Resonant Tidal Excitations of Rotating Neutron Stars in Coalescing Binaries

In a coalescing neutron star-neutron star or neutron star-black hole binary, oscillation modes of the neutron star can be resonantly excited by the companion during the final minutes of the binary inspiral, when the orbital frequency sweeps up from a few Hertz to a few thousand Hertz. The resulting resonant energy transfer between the orbit and the neutron star speeds up or slows down the inspiral, depending on whether the resonant mode has positive or negative energy, and induces a phase change in the emitted gravitational waves from the binary. While only g-modes can be excited for a nonrotating neutron star, f-modes and r-modes can also be excited when the neutron star is spinning. A tidal resonance, designated by the index (jk,m) ({jk} specifies the angular order of the mode, as in the spherical harmonic Yjk), occurs when the mode frequency equals m times the orbital frequency. For the f-mode resonance to occur before coalescence, the neutron star must have rapid rotation, with spin frequency νs >∼ 710 Hz for the (22, 2)-resonance and νs >∼ 570 Hz for the (33, 3)-resonance (assuming canonical neutron star mass, M = 1.4 M⊙, and radius, R = 10 km; however, for R = 15 km, these critical spin frequencies become 330 Hz and 250 Hz, respectively). Because of their strong tidal coupling, the f-mode resonances induce a large change in the number of orbital cycles of coalescence, ∆Norb, with the maximum ∆Norb ∼ 10− 1000 for the (22, 2)-resonance and ∆Norb ∼ 1 for the (33, 3)-resonance. Such f-mode resonant effects, if present, must be included in constructing the templates of waveforms used in searching for gravitational wave signals. Higher order f-mode resonances can occur at slower rotation rates, but the induced orbital change is much smaller (∆Norb <∼ 0.1). For the dominant g-mode (22, 2)-resonance, even modest rotation (νs <∼ 100 Hz) can enhance the resonant effect on the orbit by shifting the resonance to a smaller orbital frequency. However, because of the weak coupling between the g-mode and the tidal potential, ∆Norb lies in the range 10 −10 (depending strongly on the neutron star equation of state) and is probably negligible for the purpose of detecting gravitational waves. Resonant excitations of r-modes require misaligned spin-orbit inclinations, and the dominant resonances correspond to octopolar excitations of the j = k = 2 mode, with (jk,m) = (22, 3) and (22, 1). Since the tidal coupling of the r-mode depends strongly on rotation rate, ∆Norb <∼ 10 (R/10 km)10(M/1.4M⊙) −20/3 is negligible for canonical neutron star parameters but can be appreciable if the neutron star radius is larger.