The Thermal Evolution of the Donors in AM CVn Binaries

(Abridged) We calculate the full stellar-structural evolution of donors in AM CVn systems formed through the WD channel coupled to the binary's evolution. Contrary to assumptions made in prior modelling, these donors are not fully convective over much of the AM CVn phase and do not evolve adiabatically under mass loss indefinitely. Instead, we identify three distinct phases of evolution: a mass transfer turn-on phase (during which the orbital period continues to decrease even after contact, the donor contracts, and the mass transfer rate accelerates to its maximum), a phase in which the donor expands adiabatically in response to mass loss, and a cooling phase beginning at orbital periods of approximately 45--55 minutes during which the donor contracts. The physics that determines the behaviour in the first and third phases, both of which are new outcomes of this study, are discussed in some detail. We find the overall duration of the turn-on phase to be between $\sim 10^4$-$10^6$ yrs, significantly longer than prior estimates. We predict the donor's luminosity and effective temperature. During the adiabatic expansion phase (ignoring irradiation effects), the luminosity is approximately $10^{-6}$--$10^{-4} L_\odot$ and the effective temperature is approximately 1000--1800 K. However, the flux generated in the accretion flow dominates the donor's intrinsic light at all times. The impact of irradiation on the donor extends the phase of adiabatic expansion to longer orbital periods and alters the donor's observational characteristics. Irradiated donors during the adiabatic phase can attain a surface luminosity of up to $\approx10^{-2} L_\odot$. We argue that the turn-on and cooling phases both will leave significant imprints on the AM CVn population's orbital period distribution.Comment: (20 pages, 20 figures, accepted to the Monthly Notices of the Royal Astronomical Society