Shape of higher-order images of equatorial emission rings around a Schwarzschild black hole: Analytical description with polar curves

Higher-order photon rings can be expected to detected in a more detailed image of the black hole found future observations. These are lensed images luminous matter surrounding and formed by photons that loop around it. In this paper we have succeeded derive an analytical expression for shape higher-order form is most convenient application: explicit equation curve polar coordinates. The formula describes apparent circular ring with given radius Schwarzschild as viewed distant observer arbitrary inclination. For derivation, strong deflection limit gravitational used. Our simple efficient alternative numerical calculation ray trajectories, main application studying $n=2$ $n=3$ rings.