Ultra-relativistic, neutrino-driven flows in gamma-ray bursts: A double transonic flow solution in a Schwarzschild spacetime

The structure of a hydrodynamic, double transonic flow driven by neutrino annihilation in the polar region of a Schwarzschild black hole is computed for different energy deposition profiles. The requirement that both the inflow into the black hole and the outflow to infinity pass smoothly through their sonic points fixes the stagnation radius and stagnation pressure. The asymptotic power of the outflow is shown to be the integral of the energy deposition rate above the stagnation radius. The outflow production efficiency depends on the energy deposition profile and is generally higher for shallower profiles. Using recent calculations of the neutrino annihilation rate, we estimate that over 50% of the total energy deposited above the horizon can emerge in the form of a relativistic outflow at infinity. The continuous creation of plasma during the expansion of the outflow leads to generation of a large specific entropy. This has important implications for the prompt photospheric emission in gamma-ray bursts.