Electron $\nu \bar{\nu}$ Bremsstrahlung in a Liquid Phase of Neutron Star Crusts

Neutrino emissivity from the electron $\nu\bar\nu$ bremsstrahlung in the liquid layers of the neutron star crusts is studied. Nuclear composition of matter in neutron star crusts is considered for various scenarios of neutron star evolution. For the deep layers of the crust, the compositions of cold catalyzed matter, accreted matter and hot matter ($T \ga 5 \times 10^9$~K) are shown to be very different, and this implies differences in the neutrino emissivity at given density and temperature. Neutrino-pair bremsstrahlung, due to collisions of relativistic degenerate electrons in a Coulomb liquid of atomic nuclei, is considered. The neutrino energy loss rate is expressed in a simple form through a Coulomb logarithm $L$ -- a slowly varying function of density, temperature, and nuclear composition. Non-Born corrections, thermal width of electron Fermi sphere, and finite sizes of nuclei are taken into account. Implications for cooling of neutron stars are discussed.