Page - 59 - 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

Now, due to‖u‖= (uTu)1/2 ∂ ∂u 1 ‖u‖=− 1 ‖u‖3u T (6.4.40) and ∂ ∂u uTu˙= ∂ ∂u u˙Tu= u˙T . (6.4.41) Putting together the chain rule, eq. (6.4.39) is ∂ρ˙ ∂u = 1 ‖u‖u˙ T− 1‖u‖3u T · ( uTu˙ ) = 1 ρ u˙T− 1‖u‖e T · ( eTu˙ ) = 1 ρ u˙T− ρ˙ ρ eT (6.4.42) The derivative of the range rate w.r.t. the differential velocity is ∂ρ˙ ∂u˙ = ∂ ∂u˙ eTu˙=eT (6.4.43) In eq. (6.4.38), the derivatives of the baseline and differential velocity can be further expanded to ∂u ∂x = ∂u ∂rA ∂rA ∂x + ∂u ∂rB ∂rB ∂x , (6.4.44) ∂u˙ ∂x = ∂u˙ ∂r˙A ∂ r˙A ∂x + ∂u˙ ∂r˙B ∂ r˙B ∂x . (6.4.45) With ∂u ∂rA = ∂u˙ ∂r˙A =−1 , (6.4.46) ∂u ∂rB = ∂u˙ ∂r˙B =1 , (6.4.47) this simplifies to ∂u ∂x = ∂rB ∂x − ∂rA ∂x , (6.4.48) ∂u˙ ∂x = ∂ r˙B ∂x − ∂ r˙B ∂x . (6.4.49) Going from generic parametersx to the specific parameters set up in ITSG-Grace2016, the derivatives of the positions and velocities of the spacecraft w.r.t. the gravity field parameters, satellite parameters, and the initial state are the variational equations from 6.4 Functional Models 59