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 } \buildrel \scriptstyle < \over { \scriptstyle \sim } 8.5 gravitational radii ( r _ { g } ) and i \buildrel \scriptstyle > \over { \scriptstyle \sim } 76 ^ { \circ } .
We detect two shifted Fe lines , in several observations , as predicted in our numerical simulations of caustic crossings .
The current { \Delta } E -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 .