Mass Transfer and Stellar Evolution of the White Dwarfs in AM CVn Binaries

We calculate the stellar evolution of both white dwarfs (WDs) in AM CVn binaries with orbital periods of $P_{\mathrm{orb}} \approx 5-70$ minutes. We focus on the cases where the donor starts as a $M_{\mathrm{He}} < 0.2 \, M_{\odot}$ Helium WD and the accretor is a $M_{\mathrm{WD}} > 0.6 \, M_{\odot}$ WD. Using Modules for Experiments in Stellar Astrophysics (MESA), we simultaneously evolve both WDs assuming conservative mass transfer and angular momentum loss from gravitational radiation. This self-consistent evolution yields the important feedback of the properties of the donor on the mass transfer rate, $\dot{M}$, as well as the thermal evolution of the accreting WD. Consistent with earlier work, we find that the high $\dot{M}$'s at early times forces an adiabatic evolution of the donor for $P_{\mathrm{orb}} < 30$ minutes so that its mass-radius relation depends primarily on its initial entropy. As the donor reaches $ M_{\mathrm{He}} \approx 0.02-0.03 \, M_{\odot}$ at $P_{\mathrm{orb}} \simeq 30 $ minutes, it becomes fully convective and could lose entropy and expand much less than expected under further mass loss. However, we show that the lack of reliable opacities for the donor's surface inhibit a secure prediction for this possible cooling. Our calculations capture the core heating that occurs during the first $\approx 10^7$ years of accretion and continue the evolution into the phase of WD cooling that follows. When compared to existing data for accreting WDs, as seen by Cheng and collaborators for isolated WDs, we also find that the accreting WDs are not as cool as we would expect given the amount of time they have had to cool.