On the approximation of the black hole shadow with a simple polar curve

Abstract

A black hole embedded within a bright, optically thin emitting region imprints a nearly circular "shadow" on its image, corresponding to the observer's line-of-sight into the black hole. The shadow boundary depends on the black hole's mass and spin, providing an observable signature of both properties via high resolution images. However, standard expressions for the shadow boundary are most naturally parametrized by Boyer-Lindquist radii rather than by image coordinates. We explore simple, approximate parameterizations for the shadow boundary using ellipses and a family of curves known as limacons. We demonstrate that these curves provide excellent and efficient approximations for all black hole spins and inclinations. In particular, we show that the two parameters of the limacon naturally account for the three primary shadow deformations resulting from mass and spin: size, displacement, and asymmetry. These curves are convenient for parametric model fitting directly to interferometric data, they reveal the degeneracies expected when estimating black hole properties from images with practical measurement limitations, and they provide a natural framework for parametric tests of the Kerr metric using black hole images.

9 pages, 9 figures; accepted for publication in ApJ

References in corpus (10)

• First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
• 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
• Black Hole Shadows, Photon Rings, and Lensing Rings
• The Shadow of a Spherically Accreting Black Hole
• Lensing by Kerr Black Holes
• Reconstruction of Static Black Hole Images Using Simple Geometric Forms

Cited by in corpus (18)

• Horizon-scale tests of gravity theories and fundamental physics from the Event Horizon Telescope image of Sagittarius A${}^{\ast }$
• Calculating black hole shadows: Review of analytical studies
• The Shape of the Black Hole Photon Ring: A Precise Test of Strong-Field General Relativity
• Observing the Inner Shadow of a Black Hole: A Direct View of the Event Horizon
• Photon Ring Autocorrelations
• On the Observable Shape of Black Hole Photon Rings
• The Black Hole Explorer: Motivation and Vision
• Analytical study of higher-order ring images of accretion disk around black hole
• Photon ring test of the Kerr hypothesis: Variation in the ring shape
• Shape of higher-order images of equatorial emission rings around a Schwarzschild black hole: Analytical description with polar curves
• Deflection of light rays by spherically symmetric black hole in dispersive medium
• Superentropic AdS Black Hole Shadows
• Light Deflection by Rotating Regular Black Holes with a Cosmological Constant
• Light Deflection Angle by Superentropic Black Holes
• Optical Shadows of Rotating Bardeen AdS Black Holes
• Superentropic Black Hole Shadows in Arbitrary Dimensions
• Inferring Black Hole Spin from Interferometric Measurements of the First Photon Ring: A Geometric Approach
• Decoding a black hole metric from the interferometric pattern of the relativistic images of a compact source