tokens: coordinate. The photon ring is the gravitationally lensed image of the (unstable) photon orbit. Because the absolute values of angular momenta of the co- and counter-rotating photons on the photon orbit around highly spinning BHs (i.e., blue and red curves in Figure 2 (a)) are distinctly different, the impact parameter is significantly different between them. This results in the positional shift of the photon-ring center in the observer screen (Bardeen 1973 ). On the other hand, photons emitted by the innermost accretion flow produce an SSA-thick ISCO ring and the ISCO ring does not show such a significant positional shift of its center (green and purple curves in Figure 2 (b)). This is because these rays deviate from the photon orbit near the BH and the resultant impact parameter does not strongly depend on the sign of angular momentum of photons. Then, the positional offset between the center of the photon ring and that of the SSA-thick innermost region forms the “dark crescent” in the resultant image. The dark crescent appears in the opposite side of relativistically beamed bright region. Figure 2. Trajectories of photons forming the photon ring (top) and the SSA-thick ISCO ring (bottom), in the vicinity of the Kerr BH with a * = 0.998. The viewing angle is i = 30°. Each trajectory is projected onto the equatorial plane of the Boyer–Lindquist coordinate (we note that we define ( x eq , y eq ) ≡ ( r sin θ cos φ , r sin θ sin φ )). The solid/dashed curves represent the SSA-thin/SSA-thick regions along the rays. The positions of the rays on the observer screen are represented by the dashed lines in Figure 4 . Because the horizontal shift of the photon ring increases with BH spin (as well as the ISCO radius for matter decreasing with BH spin), the larger and clearer dark crescent appears when the BH spin is higher: the dark crescent appears in the models with a * = 0.998 and 0.75, while it is not found around the BH with a * = 0.5 in our shadow images. We can find a clearer dark crescent for the model with i = 30° than for i = 15°, because of a more prominent horizontal shift of the photon ring by the frame-dragging effect in the case with larger viewing angle. In Figure 3 , we present the intensity images (top) and τ SSA (bottom) in the observer’s screen at 230 and 350 GHz for i = 30° and 15° in the model with a * = 0.998. This is the first example explicitly displaying τ SSA in the observer’s screen. These spatial distribution maps of τ SSA on the observer screen enable us to understand the formation of the dark crescent in the BH shadow image. By comparing the intensity and τ SSA map at 230 GHz, we can confirm that the bright inner ring is formed via the emission from the SSA-thick innermost accretion flow. We note that the photon ring is τ SSA ≳ 1, because the rays pass close to the photon orbit, which results in the large emission gain and the formation of the bright photon ring. At ν = 350 GHz, the right side of both the photon ring and
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