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 $β$ equilibrium, quantified by the chemical potential difference $δμ\equivμ_n-μ_p-μ_e$. 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 $β$ decay and capture rates), $Γ_{\rm net}(T,δμ)$, and the total emissivity (total energy emission rate in the form of neutrinos and antineutrinos), $ε_{\rm tot}(T,δμ)$. Here, we present a simple and general relationship between these variables, ${\partialε_{\rm tot}/\partialδμ=3Γ_{\rm net}}$, 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.