arXiv:gr-qc/0703054v1 8 Mar 2007 February 8, 2020 6:36 WSPC - Proceedings Trim Size: 9.75in x 6.5in main ORBITAL PHASE IN INSPIRALLING COMPACT BINARIES * M ´ ATY ´ AS VAS ´ UTH † , BAL ´ AZS MIK ´ OCZI ‡ and L ´ ASZL ´ O ´ A. GERGELY † † KFKI Research Institute for Particle and Nuclear Physics Budapest 114, P.O.Box 49, H-1525, Hungary ‡ Departments of Theoretical and Experimental Physics, University of Szeged D´ om t´ er 9, Szeged H-6720, Hungary vasuth@rmki.kfki.hu, mikoczi@titan.physx.u-szeged.hu, gergely@physx.u-szeged.hu We derive the rate of increase of the orbital frequency up to the second post-Newtonian order for inspiralling compact binaries with spin, mass quadrupole and magnetic dipole moments on eccentric orbits. We give this result in terms of orbital elements. Keywords : compact binaries, post-Newtonian expansion, spin, quadrupole moment Observations by Earth-based gravitational wave observatories are under way aiming to detect gravitational radiation. Upper limits from interferometer data were already set on inspiral event rates for both binary neutron stars 1 and binaries of 3 − 20 solar mass black holes. 2 The parameters of spinning compact binaries can be estimated and alternative theories of gravity can also be tested from these measurements. 3 An important characteristic of these binaries is the rate of decrease of the orbital period T due to the energy and angular momentum carried away by gravitational waves. Here we give the radiative change of the mean motion n =2π/T (for eccentric orbits). We also present the related change occured in the orbital phase (for circular orbits). In both expressions we include all known linear perturbations for an isolated compact binary. These are the post-Newtonian (PN), spin-orbit (SO), spin-spin (SS), self spin (Self, quadratic in the single spins), quadrupole-monopole (QM) and magnetic dipole-magnetic dipole (DD) contributions. The expression of the radial period, defined as half of the time elapsed between the turning points, emerges from generic considerations on the perturbed Keplerian motion. 4,5 Collecting all linear contributions the mean motion has the following form n = (2E ) 3/2 Gm  1 − (15 − η) E 4c 2  , (1) where η = µ/m is the ratio of the reduced mass µ to the total mass m of the binary system, and E = −E/µ where E is the conserved energy. Remarkably there are no explicit spin, quadrupolar and magnetic dipolar contributions in the functional form of the mean motion. These however contribute implicitly to n through E . Since the mean motion is a function of E alone, its evolution can be computed as * Research supported by OTKA grants nos. T046939, TS044665, F049429 and the J´ anos Bolyai Fellowships of the Hungarian Academy of Sciences. M.V. and L. ´ A.G. wish to thank the organizers of the 11th Marcel Grossmann Meeting for support. 1