Page - 33 - in Contributions to GRACE Gravity Field Recovery - Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations

This integration, and the determination of the dynamic orbit, requires the equations of motion for the satellite to be solved. For the simple potential of a point mass or uniform sphere the equations of motion have a closed solution, described by Kepler’s laws of planetary motion. For more complex potential fields, no closed solution exists, and numerical integration methods are used to determine the satellite motion. Classical examples of such numerical methods are Euler’s algorithm, the family of Runge-Kutta methods, and multi-step methods of which the Gauss-Jackson family is a prominent example (Berry and Healy, 2004; Beutler and Mervart, 2010). All of these methods have one important characteristic in common: The satellite state is propagated from one epoch to the next. This terminology implies that the integral of the acting accelerations over one time interval is formed, and then used to determine the updated satellite state at the end of the interval. In other terms, the new position at the end of the interval is arrived at by extrapolation. For some methods, such as the aforementioned Gauss- Jackson method, the extrapolated state is then corrected through an iterative procedure. This refinement then relies on the previously extrapolated state, and converges on an estimate of the new state at the end of the interval. After completion of this refinement, the integration algorithm moves forward in time to the next interval, and continues the integration following the same procedure. In the implementation described here, the dynamic orbit is instead determined by continuous numerical integration of all accelerations along an orbit arc. In this in- tegration step the satellite state is not fixed at each epoch. Rather only the changes from one epoch to the next, the integrals in eqs. (5.1.7) and (5.1.8) are computed. In a sense, the orbit is not extrapolated from epoch to epoch, but rather its complete geometry is determined at once. This is achieved by taking advantage of a priori knowledge of all accelerations along the arc, as evaluated from an initial approximate orbit and, depending on the satellite, observed by an on-board accelerometer. For this implementation the definite integrals of the accelerations are efficiently determined using an integration polynomial, as described in section 2.7. The definite integrals for the new positions are then used in conjunction with the satellite states given by the initial approximate orbit to estimate the initial state of the integrated dynamic orbit. The initial state, together with the definite integrals, then completely describes the satellite state for the integrated arc. The following sections describe this algorithm in more detail. 5.2.1 Coarse Approximation The first iteration of dynamic orbit integration requires a coarse approximation of the spacecraft’s orbit, for example a Kepler ellipsis determined from some mean ephemerides, or purely kinematic positions determined from GPS observations. These are the approximate positionsre. The true positions of the spacecraftrare unknown, and deviate from the approximate positions by some value . The employed background models are evaluated at the approximate positions, giving the accelerations from 5.2 Orbit Integration and State Transition Matrix 33