Post-Newtonian Reference Frames for Advanced Theory of the Lunar Motion and a New Generation of Lunar Laser Ranging

Abstract

We construct a set of post-Newtonian reference frames for a comprehensive study of the orbital dynamics and rotational motion of Moon and Earth by means of lunar laser ranging (LLR) with the precision of one millimeter. We work in the framework of a scalar-tensor theory of gravity depending on two parameters, and, of the parameterized post-Newtonian (PPN) formalism and utilize the concepts of the relativistic resolutions on reference frames adopted by the International Astronomical Union (IAU) in 2000. We assume that the solar system is isolated and space-time is asymptotically flat at infinity. The primary reference frame covers the entire space-time, has its origin at the solarsystem barycenter (SSB) and spatial axes stretching up to infinity. The SSB frame is not rotating with respect to a set of distant quasars that are assumed to be at rest on the sky forming the International Celestial Reference Frame (ICRF). The secondary reference frame has its origin at the Earth-Moon barycenter (EMB). The EMB frame is locally-inertial with its spatial axes spreading out to the orbits of Venus and Mars, and is not rotating dynamically in the sense that equation of motion of a test particle moving with respect to the EMB frame, does not contain the Coriolis and centripetal forces. Two other local frames - the geocentric (GRF) and the selenocentric (SRF) frames - have their origins at the center of mass of Earth and Moon respectively and do not rotate dynamically. Each local frame is subject to the geodetic precession both with respect to other local frames and with respect to the ICRF because of the relative motion. The advantage of dynamically non-rotating local frames is in a more simple mathematical description. The set of the global and three local frames is introduced in order to fully decouple the relative motion of Moon with respect to Earth from the orbital motion of the Earth-Moon barycenter as well as to connect the coordinate description of the lunar motion, an observer on Earth, and a retro-reflector on Moon to directly measurable quantities such as the proper time and the round-trip laser-light distance. We solve the gravity field equations and find out the metric tensor and the scalar field in all frames, which description includes the post-Newtonian definition of the multipole moments of the gravitational field of Earth and Moon. We also derive the post-Newtonian coordinate transformations between the frames and analyze the residual gauge freedom imposed by the scalar-tensor theory on the metric tensor. The residual gauge freedom is used for removal spurious, coordinate-dependent post-Newtonian effects from the equations of motion of Earth and Moon.