Hydrodynamical Neutron-star Kicks in Electron-capture Supernovae and Implications for the CRAB Supernova

Neutron stars (NSs) obtain kicks of typically several 100 km/s at birth. The gravitational tug-boat mechanism can explain these kicks as consequences of asymmetric mass ejection during the supernova (SN) explosion. Support for this hydrodynamic explanation is provided by observations of SN remnants with associated NSs, which confirm the prediction that the bulk of the explosion ejecta, in particular chemical elements between silicon and the iron group, are dominantly expelled in the hemisphere opposite to the direction of the NS kick. Here, we present a large set of two- and three-dimensional explosion simulations of electron-capture SNe, considering explosion energies between ~3x10^49 erg and ~1.6x10^50 erg. We find that the fast acceleration of the SN shock in the steep density gradient delimiting the O-Ne-Mg core of the progenitor enables such a rapid expansion of neutrino-heated matter that the growth of neutrino-driven convection freezes out quickly in a high-mode spherical harmonics pattern. Since the corresponding momentum asymmetry of the ejecta is very small and the gravitational acceleration by the fast-expanding ejecta abates rapidly, the NS kick velocities are at most a few km/s. The extremely low core compactness of O-Ne-Mg-core progenitors therefore favors hydrodynamic NS kicks much below the ~160 km/s measured for the Crab pulsar. This suggests either that the Crab Nebula is not the remnant of an electron-capture SN, but of a low-mass iron-core progenitor, or that the Crab pulsar was not accelerated by the gravitational tug-boat mechanism but received its kick by a non-hydrodynamic mechanism such as, e.g., anisotropic neutrino emission.