograde equatorial, and the parameters associated with the perturbations. These perturbations include atmospheric drag, thrust periods, impulses, direct solar radiation pressure, oxidizer outgassing, and the geopotential. The method, however, can be extended to any perturbation, nor is it restricted to geocentric orbit calculations. Results are presented in tabular form. Partial derivatives of the satellite's position with respect to the orbital elements and other vehicular parameters, such as the ballistic parameter Cz>/4/m, are listed in the tables as obtained by each of three methods: by the analytical procedure, by integrating the variational equations, and by differencing neighboring trajectories. The comparisons are made on a satellite with a period of 90 min, over complete revolutions one day after epoch and one week after epoch.