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

TL;DR

This paper introduces simple polar curve approximations, including ellipses and limacons, to model black hole shadows, enabling efficient analysis of black hole properties from high-resolution images.

Contribution

It proposes novel, accurate parametric models for black hole shadows using limacons, facilitating better data fitting and tests of the Kerr metric.

Findings

Limacons effectively approximate shadow boundaries across all spins and inclinations.

The parameters of limacons capture size, displacement, and asymmetry of shadows.

The models assist in understanding degeneracies and testing the Kerr metric.

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.