Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits

It is believed that rays bent inside and outside photon spheres could affect partially the black hole shadow images by the Event Horizon Telescope and the rays near photon spheres would be detected by near-future space observations. The investigation of the rays near the photon spheres in not only black hole spacetimes but also exotic spacetimes would be important since one will need them to exclude black hole mimickers. The deflection angles of the rays deflected by the photon spheres diverge logarithmically and we can treat them by a strong-deflection-limit analysis. The error of the strong-deflection-limit analysis becomes large if antiphoton spheres exist in the spacetimes and the analysis breaks down when the photon spheres and the antiphoton spheres degenerate to form a marginally unstable photon sphere. This is because the deflection angles of the rays bent by the marginally unstable photon sphere diverge in powers. In this paper, we extend Eiroa, Romero, and Torres's method to gravitational lensing of rays inside and outside of the marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits and we apply it to a Reissner-Nordstr\"{o}m spacetime and a Hayward spacetime with the marginally unstable photon sphere. We have also confirmed that the deflection angles in the strong deflection limits by the method converge correctly to the deflection angle without approximations, while there are the mismatches of the coefficient of the power-divergent term of the deflection angles of the rays deflected just outside of the marginally unstable photon sphere in a semianalytic calculation by the author previously.