Collapse of a Rotating Supermassive Star to a Supermassive Black Hole: Analytic Determination of the Black Hole Mass and Spin

摘要

The collapse of a uniformaly rotating, supermassive star (SMS) to asupermassive black hole (SMBH) has been followed recently by means ofhydrodynamic simulations in full general relativity. The initial SMS ofarbitrary mass M in these simulations rotates uniformly at the mass--sheddinglimit and is marginally unstable to radial collapse. The final black hole hasmass M_h/M = 0.9 and and spin J_h/M_h^2 = 0.75, approximately. The remainingmass goes into a disk of mass M_disk/M = 0.1, also approximately. Here we showthat these black hole and disk parameters can be calculated analytically fromthe initial stellar density and angular momentum distribution. The analyticcalculation thereby corroborates and provides a simple physical explanation forthe computational discovery that SMS collapse inevitably terminates in thesimultaneous formation of a SMBH and a rather substantial ambient disk. Thisdisk arises even though the total spin of the progenitor star, J/M^2 = 0.97, issafely below the Kerr limit. The calculation performed here applies to anymarginally unstable n = 3 polytrope uniformly rotating at the break--up speed,independent of stellar mass or the source of internal pressure. It illustrateshow the black hole and disk parameters can be determined for the collapse ofother types of stars with different initial density and rotation profiles.