Using a two-dimensional hydrodynamics code (PROMETHEUS), we explore the continued evolution of rotating helium stars, M-proportional to greater than or similar to 10 M-., in which iron-core collapse does not produce a successful outgoing shock but instead forms a black hole of 2-3 M-.. The model explored in greatest detail is the 14 M, helium core of a 35 M-. main-sequence star. The outcome is sensitive to the angular momentum. For j(16) = f/(10(16) cm(2) s(-1)) less than or similar to 3, material falls into the black hole almost uninhibited. No outflows are expected. For j(16) greater than or similar to 20, the infalling matter is halted by centrifugal force outside 1000 km where neutrino losses are negligible. The equatorial accretion rate is very low, and explosive oxygen burning may power a weak equatorial explosion. For 3 less than or similar to j(16) less than or similar to 20, however, a reasonable value for such stars, a compact disk forms at a radius at which the gravitational binding energy can be efficiently radiated as neutrinos or converted to beamed outflow by magnetohydrodynamical (MHD) processes. These are the best candidates for producing gamma-ray bursts (GRBs). Here we study the formation of such a disk, the associated flow patterns, and the accretion rate for disk viscosity parameter alpha approximate to 0.001 and 0.1. Infall along the rotational axis is initially uninhibited, and an evacuated channel opens during the first few seconds. Meanwhile the black hole is spun up by the accretion (to a approximate to 0.9), and energy is dissipated in the disk by MHD processes: and radiated by neutrinos. For the alpha = 0.1 model, appreciable energetic outflows develop between polar angles of 30 degrees and 45 degrees. These outflows, powered by viscous dissipation in the disk, have an energy of up to a few times 10(51) ergs and a mass similar to 1 M-. and are rich in Ni-56. They constitute a supernova-like explosion by themselves. Meanwhile accretion through the disk is maintained for approximately 10-20 s but is time variable (+/- 30%) because of hydrodynamical instabilities at the outer edge in a region where nuclei are experiencing photodisintegration. Because the efficiency of neutrino energy deposition is sensitive to the accretion rate, this instability leads to highly variable energy deposition in the polar regions. Some of this variability, which has significant power at 50 ms and overtones, may persist in the time structure of the burst. During the time followed, the average accretion rate for the standard alpha = 0.1 and j(16) = 10 model is 0.07 M-. s(-1). The total energy deposited along the rotational axes by neutrino annihilation is (1-14) x 10(51) ergs, depending upon the evolution of the Kerr parameter and uncertain neutrino efficiencies. Simulated deposition of energy in the polar regions, at a constant rate of 5 x 10(50) ergs s(-1) per pole, results in strong relativistic outflow jets beamed to about 1% of the sky. These jets may be additionally modulated by instabilities-in the sides of the "nozzle" through which they flow. The jets blow aside the accreting material, remain highly focused, and are capable of penetrating the star in similar to 10 s. After the jet breaks through the surface of the star, highly relativistic how can emerge. Because of the sensitivity of the mass ejection and jets to accretion rate, angular momentum, and disk viscosity, and the variation of observational consequences with viewing angle, a large range of outcomes is possible, ranging from bright GRBs like GRB 971214 to faint GRB-supernovae like SN 1998bw. X-ray precursors are also possible as the jet first breaks out of the star. While only a small fraction of supernovae make GRBs, we predict that collapsars will always make supernovae similar to SN 1998bw. However, hard, energetic GRBs shorter than a few seconds will be difficult to produce in this model and may require merging neutron stars and black holes for their explanation.
