Collision of an innermost stable circular orbit particle around a Kerr black hole

Abstract

We derive a general formula for the center-of-mass (CM) energy for the near-horizon collision of two particles of the same rest mass on the equatorial plane around a Kerr black hole. We then apply this formula to a particle which plunges from the innermost stable circular orbit (ISCO) and collides with another particle near the horizon. It is found that the maximum value of the CM energy ${E}_{\mathrm{cm}}$ is given by ${E}_{\mathrm{cm}}/(2{m}_{0})\ensuremath{\simeq}1.40/\sqrt[4]{1\ensuremath{-}{a}_{*}^{2}}$ for a nearly maximally rotating black hole, where ${m}_{0}$ is the rest mass of each particle and ${a}_{*}$ is the nondimensional Kerr parameter. This coincides with the known upper bound for a particle which begins at rest at infinity within a factor of 2. Moreover, we also consider the collision of a particle orbiting the ISCO with another particle on the ISCO and find that the maximum CM energy is then given by ${E}_{\mathrm{cm}}/(2{m}_{0})\ensuremath{\simeq}1.77/\sqrt[6]{1\ensuremath{-}{a}_{*}^{2}}$. In view of the astrophysical significance of the ISCO, this result implies that particles can collide around a rotating black hole with an arbitrarily high CM energy without any artificial fine-tuning in an astrophysical context if we can take the maximal limit of the black hole spin or ${a}_{*}\ensuremath{\rightarrow}1$. On the other hand, even if we take Thorne's bound on the spin parameter into account, highly or moderately relativistic collisions are expected to occur quite naturally, for ${E}_{\mathrm{cm}}/(2{m}_{0})$ takes 6.95 (maximum) and 3.86 (generic) near the horizon and 4.11 (maximum) and 2.43 (generic) on the ISCO for ${a}_{*}=0.998$. This implies that high-velocity collisions of compact objects are naturally expected around a rapidly rotating supermassive black hole. Implications to accretion flows onto a rapidly rotating black hole are also discussed.