Energy Relations for the Motion of two Satellites within the Gravity Field of the Earth

Energy exchange relations of the satellites’ motion within the gravity field of the Earth as well as integrals of motion seem to be useful as basic models for the determination of field parameters of the gravity field as well as for the validation and consistency check of gravity field models. The balance equations are adapted especially to the characteristics of accurate satellite-to-satellite tracking techniques either in the high-low or in the low-low applications. The reason is that these methods are based on local measurement principles while the classical techniques exploit accumulated disturbing effects of the inhomogeneous gravity field onto the satellite’s motion. The energy and motion integrals, respectively, show at every instant the energy of the relative and centre of mass motion while changes of the kinematic observables at subsequent instants are directly related to the changes of the potential energy of the gravity field. Alternative procedures for validation of gravity field recovery results, e.g., based on integration techniques as well as downward continuation procedures, frequently applied for this task are susceptible for numerical errors and instability effects. In the following, the balance equations for the energy exchange as well as motion and energy integrals are derived for the satellite-to-satellite motion within the gravity field of the Earth. The formulation is based in principle on the Hamiltonian formulation of classical mechanics. Only mechanical energy forms are considered, but an extension to other energy forms is basically possible but not presented here. The relations are derived in an quasi-inertial system as well as in an earth-fixed reference system. The energy relations are demonstrated in case of high-low and low-low SST scenarios realised with the gravity field missions CHAMP and GRACE.