Neutrinos decoupled from $β$-processes and supernova explosion
Based on the gravitational collapse time-scale is larger than the weak interaction time-scale at core densities $ρ> 10^{11} {gr}/ {cm}^{3}$, we approximately use the $β$-equilibrium condition and particle number conservations to calculate the number and energy densities of neutrino sphere in the process of gravitational core collapse towards the formation of a proto-neutron star. We find that at core densities $ρ_{dec} > 10^{12} {gr}/ {cm}^{3}$, the $β$-equilibrium condition cannot be satisfied consistently with charge, baryon and lepton number conservations, leading to the presence of excess neutrinos decoupling from the $β$-equilibrium. These excess neutrinos interact with nucleons and electrons via the neutral current channel only and their diffusion time is about $10^{-2}$ sec. The excess neutrino flux could play an important role in an Supernova explosion, provided the fraction of excess neutrinos over all neutrinos is at least one present.