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

Abstract

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

Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A ∗

“…In fact, the EHT observations set rather weak limits on ν, with ν ≲ 0.4M (1σ) and ν ≲ 0.45M (2σ), barely excluding the extremal JNW NS, ν = 0.5. These limits can of course be translated into rather weak limits on the strength of the scalar field from equation (40). More importantly, among all the examples of BH mimickers we have considered so far, the JNW NS is one of the few which is in excellent agreement with the EHT observations given the metric tests we have performed in this work.…”

“…The main features observed in VLBI horizon-scale images of BHs are a bright emission ring surrounding a central brightness depression, with the latter related to the BH shadow (see e.g. [25][26][27][28][29][30][31][32][33][34] for some of the earliest calculations in this sense [35][36][37][38][39], for more recent studies on the shapes of BH shadows and related observables, and higher-order photon rings in [40]). On the plane of a distant observer, the boundary of the BH shadow marks the apparent image of the photon region (the boundary of the region of space-time which supports closed spherical photon orbits) 22 , and separates capture orbits from scattering orbits: for detailed reviews on BH shadows, see e.g.…”

Shadows and photon rings of regular black holes and geonic horizonless compact objects

“…Indeed, such new contributions are associated to those rays circulating between the location of the maximum of the effective potential, its minimum (dubbed as the anti-photon sphere) and the infinite potential slope (recall figure 3, green curve). The net effect is that the corresponding object will show a multi-ring structure, namely, a cascade of additional non-negligible higherorder photon ring contributions (for an overly general analytic analysis of this problem we refer the reader to references [96,97]). These appear thanks to the enhancing effect of the region between the photon sphere and the internal part of the potential such that every photon with low enough impact parameter hits the infinite potential slope, allowing for further transits of light trajectories around the photon sphere on its way back, this way collecting additional luminosities 7 .…”

Shadows and photon rings of regular black holes and geonic horizonless compact objects

“…The net effect is that the corresponding object will show a multi-ring structure, namely, a cascade of additional non-negligible higher-order photon ring contributions (for an overly general analytic analysis of this problem we refer the reader to Refs. [80,81]). These appear thanks to the enhancing effect of the region between the photon sphere and the internal part of the potential such that every photon with low enough impact parameter hits the infinite potential slope, allowing for further transits of light trajectories around the photon sphere on its way back, this way collecting additional luminosi-ties 1 .…”