Circular light orbits of a general, static, and spherical symmetrical wormhole with $Z_2$ symmetry

Abstract: Recently, the ring images or the shadow images of the centers of the galaxy M87 and the Milky way have been reported by Event Horizon Telescope Collaboration. It is believed that the ring images imply that the central objects form unstable light circular orbits. Some of wormholes with $Z_2$ symmetry against a throat are wrongly excluded from the candidates at the centers of M87 and the Milky way due to the overlooking the unstable light circular orbits on the throat. A general asymptotically-flat, static, and spherical symmetrical wormhole without a thin shell has at least one unstable circular light orbit at the throat or elsewhere. If the wormhole has $Z_2$ symmetry against the throat, it has the unstable circular light orbits on the throat or it has stable circular light orbits on the throat and unstable ones near the throat. We need to analyze the throat carefully to make sure we do not unfairly rule out the $Z_2$-symmetrical wormholes. In this study, we categorize the numbers of the circular light orbits of the $Z_2$-symmetrical wormhole and their stability from the derivatives of an effective potential at the throat and we investigate the circular light orbits around a Simpson-Visser black-bounce spacetime, a Damour-Solodukhin wormhole spacetime, a Reissner-Nordstr\"{o}m black-hole-like wormhole spacetime or a charged Damour-Solodukhin wormhole spacetime as examples. We give complete treatments including degenerated circular light orbits made from more than one stable and unstable circular light orbits on and off the throat.