Black hole spin estimation with time-variable image of M 87 during the flaring state

Abstract By investigating the time-variable 230 GHz images using ray-tracing general relativistic radiative transfer calculation, we propose a novel method for estimating the spin parameter of the supermassive black hole at the M 87 center by utilizing the sudden and short-term increase in emissivity in the innermost region of the accretion disk. It is found that the flux of the photon ring increases simultaneously as the flux of the direct ring, which brightens first, decreases, and then gradually diminishes, when the increase in emissivity persists for $15 t_{\rm g}$ with $t_{\rm g}$ being the light crossing time of the gravitational radius. The direct ring is formed by photons emitted from the vicinity of the innermost region of the disk and traveling directly to the observer without orbiting around the black hole, while the photon ring is formed by photons passing near the spherical photon orbit. The time-averaged width of the dark region between the direct ring and the photon ring (dark crescent) becomes thinner when the spin parameter is higher and the increase in the emissivity of the accretion disk is greater. The time variation of two rings also causes the intensity-weighted center to oscillate both in the direction of the black hole’s angular momentum vector projected on to the screen (Y-direction) and in the perpendicular direction (X-direction). The amplitude of oscillatory time variation in the X-direction becomes large when the spin parameter is higher, and that in the Y-direction becomes large when the increase in the emissivity of the disk is large. The spin parameter can be estimated by combining the time-averaged dark crescent width and the ratio of the amplitudes in the X- and Y-directions. This method is applicable when the duration of the increase in emissivity of the accretion disk ranges at least from approximately 10–$20t_{\rm g}$.