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

TL;DR

This paper derives an analytical polar coordinate formula for higher-order photon rings around Schwarzschild black holes, aiding future observational analysis of black hole images.

Contribution

It provides a new explicit analytical expression for the shape of higher-order photon rings, simplifying the study of their appearance without numerical ray-tracing.

Findings

Derived an explicit polar coordinate formula for photon rings.

Applicable to rings with arbitrary inclination angles.

Simplifies analysis compared to numerical methods.

Abstract

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. For the derivation, the strong deflection limit of the gravitational deflection is used. Our formula is a simple and efficient alternative to the numerical calculation of ray trajectories, with the main application to studying the shape of $n=2$ and $n=3$ photon rings.