Page - 85 - 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

modification of the Kepler parameters due to higher order terms or external perturba- tions proofs completely sufficient. Instead, the choice of the initial parameters of the reference orbityref,0 is reconsidered. To formulate a constraint for the initial state of the reference orbit, consider that the Encke ratio e(τ) is small if the Encke vector∆r(τ) is small. The goal must thus be to minimize the magnitude of the Encke vectors over the whole arc. This can be written as N ∑ n=0 ‖∆r(τn)‖2→min . (7.2.25) Knowing that the hypothetical true position of the spacecraft is arrived at by taking the sum of the reference position as a function of its initial values and the Encke vector, these positions can be written as r=rref(yref,0)+∆r . (7.2.26) Recognizing eq. (7.2.25) as the minimisation criteria of a classical least squares ad- justment, eq. (7.2.26) can be solved in a least squares sense. To this end, the Encke vector∆r is treated as if it represented the residualseof the least squares fit. The true positions are however not available, instead the approximate positions are used as observations for the spacecraft state at each epoch, giving re=rref(yref,0)+e . (7.2.27) This finds the initial values of the reference ellipse yˆref,0= [ rˆref,0 ˆ˙rref,0 ] (7.2.28) that lead to the minimal square sum∆rT∆r, fulfilling the condition in eq. (7.2.25). The resulting differential initial values are then ∆y0=y0− yˆref,0= [ r0 r˙0 ] − [ rˆref,0 ˆ˙rref,0 ] . (7.2.29) As∆rT∆r is minimized and the Encke ratio is by definition positive, the solution yˆref,0 also minimizes the sum of all Encke ratios over the complete orbit arc. Figure 7.2 illustrates this optimized best-fit reference ellipse. [r0, r˙0] ∆y0 Osculating Best-fit Figure 7.2: Osculating ellipse (in pink) and best-fit ellipse (in green). 7.2 Improved Algorithm 85