Non-dissipative tidal synchronization in accreting binary white dwarf systems

We study a non-dissipative hydrodynamical mechanism that can stabilize the spin of the accretor in an ultracompact double white dwarf (WD) binary. This novel synchronization mechanism relies on a non-linear coupling between tides and the uniform (or rigid) rotation mode, which spins down the background star. The essential physics of the synchronization mechanism is summarized as follows. As the compact binary coalesces due to gravitational wave emission, the largest star eventually fills its Roche lobe and accretion starts. The accretor then spins up due to infalling material and eventually reaches a spin frequency where a normal mode of the star is resonantly driven by the gravitational tidal field of the companion. If the resonating mode satisfies a set of specific criteria, which we elucidate in this paper, it exchanges angular momentum with the background star at a rate such that the spin of the accretor locks at this resonant frequency, even though accretion is ongoing. Some of the accreted angular momentum that would otherwise spin up the accretor is fed back to the orbit through this resonant tidal interaction. In this paper we solve analytically a simple dynamical system that captures the essential features of this mechanism. Our analytical study allows us to identify two candidate Rossby modes that may stabilize the spin of an accreting WD in an ultracompact binary. These two modes are the l = 4, m = 2 and l = 5, m = 3 Chandrasekhar–Friedman–Schutz (CFS) unstable hybrid r modes, which, for an incompressible equation of state, stabilize the spin of the accretor at frequency 2.6 ωorb and 1.54 ωorb, respectively, where ωorb is the binary’s orbital frequency. For an n = 3/2 polytrope, the accretor’s spin frequency is stabilized at 2.13 ωorb and 1.41 ωorb, respectively. Since the stabilization mechanism relies on continuously driving a mode at resonance, its lifetime is limited since eventually the mode amplitude saturates due to non-linear mode– mode coupling. Rough estimates of the lifetime of the effect lie from a few orbits to possibly millions of years. We argue that one must include this hydrodynamical stabilization effect to understand stability and survival rate of ultracompact binaries, which is relevant in predicting the Galactic WD gravitational background that LISA will observe.