Gravitomagnetic resonant excitation of Rossby modes in coalescing neutron star binaries

In coalescing neutron star binaries, r-modes in one of the stars can be resonantly excited by the gravitomagnetic tidal field of its companion. This post-Newtonian gravitomagnetic driving of these modes dominates over the Newtonian tidal driving previously computed by Ho and Lai. To leading order in the tidal expansion parameter R/r (where R is the radius of the neutron star and r is the orbital separation), only the l=2, vertical bar m vertical bar=1, and vertical bar m vertical bar=2 r-modes are excited. The tidal work done on the star through this driving has an effect on the evolution of the inspiral and on the phasing of the emitted gravitational wave signal. For a neutron star of mass M, radius R, spin frequency f(spin), modeled as a Gamma=2 polytrope, with a companion also of mass M, the gravitational wave phase shift for the m=2 mode is similar to 0.1 radians (R/10 km)(4)(M/1.4M)(-10/3)(f(spin)/100 Hz)(2/3) for optimal spin orientation. For canonical neutron star parameters this phase shift will likely not be detectable by gravitational wave detectors such as LIGO, but if the neutron star radius is larger it may be detectable if the signal-to-noise ratio is moderately large. The energy transfer is large enough to drive the mode into the nonlinear regime if f(spin)greater than or similar to 100 Hz. For neutron star-black hole binaries, the effect is smaller; the phase shift scales as companion mass to the -4/3 power for large companion masses. The net energy transfer from the orbit into the star is negative corresponding to a slowing down of the inspiral. This occurs because the interaction reduces the spin of the star, and occurs only for modes which satisfy the Chandrasekhar-Friedman-Schutz instability criterion. A large portion of the paper is devoted to developing a general formalism to treat mode driving in rotating stars to post-Newtonian order, which may be useful for other applications. We also correct some conceptual errors in the literature on the use of energy conservation to deduce the effect of the mode driving on the gravitational wave signal.