  Asymmetric mass ejection in the early phase of supernova (SN) explosions can
impart a kick velocity to the new-born neutron star (NS). For neutrino-driven
explosions the NS acceleration was shown to be mainly caused by the
gravitational attraction of the anisotropically expelled inner ejecta, while
hydrodynamic forces contribute on a subdominant level, and asymmetric neutrino
emission plays only a secondary role. Two- and three-dimensional hydrodynamic
simulations demonstrated that this gravitational tug-boat mechanism can explain
the observed space velocities of young NSs up to more than 1000 km/s. Here, we
discuss how the NS kick depends on the energy, ejecta mass, and asymmetry of
the SN explosion, and which role the compactness of the pre-collapse stellar
core plays for the momentum transfer to the NS. We also provide simple analytic
expressions for the NS velocity in terms of these quantities. Referring to
results of hydrodynamic simulations in the literature, we argue why within the
discussed scenario of NS acceleration, electron-capture SNe, low-mass Fe-core
SNe and ultra-stripped SNe can be expected to have considerably lower intrinsic
NS kicks than core-collapse SNe of massive stellar cores. Our basic arguments
remain valid also if progenitor stars possess large-scale asymmetries in their
convective silicon and oxygen burning layers. Possible scenarios for spin-kick
alignment are sketched. Much of our discussion remains on a conceptual and
qualitative level and more work is needed on the numerical modeling side to
determine the dependences of involved parameters, whose prescriptions will be
needed for recipes that can be used to better describe NS kicks in binary
evolution and population synthesis studies.
