Tidal excitations of oscillation modes in compact white dwarf binaries - I. Linear theory

We study the tidal excitation of gravity modes (g-modes) in compact white dwarf binary systems with periods ranging from minutes to hours. As the orbit of the system decays via gravitational radiation, the orbital frequency increases and sweeps through a series of resonances with the g-modes of the white dwarf. At each resonance, the tidal force excites the g-mode to a relatively large amplitude, transferring the orbital energy to the stellar oscillation. We calculate the eigenfrequencies of g-modes and their coupling coefficients with the tidal field for realistic non-rotating white dwarf models. Using these mode properties, we numerically compute the excited mode amplitude in the linear approximation as the orbit passes though the resonance, including the back reaction of the mode on the orbit. We also derive analytical estimates for the mode amplitude and the duration of the resonance, which accurately reproduce our numerical results for most binary parameters. We find that the g-modes can be excited to a dimensionless (mass-weighted) amplitude up to 0.1, with the mode energy approaching 10-3 of the gravitational binding energy of the star. Therefore the low-frequency (less than or similar to 10-2 Hz) gravitational waveforms produced by the binaries, detectable by LISA, are strongly affected by the tidal resonances. Our results also suggest that thousands of years prior to the binary merger, the white dwarf may be heated up significantly by tidal interactions. However, more study is needed since the physical amplitudes of the excited oscillation modes become highly non-linear in the outer layer of the star, which can reduce the mode amplitude attained by tidal excitation.