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Neutron stars (NSs) obtain kicks of typically several 100 km/s at birth. Thegravitational tug-boat mechanism can explain these kicks as consequences ofasymmetric mass ejection during the supernova (SN) explosion. Support for thishydrodynamic explanation is provided by observations of SN remnants withassociated NSs, which confirm the prediction that the bulk of the explosionejecta, in particular chemical elements between silicon and the iron group, aredominantly expelled in the hemisphere opposite to the direction of the NS kick.Here, we present a large set of two- and three-dimensional explosionsimulations of electron-capture SNe, considering explosion energies between~3x10^49 erg and ~1.6x10^50 erg. We find that the fast acceleration of the SNshock in the steep density gradient delimiting the O-Ne-Mg core of theprogenitor enables such a rapid expansion of neutrino-heated matter that thegrowth of neutrino-driven convection freezes out quickly in a high-modespherical harmonics pattern. Since the corresponding momentum asymmetry of theejecta is very small and the gravitational acceleration by the fast-expandingejecta abates rapidly, the NS kick velocities are at most a few km/s. Theextremely low core compactness of O-Ne-Mg-core progenitors therefore favorshydrodynamic 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 anelectron-capture SN, but of a low-mass iron-core progenitor, or that the Crabpulsar was not accelerated by the gravitational tug-boat mechanism but receivedits kick by a non-hydrodynamic mechanism such as, e.g., anisotropic neutrinoemission.

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