acta physica slovaca vol. 60 No. 4, 393 – 495 August 2010 POST-NEWTONIAN REFERENCE FRAMES FOR ADVANCED THEORY OF THE LUNAR MOTION AND FOR A NEW GENERATION OF LUNAR LASER RANGING Yi Xie∗y and Sergei Kopeikin1y ∗Astronomy Department, Nanjing University, Nanjing, Jiangsu 210093, China yDepartment of Physics & Astronomy, University of Missouri, Columbia, Missouri 65211, USA Received 3 June 2010, accepted 10 June 2010 We overview 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). We employ a scalar-tensor theory of gravity depending on two post-Newtonian pa- rameters, β and γ, and utilize 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 en- tire space-time, has its origin at the solar-system 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 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 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 – geocentric (GRF) and selenocentric (SRF) – 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 their relative motion with respect to each other. Theoretical advantage of the dynamically non-rotating local frames is in a more simple mathematical description. Each local frame can be aligned with the axes of ICRF after applying the matrix of the rela- tivistic precession. The set of one 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 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. DOI: 10.2478/v10155-010-0004-0 PACS: 04.20.Gz, 04.80.-y, 95.55.Br, 96.15.Vx Relativity, Gravitation, Gravitomagnetism, Gauge invariance, Brans- KEYWORDS: Dicke theory, Post-Newtonian celestial mechanics, Reference frames, Earth-Moon system, Lunar laser ranging 1E-mail address: [email protected] 393 394 Reference Frames for Advanced Theory of the Lunar Motion Contents 1 Introduction 396 1.1 Background..................................... 396 1.2 Lunar Laser Ranging................................ 397 1.3 EIH Equations of Motion in N-body Problem................... 398 1.4 Gravitoelectric and Gravitomagnetic Forces.................... 402 1.5 The Principle of Equivalence in the Earth-Moon System............. 403 1.6 The Residual Gauge Freedom........................... 405 1.7 Towards a New Lunar Ephemeris.......................... 408 1.8 Main Objectives of The Present Paper....................... 410 2 The Scalar-Tensor Theory of Gravity 412 2.1 The Field Equations................................. 412 2.2 The Energy-Momentum Tensor........................... 414 3 Theoretical Principles of the Post-Newtonian Celestial Mechanics 416 3.1 External and Internal Problems of Motion..................... 416 3.2 Post-Newtonian Approximations.......................... 417 3.2.1 Small Parameters.............................. 417 3.2.2 The Post-Newtonian Series......................... 419 3.3 The Post-Newtonian Field Equations........................ 421 3.4 Conformal Harmonic Coordinates......................... 422 3.5 Microscopic Post-Newtonian Equations of Motion................. 424 4 Post-Newtonian Reference Frames 425 4.1 Coordinates and Observables............................ 425 4.2 The Solar System Barycentric Frame........................ 426 4.2.1 Boundary Conditions and Kinematic Properties.............. 426 4.2.2 The Metric Tensor and Scalar Field.................... 428 4.2.3 The Post-Newtonian Conservation Laws.................. 430 4.3 The Earth-Moon Barycentric Frame........................ 432 4.3.1 The Boundary Conditions and Dynamic Properties............ 432 4.3.2 The Metric Tensor and Scalar Field.................... 434 4.3.3 Internal Multipoles of the Earth-Moon System.............. 441 4.4 The Geocentric Frame............................... 443 4.4.1 The Boundary Conditions and Dynamic Properties............ 443 4.4.2 The Metric Tensor and Scalar Field.................... 443 4.4.3 Gravitational Multipoles of Earth..................... 447 4.5 The Selenocentric Frame.............................. 449 4.5.1 The Boundary Conditions and Dynamic Properties............ 449 4.5.2 The Metric Tensor and Scalar Field.................... 450 4.5.3 Gravitational Multipoles of Moon..................... 454 CONTENTS 395 5 Post-Newtonian Transformations Between Reference Frames 456 5.1 Transformation from the Earth-Moon to the Solar-System Frame......... 456 5.1.1 General Structure of the Transformation.................. 456 5.1.2 Matching the Post-Newtonian Expansions................. 461 5.1.3 Post-Newtonian Coordinate Transformation................ 463 5.1.4 The External Multipoles.......................... 464 5.2 Transformation from the Geocentric to the Earth-Moon Frame.......... 466 5.2.1 Matching Procedure............................ 466 5.2.2 Post-Newtonian Coordinate Transformation................ 470 5.2.3 The External Multipoles.......................... 472 5.3 Transformation from the Selenocentric to the Earth-Moon Frame......... 475 5.3.1 Matching Procedure............................ 475 5.3.2 Post-Newtonian Coordinate Transformation................ 476 5.3.3 The External Multipoles.......................... 478 6 Conclusion 482 Acknowledgment 482 References 483 396 Reference Frames for Advanced Theory of the Lunar Motion 1 Introduction 1.1 Background The tremendous progress in technology, which we have witnessed during the last 30 years, has led to enormous improvements of precision in the measuring time and distances within the boundaries of the solar system. Further significant growth of the accuracy of astronomical obser- vations is expected in the course of time. Observational techniques like lunar and satellite laser ranging, radar and Doppler ranging, very long baseline interferometry, high-precision atomic clocks, gyroscopes, etc. have made it possible to start probing not only the static but also kine- matic and dynamic effects in motion of celestial bodies to unprecedented level of fundamental interest. Current accuracy requirements make it inevitable to formulate the most critical astro- nomical data-processing procedures in the framework of Einstein’s general theory of relativity. This is because major relativistic effects are several orders of magnitude larger than the technical threshold of practical observations and in order to interpret the results of such observations, one has to build physically-adequate relativistic models. Many current and planned observational projects and specialized space missions can not achieve their goals unless the relativity is taken into account properly. The future projects will require introduction of higher-order relativistic models supplemented with the corresponding parametrization of the relativistic effects, which will affect the observations. The dynamical modeling for the solar system (major and minor planets), for deep space nav- igation, and for the dynamics of Earth’s satellites and Moon must be consistent with general relativity. Lunar laser ranging (LLR) measurements are particularly important for testing general relativistic predictions and for advanced exploration of other laws of fundamental gravitational physics. Current LLR technologies allow us to arrange the measurement of the distance from a laser on Earth to a corner-cube reflector (CCR) on Moon with a precision approaching 1 mil- limeter (Battat et al, 2007; Murphy et al, 2008). There is a proposal to place a new CCR array on Moon (Currie et al, 2008), and possibly to install other devices such as microwave transponders (Bender et al, 1990) for multiple scientific and technical purposes. Successful human exploration of Moon strongly demands further significant improvement of the theoretical model of the orbital and rotational dynamics of the Earth-Moon system. This model should inevitably be based on the theory of general relativity, fully incorporate the relevant geophysical processes, lunar libration, tides, and should rely upon the most recent standards and recommendations of the IAU for data analysis (Soffel et al, 2003). The present paper discusses relativistic reference frames in construction of the high-precise dynamical model of motion of Moon and Earth. The model will take into account all the clas- sical