Rings and images of Horndeski hairy black hole illuminated by various thin accretions

We analyze the light rays around a static hairy black hole in Horndeski gravity with the use of ray-tracing procedure. We find that a stronger Horndeski hairy parameter corresponds to larger photon sphere as well as critical impact parameter, and wider ranges of photon ring and lensed ring emissions. These influences can be robustly interpreted from the shape of the effective potential of the photon's radial motion. Based on the distribution of the light rays, we then investigate the optical appearances of the Horndeski hairy black hole surrounded by various thin accretions. Firstly, we consider that the Horndeski hairy black hole is illuminated by the optically and geometrically thin accretion disk. We carefully clarify the contributions from the direct, lensed ring and photon ring intensities to the total observed intensity via the transfer function, which is rarely discussed in this scenario. We find that the Horndeski hair has significant influences on both shadow size and distributions of direct, lensed ring and photon ring brightness in three standard emission profiles. As a result, the rings and images of Horndeski hairy black hole and the origination of their brightness differentiate from those of Schwarzschild black hole (SBH). Then, when the Horndeski hairy black hole is illuminated by thin spherical accretions, the hairy black hole's shadow surrounded by a bright ring is larger than that of SBH, but the brightness of ring is fainter. Similar to that of SBH, the size of hairy black hole shadow does not change as the radial moving of the spherical accretion, and the brightness for the infalling accretion is fainter than that for the static accretion due to the Doppler effect. Therefore, we argue that the black hole image consisting of the shadow and accretion construction could, in theory, reflect the observational differences between the Horndeski hairy black hole and SBH.