Multidimensional supernova simulations with approximative neutrino transport

We study hydrodynamic instabilities during the first seconds of core-collapse supernovae by means of 2D simulations with approximative neutrino transport and boundary conditions that parameterize the effects of the contracting neutron star and allow us to obtain sufficiently strong neutrino heating and, hence, neutrino-driven explosions. Confirming more idealised studies, as well as supernova simulations with spectral transport, we find that random seed perturbations can grow by hydrodynamic instabilities to a globally asymmetric mass distribution in the region between the nascent neutron star and the accretion shock, leading to a dominance of dipole ($l=1$) and quadrupole ($l=2$) modes in the explosion ejecta, provided the onset of the supernova explosion is sufficiently slower than the growth time scale of the low-mode instability. By gravitational and hydrodynamic forces, the anisotropic mass distribution causes an acceleration of the nascent neutron star, which lasts for several seconds and can propel the neutron star to velocities of more than 1000 km s-1. Because the explosion anisotropies develop chaotically and change by small differences in the fluid flow, the magnitude of the kick varies stochastically. No systematic dependence of the average neutron star velocity on the explosion energy or the properties of the considered progenitors is found. Instead, the anisotropy of the mass ejection, and hence of the kick, seems to increase when the nascent neutron star contracts more quickly, and thus low-mode instabilities can grow more rapidly. Our more than 70 models separate into two groups, one with high and the other with low neutron star velocities and accelerations after one second of post-bounce evolution, depending on whether the $l=1$ mode is dominant in the ejecta or not. This leads to a bimodality of the distribution when the neutron star velocities are extrapolated to their terminal values. Establishing a link to the measured distribution of pulsar velocities, however, requires a much larger set of calculations and ultimately 3D modelling.