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

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 $E_{\rm cm}$ is given by $E_{\rm cm}/(2m_{0})\simeq 1.40/\sqrt[4]{1-a_{*}^{2}}$ 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 $E_{\rm cm}/(2m_{0})\simeq 1.77/\sqrt[6]{1-a_{*}^{2}}$. 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 $a_{*}\to 1$. 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 $E_{\rm cm}/(2m_{0})$ 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.