Page - 83 - 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

7. Integrate the corrected accelerations as in eqs. (5.2.38) and (5.2.39) with ∆r˙intc =Kr˙∆r¨c , (7.2.17) ∆rintc =Kr∆r¨c , (7.2.18) and, similar to eq. (5.2.40), compute a new estimate∆yˆ0 of the differential state from re−rref−∆rintc =Φr∆y0 . (7.2.19) 8. Compute the final dynamic orbit as in eqs. (5.2.41) and (5.2.42) with r˙= r˙ref+Φr˙∆yˆ0+∆r˙ int c (7.2.20) r=rref+Φr∆yˆ0+∆r int c (7.2.21) This is a general formulation of what could be termed a reduced initial value approach to dynamic orbit determination. 7.2.1 Reference Motion These derivations are fully independent of the choice of reference forcef0. The only prerequisite is that the equation of motion due to the reference force should be analytically solvable. If this condition were not fulfilled, the given formulations would still hold, but the numerical advantages attributed to the method might disappear. If, for example, one were to choosef0(τ)=0and∆y0=y0, the approach would simplify into the same apparatus as presented in sections 5.2.1 to 5.2.4. In this case, the reference motion is a linear unperturbed motion through space, tangent to the satellite orbit at the first epoch. The classical choice for an analytically solvable reference motion is the Kepler ellipse, with the reference force that of a point-like or spherical Earth, or more generally the acceleration due to the central term of a more complex gravitational potential: f0(τ)=−GM r(τ)‖r(τ)‖3 (7.2.22) In his work on the perturbation of planets, Encke suggests to compute the perturbed orbit relative to such a Keplerian reference motion (Encke, 1852, 1857). Encke defines the ellipse by requiring that the position and velocity at the first epoch of the reference trajectory are identical to that of the perturbed orbit at the first epoch, or rref,0=re(0) and r˙ref,0= r˙e(0) . (7.2.23) Such an ellipse is termed an osculating ellipse. Setting the reference motion to an osculating ellipse has the undesirable effect that the reference motion and the per- turbed orbit will diverge significantly, usually after only a short period of integration. This leads to the integrals of the perturbing accelerations in eqs. (7.2.10) and (7.2.11) 7.2 Improved Algorithm 83