Analytic toy-model for the ISCO shift

Abstract

A simple black-hole-ring system is proposed as a toy model for the two-body problem in general relativity. This toy-model yields the fractional shift $\Delta\Omega_{\text{isco}}/\Omega_{\text{isco}}={{29}\over{81\sqrt{2}}}\eta$ in the Schwarzschild ISCO (innermost stable circular orbit) frequency, where $\eta\equiv m/M_{\text{ir}}\ll 1$ is the dimensionless ratio between the mass of the particle and the irreducible mass of the black hole. Our model suggests that the second-order spin-orbit interaction between the black hole and the orbiting particle (the dragging of inertial frames) is the main element determining the observed value of the ISCO shift.