Axisymmetric collapse simulations of rotating massive stellar cores in full general relativity: Numerical study for prompt black hole formation

We perform axisymmetric simulations for gravitational collapse of a massive iron core to a black hole in full general relativity. The iron cores are modeled by Gamma=4/3 equilibrium polytrope for simplicity. The hydrodynamic equations are solved using a high-resolution shock-capturing scheme with a parametric equation of state. The Cartoon method is adopted for solving the Einstein equations. Simulations are performed for a wide variety of initial conditions changing the mass (approximate to 2.0-3.0M), the angular momentum, the rotational velocity profile of the core, and the parameters of the equations of state which are chosen so that the maximum mass of the cold spherical polytrope is approximate to 1.6M. Then, the criterion for the prompt black hole formation is clarified in terms of the mass and the angular momentum for several rotational velocity profile of the core and equations of state. It is found that (i) with the increase of the thermal energy generated by shocks, the threshold mass for the prompt black hole formation is increased by 20-40%, (ii) the rotational centrifugal force increases the threshold mass by less than or similar to 25%, (iii) with the increase of the degree of differential rotation, the threshold mass is also increased, and (iv) the amplification factors shown in the results (i)-(iii) depend sensitively on the equation of state. We also find that the collapse dynamics and the structure of the shock formed at the bounce depend strongly on the stiffness of the adopted equation of state. In particular, as a new feature, a strong bipolar explosion is observed for the collapse of rapidly rotating iron cores with an equation of state which is stiff in subnuclear density and soft in supranuclear density. Gravitational waves are computed in terms of a quadrupole formula. It is also found that the waveform depends sensitively on the equations of state.