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arxiv: 2604.21974 · v1 · submitted 2026-04-23 · 🌌 astro-ph.HE

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The Effects of Complex Accretion Disk Geometry on Broadened Iron Kα Lines

William Surgent , Daniel R. Wilkins
Authors on Pith no claims yet

Pith reviewed 2026-05-09 20:18 UTC · model grok-4.3

classification 🌌 astro-ph.HE
keywords accretion diskiron K alphablack hole spinX-ray reflectionray tracingXRISMwarped diskcorona
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The pith

Non-negligible thickness in black hole accretion disks leads flat models to underestimate spin, corona height, and inclination.

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

This paper creates general relativistic ray tracing simulations to study how the shape of accretion disks around black holes changes the iron K alpha line in X-ray spectra. It shows that disks with significant thickness produce line shapes that cause flat disk models to report lower black hole spin, lower corona height, and lower inclination than the true values. Warped disks produce profiles that flat models cannot fit at all. These findings are important because they indicate that current and future precise measurements of black hole properties from X-ray lines may contain systematic biases if disk geometry is not accounted for.

Core claim

Using novel general relativistic ray tracing simulations, we investigate the effects of complex accretion disk geometries, including constant aspect ratio, radiation-pressure-dominated Shakura-Sunyaev, expanded inner disk, and warped disks, on the iron Kα line. We find that non-negligible thickness underestimates black hole spin, corona height, and inclination angle when fitted with a flat disk model using XRISM uncertainties, and that warped disk models cannot be fit with the flat disk approximation.

What carries the argument

General relativistic ray tracing simulations of X-ray illumination and reflection off accretion disks with varying geometries to model the broadened iron Kα line.

Load-bearing premise

That the differences in the iron line arise solely from the disk geometry and not from other factors such as turbulence, ionization structure, or the corona not being a point source.

What would settle it

Observe the iron Kα line profile in a source with independently measured disk thickness and compare the spin value from a flat model fit to the true spin to see if it matches the underestimation predicted by the simulations.

Figures

Figures reproduced from arXiv: 2604.21974 by Daniel R. Wilkins, William Surgent.

Figure 1
Figure 1. Figure 1: Cross-section of the constant aspect ratio accre￾tion disk geometry viewed edge-on (θ = π 2 and ϕ = 0). The black hole is shown at the center of the disk geometry, and hc is the height of the lamppost corona above the disk. Note that this plot has no associated units and is meant to give an understanding of the basic configuration of the system [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Emissivity profiles associated with the constant aspect ratio disk with h ρ = 0.1 and h ρ = 1 with corona heights minimum and maximum corona heights explored (hc = 2.5 rg and hc = 20 rg). The emissivity profiles are calculated for four values of corona height, hc = 2.5 rg, 5 rg, 10 rg, and 20 rg. We in￾vestigate this range of corona heights to span the range commonly inferred for Seyfert AGN as in Cackett … view at source ↗
Figure 3
Figure 3. Figure 3: Line profiles from the constant aspect ratio accretion disk. The top row shows the profiles associated with a corona height of 2.5 rg, the middle rows are the same but for corona heights of 5 rg and 10 rg, and the last is for 20 rg. The columns correspond to different inclination angles of observation, i. The blue line corresponds to the flat disk, the orange line corresponds to the disk with h ρ = 0.1, th… view at source ↗
Figure 4
Figure 4. Figure 4: Ratio plots of the constant aspect ratio disk line profiles and those from the flat disk. Each line is the division of the constant aspect ratio disk line by the flat disk line with both having the same corona height and observation inclination, i [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematic of the constant aspect ratio disk geometries with a lamppost corona emitting constant emissivity over a boundary (circle in each panel). The green disk has h ρ = 1, the blue disk has h ρ = 0.5, and the orange disk has h ρ = 0.1. On the far left, with hc = 5 rg, the region of constant emissivity makes first contact with the inner radii of each constant aspect ratio disk. In the second panel from t… view at source ↗
Figure 6
Figure 6. Figure 6: Emissivity profiles associated with the delayed wedged accretion disk with h ρ = 1. Each line corresponds to an accretion disk with a corona height of 5 rg but with different break radii, rb. shifted photons. This reduction in the most redshifted photons is greater in this disk geometry than it was for the constant aspect ratio disk. This additional reduc￾tion in photons that hit the inner accretion flow i… view at source ↗
Figure 7
Figure 7. Figure 7: Line profiles from the compressed inner accretion disk with hc = 5 rg. The top row shows the profiles associated with a break radius of rb = 5 rg, the middle row is the same but for rb = 10 rg, and the last is for rb = 20 rg. The columns correspond to different inclination angles of observation, i. A key labeling the corresponding colors for each disk is shown at the bottom of the plot. tably, as with the … view at source ↗
Figure 8
Figure 8. Figure 8: Cross-section of the Shakura-Sunyaev accretion disk geometry viewed edge-on, θ = π 2 and ϕ = 0. Black hole shown at center of the disk geometry and hc is the height of the lamppost corona above the disk. Note this plot has no associated units and is meant to give an understanding of the basic configuration of the system [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Emissivity profiles associated with the Shakura￾Sunyaev accretion disk with a corona height of 5 rg and Ed￾dington ratios M /˙ M˙ Edd = 0.3, 0.7, 1.1, and 17. Each line corresponds to the respective accretion disk geometry, but with a different Eddington ratio. We investigate a Shakura-Sunyaev disk in an extreme accretion scenario in which its Eddington ratio is 17. We choose this Eddington rate as it appr… view at source ↗
Figure 10
Figure 10. Figure 10: Line profiles from the Shakura-Sunyaev accretion disk. The top row shows the profiles associated with a corona height of 2.5 rg, the middle rows are the same but for corona heights of 5 rg and 10 rg, and the last is for 20 rg. The columns correspond to different inclination angles of observation, i. The blue line corresponds to the flat disk, the orange line corresponds to the disk with M /˙ M˙ Edd = 0.3,… view at source ↗
Figure 11
Figure 11. Figure 11: Ratio plots of the Shakura-Sunyaev disk line profiles and those from the flat disk. Each line is the division of the Shakura-Sunyaev disk line by the flat disk line at the same corona height and observation inclination, i. 6. THE WARPED DISK We investigated a warped disk geometry in which the accretion disk consists of a flat (θ = π 2 ) inner accretion disk that is aligned with the spin axis of the black … view at source ↗
Figure 12
Figure 12. Figure 12: Line profiles from the Shakura-Sunyaev accretion disk with super-Eddington rate of M /˙ M˙ Edd = 17. The top row shows the profiles associated with a corona height in decreasing order. The columns correspond to different inclination angles of observation, i. The blue line corresponds to the flat disk and the brown line corresponds to the disk with M /˙ M˙ Edd = 17 [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: Emissivity profiles associated with the expanded inner accretion disk with Eddington ratio M /˙ M˙ Edd = 1.1. Each line corresponds to the same accretion disk geometry but with a different corona height, hc, above the disk. the flat inner accretion flow on the misaligned position of the disk. For warped disk configurations that are [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Line profiles from the expanded inner accretion disk. The top row shows the profiles associated with a corona height of 2.5 rg, the middle rows are the same but for corona heights of 5 rg and 10 rg, and the last is for 20 rg. The columns correspond to different inclination angles of observation, i. The blue line corresponds to the flat disk, the orange line corresponds to the disk with M /˙ M˙ Edd = 0.3, … view at source ↗
Figure 16
Figure 16. Figure 16: Cross-section of the warped accretion disk ge￾ometry viewed edge-on, θ = π 2 and ϕ = 0. Black hole shown at center of the disk geometry and hc is the height of the lamppost corona above the disk; rb is the break radius of the inner accretion disk from the outer disk; and α is the angle of misalignment between the inner and outer disk. Note this plot has no associated units and is meant to give an under￾st… view at source ↗
Figure 17
Figure 17. Figure 17: Depiction of the effect of the azimuthal angle of observation on the warped disk. From the observer’s view, the effective area of the accretion disk is the same. How￾ever, because the material in the disk is orbiting in the same direction in both observations (shown by black arrows), the energy shifts change as a result of the shadowing of the inner disk on the outer disk. On the right, the emitting mater… view at source ↗
Figure 19
Figure 19. Figure 19: In gray, the line profile from a warped disk with hc = 10 rg, rb = 10 rg, and α = 30◦ viewed from an inclination of 10◦ and ϕ = 0◦ . In green, the line profile from the addition of the line profile from a flat disk with radius 10 rg and a misaligned outer disk (the outer portion of the warped disk). Simulating the two disks that make up the warped disk does not give the same line profile as simulating the… view at source ↗
Figure 21
Figure 21. Figure 21: Plot showing the effect of varying the break radius on the warped disk iron Kα line. Shown are the line profiles associated with the warped disk with α = 15◦ , hc = 5 rg viewed at i = 10◦ and ϕ = 0. Each line corresponds to a warped disk with a different value of rb. An increase in rb makes the inner disk more dominant in the line profile. Additionally, the angle of misalignment, α, of the outer disk and … view at source ↗
Figure 20
Figure 20. Figure 20: The warped disk line profiles from a warped disk with rb = 10 rg, hc = 5 rg, and α = 30◦ viewed from i = 10◦ with varying values of ϕ. We plot each of the line profiles for each value of ϕ from 30◦ to 330◦ on top of each other. We see that the most blueshifted spectra occur around ϕ = 180◦ and the most redshifted line profiles are near ϕ = 0◦ . Changes in the azimuthal angle of observation shift the locat… view at source ↗
Figure 22
Figure 22. Figure 22: Line profiles from warped disks with rb = 10 rg, hc = 5 rg, viewed from i = 10◦ and ϕ = 0◦ and varying angles of misalignment, α. Increasing α increases the shadowing effect from the inner disk on the misaligned disk, as seen in the reduction in the counts of photons between the doubly peaked reflection spectra. In addition to this, the effects detailed in [PITH_FULL_IMAGE:figures/full_fig_p016_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The flat disk line profile (gray line) plotted against all the line profiles from the warped disk with rb = 20 rg, α = 30◦ , and hc = 5 rg across all values of ϕ. Each panel corresponds to a different set of ϕ values. 1, we see that the accepted value (parameter value of constant aspect ratio disk model) falls within the uncer￾tainty, making the values consistent of spin and corona height consistent betwe… view at source ↗
read the original abstract

X-rays are emitted from the corona above the orbiting matter of the accretion disk and travel either directly to us or illuminate the disk. This illumination of the inner disk is enhanced by gravitational light bending, which focuses the rays towards the black hole and therefore towards the inner radii of the disk. These rays that hit the inner radii are reflected back to us, and we observe them in the X-ray reflection spectrum. In this work, we create novel general relativistic ray tracing simulations to investigate the effects of altering the geometry of the accretion disks of black holes on the most dominant part of the reflection spectrum, the iron K$\alpha$ line. Work demonstrating the effect of disk geometry on the iron line has been performed, though many previous analyses have assumed a simplistic system, consisting of a point-source corona with a flat and infinitesimally thin accretion disk. We extend these models to more realistic accretion disk approximations. These include a constant aspect ratio disk, a radiation-pressure-dominated Shakura-Sunyaev disk, an expanded inner disk that has a non-negligible scale height in its inner regions due to radiation pressure, as well as various warped disks. Using measurement uncertainties from XRISM, we find that non-negligible thickness in accretion disks underestimates the black hole spin, corona height, and inclination angle if fitted with a flat disk model. The warped disk model could not be fit with the flat disk approximation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper uses general relativistic ray-tracing simulations to model iron Kα line profiles from black hole accretion disks with non-flat geometries (constant aspect ratio, Shakura-Sunyaev, expanded inner, and warped disks) illuminated by a point-source corona. It reports that fitting the resulting spectra to a standard flat thin-disk model with XRISM-like uncertainties underestimates black hole spin, corona height, and inclination for thick disks, while warped disks cannot be adequately fit by the flat model.

Significance. If the results hold, the work demonstrates concrete biases in spin and geometry inferences from iron line profiles when disk thickness is ignored, with direct relevance to XRISM observations. The forward ray-tracing of multiple realistic geometries is a strength that provides falsifiable predictions for how thickness affects fitted parameters.

major comments (3)
  1. [§3] §3 (Simulation Setup): The ray-tracing retains a point-source corona and omits disk turbulence, radial ionization gradients, and non-point-source coronal illumination. This assumption is load-bearing for the central claim, as these effects can reshape the iron Kα profile in ways degenerate with geometric thickness, preventing attribution of the reported underestimates solely to disk geometry.
  2. [Results] Results (XRISM uncertainty fits): No error budgets or Monte Carlo realizations on the simulated spectra are shown, and there is no test that the biases in spin, height, and inclination persist when other assumptions (e.g., emissivity law or ionization) are varied. This limits the robustness of the observational implication stated in the abstract.
  3. [Warped disk results] Warped disk subsection: The statement that the warped disk 'could not be fit' with the flat model lacks quantitative details on the fitting statistic, parameter covariances, or residual structure, making it impossible to determine whether the failure is due to geometry or to other unmodeled aspects of the simulation.
minor comments (2)
  1. [Abstract] Abstract and §2: The specific warp parameters (radius, amplitude, orientation) and the exact definition of 'expanded inner disk' scale height should be stated explicitly for reproducibility.
  2. [Figures] Figure captions: Axis labels and units on the simulated spectra and residual plots should be clarified to distinguish between model and data.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have helped us identify areas where the manuscript can be strengthened. We address each major comment below and describe the revisions we will implement.

read point-by-point responses

  1. Referee: [§3] §3 (Simulation Setup): The ray-tracing retains a point-source corona and omits disk turbulence, radial ionization gradients, and non-point-source coronal illumination. This assumption is load-bearing for the central claim, as these effects can reshape the iron Kα profile in ways degenerate with geometric thickness, preventing attribution of the reported underestimates solely to disk geometry.

    Authors: We acknowledge that the simulations employ a point-source corona and exclude turbulence, radial ionization gradients, and extended coronal illumination. These simplifications were chosen to isolate the effects of disk geometry, which is the primary focus of the work. We agree that the omitted physics can produce line-profile changes that are potentially degenerate with thickness effects. In the revised manuscript we will expand the discussion section to explicitly address these degeneracies, state the limitations of the current setup, and note that future modeling incorporating turbulence and ionization gradients will be required to fully disentangle the contributions. This addition will clarify the scope of our central claim without altering the reported geometric biases. revision: partial

  2. Referee: [Results] Results (XRISM uncertainty fits): No error budgets or Monte Carlo realizations on the simulated spectra are shown, and there is no test that the biases in spin, height, and inclination persist when other assumptions (e.g., emissivity law or ionization) are varied. This limits the robustness of the observational implication stated in the abstract.

    Authors: We thank the referee for highlighting the need for quantitative robustness checks. We will add Monte Carlo error analysis by generating multiple noisy realizations of each simulated spectrum using the reported XRISM uncertainties, refitting them, and reporting the resulting distributions and error budgets on the recovered spin, height, and inclination. We will also repeat the fitting exercise for a range of emissivity indices and ionization states to test whether the reported underestimates remain. These new results will be incorporated into the results section and will directly support the observational implications in the abstract. revision: yes

  3. Referee: [Warped disk results] Warped disk subsection: The statement that the warped disk 'could not be fit' with the flat model lacks quantitative details on the fitting statistic, parameter covariances, or residual structure, making it impossible to determine whether the failure is due to geometry or to other unmodeled aspects of the simulation.

    Authors: We agree that the current description lacks the necessary quantitative support. In the revised manuscript we will report the best-fit χ²/dof values, the recovered parameter values with 1σ uncertainties, and the covariance information for the attempted flat-disk fits to the warped-disk spectra. We will also include residual plots that illustrate the systematic mismatches. These additions will demonstrate that the poor fits arise from the inability of the flat-disk model to reproduce the complex line profile produced by the warp. revision: yes

Circularity Check

0 steps flagged

No circularity: results from independent forward simulations fitted to separate model

full rationale

The paper generates iron Kα line profiles via general relativistic ray-tracing simulations for thick, Shakura-Sunyaev, expanded-inner, and warped disks, then fits those synthetic spectra with an independent flat thin-disk model under XRISM uncertainties. No equation, ansatz, or self-citation reduces the reported biases in spin, height, or inclination to a fitted parameter by construction; the central claim remains an empirical outcome of the simulation-plus-fitting pipeline rather than a definitional or self-referential tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard general-relativistic ray tracing and established thin-disk solutions; no new free parameters are introduced beyond the geometric choices already present in the cited disk models.

axioms (2)
  • standard math General relativity governs photon trajectories and gravitational redshift in the strong-field regime near the black hole.
    Invoked throughout the ray-tracing description in the abstract.
  • domain assumption The corona can be approximated as a point source whose illumination pattern is modified only by light bending.
    Implicit in the comparison between simulated disks and the flat-disk fitting model.

pith-pipeline@v0.9.0 · 5556 in / 1440 out tokens · 57564 ms · 2026-05-09T20:18:44.606113+00:00 · methodology

discussion (0)

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