Non-linear dynamical tides in white dwarf binaries

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ABSTRACT Compact white dwarf (WD) binaries are important sources for space-based gravitational-wave (GW) observatories, and an increasing number of them are being identified by surveys like Extremely Low Mass (ELM) and Zwicky Transient Facility (ZTF). We study the effects of non-linear dynamical tides in such binaries. We focus on the global three-mode parametric instability and show that it has a much lower threshold energy than the local wave-breaking condition studied previously. By integrating networks of coupled modes, we calculate the tidal dissipation rate as a function of orbital period. We construct phenomenological models that match these numerical results and use them to evaluate the spin and luminosity evolution of a WD binary. While in linear theory the WD’s spin frequency can lock to the orbital frequency, we find that such a lock cannot be maintained when non-linear effects are taken into account. Instead, as the orbit decays, the spin and orbit go in and out of synchronization. Each time they go out of synchronization, there is a brief but significant dip in the tidal heating rate. While most WDs in compact binaries should have luminosities that are similar to previous traveling-wave estimates, a few per cent should be about 10 times dimmer because they reside in heating rate dips. This offers a potential explanation for the low luminosity of the CO WD in J0651. Lastly, we consider the impact of tides on the GW signal and show that the Laser Interferometer Space Antenna (LISA) and TianGO can constrain the WD’s moment of inertia to better than $1{{\ \rm per\ cent}}$ for centi-Hz systems.