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

Understand

Higher-order photon rings can be expected to be detected in a more detailed image of the black hole found in future observations.

• These rings are lensed images of the luminous matter surrounding the black hole and are formed by photons that loop around it.
• In this paper we have succeeded to derive an analytical expression for the shape of the higher-order rings in the form that is most convenient for application: the explicit equation of the curve in polar coordinates.
• The formula describes the apparent shape of the higher-order image of the circular ring with the given radius around Schwarzschild black hole as viewed by distant observer with an arbitrary inclination.