The ring-shaped shadow of a rotating naked singularity with a complete photon sphere

We investigate the shadows of the Konoplya-Zhidenko naked singularity. In the spacetime of the Konoplya-Zhidenko naked singularity, not only an unstable retrograde light ring (LR) but also an unstable prograde LR exists, leading to the formation of a complete photon sphere (PS). Due to the absence of an event horizon, a dark disc-shaped shadow does not appear; instead, a ring-shaped shadow is observed. The ring-shaped shadow appears as an infinite number of relativistic Einstein rings in the image of the naked singularity. For some parameter values, only the unstable retrograde LR exists, resulting in an incomplete unstable PS and thus giving rise to an arc-shaped shadow for the Konoplya-Zhidenko naked singularity. The shadow of the Konoplya-Zhidenko naked singularity gradually shifts to the right as the rotation parameter a increases and gradually becomes smaller as the deformation parameter $ |\eta| $ increases. Moreover, stable LRs and stable photon spherical orbits can exist in the Konoplya-Zhidenko naked singularity spacetime, but they have no effect on the image of the naked singularity. This study demonstrates that a rotating naked singularity can exhibit not only an arc-shaped shadow but also a ring-shaped shadow. *Supported by the National Natural Science Foundation of China (12105151), the Natural Science Foundation of Shandong Province, China (ZR2020QA080) and partially Supported by the National Natural Science Foundation of China (11875026, 11875025, 12035005)