Strong gravitational lensing by a Reissner-Nordstr\"{o}m naked singularity with a marginally unstable photon sphere

Abstract

We investigate strong gravitational lensing by a marginally unstable photon sphere in a Reissner-Nordstr\"{o}m naked singularity spacetime. Using the Picard-Fuchs equation, we derive full-order power series expressions for the deflection angle in various regimes, including the strong deflection limits from both outside and inside the photon sphere. We show that the deflection angle diverges non-logarithmically in both cases, refining existing asymptotic formulae. Comparing truncated approximations with numerical results, we find that higher-order corrections are essential to achieve comparable accuracy to logarithmic divergence cases. Using these improved formulae, we also derive precise approximations for image positions that are not restricted to the almost perfectly aligned cases.