Shapes and Positions of Black Hole Shadows in Accretion Disks and Spin Parameters of Black Holes

Rohta Takahashi

REVIEW

Shapes and Positions of Black Hole Shadows in Accretion Disks and Spin Parameters of Black Holes

Rohta Takahashi

Astrophysics · Submitted 6 May 2004 · 41 references

Pith's verdict: The shadow-axis construct is the real contribution; the negative claim about size and shape is correct but stated more broadly in the abstract than figure 5 supports.

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CitationRohta Takahashi. "Shapes and Positions of Black Hole Shadows in Accretion Disks and Spin Parameters of Black Holes." arXiv:astro-ph/0405099 (2004). https://pith.science/paper/3EL4POJD

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Editor's note

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Bottom line: the useful part of this paper is not the negative result, it is a small clean geometric construct — the perpendicular bisector of the shadow's maximum-width chord, which the author calls the shadow axis — plus the observation that its offset from the black hole mass center grows roughly linearly with spin. The negative claim that size and shape alone cannot pin down spin is right, but narrower than the abstract makes it sound. What is new: the ray tracing is standard Kerr (Cunningham & Bardeen, Rauch & Blandford) and the author says so. The shadow-axis offset is genuinely new, and unlike the horizontal shift from the rotation axis it is measurable from an image alone. That is the piece later shadow-fitting work would actually use. The numbers check out where I can check them: the optically thin shadow diameter of about 10.4 rg, and the lensed 13.85 rg for a 12 rg disk edge. Soft spots, in order of size. First, the mass center. The diagnostic is a difference of two positions, so it needs the mass center localized to roughly 1.5 rg. The author states this plainly in §4.2 and again in §5.1, and offers stellar proper motions or fitting the outer brightness distribution. That is honest, but it means the diagnostic inherits much of the practical difficulty it diagnoses for the horizontal shift. Anyone citing this should carry that along. Second, the scope of the abstract. Figure 5's degeneracy is shown with inner edges all near 2 rg, including rin = 1.05 rh+ for a/m = 0.5, which sits inside the marginally stable orbit. The body lists the parameters and §1 justifies such edges via super-Eddington disks, so this is not sleight of hand. But the abstract's 'practically difficult' reads as unconditional, and for a standard disk figure 2 shows shadow size running from 4.2 to 14.7 rg across spin, which would discriminate. The stress-test note frames this as weakening the demonstration; that is wrong. The degeneracy is at fixed physical inner edge, which is the stronger statement. It is a presentation issue. Third, minor: the near-linear scaling is numerical with no analytic argument and no test across disk models. There is also a real slip in §4.2, where the worked example says 'horizontal shift ... 0.5' but reads the value off figure 6, which plots the shadow-axis interval. Worth fixing. One correction to the record: the classifier found no stated limitations, which is wrong — §4.2, §5.1 and §5.3 all state them. Who it is for: people doing shadow imaging and spin fitting, especially anyone about to fit a contour to an observed image. It deserves referees.

Referee report

Significance

The paper addresses a question that has become central to black hole astrophysics, and it does so with the right machinery: the Kerr geodesic equations are handled analytically, two physically motivated disk inner edges are carried through the whole analysis, and the results are presented as quantitative curves rather than cartoons. Two strengths deserve explicit credit. The pipeline is validated against a known limit: the optically thin shadow size quoted as roughly 9.6–10.4 r_g brackets the Schwarzschild critical diameter 2×3√3 m ≈ 10.39 r_g, which is a genuine self-consistency check. And the central scaling, interval ∝ a/m at fixed inclination, is a falsifiable prediction stated in a form that an image can test. The negative result is also useful: it warns against reading a spin off a shadow size without independent knowledge of the disk inner edge. The paper is honest about its main vulnerability, stating in §4.2 and again in §5.1 that the diagnostic requires the mass center to about 1.5 r_g.

Major objections (3)

[§4.1–§4.2, Fig. 6] The proposed diagnostic rests on measuring a small residual: the perpendicular distance from the mass center to the shadow axis is at most about 1.5 r_g, and the shadow axis is itself defined through the direction of the maximum-width chord. A small error δθ in that direction rotates the bisector by δθ, and because the bisector is anchored at the midpoint of the chord, the perpendicular distance from the mass center changes by of order |d| sin δθ, where d is the distance from the mass center to that midpoint. For d of a few r_g and δθ of a few degrees, this can be comparable to the entire signal. The manuscript asserts in §4.1 that 'We can easily determine a position of the shadow axis by observation of a black hole shadow in an accretion disk' but nowhere quantifies the conditioning of the construction. Please add an explicit sensitivity analysis: propagate an angular uncertainty in the maximum-width direction, and a positional uncertainty in the chord midpoint, into the inferred interval and hence into the recovered a/m. Without this, the paper's own criterion for dismissing the horizontal-shift method (that the required reference point cannot be located) has not been shown to be met by the replacement method.
[Abstract (s3), §4.1, Fig. 5] The negative claim in the abstract, that it is 'practically difficult to determine a spin parameter ... from a size and a shape of a black hole shadow', is stated without the condition under which Fig. 5 demonstrates it. The three degenerate models in Fig. 5 use r_in = r_h+, r_in = 1.05 r_h+ and r_in = r_ms, i.e. the inner edge is varied together with the spin so that the physical inner radius stays near 2 r_g in all three cases. That is a legitimate and interesting demonstration, but it is a demonstration about disks whose inner edge is unknown over the range [r_h+, r_ms]. If the disk is independently known to be standard with r_in = r_ms, Fig. 2 shows the shadow size sweeping roughly 4.2–14.7 r_g across the spin range, which would discriminate. The abstract should state this conditionality explicitly, so that the negative claim is not read as unconditional.
[§5.1] The paper identifies mass-center localization to about 1.5 r_g as the crux of the proposed method, and §5.1 offers two routes. The first, stellar proper motions from GAIA and JASMINE, is a genuine external constraint. The second, fitting the outer, weakly lensed brightness distribution with Luminet-type models, is asserted rather than demonstrated: no calculation is presented showing that such a fit constrains the center to the required tolerance, nor how the fit degrades with the unknown disk inclination, density profile and velocity field. Since this route is what makes the method applicable to sources without resolved stellar motions, it is load-bearing for the paper's practical claim. Either provide a demonstration (even a simplified one) or soften the claim to an untested suggestion.

Minor objections (9)

[Eq. (9)] The expression for the impact parameter y is written as y = q² + cos²i − p² cot²i, without the square root that the standard Bardeen form requires and that is dimensionally necessary to match Eq. (1). Please correct to y = ±(q² + cos²i − p² cot²i)^{1/2} and state the sign convention.
[Fig. 1 caption] The caption states that 'we can see an axisymmetric shape and position of a black hole shadow with respect to a rotation axis', which contradicts the body text and the figure itself; this should read 'non-axisymmetric'.
[§4.2] The worked example reads 'with an inclination angle of i = 45° and a horizontal shift of a black hole shadow to be 0.5, from figure 6 a spin parameter ... is constrained between 0.75 and 0.89'. Figure 6 plots the minimum interval between the mass center and the shadow axis, not the horizontal shift, which is Fig. 4. Please correct the label and re-check the quoted range against the correct curve.
[§3.1] The text alternates between 'r_ms' and 'rsm' (e.g. 'rin = rh+ (thick solid lines) and rsm (thin solid lines)'); please make the notation uniform throughout.
[Abstract] The phrase 'not symmetric with respect to a rotation axis of a black hole shadow' should read 'with respect to the rotation axis of the black hole'; the shadow axis is the new construct and should not be conflated with the rotation axis.
[§3.2] The value 'nearly 13.85' for the maximum shadow size at i = 0°, r_in = r_ms and a/m = 0 is quoted without an indication of numerical accuracy; please state the resolution or tolerance of the contour extraction so that the reader can judge the significance of the quoted figures.
[§3.1, §4.1] The construction of the shadow axis is ill-conditioned as i → 0, where the shadow becomes circular and the maximum-width chord is degenerate. The paper notes only that the interval vanishes in that limit. Please add a sentence on the behavior at small inclination and on the two-tip contours of Fig. 1, where the maximum-width chord may connect the tips and the bisector may be far from the center.
[§2] The paper considers only co-rotating disks. Please state explicitly that the counter-rotating case is not treated and indicate whether the qualitative conclusions are expected to survive it.
[§5.1] Typographical: 'than the image at large radii would be the same as produced by a non-rotating black hole' should read 'then'. Other typographical slips include 'casted' (abstract and throughout), 'adress' (§5.3), 'nealy' (§3.2), 'quantitavive' (§1) and the garbled 'greaterorsimilar' in §3.2.

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The paper

The paper claims

Two black holes spinning at very different rates can cast shadows of nearly the same size and shape, so a shadow's outline alone cannot pin down spin.

The reading

Takahashi asks a practical question about black hole imaging: if a black hole sits inside an optically thick, geometrically thin accretion disk, can the size and shape of its shadow tell us how fast it spins? By tracing null geodesics in the Kerr metric for two choices of the disk's inner edge — the event horizon and the marginally stable orbit — he finds shadows with almost identical maximum widths and flatness ratios for spin parameters as different as $a/m = 0$ and $a/m = 0.958$. That degeneracy means naive contour fitting is not a spin measurement. The paper's way out is a new image-measurable line: the shadow axis, the bisector perpendicular to the segment of the shadow's maximum width. For a non-rotating hole the mass center lies exactly on that axis; for a rotating hole it does not, and the minimum distance from the mass center to the shadow axis grows roughly in proportion to spin, reaching about $1.5r_g$ at maximum spin and near edge-on viewing. The catch the author states plainly is that the mass center must be located inside the shadow to roughly that same tolerance.

What carries the argument

The machinery is the exact contour of a black hole shadow in Kerr spacetime, obtained by integrating the $r$- and $θ$-components of the null geodesic equations with Carter's constant and the two impact parameters $p = L_z/E$ and $q^2 = C/E^2$, then mapping them to sky coordinates $x = -p/λsin i$ and $y^2 = q^2 + λcos^2 i - p^2λcot^2 i$. The disk's inner edge — event horizon $r_{h+} = m + (m^2-a^2)^{1/2}$ or marginally stable orbit $r_{ms}$ — sets where the occulting luminous matter stops and therefore where the shadow boundary falls. On top of this contour the paper defines the shadow axis: the perpendicular bisector of the segment of maximum shadow width, a line an observer can in principle draw from the image alone, in contrast to the black hole's rotation axis.

Extended reading notes

Core claim

The paper establishes a negative result and a repair. The negative result: for a black hole embedded in an optically thick accretion disk, shadows computed for $a/m = 0$, $0.5$ and $0.958$ can share a maximum width (about $6.2r_g$ in one group, about $7.7r_g$ in another) and a flatness ratio (about $0.86$ and $0.60$ respectively), so size and shape alone leave spin badly underdetermined. The repair: define the shadow axis as the bisector perpendicular to the line segment of the shadow's maximum width. Because frame dragging makes the shadow asymmetric about the black hole's rotation axis, the mass center is offset from this bisector, and that minimum interval is zero at $a/m = 0$ and rises roughly linearly with spin for fixed inclination, reaching about $1.5r_g$ for maximum spin, near-edge-on inclination, and a disk whose inner edge is the event horizon. The offset is larger when the inner edge is the horizon than when it is the marginally stable orbit, since frame dragging matters more close in. The author also notes a separate, weaker diagnostic — the horizontal shift of the shadow center from the rotation axis, up to about $1.45r_g$ — but sets it aside because the rotation axis cannot be located observationally.

Load-bearing premise

The spin estimate is a difference between two positions, so it only works if the black hole's mass center can be located inside the shadow to within roughly the size of the effect, about $1.5r_g$ at worst.

Editorial extensions

If this is right

Fitting a single observed shadow contour with a disk model cannot by itself yield a spin; only a positional offset relative to a known mass center does.
The mass center-to-shadow-axis interval gives a roughly linear spin estimator for a fixed inclination, with a maximum of about $1.5r_g$ to work with.
Because the offset is larger for a disk reaching the event horizon than for one truncated at the marginally stable orbit, the disk's inner boundary must be modelled jointly with spin.
Shadows with visible tips — one or two cusps, seen at large inner radii and high inclination — are the limited cases where direct contour fitting remains informative.
The flatness ratio (minimum width over maximum width) is a good inclination indicator but a poor spin indicator, hovering near $0.5$ at $i = 89°$ across most of the spin range.

Reading between the lines

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

If the mass center can be fixed by stellar proper motions at the tens-of-microarcsecond level, the shadow-axis interval becomes a genuinely model-light spin estimator, since it needs no assumption about the rotation axis and only weakly involves the disk's radiative properties.
A finite disk thickness would let the disk eclipse its own far side at high inclination, shifting the apparent maximum-width segment and hence the bisector; that systematic could bias the spin estimate in the same direction as high spin, and the paper flags it as future work.
The degeneracy reported here is a statement about a single broadband image; adding spectral or polarization information that weights the approaching and receding sides differently could break it without any new spatial resolution.
The same logic — a symmetry axis recoverable from the image versus a symmetry axis that is not — should apply to any optically thin-plus-thick hybrid source, so the diagnostic may transfer to jet-dominated nuclei where a thin disk is doubtful.

Original abstract

Can we determine a spin parameter of a black hole by observation of a black hole shadow in an accretion disk? In order to answer this question, we make a qualitative analysis and a quantitative analysis of a shape and a position of a black hole shadow casted by a rotating black hole on an optically thick accretion disk and its dependence on an angular momentum of a black hole. We have found black hole shadows with a quite similar size and a shape for largely different black hole spin parameters and a same black hole mass. Thus, it is practically difficult to determine a spin parameter of a black hole from a size and a shape of a black hole shadow in an accretion disk. We newly introduce a bisector axis of a black hole shadow named a shadow axis. For a rotating black hole a shape and a position of a black hole shadow are not symmetric with respect to a rotation axis of a black hole shadow. So, in this case the minimum interval between a mass center of a black hole and a shadow axis is finite. An extent of this minimum interval is roughly proportional to a spin parameter of a black hole for a fixed inclination angle between a rotation axis of a black hole and a direction of an observer. In order to measure a spin parameter of a black hole, if a shadow axis is determined observationally, it is crucially important to determine a position of a mass center of a black hole in a region of a black hole shadow.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,998 input tokens.

What the abstract itself states, as probabilities: claims first 0.14 · states limitation 0.85 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.10 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Can we determine a spin parameter of a black hole by observation of a black hole shadow in an accretion disk? In order to answer this question, we make a qualitative analysis and a quantitative analysis of a shape and a position of a black hole shadow casted by a rotating black hole on an optically thick accretion disk and its dependence on an angular momentum of a black hole. We have found black hole shadows with a quite similar size and a shape for largely different black hole spin parameters and a same black hole mass. Thus, it is practically difficult to determine a spin parameter of a black hole from a size and a shape of a black hole shadow in an accretion disk. We newly introduce a bisector axis of a black hole shadow named a shadow axis. For a rotating black hole a shape and a position of a black hole shadow are not symmetric with respect to a rotation axis of a black hole shadow. So, in this case the minimum interval between a mass center of a black hole and a shadow axis is finite. An extent of this minimum interval is roughly proportional to a spin parameter of a black hole for a fixed inclination angle between a rotation axis of a black hole and a direction of an observer. In order to measure a spin parameter of a black hole, if a shadow axis is determined observationally, it is crucially important to determine a position of a mass center of a black hole in a region of a black hole shadow.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Can we determine a spin parameter of a black hole by observation of a black hole shadow in an accretion disk?	background 0.84	0.03	0.03	unclear 0.68
s1	In order to answer this question, we make a qualitative analysis and a quantitative analysis of a shape and a position of a black hole shadow casted by a rotating black hole on an optically thick accretion disk and its dependence on an angular momentum of a black hole.	method 0.76	0.87	0.03	asserted 0.66
s2	We have found black hole shadows with a quite similar size and a shape for largely different black hole spin parameters and a same black hole mass.	result 1.00	0.94	0.15	asserted 1.00
s3	Thus, it is practically difficult to determine a spin parameter of a black hole from a size and a shape of a black hole shadow in an accretion disk.	interpretation 0.84	0.85	0.63	asserted 0.94
s4	We newly introduce a bisector axis of a black hole shadow named a shadow axis.	contribution 0.99	0.91	0.04	asserted 1.00
s5	For a rotating black hole a shape and a position of a black hole shadow are not symmetric with respect to a rotation axis of a black hole shadow.	result 0.96	0.77	0.05	asserted 1.00
s6	So, in this case the minimum interval between a mass center of a black hole and a shadow axis is finite.	result 0.93	0.80	0.06	asserted 1.00
s7	An extent of this minimum interval is roughly proportional to a spin parameter of a black hole for a fixed inclination angle between a rotation axis of a black hole and a direction of an observer.	result 0.99	0.87	0.14	asserted 0.89
s8	In order to measure a spin parameter of a black hole, if a shadow axis is determined observationally, it is crucially important to determine a position of a mass center of a black hole in a region of a black hole shadow.	interpretation 0.43	0.34	0.29	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 0.65 · evidence from proof or derivation 0.60.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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astro-ph 2008-12 conditional novelty 5.0 of 10

A black hole violating the Kerr bound (J > M) would cast a shadow roughly an order of magnitude smaller than a Kerr black hole, providing a potential observational signature.
Hidden symmetries, null geodesics, and photon capture in the Sen black hole
gr-qc 2008-05 accept novelty 5.0 of 10

The Sen black hole admits an explicit irreducible Killing tensor and conformal Killing tensor that explain its separability, and its photon shadow, with those of related Kerr-Newman solutions, takes disk, circle, dot,...
Observing the shadow of Einstein-Maxwell-Dilaton-Axion black hole
gr-qc 2013-11 conditional novelty 4.0 of 10

The shadow of the Einstein-Maxwell-Dilaton-Axion black hole shrinks as the dilaton parameter grows and distorts most near extremality, with a matching arc-plus-spot shadow for the naked singularity.
Measuring the Kerr spin parameter of regular black holes from their shadow
gr-qc 2013-09 conditional novelty 4.0 of 10

For non-rotating or slowly rotating regular black holes the shadow, iron line, and ISCO methods yield different inferred Kerr spins, so combining them can test the Kerr hypothesis; for near-extremal objects all three ...
Can the supermassive objects at the centers of galaxies be traversable wormholes? The first test of strong gravity for mm/sub-mm VLBI facilities
gr-qc 2013-04 conditional novelty 4.0 of 10

A traversable wormhole at the Galactic center would cast a shadow about half the angular size of a black hole's, a gap large enough for near-future mm/sub-mm VLBI to resolve.
Testing Extended Theories of Gravity via Black Hole Photon Rings
gr-qc 2025-05 conditional novelty 3.0 of 10

For Konoplya-Zhidenko deformed Schwarzschild black holes, epsilon controls photon sphere, shadow, and photon ring size while a2 and b2 are observationally degenerate, and EHT data constrain epsilon to about -0.09 to 0...
Probing a nonlinear electrodynamics black hole with thin accretion disk, shadow and deflection angle with M87* and Sgr A* from EHT
gr-qc 2022-05 conditional novelty 3.0 of 10

A nonlinear-electrodynamics black hole's shadow, deflection angle, and thin-disk emission are computed, and the coupling parameter is bounded loosely by EHT shadow sizes for M87* and Sagittarius A*.
Probes and Tests of Strong-Field Gravity with Observations in the Electromagnetic Spectrum
astro-ph 2008-06 accept novelty 3.0 of 10

Electromagnetic observations of neutron stars already exclude large regions of scalar-tensor gravity parameter space, while observations near black hole horizons cannot discriminate between theories at all.
M87: a cosmic laboratory for deciphering black hole accretion and jet formation
astro-ph.HE 2024-12 unverdicted novelty 1.0 of 10

A comprehensive review of M87's black hole, accretion flow, and jet observations from event-horizon to kiloparsec scales, with no new measurements.

How later papers used it

77 later papers cite it

Evidence: some. 77 of the citing papers share no author with it, and 2 of those carry a confident label from Pith's citation classifier.

Pith read 6 of the 90 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 built on it.

Builds on it 1 paper

Our results extend earlier work by [ref] and [ref] .

Jets and Rings in Images of Spinning Black Holes (2022) · Conclusion · 1.00

How the literature uses it

51 arXiv papers cite this in 56 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 32 sentences without a confident label

Uses its data or values 1 paper

Equivalently, this corresponds to $≈0.14$ times the BHS diameter [ref] .

When the Shadow Meets Its Measure: Assessing the Feasibility of Submillimeter Black Hole Shadow Imaging in Megamaser Disk AGN · Constraining the SMBH Spin Parameter using SVLBI · 0.97

Cites it as background 23 papers

This feature, originally termed the “central shadow” [ref] , and whose morphology was subsequently further investigated by [ref] , has now become widely recognized as the “inner shadow” [ref] .

Reshaping the inner shadow of a Kerr black hole by a torn accretion disk · Introduction · 0.93
However SMBHs do rotate and the effect of their spin on the surrounding environment alters the shape and the size of their shadows, quantities that now need to be computed in the Kerr metric, describing rotating BHs [ref] .

Climbing out of the shadows: building the distance ladder with black hole images · Spinning SMBHs and their shadows · 0.85
There exists several works in the literature which explored the optical properties of the Janis-Newman-Winicour spacetime, e.g. gravitational lensing and relativistic images [ref] , accretion and shadow [ref] .

Imprints of the Janis-Newman-Winicour spacetime on observations related to shadow and accretion · Introduction · 0.98

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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