Inferring Black Hole Spin from Interferometric Measurements of the First Photon Ring: A Geometric Approach

TL;DR

The paper addresses the challenge of measuring black hole spin from electromagnetic observations by proposing a geometry-driven approach that uses the interferometric shape of the first photon ring ($n=1$) as a proxy for the Kerr critical curve. It develops two inference pipelines—least-squares circlipse fitting and Bayesian MCMC—to extract the ring’s angular-dependent diameters and its asymmetry $f_A$ from visibility data, then maps this shape to spin via a dense grid of critical-curve shapes parameterized by $(a_{*}, i_{*})$ with $i_{*}$ assumed or constrained. In noiseless, time-averaged, high-spin cases at low inclination, the method can recover $a_{*}$ within about $5 ext{\%}$ (2σ) when baselines are long and the $n=2$ ring dominates; however, noise and source variability broaden posteriors and often yield only lower bounds, especially for space-based observational scenarios. The results demonstrate a viable, geometry-based spin diagnostic for future space VLBI missions (e.g., BHEX), while highlighting the need for joint modeling of $n=0$/$n=1$ morphology, multi-frequency data, and robust emission-model priors to improve accuracy and reliability in realistic, noisy data sets.

Abstract

Accurately inferring black hole spin is crucial for understanding black hole dynamics and their astrophysical environments. In this work, we outline a geometric method for spin estimation by using the interferometric shape of the first photon ring ($n=1$) as an approximation to the critical curve, which, given an assumed value of the black hole inclination, is then mapped to a spin value. While future space-based missions will capture a wealth of data on the first photon ring--including the full angle-dependent diameter, angular brightness profile, and astrometric offset from the $n=0$ ring--our analysis is restricted to using only two angle-dependent diameters to compute its shape asymmetry and infer spin. Focusing on low inclinations and moderate-to-high spins, we test the method across various emission models, baselines, and noise sources, including a mock space-based observation. Although the size of the $n=1$ ring depends on the emission model, its interferometric shape remains a robust spin probe at low inclinations. We find that the inferred asymmetry of the $n=1$ image may be biased by the critical curve morphology, and it can be heavily skewed by the presence of noise, whether astrophysical or instrumental. In low-noise limits at low viewing inclination, significant contributions from the $n=0$ image at short baselines may lead to a downward bias in asymmetry estimates. While our method can estimate high spins in noise-free time-averaged images, increasing the noise and astrophysical variability degrades the resulting constraints, providing only lower bounds on the spin when applied to synthetic observed data. Remarkably, even using only the ring's asymmetry, we can establish lower bounds on the spin, underscoring the promise of photon ring-based spin inference in future space-based very long baseline interferometry missions, such as the proposed Black Hole Explorer.

Figures (12)

• Figure 1: Fractional shape asymmetry (\ref{['eq:FractionalAsymmetry']}) contours interpolated from critical curves with spin-inclination pairs $(a_{*},i_{*})$, where $a_{*}$ and $i_{*}$ take on $2,000$ equally-spaced values in the ranges $[0.00001, 0.999999]$, and $[1^{\circ}, 90^{\circ}]$, respectively. The first (leftmost) contour corresponds to a critical curve fractional asymmetry of $0.1\%$ while the respective asymmetry of the subsequent contours follow the arithmetic sequence $0.5\%$, $1.0\%,\ldots,13.0\%$. The $0.0\%$ asymmetry contour lies on the $y$-axis, since the critical curve is circular for $a_{*}=0$ at all inclinations $0^{\circ}\leq i\leq90^{\circ}$.
• Figure 2: Contour bands in the spin-inclination plane for the photon ring shape asymmetries inferred across the baseline windows $[20, 40]\,\mathrm{G}\lambda$ (green and where the $n=1$ photon ring typically dominates the signal) and $[280, 300]\,\mathrm{G}\lambda$ (orange and where the $n=2$ photon ring typically dominates the signal) with underlying spins $a_{*}=0.5$ (left) and $a_{*}=0.94$ (right). Each band indicates the range of inferred asymmetries: the left and right edge of each band is given by the minimum and maximum inferred asymmetry, respectively, while the region in between is filled in for visual clarity. Here (and in subsequent figures) "Truth" denotes the point $(a_{*}, i_{*})$. The black dashed lines denote the fractional asymmetry of the underlying critical curve circlipses---by construction, the closer any inferred asymmetry is to this value, the more accurate the spin inference. For the underlying spin $a_{*}=0.5$, the critical curve asymmetry is $0.13\%$ and the minimum and maximum inferred asymmetries are $(0.21\%,1.30\%)$ across $[20, 40]\,\mathrm{G}\lambda$, and $(0.05\%,0.52\%)$ across $[280, 300]\,\mathrm{G}\lambda$. For $a_{*}=0.94$, the critical curve asymmetry is $1.02\%$, and the minimum and maximum inferred asymmetries are $(0.56\%,1.13\%)$ across $[20, 40]\,\mathrm{G}\lambda$, and $(0.75\%,1.18\%)$ across $[280, 300]\,\mathrm{G}\lambda$.
• Figure 3: Distributions of the spin inferred across the baseline windows $[20, 40]\,\mathrm{G}\lambda$ (green), $[80, 100]\,\mathrm{G}\lambda$ (purple), and $[280, 300]\,\mathrm{G}\lambda$ (orange) with underlying spins $a_{*}=0.5$ (left) and $a_{*}=0.94$ (right). Each histogram has a corresponding set of inferred asymmetries (which lie within bands like those depicted in Fig. \ref{['fig:AsymmetryBands']}) from which the spin distribution is obtained by finding the point of intersection of each asymmetry contour with the horizontal line $i=17^{\circ}$ in the spin-inclination plane. Estimated $2\sigma$ error intervals for each distribution are listed in Tab. \ref{['tbl:AstroSpinInferenceStatistics']}.
• Figure 4: Equatorial $J_{\rm SU}(r;\mu,\vartheta,\gamma)$ emission profiles as a function of the Boyer-Lindquist radius for the $140$ profiles listed in Tab. \ref{['tbl:Parameters']}. The blue curves correspond to profiles for which spin inference from the corresponding visibility amplitude across the baseline windows $[20, 40]\,\mathrm{G}\lambda$ (left) and $[80, 100]\,\mathrm{G}\lambda$ (right) was successful, while red denotes unsuccessful inferences. In all cases, the underlying black hole has spin $a_{*}=0.94$ and inclination $i_{*}=17^{\circ}$. The dash-dotted grey lines denote the location of the inner/outer event horizons $r_{\pm}$ and the inner-most stable circle orbit $r_{\mathrm{ms}}$. As listed in Tab. \ref{['tbl:AstroSpinInferenceStatistics']}, spin inference was successful across $[20, 40]\,\mathrm{G}\lambda$ for $38\%$ of the emission profiles, compared to $69\%$ across $[80, 100]\,\mathrm{G}\lambda$.
• Figure 5: Left: Inferred diameters and their $1\sigma$ error bars (blue), obtained from MCMC-based fits of AART-simulated visibility amplitudes along the $36$ cuts $\varphi=\{0^{\circ}, 5^{\circ},\ldots,175^{\circ}\}$ across the baseline window $[20, 40]\,\mathrm{G}\lambda$ (with artificially-prescribed errors as explained in Sec. \ref{['ssec:SpinInferenceGLM']}). In dashed red we plot the best-fit circlipse \ref{['eq:Circlipse']}. The underlying black hole has spin $a_{*}=0.94$, inclination $i_{*}=17^{\circ}$ and $J_{\mathrm{SU}}$ emission profile parameterized by $\mu=r_{-}$, $\sigma=0.5\,M$, and $\gamma=-1.5\,M$. Right: Posterior distribution (blue) of the fractional asymmetry, obtained via pointwise evaluation of Eq. \ref{['eq:FractionalAsymmetry']} using the posterior distributions of the circlipse parameters $(R_{0}, R_{1}, R_{2})$. The black dashed lines correspond to the $2.5\%$, $50\%$ and $97.5\%$ percentiles of the posterior distribution, given by $0.48\%$, $0.70\%$, and $0.92\%$, respectively.