On the Observable Shape of Black Hole Photon Rings

TL;DR

This paper demonstrates that the detailed shape of black hole photon rings can be reconstructed from interferometric measurements, explores the shape information obtainable from current and future observations, and analyzes the Kerr critical curve's shape.

Contribution

It provides a method to reconstruct photon ring shapes from projected position data and analyzes the Kerr critical curve's shape across different parameters.

Findings

Photon ring shape can be reconstructed from projected position functions.

Visibility amplitude alone encodes limited shape information, mainly widths for convex curves.

The Kerr critical curve is an ellipse at low spin and inclination, and a convex hull of a Cartesian oval at extremal spin.

Abstract

Motivated by the prospect of measuring a black hole photon ring, in previous work we explored the interferometric signature produced by a bright, narrow curve in the sky. Interferometric observations of such a curve measure its "projected position function" $\mathbf{r}\cdot\hat{\mathbf{n}}$, where $\mathbf{r}$ parameterizes the curve and $\hat{\mathbf{n}}$ denotes its unit normal vector. In this paper, we show by explicit construction that a curve can be fully reconstructed from its projected position, completing the argument that space interferometry can in principle determine the detailed photon ring shape. In practice, near-term observations may be limited to the visibility amplitude alone, which contains incomplete shape information: for convex curves, the amplitude only encodes the set of projected diameters (or "widths") of the shape. We explore the freedom in reconstructing a convex curve from its widths, giving insight into the shape information probed by technically plausible future astronomical measurements. Finally, we consider the Kerr "critical curve" in this framework and present some new results on its shape. We analytically show that the critical curve is an ellipse at small spin or inclination, while at extremal spin it becomes the convex hull of a Cartesian oval. We find a simple oval shape, the "phoval", which reproduces the critical curve with high fidelity over the whole parameter range.