Accretion Rates and Thermohaline Convection in Polluted White Dwarfs
Polluted white dwarfs provide a unique window into the composition of exoplanetary material, but interpreting their surface abundances requires a quantitative understanding of the mixing processes in their envelopes. Accretion of metal-rich debris onto hydrogen-rich white dwarfs drives thermohaline (fingering) convection in the underlying radiative zone. Previous studies have modeled this as enhanced diffusive transport, often inferring mass accretion rates, $\dot{M}_{\rm acc}$, that exceed constraints from X-ray observations. Here 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 marginal stability. Using this condition, we derive analytic solutions in shallow and deep stratification regimes, finding that the mixed layer depth, $h$, and surface abundance, $X_{\rm surf}$, 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. The mixed layer reaches depths of only $\sim 100$ km over $10^5$ yr and, without gravitational settling, never reaches a steady state. Including settling leads to an equilibrium surface abundance scaling as $X_{\rm surf, eq} \propto \dot{M}_{\rm acc}^{30/43}$. Applied to G 29--38, the model matches the observed heavy-element abundance for $\dot{M}_{\rm acc} \sim 2\times 10^9$--$4\times 10^{9}~\mathrm{g~s^{-1}}$, in better agreement with X-ray constraints than prior estimates incorporating thermohaline mixing. We also present a general scaling relation for inferring accretion rates directly from observed surface abundances and stellar properties, giving a practical prescription for other polluted white dwarfs.