Non-Equilibrium Beta Processes in Neutron Stars: A Relationship between the Net Reaction Rate and the Total Emissivity of Neutrinos

Several different processes could be changing the density in the core of a neutron star, leading to a departure from $\beta$ equilibrium, quantified by the chemical potential difference $\delta\mu\equiv\mu_n-\mu_p-\mu_e$. The evolution of this quantity is coupled to that of the star's interior temperature $T$ by two functions that quantify the rate at which neutrino-emitting reactions proceed: the net reaction rate (difference between $\beta$ decay and capture rates), $\Gamma_{\rm net}(T,\delta\mu)$, and the total emissivity (total energy emission rate in the form of neutrinos and antineutrinos), $\epsilon_{\rm tot}(T,\delta\mu)$. Here, we present a simple and general relationship between these variables, ${\partial\epsilon_{\rm tot}/\partial\delta\mu=3\Gamma_{\rm net}}$, and show that it holds even in the case of superfluid nucleons. This relation may simplify the numerical calculation of these quantities, including superfluid reduction factors.