The images of a rotating naked singularity and its photon spheres

For a black hole, the appearance of a shadow is due to the light rays entering the event horizon, and the unstable photon sphere determines the boundary of shadow. We present the images of Konoplya-Zhidenko compact object with or without complete photon sphere, and find that even in the presence of a complete photon sphere without an event horizon, a dark shadow will not be formed. When the event horizon is absent, the unstable prograde and retrograde light rings can also exist, so does the complete photon sphere. But the dark shadow doesn't emerge, and the image of the unstable complete photon sphere appears as an infinite number of relativistic Einstein rings. For this case, the light rays pass through the photon sphere, but eventually escape to infinity. For some parameter values, only the unstable retrograde light ring can exist, which leads to an incomplete photon sphere. In this case, the dark shadow also doesn't emerge, and the image of the incomplete unstable photon sphere appears as an infinite number of relativistic Einstein arcs. Furthermore, in Konoplya-Zhidenko naked singularity spacetime, the stable light rings and stable photon circular orbits can also exist, but they have no effect on the naked singularity image. In this paper, we demonstrate the effects of complete and incomplete photon sphere on the image of a compact object. It may contribute to future astronomical observations, and aid in verifying the cosmic censorship conjecture and various gravitational theories.