Sinking and spreading of metal pollution in magnetic white dwarfs

Abstract

Observations of magnetic polluted white dwarfs indicate that most of them have higher concentrations of metals near the poles compared to lower latitudes. Maintaining such abundance gradients requires gravitational sinking times $t_\downarrow$ below the convection zone that are shorter than the horizontal spreading time $t_\leftrightarrow$ across the surface by convective eddies. We show analytically that $t_\downarrow/t_\leftrightarrow\propto T^{8/3}$, where $T$ is the temperature at the base of the convection zone, which rises by more than an order of magnitude as convection penetrates deeper into the atmosphere. Correspondingly, $t_\downarrow/t_\leftrightarrow$ jumps by several orders of magnitude, clearly delineating between hot white dwarfs with abundance variations and cold ones with homogenous surfaces. We incorporate magnetic fields self-consistently into the stellar structure and compute for the first time the sinking to spreading time-scale ratio as a function of $B$. Magnetic white dwarfs have shallower convection zones at a given $T_{\rm eff}$, and the inward penetration of convection shifts to lower $T_{\rm eff}$. Quantitatively, $B\sim 10^5\textrm{ G}$ extends the range for enhanced abundance patches from $T_{\rm eff}\gtrsim 30\,000 \textrm{ K}$ ($13\,000\textrm{ K}$) to $T_{\rm eff}\gtrsim 15\,000 \textrm{ K}$ ($6000\textrm{ K}$) for helium (hydrogen) dominated atmospheres, which is insufficient to explain the observed patchy white dwarfs, which are even colder. A possible solution is variability in the accretion rate on a time-scale of a decade, as recently detected in another white dwarf.