The Equations of Motion for Binary Systems with Relativistic Quadrupole-Quadrupole Moments Interaction

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The study of binary systems is one of the most important problems in astronomy. Especially recently, gravitational wave detection made possible by Laser Interferometer Gravitational wave Observatory - LIGO and VIRGO, LISA opens up a completely new window for the observation of our universe, it becomes one of the most important and forward area in the modern general relativistic astrophysics. Coalescing binary neutron star (NS) systems are believed to be the most important source emitted high-frequency gravitational wave. Therefore the study of NS coalescence is regarded as a major challenge in modern relativistic astrophysics. Indeed, if the two-body problem could be solved with a sufficient accuracy, the wealth of information might be extracted from the waveforms of coalescing binaries. Many early works to derive and investigate the gravitational two-body system with spin and quadrupole moment interaction have been done already. A detail appraisal of their works has been made by Xu, Wu and Schäfer where they derived the first post-Newtonian equations of motion for binary systems with monopole, spin and quadrupole interaction by making use of the scheme developed by Damour, Soffel and Xu (DSX)As we know, in the last stages of coalescence in binary system, the distance between two stars is closer, the tidal force is stronger, so the nonspherical size l is larger, l²/r² can reach the level of v²/c², where r is the distance between two bodies. In this case, the relativistic qudrupole-quadrupole term is of 3-PN order, so one can not neglect the 1-PN contribution of the q-q terms when 3-PN equations of motion (for mass-monopole) are considered. In order to fit the requirement of more accurate solution for binary system, the relativistic q-q terms in the post Newtonian equation of motion have been calculated in this paper. Our work is the first to obtain explicit 1-PN equations of motion for binary systems with relativistic quadrupole-quadrupole interaction in terms of only collective coordinates and B-D moments.