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REVIEW 4 major objections 5 minor 18 references

Method for Determining the Parameters of a Ring-like Structure from the Visibility Function Shape

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The radius, width, and angular asymmetry of a black-hole ring can be recovered from the zeros and envelope of its VLBI visibility function.

desk verdict Solid visibility math, but the M87 asymmetry measurement is not supported because the fit ignores the model's own position angle. read the letter →

arxiv 2502.03026 v1 pith:JA4OIATS submitted 2025-02-05 astro-ph.IM

S. V. Chernov , M. A. Shchurov , I. I. Bulygin , A. G. Rudnitskiy This is my paper · ORCID
classification astro-ph.IM
keywords blackholeshadowvisibilityfunctionVLBIringparametersEventHorizonTelescopeM87*Besselzerosasymmetryparameter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a direct analytical route from VLBI visibility measurements to the basic parameters of a ring-like black-hole image: radius, width, and angular asymmetry. The key idea is that each parameter leaves a distinctive fingerprint in the visibility function—the ring radius sets the positions of the visibility minima, the ring width sets the first zero of the oscillation envelope, and the asymmetry makes the visibility complex and different in two perpendicular directions. The authors test the recipe on synthetic rings of various widths and on two-ring configurations, then apply it to the Event Horizon Telescope's M87* data, obtaining a ring radius r≈23 μas and an asymmetry parameter B=0.48. If the method holds, it gives a fast, imaging-free way to extract black-hole shadow parameters from current and future interferometric data, and a step toward estimating black-hole spin.

What carries the argument

The load-bearing object is the analytic visibility function of a Gaussian asymmetric ring, obtained by substituting the brightness model $I(r,\phi_r)=I_r(r)I_\phi(\phi_r)$ into the Fourier-transform definition of the visibility and integrating over the polar angle. That angular integration collapses the brightness asymmetry $I_\phi(\phi_r)=(1-B\sin^2((\phi_r-\phi_0)/2))^n$ into the compact expression $V(u,\phi_u)=\pi\int[(2-B)J_0(2\pi u r)-iB J_1(2\pi u r)\cos(\phi_u-\phi_0)]I_r(r)r\,dr$. The Bessel zeros $j_{0,n}$ then mark the visibility minima that fix the radius; the approximate large-baseline form shows a low-frequency envelope $\sin(\pi u w)$ whose first zero gives the width $w\approx 1/u$; and the imaginary part proportional to $B$ carries the asymmetry. This reduction turns a two-dimensional imaging problem into a one-dimensional curve-fitting problem.

What would settle it

Take a VLBI dataset synthesized from a simulated black-hole image with known radius, width, and asymmetry but with ellipticity or non-Gaussian radial brightness; if the zero-based recipe returns a radius or B that differs from the known input by more than the scatter in the paper's own Table I, the central circular-Gaussian assumption fails.

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Extended reading notes

Core claim

On its own terms, the paper's discovery is that a black-hole shadow's ring can be parameterized without image reconstruction: the radius sits in the zeros of the visibility function, the width sits in the zeros of its envelope, and the asymmetry sits in the complex part of the visibility. The authors show this for an asymmetric Gaussian ring by reducing the two-dimensional Fourier integral to a one-dimensional combination of $J_0$ and $J_1$ Bessel functions, give exact expressions for thin, thick, and parabolic rings, and verify the zero-based recipe on synthetic rings and on two-ring configurations. Applied to the EHT observations of M87*, the recipe yields $r\approx 23\,\mu\text{as}$ and $B=0.48$, with the width left undetermined because the relevant envelope zero falls outside the available baseline coverage.

Load-bearing premise

The load-bearing premise is that the true image is a perfectly circular, centered ring whose radial profile is Gaussian and whose angular asymmetry has the fixed form of Eq. (2); if M87*'s actual brightness is elliptical, has jet emission, or falls off non-Gaussianly, the fitted radius and B will be biased.

Editorial extensions

If this is right

  • The shadow radius of a black hole can be estimated from the minima of the visibility amplitude alone, without reconstructing an image; in the paper's synthetic tests the first three minima put the radius of rings of width 0.1–5 μas at 20.0–20.2 μas against the input 20 μas.
  • The ring width can be read from the first zero of the visibility envelope whenever the baseline coverage reaches $u\approx 1/w$; the paper's estimates for widths of 5.0, 7.5, and 10.0 μas are 3.41, 6.91, and 10.5 μas, with accuracy improving for wider rings.
  • For a bright thin ring superimposed on a thicker ring, both radii come from Bessel zeros and the thick ring's width is recovered to better than 7% by fitting the approximate Gaussian-ring visibility.
  • Applied to the EHT M87* observations, the method gives $r\approx 23$ μas and, the paper reports, the first estimate of the asymmetry parameter $B=0.48$; the ring width could not be determined because the envelope zero lies beyond the available baselines.
  • Because the visibility is complex only when the ring is asymmetric, measuring two perpendicular cuts through the visibility plane is a direct way to detect and quantify angular brightness asymmetry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the $B=0.48$ value should be read as the asymmetry of the best-fitting circular Gaussian model, not as a direct spin measurement; connecting $B$ to spin and inclination requires comparison with general-relativistic radiative-transfer simulations.
  • Editorial inference: the recipe's reliance on a single circular radius means it will be biased by ellipticity or off-center structure; checking whether visibility zero positions vary with position angle would be a cheap test of that assumption on real data.
  • Editorial inference: the same zero-and-envelope logic could be applied to closure amplitudes instead of complex visibilities, which would make the extraction less sensitive to phase-calibration errors in VLBI.
  • Editorial inference: if future space-VLBI baselines reach the first envelope zero for M87*, the currently unresolved ring width could be measured and the radius estimate cross-checked, potentially tightening black-hole mass estimates.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops an analytical model of an asymmetric Gaussian ring, derives closed-form visibility expressions for thin, thick, parabolic, and Gaussian-profile rings (Eqs. 5–11 and 30 in the Appendix), and proposes to estimate the ring radius from the positions of visibility-function zeros and the ring width from the first envelope zero. The method is tested on simulated rings of the same model family and on two-ring configurations. In the final section, the thin-ring formula (7) is fitted to two perpendicular cuts of the M87* EHT visibility amplitudes, yielding a ring radius r ≈ 23 μas and an asymmetry parameter B = 0.48.

Significance. If made robust, the paper would provide a simple and transparent frequency-domain route to ring parameters from VLBI visibilities, and the explicit Fourier transforms and the zero/envelope relations are useful additions to the methods toolkit. The analytical derivation is a genuine strength: the visibility formulas are obtained directly from a stated brightness model, and the zero-locating approach is standard. However, the headline M87* inference currently depends on an unstated alignment assumption and lacks uncertainty propagation, so the claimed B = 0.48 is not yet a reliable measurement.

major comments (4)
  1. [Section V, Eq. (7), Section III.A] The extraction of B = 0.48 from two fixed perpendicular cuts is not invertible as presented. For the thin-ring model, |V(u,φu)| = 1/2 sqrt((2−B)^2 J0^2(2πur0) + B^2 J1^2(2πur0) cos^2(φu−φ0)). Two cuts at φu = 0 and φu = π/2 therefore constrain two projections of the three parameters (r0, B, φ0), and B is determined only after fixing φ0. Section III.A explicitly states that the model is rotated so that the brightness maximum lies on the negative X-axis before sections are taken, but Section V does not report or perform such an alignment for M87*; it simply selects left–right and up–down axes. Unless φ0 is fitted jointly or fixed with a justified position angle, the “left–right” cut is generally not a pure J0 section, the zero-derived radius is biased, and the “up–down” cut gives only a projection of the asymmetry rather than the intrinsic parameter B. A concrete remedy is to fit Eq. (7) to the full visibility amplitudes with (r0, B, φ0) free and to compare that fit with the two-axis result.
  2. [Section V and Fig. 8] The M87* estimates are based on visibility points digitized from published figures without uncertainty estimates, and the fitted parameters are quoted without error bars. The statement that the asymmetry parameter was “estimated for the first time” therefore cannot be evaluated: there is no variance from the digitization, no systematic from the choice of the two cuts, and no goodness-of-fit. The paper should report parameter uncertainties and, ideally, fit the model to the calibrated visibility data; at minimum it should provide a Monte Carlo sensitivity test that randomizes the digitized points and the section orientations.
  3. [Section V and Table I] The zero-location radius estimator carries a finite-width bias that is visible in the paper's own Table I: for a true ring width w = 10 μas, the first three zero-based radii are 21.0–21.4 μas instead of 20.0 μas, i.e. a 5–7% bias. Section V applies the infinitely thin formula (7) to M87* and quotes r ≈ 23 μas without applying a finite-width correction or adding this systematic error. Since the fit does not constrain the M87* width, this bias should be propagated into the quoted radius before comparing with EHT's r ≈ 21 μas estimate.
  4. [Sections II–IV] The validation is performed only on simulated images generated from the same Gaussian asymmetric-ring family used to construct the estimators; the method has not been tested on GRMHD simulation images or on models with ellipticity, jet emission, or n > 1 angular profiles. This is a correctness-risk for the M87* application, which interprets B = 0.48 as a physical asymmetry. A concrete test would be to run the estimator on a set of GRMHD images with EHT-like (u,v) coverage and report the resulting scatter and bias.
minor comments (5)
  1. [Section VI] The text says “for a thin ring (w ≫ r0)”; this should be w ≪ r0.
  2. [Fig. 4 caption] The caption states that both the red and black dots correspond to φu = π; the second direction should presumably be φu = π/2.
  3. [Table I] The three rows labeled “r0, μas” are not identified with their zero orders; adding column headers such as u0, u1, u2 would make the bias trend clearer.
  4. [Section V] The use of “plotdigitizer” and “curvefit” is informal; the paper should specify the digitization procedure, the number of points, the fitting algorithm, and the initial values used.
  5. [General] There are several typographical errors, including “obtaoined”, “t M87*”, and inconsistent use of commas in numerical values; these should be corrected in a careful revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the visibility model is derived by direct Fourier transformation, and the M87 radius/asymmetry are fitted parameters compared with EHT, not predictions equal to inputs by construction.

full rationale

The paper's analytical chain is self-contained: Eq. (5) follows from substituting Eqs. (1)-(3) into the Fourier-transform definition Eq. (4); Eqs. (7), (8), (10), (11), and (30) are subsequent specializations or approximations of that integral, none of which reintroduce the fitted M87 parameters as assumptions. In Sec. V, the authors fit the free parameters r and B of Eq. (7) to the EHT visibility data; reporting the fitted values r≈23 μas and B=0.48 is parameter estimation, not a forced prediction. The comparison with the EHT image-domain radius is an independent congruency check. The only self-citation, Ref. [17] (same first author) for the direct evaluation of the thick-ring integral, is not load-bearing: the integral is a standard Bessel/Struve evaluation that can be independently checked, and no uniqueness or modeling choice is imported from it. The asymmetric brightness ansatz (2) is explicitly adopted from the literature as a model assumption, not presented as a derived conclusion. The potential degeneracy between B and the unquoted position angle ϕ0 in the M87 fits is an identifiability/model-misspecification concern about the application, not a circular derivation. Overall, no target result is equal by construction to an input or to a self-citation.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivation relies on the choice of a specific brightness model (Gaussian radial profile with angular asymmetry) and on asymptotic approximations for the Bessel functions. The only numbers fitted to real data are the ring radius and asymmetry parameter; all other inputs are model choices.

free parameters (2)
  • ring radius r0 = 23.01 μas (M87*)
    Free parameter in the thin-ring fit (Eq. 7) to the M87* visibility curve in Sec. V.
  • asymmetry parameter B = 0.48 (M87*)
    Free parameter in the same fit; fixed n=1 and the angular profile (Eq. 2).
assumptions (6)
  • domain assumption Brightness distribution separates radially and angularly: I(r,φ)=Ir(r)Iφ(φ) (Eq. 1).
    The black hole image is assumed to factor into radial and angular functions, which is an idealization of GRMHD images.
  • domain assumption The ring is circular with zero eccentricity and centered at origin (Sec. II).
    Elliptical or off-center features are ignored.
  • domain assumption Angular brightness profile is Iφ=(1 - B sin²((φ-φ0)/2))^n with n=1 and constant B (Eq. 2).
    This is a chosen model, not derived from general relativity; the asymmetry parameter B is then fitted to data.
  • domain assumption Radial brightness profile is Gaussian (Eq. 3).
    A convenient profile; real images may differ.
  • standard math Bessel functions expanded asymptotically for 2πur ≫ 1 (Eqs. 14, 16-22).
    Used to derive the width estimate from the envelope. Fails at short baselines, but the paper claims large baselines.
  • standard math Saddle-point approximation with √r ≈ (√r0 + r/√r0)/2 and r0 ≫ Δr (Appendix).
    Used to derive approximate visibility (30).

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Cite this review

Pith. "Pith review of Method for Determining the Parameters of a Ring-like Structure from the Visibility Function Shape." pith.science (2026). https://pith.science/paper/JA4OIATS

@misc{pith2026250203026,
  author       = {Pith},
  title        = {Pith review of: Method for Determining the Parameters of a Ring-like Structure from the Visibility Function Shape},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JA4OIATS}},
  note         = {Machine review of arXiv:2502.03026}
}
abstract

Black hole images obtained by very long baseline interferometry (VLBI) by the Event Horizon Telescope are a new tool for testing general relativity in super-strong gravitational fields. These images demonstrated a ring-like structure which can be explained as the black hole shadow image. To date, there are no reliable methods for determining the parameters of these ring-like structures, such as diameter, width, and asymmetry. In this paper, an algorithm for determining black hole image parameters is proposed using a Gaussian asymmetric ring as an example. Using the proposed method, the diameter and asymmetry parameters of the image of a supermassive black hole in the galaxy M87$^{*}$ were estimated based on observational data obtained by the Event Horizon Telescope group.

Figures

Figures reproduced from arXiv: 2502.03026 by the authors.

Figure 1
Figure 1. FIG. 1. Brightness distribution in the ring depending on the ring width ∆ [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Visibility functions in two perpendicular directions obtained for the images shown in Fig. 1: [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. An example of images with two thin rings with different maximum brightness positions. [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Visibility functions for images consisting of two thin rings. Blue stars show the position of [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. An example of images consisting of one thick and one thin ring. The radius and width of [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Visibility function modulus in two perpendicular directions for the case of one thin ring [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Upper panel: ( [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Approximation of the visibility function of M87 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

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Reference graph

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