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. 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.

See also our previous paper arXiv:2201.01716

References in corpus (75)

• First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
• First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way
• First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole
• First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole
• First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring
• First M87 Event Horizon Telescope Results. II. Array and Instrumentation
• First M87 Event Horizon Telescope Results. III. Data Processing and Calibration
• First Sagittarius A* Event Horizon Telescope Results. VI: Testing the Black Hole Metric
• Black Hole Shadows, Photon Rings, and Lensing Rings
• Calculating black hole shadows: Review of analytical studies
• First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole
• Measurement of the Kerr Spin Parameter by Observation of a Compact Object's Shadow
• Relativistic images of Schwarzschild black hole lensing
• First Sagittarius A* Event Horizon Telescope Results. III: Imaging of the Galactic Center Supermassive Black Hole
• First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multi-wavelength Observations, Data Processing, and Calibration
• First Sagittarius A* Event Horizon Telescope Results. IV. Variability, Morphology, and Black Hole Mass
• Constraints on black-hole charges with the 2017 EHT observations of M87*
• Gravitational Test Beyond the First Post-Newtonian Order with the Shadow of the M87 Black Hole
• Stationary black holes and light rings
• The Shadow of a Spherically Accreting Black Hole
• The Shape of the Black Hole Photon Ring: A Precise Test of Strong-Field General Relativity
• Photon Rings around Kerr and Kerr-like Black Holes
• Lensing by Kerr Black Holes
• Strong deflection limit analysis and gravitational lensing of an Ellis wormhole
• Strong field gravitational lensing in the deformed Hǒrava-Lifshitz black hole
• Shadow of a Spinning Black Hole in an Expanding Universe
• Kerr black hole lensing for generic observers in the strong deflection limit
• A comparison of approximate gravitational lens equations and a proposal for an improved new one
• Gravitational Lensing in presence of Plasma: Strong Lens Systems, Black Hole Lensing and Shadow
• Concerns regarding the use of black hole shadows as standard rulers
• Strong deflection limit of black hole gravitational lensing with arbitrary source distances
• Black hole mimicker hiding in the shadow: Optical properties of the $\gamma$-metric
• Photon Ring and Observational Appearance of a Hairy Black Hole
• Superradiance evolution of black hole shadows revisited
• Light's Bending Angle due to Black Holes: From the Photon Sphere to Infinity
• Braneworld black hole gravitational lens: Strong field limit analysis
• Kerr-Sen dilaton-axion black hole lensing in the strong deflection limit
• No-Hair Theorem in the Wake of Event Horizon Telescope
• Can different black holes cast the same shadow?
• Gravitational lensing in the Simpson-Visser black-bounce spacetime in a strong deflection limit
• Gravitational Lensing by Spinning Black Holes in Astrophysics, and in the Movie Interstellar
• Geometric modeling of M87* as a Kerr black hole or a non-Kerr compact object
• Photon rings of spherically symmetric black holes and robust tests of non-Kerr metrics
• Shadow and weak deflection angle of extended uncertainty principle black hole surrounded with dark matter
• Higher order corrections to deflection angle of massive particles and light rays in plasma media for stationary spacetimes using the Gauss-Bonnet theorem
• Superradiant evolution of the shadow and photon ring of Sgr A${}^{\star }$
• Photon Ring Autocorrelations
• On the Observable Shape of Black Hole Photon Rings
• Light rings of stationary spacetimes
• Influence of a plasma on the observational signature of a high-spin Kerr black hole
• Toward determining the number of observable supermassive black hole shadows
• Analytical study of higher-order ring images of accretion disk around black hole
• Strong Gravitational Lensing by Schwarzschild Black Holes
• Constraining the tidal charge of brane black holes using their shadows
• Strong field gravitational lensing by hairy Kerr black holes
• Strong gravitational lensing across dilaton anti-de Sitter black hole
• Strong gravitational lensing in a noncommutative black-hole spacetime
• The weakly perturbed Schwarzschild lens in the strong deflection limit
• Deflection of light rays by spherically symmetric black hole in dispersive medium
• First analytical calculation of black hole shadow in McVittie metric
• Unbound motion of massive particles in the Schwarzschild metric: Analytical description in case of strong deflection
• Simulations of M87 and Sgr A* Imaging with the Millimetron Space Observatory on near-Earth orbits
• Gravitational lensing by a photon sphere in a Reissner-Nordström naked singularity spacetime in strong deflection limits
• Chaotic Shadows of Black Holes: A Short Review
• The light ring and the appearance of matter accreted by black holes
• Strong Gravitational Lensing by Sgr A*
• Testing Gravity with Black Hole Shadow Subrings
• On the approximation of the black hole shadow with a simple polar curve
• Visibility of Black Hole Shadows in Low-luminosity AGN
• Strong gravitational lensing in a spacetime with topological charge within the Eddington-inspired Born-Infeld gravity
• Relativistic iron lines in accretion disks: the contribution of higher order images in the strong deflection limit
• Flares and their echoes can help distinguish photon rings from black holes with space-Earth very long baseline interferometry
• Strong gravitational lensing from compact bodies: analytical results
• Black Hole Shadow Drift and Photon Ring Frequency Drift
• Strong Lensing and Nonminimally Coupled Electromagnetism

Cited by in corpus (30)

• Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A${}^{\ast }$
• Shadows and photon rings of regular black holes and geonic horizonless compact objects
• Adaptive Analytical Ray Tracing of Black Hole Photon Rings
• Prediction for the interferometric shape of the first black hole photon ring
• Gravitational lensing by using the 0th order of affine perturbation series of the deflection angle of a ray near a photon sphere
• Strong deflection limit analysis of black hole lensing in inhomogeneous plasma
• General Approach on Shadow Radius and Photon Spheres in Asymptotically Flat Spacetimes and the Impact of Mass-Dependent Variations
• Constraining spherically symmetric metrics by the gap between photon rings
• Analyzing the Influence of Geometrical Deformation on Photon Sphere and Shadow Radius: A New Analytical Approach -- Spherically Symmetric Spacetimes
• A Lensing-Band Approach to Spacetime Constraints
• Strong deflection of massive particles in spherically symmetric spacetimes
• Affine perturbation series of the deflection angle of a ray near the photon sphere of a Reissner-Nordström black hole
• Constraints on the black-hole charges of M87* and Sagittarius A* by changing rates of photon spheres can be relaxed
• Shadows aspect of rotating black holes in the Einstein-AdS SU(N)-nonlinear sigma model
• Gravitational lensing and accretion disk imaging of a Buchdahl dense core
• Interferometric signature of higher-order images in a parametrized framework
• Accurate analytical modeling of light rays in spherically symmetric spacetimes: Applications in the study of black hole accretion disks and polarimetry
• Observational signatures from higher-order images of moving hotspots in accretion disks
• Analyzing the Influence of Geometrical Deformation on Photon Sphere and Shadow Radius: A New Analytical Approach -Stationary, and Axisymmetric Spacetime
• Optical images of the Kerr-Sen black hole and thin accretion disk
• Photon ring autocorrelations from gravitational fluctuations around a black hole
• Overlapping of photon rings in black hole imaging
• Shadow analysis of an approximate rotating black hole solution with weakly coupled global monopole charge
• Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits
• Lensing, Shadow and Photon Rings in a Magnetized Black Hole Spacetime
• Auto- and cross-correlations for multiple images of corotating hotspots in accretion disks
• Effect of primary scalar hair on black hole's strong lensing in Beyond Horndeski Gravity
• Light deflection in axially symmetric stationary spacetimes filled with a moving medium
• Fast and accurate analytical formulas for light propagation in general static, spherically symmetric spacetimes
• Light Rings, Accretion Disks and Shadows of Hayward Boson Stars