Resupplying planetary debris to old white dwarfs with supernova blast waves

Abstract

One challenge with explaining how high levels of planetary debris can enrich, or "pollute", old ($\sim$3 Gyr) and very old ($\sim$10 Gyr) white dwarfs is that debris reservoirs deplete on shorter timescales, akin to the solar system's already eviscerated Main Belt and Kuiper Belt. Here, I explore how these extrasolar reservoirs can be resupplied through supernovae that propel distant ($\gtrsim 10^4$ au) dust, sand and pebbles, and potentially boulders and comets, into the inner ($\lesssim 10^2$ au) planetary system. I analytically constrain the geometry of these blast waves, and derive expressions for the probability of apt blast configurations occurring. I then derive the minimum kick magnitudes needed to generate stable, leaky and broken post-blast orbits, and prove that within this formalism, at most 23 per cent of true anomalies along an eccentric orbit could allow for resupplied planetary debris to experience repeated pericentre passages. By linking these kick magnitudes with debris sizes and relating these quantities to the local neighbourhood supernova rate, I conclude that the probabilities for ejection or resupply per supernova blast are $\approx$100 per cent for micron-sized dust and millimetre-sized pebbles and sand, and $\approx$0 per cent for asteroids larger than $\sim$10 km. In between these extremes, I expect metre-sized boulders to be resupplied at least once to very old white dwarfs over their cooling ages. The efficacy of this debris delivery mechanism is dependent on the time-varying sources and sinks in an exo-Oort cloud and how its parent white dwarf has, throughout its cooling age, traversed the Milky Way.