Approximate Approaches for Nuclear Weak Interaction Rates in Astrophysics

Nuclear weak interactions, like beta decay, are important inputs for modeling astrophysical explosions. In the allowed approximation, these processes proceed as Fermi or Gamow Teller (GT) processes where the spins of the electron and neutrino are anti-parallel or parallel, respectively. In the GT case, transition probability is spread over many final states in the daughter nucleus, with each probability determination requiring numerical integration of the available phase space. Developing a fast and accurate method for calculating each contribution to the total decay rate would provide reliable weak rate libraries for astrophysical modelers. The phase space integrand includes the classical statistical factor, a Coulomb correction, and the Fermi Dirac distribution of continuum electrons in the stellar material. In this paper, we specifically examine the phase space integration and discuss various approximations to the Coulomb correction, comparing computational speed and numerical accuracy. An approximate approach that is fast and accurate is introduced