  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.
