Measuring the Innermost Stable Circular Orbits of Supermassive Black Holes

We present a promising new technique, the g-distribution method, for measuring the inclination angle (i), the innermost stable circular orbit (ISCO), and the spin of a supermassive black hole. The g-distribution method uses measurements of the energy shifts in the relativistic iron line emitted by the accretion disk of a supermassive black hole due to microlensing by stars in a foreground galaxy relative to the g-distribution shifts predicted from microlensing caustic calculations. We apply the method to the gravitationally lensed quasars RX J1131-1231 (z _s = 0.658, z _l = 0.295), QJ 0158-4325 (z _s = 1.294, z _l = 0.317), and SDSS 1004+4112 (z _s = 1.734, z _l = 0.68). For RX J1131-1231, our initial results indicate that r _ISCO ≲ 8.5 gravitational radii (r _g) and i 55 (99% confidence level). We detect two shifted Fe lines in several observations, as predicted in our numerical simulations of caustic crossings. The current ΔE distribution of RX J1131-1231 is sparsely sampled, but further X-ray monitoring of RX J1131-1231 and other lensed quasars will provide improved constraints on the inclination angles, ISCO radii, and spins of the black holes of distant quasars.