Analytic toy-model for the ISCO shift

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 $ΔΩ_{\text{isco}}/Ω_{\text{isco}}={{29}\over{81\sqrt{2}}}η$ in the Schwarzschild ISCO (innermost stable circular orbit) frequency, where $η\equiv m/M_{\text{ir}}\ll 1$ 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.

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