Analytical description for light propagation with the source inside the black hole photon sphere

Abstract: The photon sphere defines the unstable circular orbit of photons in a black hole spacetime. Photons emitted by a source located inside the photon sphere can be gravitationally lensed by the black hole and have time delays when reaching the observer. These delays may lead to light echoes produced in the light curve if an accretion event in the vicinity of the horizon can be observed. In this work, we present fully analytical formulas with high accuracy to describe the change of the azimuthal angle and the travel time of those photons. By employing the analytical approaches, we find that the time delay between photons emitted from the interior of the photon sphere has a typical time scale of $2(\pi - \phi_\mathrm{S}) u_\mathrm{m}$ with $\phi_\mathrm{S}$ and $u_\mathrm{m}$ being respectively the azimuthal angle of the source and the impact parameter evaluated at the photon sphere, which can provide some clues on the future search for gravitational lensing signatures in the accretion inflow event.