Routing protocol

Physics

Gravitation

Photon sphere

Deflection (physics)

Deflection angle

Computer network

Mathematics

Black hole (networking)

Link-state routing protocol

Optics

Schwarzschild radius

General relativity

Schwarzschild metric

RADIUS

Photon

Charged black hole

Geometry

Polar

Polar coordinate system

Computer science

Quantum mechanics

Routing (electronic design automation)

Computer security

Classical mechanics

Physical review. D/Physical review. D.

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.

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