Constraining white dwarf viscosity through tidal heating in detached binary systems

Abstract

Although the internal structure of white dwarfs is considered to be generally well understood, the source and entity of their viscosity is still very uncertain. We propose here to study white dwarf viscous properties using short-period (<1 h), detached white dwarf binaries, such as the newly discovered ∼12.8 min system (J0651). These binaries are wide enough that mass transfer has not yet started but close enough that the secondary (least massive) component is subject to a measurable tidal deformation. The associated tidal torque transfers orbital energy, which is partially converted into heat by the action of viscosity as the secondary gets spun up. As a consequence, its outer non-degenerate layers expand, and the star puffs up. We selfconsistently calculate the fractional change in radius, and the degree of synchronization (ratio of stellar spin to orbital period) as a function of the viscous time. Specializing to the case of J0651, we find that an ∼10 per cent discrepancy between the measured radius of the secondary star and predictions of He white dwarf models can be interpreted as tidal inflation if the viscous time-scale is ∼4 × 104 yr. Such value is well in the range of various non-microscopic viscosities proposed in the literature like, e.g. tidally induced turbulence, non-linear damping of dynamical tides or internal magnetic stresses with a magnetic field strength ∼10-100 G. A 10 per cent tidal inflation is the maximum possible effect in J0651, at its current orbital separation, hence it selects a single value of the viscous time-scale: the latter implies that the system is still far from synchronization. Smaller effects of tidal inflation - well consistent with current uncertainties - would instead correspond to two different viscous time-scales, one longer and one shorter than 4 × 104 yr. In this more general case, the degeneracy can be broken by a joint measurement of the secondary's spin, since the two time-scales imply very different degrees of synchronization. Extrapolating the secondary's expansion into the future, we find that the star will fill its Roche lobe at a separation which is ∼1.2-1.5 smaller than the current one. Applying this method to a future sample of systems can allow us to learn whether viscosity changes with mass and/or nuclear composition. © 2014 The Authors. Published by Oxford University Press on behalf of the Royal Astronomical Society.