General relativistic observables of the GRAIL mission

We present a realization of astronomical relativistic reference frames in the Solar System and its application to the Gravity Recovery and Interior Laboratory (GRAIL) mission. We model the necessary space-time coordinate transformations for light-trip time computations and address some practical aspects of the implementation of the resulting model. We develop all the relevant relativistic coordinate transformations that are needed to describe the motion of the GRAIL spacecraft and to compute all observable quantities. We take into account major relativistic effects contributing to the dual one-way range observable, which is derived from one-way signal travel times between the two GRAIL spacecrafts. We develop a general relativistic model for this fundamental observable of GRAIL, accurate to $1\mu \mathrm{m}$. We develop and present a relativistic model for another key observable of this experiment, the dual one-way range rate, accurate to $1\mu \mathrm{m}/\mathrm{s}$. The presented formulation justifies the basic assumptions behind the design of the GRAIL mission. It may also be used to further improve the already impressive results of this lunar gravity recovery experiment after the mission is complete. Finally, we present transformation rules for frequencies and gravitational potentials and their application to GRAIL.

Figure 1

Representative geometry (not to scale) of the vectors involved in the computation of the GRAIL observables. “SSB” is the Solar System barycenter, “E” is the center of the Earth, “M” is the center of Moon, “EMB” is the Earth-Moon Barycenter. “A” and “B” are the positions of the GRAIL-A and GRAIL-B spacecraft, respectively, and “C” is the position of the DSN tracking antenna on the surface of the Earth.

Figure 2

Timing events on GRAIL: Depicted (not to scale) are the worldlines of the GRAIL-A and GRAIL-B spacecrafts with corresponding proper times ${\tau }_{\mathrm{A}}$ and ${\tau }_{\mathrm{B}}$. Ka-band signals that are emitted from points $A_0$ and $B_0$ are received at $B$ and $A$, respectively. The times $t_{\mathrm{A}0}$, $t_{\mathrm{B}0}$, $t_{\mathrm{A}}$ and $t_{\mathrm{B}}$ are the coordinate times at these points, as measured in the BCRS.