Collapse of a rotating supermassive star to a supermassive black hole: Analytic determination of the black hole mass and spin

The collapse of a uniformly rotating, supermassive star (SMS) to a supermassive black hole (SMBH) has been followed recently by means of hydrodynamic simulations in full general relativity. The initial SMS of arbitrary mass M in these simulations rotates uniformly at the mass-shedding limit and is marginally unstable to radial collapse. The final black hole mass and spin have been determined to be M(h)/M approximate to 0.9 and J(h)/M(h)(2) approximate to 0.75. The remaining mass goes into a disk of mass M(disk)/M approximate to 0.1. Here we show that these black hole and disk parameters can be calculated analytically from the initial stellar density and angular momentum distribution. The analytic calculation thereby corroborates and provides a simple physical explanation for the computational discovery that SMS collapse inevitably terminates in the simultaneous formation of a SMBH and a rather substantial ambient disk. This disk arises even though the total spin of the progenitor star, J/M(2) = 0.97, is safely below the Kerr limit. The calculation performed here applies to any marginally unstable n = 3 polytrope uniformly rotating at the breakup speed, independent of stellar mass or the source of internal pressure. It illustrates how the black hole and disk parameters can be determined for the collapse of other types of stars with different initial density and rotation profiles.