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The spin of a neutron star at birth may be impacted by the asymmetriccharacter of the explosion of its massive progenitor. During the first secondafter bounce, the spiral mode of the Standing Accretion Shock Instability(SASI) is able to redistribute angular momentum and spin-up a neutron star bornfrom a non-rotating progenitor. Our aim is to assess the robustness of thisprocess. We perform 2D numerical simulations of a simplified setup incylindrical geometry to investigate the timescale over which the dynamics isdominated by a spiral or a sloshing mode. We observe that the spiral modeprevails only if the ratio of the initial shock radius to the neutron starradius exceeds a critical value. In that regime, both the degree of asymmetryand the average expansion of the shock induced by the spiral mode increasemonotonously with this ratio, exceeding the values obtained when a sloshingmode is artificially imposed. With a timescale of 2-3 SASI oscillations, thedynamics of SASI takes place fast enough to affect the spin of the neutron starbefore the explosion. The spin periods deduced from the simulations arecompared favorably to analytical estimates evaluated from the measuredsaturation amplitude of the SASI wave. Despite the simplicity of our setup,numerical simulations revealed unexpected stochastic variations, including areversal of the direction of rotation of the shock. Our results show that thespin up of neutron stars by SASI spiral modes is a viable mechanism even thoughit is not systematic.

New insights on the spin-up of a neutron star during core-collapse