Images from disk and spherical accretions of hairy Schwarzschild black holes

Abstract: A hairy Schwarzschild black hole describes the deformation of Schwarzschild black hole due to including additional sources. It is found that depending on the hairy parameters, the photons' configurations around this black hole can be classified into two cases, corresponding to the hairy Schwarzschild black hole with single photon sphere and double photon spheres, respectively. We focus on the shadows and images of the hairy Schwarzschild black hole under two types of static thin illuminations conditions: disk accretion and spherical accretion, respectively. Under both illuminations, the two hairy parameters ($\alpha$ and $l_o$) have competitive effects on the shadow and optical appearance image of the hairy Schwarzschild black hole with single photon sphere. This means that even though the parameters have significant influences on the rings and shadows, its images with certain groups of $\alpha$ and $l_o$ could be indistinguishable to that of Schwarzschild black hole, namely, the degeneracies of images raise between the hairy Schwarzschild black hole and Schwarzschild black hole. Moreover, the optical appearance image of the hairy Schwarzschild black hole with double photon spheres will exhibit new additional rings and accretion features, which {are not} present in the images of (hairy) Schwarzschild black hole with single photon sphere. Our theoretical studies on the rings and shadows provide a potential tool to differentiate the hairy Schwarzschild black hole with double photon spheres from Schwarzschild black hole, but they are not always helpful for the cases with single photon sphere due to the degeneracies.