Arbitrarily Degenerate Helium White Dwarfs as Donors in AM CVn Binaries

We apply the Deloye & Bildsten (2003) 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 ($M_2<0.1 \msun$) 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, $\Mdot$. For a reasonable range in donor specific entropy, $\Mdot$ can range over several orders of magnitude at fixed $\Porb$. We determine the unique relation between $\Mdot$ and $M_2$ in the AM CVn systems with known donor to accretor mass ratios, $q=M_2/M_1$. We use structural arguments, as well as each system's photometric behavior, to place limits on $\Mdot$ and $M_2$ in each. Most systems allow a factor of about 3 variation in $\Mdot$, although V803 Cen, if the current estimates of its $q$ are accurate, is an exception and must have $M_2 \approx 0.02 \msun$ and $\Mdot \approx 10^{-10} \msun$ yr$^{-1}$. Our donor models also constrain each donor's core temperature, $T_c$, range and correlate $T_c$ with $M_2$. We examine how variations in donor specific entropy across the white dwarf family \citep{nele01a} of AM CVn systems affects this population's current galactic distribution. Allowing for donors that are not fully degenerate produces a shift in systems towards longer $\Porb$ and higher $\Mdot$ 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 $\approx 2$ mHz.