Energy dissipation through quasi-static tides in white dwarf binaries

We present a formalism to study tidal interactions in white dwarf binaries in the limiting case of quasi-static tides, in which the tidal forcing frequencies are small, compared to the inverse of the white dwarf's dynamical timescale. The formalism is valid for arbitrary orbital eccentricities and therefore applicable to white dwarf binaries in the Galactic disk as well as globular clusters. In the quasi-static limit, the total perturbation of the gravitational potential shows a phase shift with respect to the position of the companion, the magnitude of which is determined primarily by the efficiency of energy dissipation through convective damping. We determine rates of secular evolution of the orbital elements and white dwarf rotational angular velocity for a 0.3 M _· helium white dwarf in binaries with orbital frequencies in the Laser Interferometer Space Antenna (LISA) gravitational wave frequency band and companion masses ranging from 0.3 M _· to 10⁵ M _·. The resulting tidal evolution timescales for the orbital semimajor axis are longer than a Hubble time, so that convective damping of quasi-static tides need not be considered in the construction of gravitational wave templates of white dwarf binaries in the LISA band. Spin-up of the white dwarf, on the other hand, can occur on timescales of less than 10Myr, provided that the white dwarf is initially rotating with a frequency much smaller than the orbital frequency. For semi-detached white dwarf binaries spin-up can occur on timescales of less than 1Myr. Nevertheless, the timescales remain longer than the orbital inspiral timescales due to gravitational radiation, so that the degree of asynchronism in these binaries increases. As a consequence, tidal forcing eventually occurs at forcing frequencies beyond the quasi-static tide approximation. For the shortest period binaries, energy dissipation is therefore expected to take place through dynamic tides and resonantly excited g-modes.