Neutrino transport in strongly magnetized proto-neutron stars and the origin of pulsar kicks: The effect of asymmetric magnetic field topology

In proto-neutron stars with strong magnetic fields, the cross section for ν_e (ν̄_e) absorption on neutrons (protons) depends on the local magnetic field strength resulting from the quantization of energy levels for the e^- (e⁺) produced in the final state. If the neutron star possesses an asymmetric magnetic field topology in the sense that the magnitude of magnetic field in the north pole is different from that in the south pole, then asymmetric neutrino emission may be generated. We calculate the absorption cross sections of ν_e and ν̄_e in strong magnetic fields as a function of the neutrino energy. These cross sections exhibit oscillatory behaviors that occur because new Landau levels for the e^- (e⁺) become accessible as the neutrino energy increases. By evaluating the appropriately averaged neutrino opacities, we demonstrate that the change in the local neutrino flux caused by the modified opacities is rather small. To generate appreciable kick velocity (∼300 km s^-1) to the newly formed neutron star, the difference between the field strengths at the two opposite poles of the star must be at least 10¹⁶ G. We also consider the magnetic field effect on the spectral neutrino energy fluxes. The oscillatory features in the absorption opacities give rise to modulations in the emergent spectra of ν_e and ν̄_e.