In order to infer the effects of rotation on the revival of a stalled shock in supernova explosions, we investigate steady accretion flows with a standing shock. We first obtain a series of solutions for equations describing nonrotating, spherically symmetric flows and confirm the results of preceding papers, that for a given mass accretion rate, there is a critical luminosity of irradiating neutrinos above which there exists no steady solution. Below the critical value, we find two branches of solutions; one is stable and the other is unstable against radial perturbations. With a simple argument based on the Riemann problem, we can identify the critical luminosity as that at which the stalled shock revives. We also obtain a condition satisfied by the flow velocity for the critical luminosity, which can easily be applied to the rotational case. If a collapsing star rotates, the accretion flow is nonspherical as a consequence of centrifugal forces. Flows are accelerated near the rotation axis, whereas they are decelerated near the equatorial plane. As a result, the critical luminosity is lowered; that is, rotation assists the revival of a stalled shock. According to our calculations, the critical luminosity is $25% lower for a mass accretion rate of 1 M s À1 and a rotational frequency of 0.1 Hz at a radius of 1000 km than that for a spherically symmetric flow with the same mass accretion rate. We find that the condition on the flow velocity at the critical luminosity is first satisfied at the rotation axis. This suggests that shock revival is triggered on the rotation axis and a jetlike explosion ensues.

We investigate the dynamics of an injected outflow propagating in a progenitor in the context of the collapsar model for gamma-ray bursts (GRBs) through two dimensional axisymmetric relativistic hydrodynamic simulations. Initially, we locally inject an outflow near the center of a progenitor. We calculate 25 models, in total, by fixing its total input energy to be 10 51 ergs s −1 and radius of the injected outflow to be 7 × 10 7 cm while varying its bulk Lorentz factor, Γ 0 = 1.05 ∼ 5, and its specific internal energy, ǫ 0 /c 2 = 0.1 ∼ 30 (with c being speed of light). The injected outflow propagates in the progenitor and drives a large-scale outflow or jet. We find a smooth but dramatic transition from a collimated jet to an expanding outflow among calculated models. The opening angle of the outflow (θ sim ) is sensitive to Γ 0 ; we find θ sim < 2 • for Γ 0 3. The maximum Lorentz factor is, on the other hand, sensitive to both of Γ 0 and ǫ 0 ; roughly Γ max ∼ Γ 0 (1 + ǫ 0 /c 2 ). In particular, a very high Lorentz factor of Γ max 100 is achieved in one model. A variety of opening angles can arise by changing ǫ 0 , even when the maximum Lorentz factor is fixed. The jet structure totally depends on Γ 0 . When Γ 0 is high, a strong bow shock appears and generates a back flow. High pressure progenitor gas heated by the bow shock collimates the outflow to form a narrow, relativistic jet. A number of internal oblique shocks within the jet are generated by the presence of the back flow and/or shear instability. When Γ 0 is low, on the contrary, the outflow expands soon after the injection, since the bow shock is weak and thus the pressure of the progenitor gas is not high enough -2to confine the flow. Our finding will explain a smooth transition between the GRBs, X-ray rich GRBs (XRRs) and X-ray Flashes (XRFs) by the same model but with different ǫ 0 values.

A perturbative analysis is used to investigate the effect of rotation on the instability of a steady accretion shock (SASI) in a simple toy model, in view of better understanding supernova explosions in which the collapsing core contains angular momentum. A cylindrical geometry is chosen for the sake of simplicity. Even when the centrifugal force is very small, rotation can have a strong effect on the nonaxisymmetric modes of SASI by increasing the growth rate of the spiral modes rotating in the same direction as the steady flow. Counterrotating spiral modes are significantly damped, while axisymmetric modes are hardly affected by rotation. The growth rates of spiral modes have a nearly linear dependence on the specific angular momentum of the flow. The fundamental one-armed spiral mode (m ¼ 1) is favored for small rotation rates, whereas stronger rotation rates favor the mode m ¼ 2. A WKB analysis of higher harmonics indicates that the efficiency of the advective-acoustic cycles associated with spiral modes is strongly affected by rotation in the same manner as low-frequency modes, whereas the purely acoustic cycles are stable. These results suggest that the linear phase of SASI in rotating core-collapse supernovae naturally selects a spiral mode rotating in the same direction as the flow, as observed in the three-dimensional numerical simulations of Blondin and Mezzacappa. This emphasizes the need for a three-dimensional approach to rotating core collapse, before conclusions on the explosion mechanisms and pulsar kicks can be drawn.