PHYSICAL REVIEW D VOLUME 45, NUMBER 4 15 FEBRUARY 1992 General-relativistic celestial mechanics II. Translational equations of motion Thibault Damour Institut des Hautes Etudes Scientigques, 91/$0 Bures su-r Yvette, France and Departement d'Astrophysique Relativiste et de Cosrnologie, Observatoire de Paris, Centre National de la Recherche Scientifique, gg195 Meudon CEDEX, France Michael Soffel and Chongming Xu' Theoretische Astrophysik, Universitat Tiibingen, Aug der Morgenstelle 10, 7/00 Tubingen, Germany (Received 18 October 1991) The translational laws of motion for gravitationally interacting systems of N arbitrarily composed and shaped, weakly self-gravitating, rotating, deformable bodies are obtained at the first post- Newtonian approximation of general relativity. The derivation uses our recently introduced multi- reference-system method and obtains the translational laws of motion by writing that, in the local center-of-mass kame of each body, relativistic inertial effects combine with post-Newtonian self- and externally generated gravitational forces to produce a global equilibrium (relativistic generalization of d Alembert s principle). Within the first post-Newtonian approximation [i.e. , neglecting terms of order (v/c) in the equations of motion], our work is the first to obtain complete and explicit results, in the form of infinite series, for the laws of motion of arbitrarily composed and shaped bodies. We first obtain the laws of motion of each body as an infinite series exhibiting the coupling of all the (Blanchet-Damour) post-Newtonian multipole moments of this body to the post-Newtonian tidal moments (recently defined by us) felt by this body. We then give the explicit expression of these tidal moments in terms of the post-Newtonian multipole moments of the other bodies. PACS number(s): 95.10.Ce, 04.20.Jb, 04.20.Me I. INTRODUCTION The present paper is a continuation of a paper by Damour, Soffel and Xu [1) (DSX, hereafter referred to as paper I), in which a new formalism was presented for treating the general-relativistic celestial mechanics of sys- tems of N arbitrarily composed and shaped, weakly self- gravitating, rotating, deformable bodies. We refer to the Introduction of paper I for the general motivation of this formalism and for references to previous works using a similar approach (notably Refs. 25 — 28 and 31 — 34 of pa- per I). The main characteristic of this formalism is to use, in a complementary manner, N+ 1 coordinate charts (or "reference systems") to treat the general-relativistic N body dynamics of the N bodies ("ephemerides"), plus N "local" charts adapted to the separate description of the structure and environment of each body. In paper I we laid the foundations of our formalism, and presented the general theory of the definition and construction of the N local reference systems [coordinates X& (cs = 0, 1, 2, 3; A = 1, ... , N)]. In particular, we were able (by freezing down in a particular way the spatial coordinate freedom) to give in closed form the transformation mapping the local coordinates, X&, onto the global ones: 'On leave from Fudan University, Shangai, People's Repub- lic of China. Present address: Department of Applied Math- ematics, University of Cape Town, Cape Town, South Africa. z" = f" (XA. , 'DA), where DA denotes some structures ("world-line data" ) which determine the precise choice of the Ath local ref- erence system. For the convenience of the reader we shall summarize in Sec. II below the notation and the main results of pa- per I that we shall need in the present paper. The full form of Eq. (1.1) will be found in Sec. II, but for our present introductory purpose we need only remark that the essential degrees of freedom among the data Bg ap- pearing in Eq. (1.1) are exactly the same as the ones needed in Newtonian mechanics, i.e. , the choice of an arbitrarily moving origin for each local reference system (technically, the choice of N "central world lines", CA), and the choice of the arbitrarily, but slowly, changing ro- tational state of the spatial coordinate grid (technically, the choice of a one-parameter family of 3x3 orthogonal matrices along each ZA). The present paper will complete the results of paper I by deriving, in explicit form, the translational equa- tions of motion of the N bodies. Let us start by clar- ifying the various meanings, within our formalism, of the expression "equations of motion, " and thereby the method we shall employ to derive them. First, let us follow Havas and Goldberg [2] in stressing the necessity to carefully distinguish between the concept of "laws of motion, " i.e. , some general mathematical expression re- lating some (more or less) specified collective coordinate for, say, body A to some, not yet specified, "external force" (e.g. , dpi'/dt = FA), and the concept of "equa- tions of motion, " i.e. , the mathematical expression o~- 45 1017 1992 The American Physical Society