A Closed-Form Surrogate for the Equivalent Diameter of the Kerr Shadow

TL;DR

This paper introduces a precise, closed-form approximation for the Kerr black-hole shadow's equivalent diameter, enabling quick size evaluations across a range of spins and inclinations without complex numerical calculations.

Contribution

The authors develop a novel analytical surrogate for the Kerr shadow diameter that combines exact face-on limits with a polynomial correction for inclination dependence.

Findings

Achieves sub-percent accuracy across the parameter space.

Median error of 0.0105% on training data, 0.023% on validation.

Provides fast, analytical size estimates for Kerr shadows.

Abstract

We present a closed-form surrogate for the equivalent diameter of the Kerr black-hole shadow, defined as the diameter of the circle with the same area as the shadow's critical curve. The construction enforces the exact face-on (polar) limit by explicitly separating an analytically computed polar contribution based on the spherical photon-orbit branch where the horizontal impact parameter vanishes. The remaining inclination dependence is captured by a compact 15-parameter polynomial placed inside an exponential correction. The coefficients are determined by ordinary least squares on a deterministic reference grid generated from the Kerr critical-curve area. Over the practical domain of dimensionless spin from 0 to 0.998 and inclination from just above 0 degrees up to 90 degrees (with the exactly polar point treated analytically), the surrogate achieves sub-percent accuracy. On the training grid the median absolute percent error is 0.0105 percent with a worst case of 0.782 percent, and on a denser out-of-sample validation set (including inclinations down to 0.5 degrees) the median, 95th-percentile, and worst-case errors are 0.023 percent, 0.471 percent, and 1.64 percent, respectively. The resulting expression provides fast evaluations of the shadow size without numerical ray tracing, making it convenient for repeated calls in parameter inference and rapid model comparisons.