Multidimensional supernova simulations with approximative neutrino transport I. Neutron star kicks and the anisotropy of neutrino-driven explosions in two spatial dimensions

By means of two-dimensional (2D) simulations we study hydrodynamic instabilities during the first seconds of neutrino-driven supernova explosions. We use a PPM hydrodynamics code, supplemented with an approximative, gray, nonequilibrium treatment of radial neutrino transport, based on an analytic integration of the neutrino number and energy equations. Tests show that this treatment reproduces basic features of more detailed transport calculations in the semitransparent and transparent regimes. It is sufficiently fast to allow us to compute a large number of models with high spatial resolution. Our simulations start with post-core bounce models of three 15 M⊙ progenitors that have markedly different density profiles, and one 15 M⊙ star that includes rotation. The dense core of the contracting, newly formed neutron star is replaced at neutrino optical depths of typically more than 100 by an inner boundary at which the incoming neutrino fluxes are imposed as a systematically varied boundary condition, thus triggering 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, but the kick seems to increase when the nascent neutron star contracts more quickly. 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.