Page - 31 - 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

keeping in mind the full breadth of influences contained therein. The analysis of the satellite orbit is limited to a specific interval of time, with start time tstart and end time tend. The duration of this interval is T= tend− tstart. Time is normalized to this interval with τ= t− tstart T , (5.1.4) giving the compact form of the equation of motion r¨(τ)=f(τ) . (5.1.5) 5.1.1 Integrating the Equation of Motion The position and velocity of the spacecraft can be determined by integrating eq. (5.1.5), giving r¨(τ)=f(τ) (5.1.6) r˙(τ)= r˙0+T ∫ τ 0 f(τ′)dτ′ (5.1.7) r(τ)=r0+ r˙0(τT)+T2 ∫ τ 0 (τ−τ′)f(τ′)dτ′ . (5.1.8) The position r(τ) and velocity r˙(τ) depend on unknown initial values r˙0 and r0. Fixing these initial values yields a position and velocity for each epoch n∈ [1,N]of the orbit arc (see e.g. Mayer-Gu¨rr, 2006). The variational equation approach is based on a linearisation of the integrated positions and velocities with regard to the sought force model parameters. In the formalism of the variational equations, the partial derivatives of the spacecraft’s position and velocity for all epochs of the orbit arc with regard to the force model parameters appear. These partials are computed efficiently through integration from some initial condition, the explicit evaluation of the complete partials at each epoch is thus avoided. Dynamic orbits then appear as the linear term in a Taylor expansion of the integrated equations of motion (eqs. (5.1.7) and (5.1.8)) necessary when setting up the variational equations. The position and velocity of the satellite for one epoch are consolidated in the state vector y(τ)= [ r(τ) r˙(τ) ] . (5.1.9) Taking the partial derivatives of the satellite state at one epoch with regard to the initial state of the satellite y0= [ r0 r˙0 ] (5.1.10) 5.1 Equation of Motion 31