Stochastic accretion of planetesimals on to white dwarfs: constraints on the mass distribution of accreted material from atmospheric pollution

This paper explores how the stochastic accretion of planetesimals on to white dwarfs would be manifested in observations of their atmospheric pollution. Archival observations of pollution levels for unbiased samples of DA and non-DA white dwarfs are used to derive the distribution of inferred accretion rates, confirming that rates become systematically lower as sinking time (assumed here to be dominated by gravitational settling) is decreased, with no discernable dependence on cooling age. The accretion rates expected from planetesimals that are all the same mass (i.e., a mono-mass distribution) are explored both analytically and using a Monte Carlo model, quantifying how measured accretion rates inevitably depend on sinking time, since different sinking times probe different times since the last accretion event. However, that dependence is so dramatic that a mono-mass distribution can be excluded within the context of this model. Consideration of accretion from a broad distribution of planetesimal masses uncovers an important conceptual difference: accretion is continuous (rather than stochastic) for planetesimals below a certain mass, and the accretion of such planetesimals determines the rate typically inferred from observations; smaller planetesimals dominate the rates for shorter sinking times. A reasonable fit to the observationally inferred accretion rate distributions is found with model parameters consistent with a collisionally evolved mass distribution up to Pluto-mass, and an underlying accretion rate distribution consistent with that expected from descendants of debris discs of main-sequence A stars. With these parameters, while both DA and non-DA white dwarfs accrete from the same broad planetesimal distribution, this model predicts that the pollution seen in DAs is dominated by the continuous accretion of < 35 km objects, and that in non-DAs by > 35 km objects (though the dominant size varies between stars by around an order of magnitude from this reference value). Furthermore, observations that characterize the dependence of inferred accretion rates on sinking time and cooling age (including a consideration of the effect of thermohaline convection on models used to derive those rates), and the decadal variability of DA accretion signatures, will improve constraints on the mass distribution of accreted material and the lifetime of the disc through which it is accreted.