Accretion Rates and Thermohaline Convection in Polluted White Dwarfs
The atmospheres of polluted white dwarfs have information about the composition of disrupted planetary material. However, interpreting the observed abundances requires a quantitative description of the mixing processes that transport such material through their envelopes. The accretion of metal-rich debris drives thermohaline convection in the star's radiative zone. Previous studies have modeled this form of turbulence as an enhanced diffusive process, often inferring mass accretion rates $\dot{M}_{\rm acc}$ that exceed X-ray constraints. We develop a time-dependent model that treats thermohaline mixing as a propagating front whose evolution is coupled self-consistently to the surface abundance of heavy elements. We perform hydrodynamical simulations which demonstrate that thermohaline mixing drives the composition gradient towards values slightly above marginal thermohaline stability. Using this condition, we derive analytic solutions for the mixed layer depth $h$ and surface abundance $X_{\rm surf}$, finding that both scale as $t^{1/2}$ at early times, transitioning to $h \propto t^{4/27}$ and $X_{\rm surf} \propto t^{10/27}$ at later times. Without gravitational settling, the thermohaline front never reaches a steady state, while including settling leads to an equilibrium surface abundance $X_{\rm surf, eq} \propto \dot{M}_{\rm acc}^{30/43}$. Applied to G 29--38, our model matches the observed heavy-element abundance for $\dot{M}_{\rm acc} \sim 1\times 10^{10}$--$2\times 10^{10}~\mathrm{g~s^{-1}}$, comparable to previous findings using stellar evolution codes. These rates exceed the X-ray estimate by about an order of magnitude, which may point to a partially suppressed X-ray luminosity from the accretion boundary layer. We present a practical prescription for inferring accretion rates directly from the observed surface abundance and stellar parameters of other polluted white dwarfs.