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The plunging region of a thin accretion disc around a Schwarzschild black hole

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A global 3D GRMHD simulation confirms the MB23 analytic geodesic model for the thermodynamics of the plunging region of a thin Schwarzschild disc, provided non-adiabatic heating is included, and measures a finite ISCO stress of δJ ≈ 5.3%.

desk verdict Solid GRMHD test of the plunging-region dynamics; the thermodynamic agreement is real but fit-dependent and should not be oversold. read the letter →

arxiv 2505.13701 v1 pith:U223HZRL submitted 2025-05-19 astro-ph.HE

Jake Rule , Andrew Mummery , Steven Balbus , James Stone , Lizhong Zhang This is my paper · ORCID
classification astro-ph.HE
keywords blackholeaccretionplungingregionISCOstressGRMHDsimulationthindiscthermalcontinuumspinmeasurementgeodesicplunge
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

The paper tests whether analytic formulas for the plunging region—the zone between the innermost stable circular orbit (ISCO) of a black hole and its event horizon, where matter falls in almost freely—can describe a realistic thin accretion disc. It reports that they can, provided the flow is not treated as adiabatic: a small radial rise in entropy, modelled as a power law in radius, captures heating attributed to magnetic reconnection. The central finding is a finite but small ISCO stress, corresponding to a ~5.3% drop in angular momentum across the plunge, which barely perturbs the geodesic infall yet keeps density, pressure, and temperature from vanishing near the ISCO. Because black-hole spin measurements from disc spectra commonly truncate the disc at the ISCO, this means the plunging fluid can radiate and should be included.

What carries the argument

The central object is the MB23 offset-geodesic plunging model: a radial 4-velocity $U^r = -c \sqrt{2 r_g/(3 r_I)} \left( r_I/r - 1 \right)^{3/2} - u_I$, where $u_I$ is the small inward speed at the ISCO, written in dimensionless form as $\epsilon = (u_I/c)\sqrt{3 r_I/(2 r_g)}$. This closes the equations through mass conservation, vertical hydrostatic equilibrium, and the entropy relation $P = K \rho^\gamma$, producing self-similar profiles for the surface density, density, pressure, temperature, and scale height. The paper adds a fitted radial power law $K = K_I (r/r_I)^{-m}$ with $m \approx 2.71$ to account for non-adiabatic heating, and uses the measured angular-momentum drop $\delta_\mathcal{J} \approx 5.3\%$ as the diagnostic of the ISCO stress. The ideal-GRMHD simulation supplies the independent numerical data against which these profiles are compared.

What would settle it

Run the same physical setup at two or more grid resolutions and with different cooling prescriptions, then compare the fitted entropy index $m$ and angular-momentum drop $\delta_\mathcal{J}$; if $m$ changes by more than the quoted uncertainties, the heating law is numerical rather than physical and the thermodynamic confirmation fails.

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Extended reading notes

Core claim

On the paper's own terms, a dedicated 3D general-relativistic magnetohydrodynamic simulation of a thin, weakly magnetised disc around a Schwarzschild black hole shows that the MB23 analytic geodesic-plunge model quantitatively reproduces the plunging-region thermodynamics. The simulated radial 4-velocity follows the offset-geodesic solution, and the angular momentum drops by δJ ≈ 5.3% from ISCO to horizon. The surface density, density, pressure, central temperature, and scale height all agree with the model's self-similar profiles once non-adiabatic heating is represented as a power-law entropy rise K = K_I (r/r_I)^(-m) with fitted index m ≈ 2.71. The authors identify the heating as grid-scale magnetic reconnection in a mid-plane current sheet. The stress is small enough that the plunge remains essentially geodesic, but large enough to dissipate energy near the ISCO and prevent the thermodynamic quantities from vanishing. The paper concludes that constant-α disc models are physically inappropriate inside the plunging region.

Load-bearing premise

The thermodynamic agreement rests on modelling the extra heating as a power-law curve in radius with an index fitted to the simulation; if the real heating profile has a different shape, the reported match for density, pressure, and temperature could be a fitting artifact rather than a confirmation.

Editorial extensions

If this is right

  • Thermal continuum spin measurements that truncate the disc at the ISCO omit a region whose thermodynamic quantities stay non-zero; including the plunging fluid should shift fitted black-hole spins and high-energy spectral tails.
  • The simulated $\delta_\mathcal{J} \approx 5.3\%$ matches the roughly 4% inferred from observations of MAXI J1820+070, supporting the reality of a finite ISCO stress.
  • Constant-$\alpha$ disc models cannot describe the plunging region, since $\alpha$ rises by an order of magnitude and the local stress-dissipation coupling fails there.
  • Because the scaled analytic profiles depend only on $r/r_I$, the same model with $r_I$ set by spin should apply to Kerr black holes as well as Schwarzschild.

Reading between the lines

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

  • The power-law entropy description is a pragmatic interpolation; a more physical sub-grid model of magnetic-reconnection heating would let the fitted index $m$ be predicted rather than measured.
  • If the heating is really grid-scale reconnection driven by flux freezing, the entropy rise should track the magnetic-field amplification set by the plunge, so simulations with different initial field geometries should yield different $m$ values and possibly spin-dependent ISCO stress.
  • The near-ISCO dissipation implied by the 5.3% stress suggests that the effective inner boundary condition for outer-disc models should be a non-zero stress carrying roughly 5% of the local angular-momentum flux, not the traditional zero-stress condition.
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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

3 major / 5 minor

Summary. The paper presents a 3D global GRMHD simulation of a thin accretion disc around a Schwarzschild black hole using the AthenaK code, and compares the simulated radial velocity, angular momentum, surface density, density, pressure, and temperature profiles in the plunging region with the analytic MB23 model. The authors find that the radial velocity is well described by an offset geodesic, that the angular momentum drops by about 5.3% across the plunge, and that the thermodynamic profiles match the MB23 solutions provided a non-adiabatic entropy rise is modeled as a power law with a fitted index. The paper argues that the results support the MB23 model and have implications for black hole spin measurements.

Significance. If the match is robust, the paper provides an independent numerical confirmation of the key assumptions of the MB23 plunging-region model: gravity-dominated geodesic dynamics and a small but finite ISCO stress that keeps thermodynamic quantities non-vanishing. This matters for thermal continuum spin measurements that currently truncate the disc at the ISCO. The simulation is a dedicated high-resolution calculation with careful attention to MRI resolution, a lower-resolution cross-check, and direct measurements of the dynamical quantities. The main weakness is that the thermodynamic agreement is obtained after fitting parameters (epsilon per profile, m for the entropy power law), which reduces the predictive power of the comparison.

major comments (3)
  1. [Section 4, Table 2, Eqs. (5)-(8)] The claim of excellent agreement for the thermodynamic quantities rests on fitting a separate epsilon for each profile. In Table 2, the best-fit values are 0.021 (Sigma), 0.015 (rho), 0.015 (P), and 0.044 (T), whereas the directly measured epsilon from the radial velocity is 0.041 +/- 0.002. Since epsilon is the same physical parameter in the MB23 model, this factor-of-three spread is not explained by the quoted error bars. The statement in Section 4 that epsilon should be treated as an effective parameter owing to non-linear averaging is plausible but is not demonstrated. Without such a demonstration, the fits to Eqs. (6)-(8) are two-parameter fits to smooth monotonic profiles and do not provide a predictive test of the model. I request a no-free-parameter check, for example using the measured epsilon to predict the Sigma, rho, P, and T profiles without fitting, or a joint fit with a single epsilon, or a synthetic-data demonstration that the averaging effect can produce the observed spread.
  2. [Section 4, Fig. 4, Table 2, entropy power law] The entropy rise is modeled as K = K_I (r/r_I)^(-m) with m fitted to the simulation (m = 2.71 in the high-resolution run, m = 2.08 in the lower-resolution run). This is an ad-hoc prescription. The agreement between the MB23 thermodynamic solutions and the simulation is conditional on this fitted profile. The paper attributes the entropy rise to magnetic reconnection but does not provide a quantitative estimate of the dissipation rate from the simulation that can be compared with the fitted m. I recommend either deriving m from a physical heating model or explicitly framing the thermodynamic test as conditional on the measured entropy profile, and additionally showing the sensitivity of the density, pressure, and temperature fits to the choice of m.
  3. [Section 4, Fig. 5, and Appendix A] The fitted epsilon values are not stable between the high-resolution and lower-resolution simulations. For example, Sigma gives epsilon = 0.021 in Table 2 but epsilon = 0.051 in Table A2, and rho gives 0.015 versus 0.031. This sensitivity suggests that the per-profile fits are not robust, reinforcing the need for a parameter-free test before the central claim can be accepted.
minor comments (5)
  1. [Abstract and Section 1] The phrase 'do not to include the plunging fluid' in the abstract and introduction should read 'do not include the plunging fluid'.
  2. [Section 3] The paper consistently misspells 'Schwarzschild' as 'Schwarzchild' in a few places, for example in Section 3 and in the Conclusions.
  3. [Figure 7 caption] The caption begins 'Theplungingregion' without a space; this is a typographical error.
  4. [Section 4, Fig. 3 caption] The text refers to 'dot-dash lines' for the geodesic solutions, but the offset and pure geodesic models are both plotted as dot-dash lines in different colors; the caption could clarify which color corresponds to which model.
  5. [Data availability] The data availability statement says numerical results will be shared upon reasonable request; for reproducibility it would be helpful to also release the analysis scripts or key derived profiles alongside the paper.

Circularity Check

2 steps flagged · score 5.0 of 10

Thermodynamic 'excellent agreement' is obtained by fitting epsilon per profile and a K(r) power law to the same simulated data; the dynamical geodesic test is independent, but the claimed thermodynamic confirmation is partly by construction.

  1. fitted input called prediction [Section 4, Table 2 and Fig. 5 caption]
    "Using the power-law model to describe the radial dependence of K, we plot the MB23 models for each quantity, fitting ε separately for each profile by minimising the error-weighted squared distance to the simulated data. The best-fit ε parameters that we find for each model and quantity are summarised in Table2."

    Equations (6)-(8) contain ε and K/K_I as free inputs. ε is fitted separately to each simulated rho, P, T (and Sigma) profile, so the reported agreement is a best-fit curve, not an independent model prediction. The single physical parameter ε takes best-fit values 0.021, 0.015, 0.015 and 0.044, while the dynamically measured value is 0.041 +/- 0.002; the paper's 'effective parameter' defence is plausible but not demonstrated. The claimed 'excellent agreement' for the thermodynamics therefore reduces partly by construction.

  2. fitted input called prediction [Section 4, Fig. 4 discussion and Table 2]
    "We find that this rise can be reasonably well-modelled as a radial power law K = K_I (r/r_I)^(-m) where m is an index and K_I is a normalisation, both of which we fit for by minimising the error-weighted squared distance between the simulated data and the model."

    The non-adiabatic heating invoked to reconcile the analytic model with the simulation is not independently measured or derived; K(r) is fitted to the simulated entropy profile and then inserted into Eqs. (6)-(8). Combined with per-profile epsilon fits, each thermodynamic comparison uses two fitted parameters on the same target curve, so the agreement is not a first-principles test of the MB23 thermodynamic scalings.

full rationale

The paper contains genuinely independent content: the 3D GRMHD simulation is an external benchmark, the radial-velocity comparison against the offset geodesic is a real test (with u_I taken from the first simulated point, not globally fitted), and the measured 5.3% angular-momentum drop is a direct simulation result. These parts are not circular, and the self-citations to MB23 and Mummery et al. are not load-bearing because the simulation provides independent evidence. However, the central thermodynamic claim — that the MB23 model quantitatively describes rho, P and T in the plunging region — is weakened by construction: the paper fits epsilon separately for each quantity and fits the entropy power-law index m to the same simulated profiles, then reports the resulting agreement. The paper is transparent about this (it calls these 'fitted MB23 models', describes the K model as 'ad-hoc', and reinterprets epsilon as an 'effective parameter'), which lowers the severity, but the 'test' of the thermodynamics is not a no-free-parameter prediction. The dynamical test is independent and supports the geodesic assumption; the thermodynamic agreement is partially circular because the fitted inputs generate the claimed match. Overall score 5 reflects this partial circularity rather than full circularity of the whole derivation chain.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model comparison rests on several domain assumptions inherited from MB23 (geodesic dominance, vertical hydrostatic equilibrium), plus a fitted, ad hoc model for non-adiabatic heating. No new physical entities are introduced; the magnetic reconnection heating is an existing mechanism, not a new particle or force.

free parameters (3)
  • epsilon (offset velocity parameter) = 0.041 +/- 0.002 (velocity), 0.021 (Sigma), 0.015 (rho, P), 0.044 (T)
    Defined in Eq. 10 of the MB23 model. Fitted separately to the simulated surface density, density, pressure, and temperature profiles (Table 2) because the model's functional forms depend nonlinearly on it.
  • m (entropy power-law index) = 2.71 (high resolution), 2.08 (low resolution)
    Index in the ad hoc radial power-law model K = K_I (r/r_I)^(-m), fitted to the simulated specific entropy profile (Fig. 4, Table 2). The non-adiabatic heating it parametrizes is not independently predicted.
  • K_I (entropy normalization) = Fitted; value not reported
    Normalization of the entropy power-law model, fitted to the entropy profile. The paper lists it under 'excluding normalisations' but the value is still determined from the simulation.
assumptions (5)
  • domain assumption Gravity dominates pressure and magnetic forces in the plunging region (a_G >> a_P, a_B), so the fluid follows a geodesic plunge.
    Core assumption of the MB23 model (Section 2.1). The paper tests it by comparing the simulated radial velocity with the geodesic prediction and finds good agreement, but the model itself requires the assumption.
  • domain assumption Vertical hydrostatic equilibrium is maintained throughout the plunge, allowing the scale height to be derived from pressure and density.
    Used by MB23 to close the thermodynamic equations (Section 2.2). The paper argues the agreement of the density, pressure, and temperature profiles supports it.
  • domain assumption The plasma is gas-pressure dominated, with T_c proportional to P/rho.
    Assumed in the temperature relation Eq. 8 (Section 2.2).
  • ad hoc to paper The non-adiabatic entropy increase can be described by a radial power law with a single index.
    Introduced in Section 4 (Fig. 4, Table 2) to model the simulated entropy rise; without this, the thermodynamic profiles do not match the MB23 model.
  • ad hoc to paper The cooling function L = -u_g ln(K/K_t)/tau_cool adequately represents radiative losses in a thin disc.
    Adopted from Penna et al. (2010), Eq. 21, to keep the disc thin. It affects the entropy and temperature profiles in the simulation.

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Pith. "Pith review of The plunging region of a thin accretion disc around a Schwarzschild black hole." pith.science (2026). https://pith.science/paper/U223HZRL

@misc{pith2026250513701,
  author       = {Pith},
  title        = {Pith review of: The plunging region of a thin accretion disc around a Schwarzschild black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U223HZRL}},
  note         = {Machine review of arXiv:2505.13701}
}
abstract

A set of analytic solutions for the plunging region thermodynamics have been developed recently under the assumption that the fluid undergoes a gravity-dominated geodesic plunge into the black hole. We test this model against a dedicated 3D global GRMHD simulation of a thin accretion disc around a Schwarzschild black hole using the code AthenaK. Provided that we account for non-adiabatic heating in the energetics, plausibly from grid-scale magnetic dissipation, we find an excellent agreement between the analytic model and the simulated quantities. These results are particularly important for existing and future electromagnetic black hole spin measurements, many of which do not to include the plunging fluid in their emission modelling. This exclusion typically stems from the assumption of a zero-stress boundary condition at the ISCO, forcing all thermodynamic quantities to vanish. Instead, we find a non-zero $\delta_\mathcal{J}\approx 5.3 \%$ drop in the angular momentum over the plunging region, which is consistent with both prior simulations and observations. We demonstrate that this stress is small enough for the dynamics of the fluid in the plunging region to be well-described by geodesic trajectories, yet large enough to cause measurable dissipation near to the ISCO - keeping thermodynamic quantities from vanishing. In the plunging region, constant $\alpha$-disc models are a physically inappropriate framework.

Figures

Figures reproduced from arXiv: 2505.13701 by the authors.

Figure 1
Figure 1. A cross-section of the initial Fishbone-Moncrief torus in the 𝑅-𝑧 plane (𝑅 = 𝑟 sin 𝜃 is the cylindrical radius). The torus extends symmetrically in the 𝜙 direction. The density is shown in colour, whilst the black lines mark the initial four-loop poloidal magnetic field. The torus has an inner edge at 𝑟in = 10 𝑟𝑔 and pressure maximum at 𝑟max = 16 𝑟𝑔 (𝑧 = 0). These fix the outer edge to 𝑟out = 29.7 𝑟𝑔. the conservati… view at source ↗
Figure 2
Figure 2. shows the time evolution of the mass accretion rate at the horizon, 𝑀¤ hor = − ∫ 𝜋 0 ∫ 2𝜋 0 𝜌𝑈𝑟√ 𝑔𝑑𝜙𝑑𝜃, (22) and a normalised magnetic horizon flux, 𝜑hor ≡ Φhor  𝑀¤ hor𝑟 2 𝑔 𝑐 −1/2 where, Φhor = 1 2 ∫ 𝜋 0 ∫ 2𝜋 0 |𝐵 𝑟 | √ 𝑔𝑑𝜙𝑑𝜃, (23) is the hemispherical magnetic horizon flux due to the factor of 1/2 (following Tchekhovskoy et al. (2011)). The mass accretion rate has a brief spike at early times, but begins to sett… view at source ↗
Figure 3
Figure 3. The plunging region radial 4-velocity (𝑈𝑟 ) (left) and angular momentum (𝑈𝜙 ) (right) profiles. The black dots in the upper left and right-hand panels show the density weighted, temporally ([15, 20]k𝑡𝑔), azimuthally ([0, 2𝜋]) and vertically ([76◦ , 104◦ ]) averaged simulated quantities, ⟨−𝑈𝑟 ⟩𝜌,𝑡 𝜙 𝜃 and ⟨𝑈𝜙 ⟩𝜌,𝑡 𝜙 𝜃 . The shaded region represents ±1𝜎 standard deviation 𝜎 = √︁ Var𝑡 𝜙 𝜃 (−𝑈𝑟 ) and 𝜎 = √︁ Var𝑡 𝜙 𝜃 (𝑈𝜙… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The plunging region specific entropy profile represented by 𝐾 = 𝑃𝜌−𝛾 (upper-left), magnetic energy profiles (upper-right) and a snapshot of the 𝑥 = 0 slice taken at 𝑡 = 20, 000 𝑡𝑔 (lower). In the upper-left, black dots show the temporally ([15, 20]k𝑡𝑔), azimuthally ([0…
Figure 5
Figure 5. Figure 5: The plunging region surface density (top-left), density (top-right), pressure (bottom-right) and temperature (bottom-left) profiles. Black dots show the temporally ([15, 20]k𝑡𝑔), azimuthally ([0, 2𝜋]) and (for the density, pressure and temperature) vertically ([76◦ , 1…
Figure 6
Figure 6. Figure 6: The inward radial velocity at the ISCO (−𝑢𝐼 ) as a function of time (top-left), vertical height (𝜃, top-right), azimuthal position (𝜙, bottom-right) and azimuthally averaged mid-plane sound speed at the ISCO (⟨𝐶𝑆,𝐼,mid⟩𝜙 , bottom-left). To marginalise over other variab…
Figure 7
Figure 7. Figure 7: The plunging region 𝛼 profile. Black dots show the density weighted temporally ([15, 20]k𝑡𝑔), azimuthally ([0, 2𝜋]) and vertically ([76◦ , 104◦ ]) averaged simulated 𝛼. A clear order of magnitude rise is observed. Gammie C. F., McKinney J. C., Tóth G., 2003, The Astrop…

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    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.stat...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.