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

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