White dwarf donors in ultracompact binaries: The stellar structure of finite-entropy objects

We discuss the mass-radius (M-R) relations for low-mass (M < 0.1 M-.) white dwarfs (WDs) of arbitrary degeneracy and evolved (He, C, O) composition. We do so with both a simple analytical model and models calculated by integration of hydrostatic balance using a modern equation of state valid for fully ionized plasmas. The M-R plane is divided into three regions where either Coulomb physics, degenerate electrons, or a classical gas dominates the WD structure. For a given M and central temperature T-c, the M-R relation has two branches differentiated by the model's entropy content. We present the M-R relations for a sequence of constant-entropy WDs of arbitrary degeneracy parameterized by M and Tc for pure He, C, and O. We discuss the applications of these models to the recently discovered accreting millisecond pulsars. We show the relationship between the orbital inclination for these binaries and the donor's composition and Tc. In particular, we find from orbital inclination constraints that the probability XTE J1807 - 294 can accommodate a He donor is approximately 15%, while for XTE J0929 - 304 it is approximately 35%. We argue that if the donors in ultracompact systems evolve adiabatically, there should be 60 - 160 more systems at orbital periods of 40 minutes than at orbital periods of 10 minutes, depending on the donor's composition. Tracks of our mass-radius relations for He, C, and O objects are available in the electronic version of this paper.