The importance of Urca-process cooling in accreting ONe white dwarfs

We study the evolution of accreting oxygen-neon (ONe) white dwarfs (WDs), with a particular emphasis on the effects of the presence of the carbon-burning products ²³Na and ²⁵Mg. These isotopes lead to substantial cooling of the WD via the ²⁵Mg-²⁵Na, ²³Na-²³Ne and ²⁵Na- ²⁵Ne Urca pairs. We derive an analytic formula for the peak Urca-process cooling rate and use it to obtain a simple expression for the temperature to which the Urca process cools the WD. Our estimates are equally applicable to accreting carbon-oxygen WDs. We use the Modules for Experiments in Stellar Astrophysics (MESA) stellar evolution code to evolve a suite of models that confirm these analytic results and demonstrate that Urca-process cooling substantially modifies the thermal evolution of accreting ONe WDs. Most importantly, we show that MESA models with lower temperatures at the onset of the ²⁴Mg and ²⁴Na electron captures develop convectively unstable regions, even when using the Ledoux criterion. We discuss the difficulties that we encounter in modelling these convective regions and outline the potential effects of this convection on the subsequent WD evolution. For models in which we do not allow convection to operate, we find that oxygen ignites around a density of log(ρc/g cm^-3) ≈ 9.95, very similar to the value without Urca cooling. Nonetheless, the inclusion of the effects of Urca-process cooling is an important step in producing progenitor models with more realistic temperature and composition profiles which are needed for the evolution of the subsequent oxygen deflagration and hence for studies of the signature of accretion-induced collapse.