Arbitrarily Degenerate Helium White Dwarfs as Donors in AM Canum Venaticorum Binaries

We apply the Deloye & Bildsten isentropic models for donors in ultracompact low-mass X-ray binaries to the AM CVn population of ultracompact, interacting binaries. The mass-radius relations of these systems' donors in the mass range of interest (M₂<0.1Msolar) are not single-valued, but parameterized by the donor's specific entropy. This produces a range in the relationships between system observables, such as orbital period Porb and mass transfer rate M˙. For a reasonable range in donor specific entropy, M˙ can range over several orders of magnitude at fixed Porb. We determine the unique relation between M˙ and M₂ in the AM CVn systems with known donor-to-accretor mass ratios q=M₂/M₁. We use structural arguments, as well as each system's photometric behavior, to place limits on M˙ and M₂ in each. Most systems allow a factor of about 3 variation in M˙, although V803 Cen, if the current estimates of its q are accurate, is an exception and must have M₂~0.02Msolar and M˙~10⁻¹⁰Msolar yr⁻¹. Our donor models also constrain each donor's core temperature (Tc) range and correlate Tc with M₂. We examine how variations in donor specific entropy across the white dwarf family of AM CVn systems affects this population's current Galactic distribution. Allowing for donors that are not fully degenerate produces a shift in systems toward longer Porb and higher M˙, increasing the parameter space in which these systems can be found. This shift increases the fraction of systems whose Porb is long enough that their gravity wave (GW) signal is obscured by the background of detached double white dwarf binaries that dominate the GW spectrum below a frequency of ~2 mHz.