Supernova Shocks Cannot Explain the Inflated State of Hypervelocity Runaways from White Dwarf Binaries

Recent observations have found a growing number of hypervelocity stars with speeds of $\approx 1500-2500\,$km\,s$^{-1}$ which could have only been produced through thermonuclear supernovae in white dwarf binaries. Most of the observed hypervelocity runaways in this class display a surprising inflated structure: their current radii are roughly an order of magnitude greater than they would have been as white dwarfs filling their Roche lobe. While many simulations exist studying the dynamical phase leading to supernova detonation in these systems, no detailed calculations of the long-term structure of the runaways have yet been performed. We use an existing \textsc{Arepo} hydrodynamical simulation of a supernova in a white dwarf binary as a starting point for the evolution of these stars with the 1 dimensional stellar evolution code MESA. We show that the supernova shock is not enough to inflate the white dwarf over timescales longer than a few thousand years, significantly shorter than the $10^{5-6}$ year lifetimes inferred for observed hypervelocity runaways. Despite experiencing a shock from a supernova less than $\approx 0.02\,R_\odot$ away, our models do not experience significant interior heating, and all contract back to radii around $0.01\,R_\odot$ within about $10^4$\,years. Explaining the observed inflated states requires either an additional source of significant heating or some other physics that is not yet accounted for in the subsequent evolution.