On the resonant detonation of sub-Chandrasekhar mass white dwarfs during binary inspiral

White dwarfs (WDs) are believed to detonate via explosive Carbon-fusion in a Type Ia supernova (SN) when their temperature and/or density reach the point where Carbon is ignited in a runaway reaction. Observations of the Type Ia SN rate imply that all WD binaries that merge through the emission of gravitational radiation within a Hubble time should result in SNe, regardless of total mass. Here we investigate the conditions under which a single WD in a binary system might extract energy from its orbit, depositing enough energy into a resonant mode such that it detonates before merger. We show that, ignoring non-linear effects in a WD binary in tidal lock at small binary separations, the sustained tidal forcing of a low-order quadrupolar g mode or a harmonic of a low-order quadrupolar p mode could, in principle, drive the average temperature of Carbon nuclei in the mode over the runaway fusion threshold. If growing mode energy is thermalized at a core/atmosphere boundary, rapid Helium burning and inwards-travelling p-waves may result in core detonation. Thermalization at a boundary in the core can also result in detonation. If energy can be efficiently transferred from the orbit to modes as the WD binary passes through resonances, the WD merger time-scale will be shortened by Myr-Gyr compared to expected time-scales from gravitational wave (GW)-emission alone and GW detectors will observe deviations from predicted chirp profiles in resolved WD binaries. Future work in this area should focus on whether tidal locking in WD binaries is naturally driven towards low-order mode frequencies.