Page - 30 - 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

of the previous iteration as an input for the next. For spacecraft equipped with an accelerometer, e.g. GRACE, accelerations due to non-conservative forces can be measured directly by this instrument. A change in the computed orbit position from one iteration of orbit determination to the next does not affect the direct accelerometer observations, and so the integrated non-conservative forces do not change between iteration steps. The conservative forces, however, do depend on the spacecraft position. The specific values of the accelerations due to conservative forces being integrated will thus change from one iteration of integration to the next. The resulting dynamic orbits are then used at multiple steps of the level 1B to level 2 processing chain to compute observation equations for high-low satellite-to-satellite tracking and low-low satellite-to-satellite tracking observations, where they are used as a Taylor point for their linearisation. The following sections give an overview of the background and process of dynamic orbit integration as implemented at IfG, as well as an introduction to the variational equations and their solution. 5.1 Equation of Motion The fundamental principle in this work is Newton’s second law of motion F=mr¨ , (5.1.1) stating that the acceleration experienced by a body is directly proportional to a forceF acting on it. Isolating the acceleration in eq. (5.1.1) yields r¨= F m =f(t,r(t),p, . . .) , (5.1.2) the equation of motion. This equation states that the acceleration experienced by the body, in this case a GRACE spacecraft, is equal to the specific force exerted on it. For GRACE, this is the sum of all conservative and non-conservative forces, as described in chapter 3. As the effective force, the superposition of all component forces, is neither uniform in space nor constant in time, the acceleration in eq. (5.1.2) depends on the time of evaluation t, as well as the position of the spacecraft at this timer(t). Further, it depends on the parameterspof the force-generating functions such as the Stokes coefficients of Earth’s gravitational field or the density of the remnant atmosphere, causing drag. Completing this thought, the occurrence of drag in this equation hints to the specific force also being dependent on more factors, namely the orientation of the spacecraft, its cross section, and its velocity r˙(t). When considering albedo and solar radiation pressure, the characteristics of the satellite surface materials and the orientation of the surface elements also become relevant. To simplify this increasingly complex notation, the function describing the specific force is abbreviated to f(t)=f(t,r(t),p, . . .) , (5.1.3) Chapter5 Variational Equations30