Neutrino Transport in Strongly Magnetized Proto-Neutron Stars and the Origin of Pulsar Kicks: I. The Effect of Parity Violation in Weak Interactions

In proto-neutron stars with strong magnetic fields, the neutrino-nucleon scattering/absorption cross sections depend asymmetrically on the direction of neutrino momentum with respect to the magnetic field axis, a manifestation of parity violation in weak interactions. We develop the moment formalism of neutrino transport in the presence of such asymmetric neutrino opacities. For a given neutrino species, there is a “drift flux” of neutrinos along the magnetic field in addition to the usual diffusive flux. Although the drift flux of νμ(τ) and that of ν̄μ(τ) completely cancel due to a crossing symmetry in the neutrino-nucleon scattering matrix element, there is a net drift flux associated with νe and ν̄e since a proto-neutron star contains more νe than ν̄e. This net drift flux induces globally asymmetric temperature and composition profiles in the star, and leads to asymmetric neutrino emission from the star. Although the asymmetry parameter ǫ (which is of order the nucleon polarization) in the neutrino opacity is small, significant asymmetry in neutrino emission can be obtained due to multiple neutrino-nucleon scatterings, with the maximum flux asymmetry proportional to ǫτ (where τ is the neutrino optical depth of the star). We also show that in local thermodynamical equilibrium, the asymmetry associated with neutrino absorption is cancelled by the asymmetry associated with neutrino emission, and therefore they do not contribute to the asymmetric neutrino flux in the bulk interior of the neutron star. We numerically evolve the asymmetric profiles of temperature and lepton number fractions of a newly-formed, magnetized neutron star through the deleptonization and thermal cooling phases. The fractional asymmetry of the neutrino flux grows during the deleptonization phase and eventually declines as the star becomes more neutronized. For an ordered magnetic field threading the neutron star interior, the fractional asymmetry in the total neutrino emission is about 0.006 (B/1014 G), corresponding to a pulsar kick velocity of about 200 (B/1014 G)km s−1 for a total radiated neutrino energy of 3× 1053 erg. This represents the most efficient way to generate pulsar kicks through asymmetric neutrino emission induced by magnetic fields.