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Analytical model of the photon ring with finite thickness

S. V. Chernov

TL;DR

The paper addresses how to extract the thickness and asymmetry of a black-hole photon ring from very long baseline interferometry data by introducing a thick, asymmetric Gaussian ring model. It represents the sky brightness as a product of a radial Gaussian and a constant azimuthal asymmetry (with $n=1$) and derives a closed-form analytic expression for the visibility $V(u,\phi_u)$ using large-argument expansions that lead to parabolic cylinder functions and Hermite polynomials. The resulting formulas reveal that the visibility is generally complex and depends on the ring parameters $(r_0,\Delta r)$ and the baseline orientation, with good agreement between the analytic expressions and numerical integrations up to baselines of $6$ Earth diameters at $690$ GHz. This framework enables direct fitting of EHT-like data and informs planning for ultra-long baseline interferometry in studies of supermassive black holes.

Abstract

An analytical model of a thick asymmetric Gaussian ring is presented for which the visibility function is calculated in two perpendicular directions for baselines up to 6 of the Earth's diameter.

Analytical model of the photon ring with finite thickness

TL;DR

The paper addresses how to extract the thickness and asymmetry of a black-hole photon ring from very long baseline interferometry data by introducing a thick, asymmetric Gaussian ring model. It represents the sky brightness as a product of a radial Gaussian and a constant azimuthal asymmetry (with ) and derives a closed-form analytic expression for the visibility using large-argument expansions that lead to parabolic cylinder functions and Hermite polynomials. The resulting formulas reveal that the visibility is generally complex and depends on the ring parameters and the baseline orientation, with good agreement between the analytic expressions and numerical integrations up to baselines of Earth diameters at GHz. This framework enables direct fitting of EHT-like data and informs planning for ultra-long baseline interferometry in studies of supermassive black holes.

Abstract

An analytical model of a thick asymmetric Gaussian ring is presented for which the visibility function is calculated in two perpendicular directions for baselines up to 6 of the Earth's diameter.

Paper Structure

This paper contains 4 sections, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. ANALYTICAL MODEL AND RESULTS
  3. Conclusion
  4. REFERENCES

Figures (2)

  • Figure 1: Brightness distribution in the ring for the following parameters: $r_0=20 \mu as$, $\Delta r=5 \mu as$, $\phi_0=\pi$, $B=1$ and $n=1$. The total flux in the ring is equal to 1 Jansky.
  • Figure 2: The visibility function as a function of the base projection u is plotted, where the lower axis represents values in wavelength units and the upper axis in Earth diameters. The green and red curves are obtained by numerical integration of the expression (1). The green curve corresponds to the direction of $\phi_u=0$, the red curve corresponds to the direction of $\phi_u=\pi/2$. Black curves are analytical curves obtained by the formula (10).