Pulsar kicks with modified Urca and electrons in Landau levels

We derive the energy asymmetry given the protoneutron star during the time when the neutrino sphere is near the surface of the protoneutron star, using the modified Urca process. The electrons produced with the antineutrinos are in Landau levels due to the strong magnetic field, and this leads to asymmetry in the neutrino momentum, and a pulsar kick. The magnetic field must be strong enough for a large fraction of the electrons to be in the lowest Landau level; however, there is no direct dependence of our pulsar velocity on the strength of the magnetic field. Our main prediction is that the large pulsar kicks start at about 10 s and last for about 10 s, with the corresponding neutrinos correlated with the direction of the magnetic field. We predict a pulsar velocity of 1.03×10^-4(T/10¹⁰   K)⁷  km/s, which reaches 1000  km/s if T≃10¹¹   K.