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

Arseny Pantsialei

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.

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

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.
Paper Structure (12 sections, 33 equations, 1 figure, 1 table)

This paper contains 12 sections, 33 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Closed-form surrogate construction
  3. Polar limit in closed form
  4. Inclination-dependent correction
  5. Reference accuracy and convergence
  6. Fitting procedure
  7. Validation and reproducibility
  8. External consistency check
  9. Accuracy on training and validation sets
  10. Reproducibility
  11. Conclusion
  12. Deterministic reference computation

Figures (1)

  • Figure 1: (a) Relative deviation $100 (y_{\rm ref}-1)$ (%) of the Kerr shadow equivalent diameter from the Schwarzschild value $6\sqrt{3}M$. (b) Surrogate error $100 (y_{\rm fit}-y_{\rm ref})/y_{\rm ref}$ (%) evaluated on the discrete training grid (no interpolation), excluding the exactly polar case $i=0^\circ$ treated analytically.