Black Hole Spin Estimation with Time-variable Image of M87 During the Flaring State

TL;DR

This work introduces a time-dependent GRRT framework to estimate the spin of the M87 black hole from flare-induced changes in inner-disk emissivity. By modeling slow-light propagation at 230 GHz and tracking two rings—the direct ring and the photon ring—the authors show that the time-averaged dark crescent width $w_{DC}$ and the elongation of the intensity-weighted center trajectory $ED$ encode the spin parameter $a$ and flare strength $f$. The proposed method relies on measuring these two observables during a flare lasting $t_d ightarrow 10-20 t_g$, with higher spins yielding larger $w_{DC}$ and larger trajectory elongation under certain conditions. The approach emphasizes the need for high-resolution, time-resolved observations and offers a pathway to constrain black hole spin through dynamic imaging, complemented by future GRMHD-informed flow models.

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 M87 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 onto 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 10-20 $t_{\rm g}$.

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