  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.
