Thermal properties of supernova matter: The bulk homogeneous phase

We investigate the thermal properties of the potential model equation of state of Akmal, Pandharipande, and Ravenhall. This equation of state approximates the microscopic model calculations of Akmal and Pandharipande, which feature a neutral pion condensate. We treat the bulk homogeneous phase for isospin asymmetries ranging from symmetric nuclear matter to pure neutron matter and for temperatures and densities relevant for simulations of core-collapse supernovae, protoneutron stars, and neutron star mergers. Numerical results of the state variables are compared with those of a typical Skyrme energy density functional with similar properties at nuclear densities but which differ substantially at supranuclear densities. Analytical formulas, which are applicable to nonrelativistic potential models such as the equations of state we are considering, are derived for all state variables and their thermodynamic derivatives. A highlight of our work is its focus on thermal response functions in the degenerate and nondegenerate situations, which allow checks of the numerical calculations for arbitrary degeneracy. These functions are sensitive to the density-dependent effective masses of neutrons and protons, which determine the thermal properties in all regimes of degeneracy. We develop the "thermal asymmetry free energy" and establish its relation to the more commonly used nuclear symmetry energy. We also explore the role of the pion condensate at supranuclear densities and temperatures. Tables of matter properties as functions of baryon density, composition (i.e., proton fraction), and temperature are being produced which are suitable for use in astrophysical simulations of supernovae and neutron stars.