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Predicting the Properties of the Fallback Rate from Tidal Disruption Events: Investigating the Maximum Gravity Model

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The maximum gravity model predicts when a star is completely disrupted by a black hole and matches hydrodynamic simulations of TDE fallback peaks.

desk verdict Systematic, honest test of the authors' own MG model: ZAMS agreement is convincing, evolved-star discrepancies are reported clearly but rest on an untested compression assumption, and the single-black-hole setup leaves the mass-scaling claim unvalidated—still worth a serious referee. read the letter →

arxiv 2507.16912 v1 pith:SJDIZFV7 submitted 2025-07-22 astro-ph.HE

Julia Fancher , Ananya Bandopadhyay , Eric R. Coughlin , C.J. Nixon This is my paper · ORCID
classification astro-ph.HE
keywords tidaldisruptioneventsfallbackratemaximumgravitymodelsupermassiveblackholeshydrodynamicalsimulationsPaczynski-Wiitapotentialstellarevolutionaccretionflares
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 tests the maximum gravity (MG) model of tidal disruption events, which predicts when a star is completely shredded by a supermassive black hole and what the resulting mass fallback rate looks like. The model says disruption happens when the black hole's tidal pull exceeds the strongest self-gravitational field inside the star, and it converts that condition into a predicted peak fallback rate and peak time. The authors run smoothed-particle hydrodynamics simulations of 24 main-sequence stars from 0.2 to 5 solar masses at three ages and compare the simulated fallback curves to the predictions. They find essentially exact agreement for zero-age main sequence stars and agreement at the 35 to 50 percent level for evolved, centrally concentrated stars, whose peaks come later and lower than predicted. A relativistic generalization using the Paczynski-Wiita potential reproduces the disruption thresholds found in general-relativistic simulation work.

What carries the argument

The load-bearing object is the maximum self-gravitational field $g_{\rm max}$ at the core radius $R_c$, the point inside the star where the inward gravitational acceleration peaks. The model equates the SMBH tidal field there with $g(R_c)$ to fix the critical complete-disruption radius $r_{t,c}$; from that radius the peak time follows from the orbital energy of material at $r_{t,c}$, $t_{\rm peak}\propto (r_{t,c}^2/2R_\star)^{3/2}(GM_\bullet)^{-1/2}$, with $\dot{M}_{\rm peak}\simeq M_\star/(4t_{\rm peak})$. Higher-order approximations to $g(r)$ near the center and surface show that the leading-order estimate $R_c\simeq R_\star(\bar{\rho}/\rho_c)^{1/3}$ is not accurate enough for the peak properties; the exact core radius from the stellar profile is required. The relativistic extension replaces the point-mass potential with the Paczynski-Wiita potential, giving $r_{t,c}^{\rm PW}=2r_g+(4GM_\bullet R_c/g(R_c))^{1/3}$.

What would settle it

Run a hydrodynamical simulation of a deep encounter of a $3\,M_\odot$ TAMS star at the MG-predicted $\beta_c$ and measure the density profile at pericenter; if the maximum of the compressed self-gravitational field exceeds the unperturbed stellar $g_{\rm max}$ by more than a factor of about 1.5 while the fallback peak still comes later and lower than predicted, then the fixed-core-radius assumption is the cause and a compression-dependent $R_c$ would be required. Observationally, a sample of TDEs from massive evolved stars with known black hole masses that shows rise times systematically longer than the MG $t_{\rm peak}$ would falsify the model's use for evolved progenitors.

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Extended reading notes

Core claim

The central claim is that the MG model is a reliable, parameter-light predictor of the two quantities that set observed TDE lightcurves: the peak fallback rate $\dot{M}_{\rm peak}$ and the time to peak $t_{\rm peak}$. For each stellar structure the model locates the 'core radius' $R_c$ where the star's self-gravity $g(r)$ is maximal, demands that the SMBH tidal field exceed $g(R_c)$ for complete disruption, and evaluates $t_{\rm peak}$ from the resulting critical radius. The hydrodynamical comparison shows that for ZAMS stars this works without tuning: predicted and simulated peaks coincide across the entire 0.2 to 5 $M_\odot$ range. For MAMS and TAMS stars the model remains within roughly 35 to 50 percent, with the discrepancy traced to tidal compression shrinking the effective core radius rather than to a wrong disruption threshold. The paper also shows that replacing the Newtonian potential with the Paczynski-Wiita potential yields critical impact parameters consistent with relativistic simulations, extending the model's reach to high black hole masses.

Load-bearing premise

The load-bearing premise is that the star's internal gravity at the moment of disruption is the same as in the star's unperturbed, pre-encounter structure, so the core radius $R_c$ does not change as the black hole's tidal field squeezes the star.

Editorial extensions

If this is right

  • For zero-age main sequence stars of 0.2 to 5 $M_\odot$ disrupted by a $10^6\,M_\odot$ black hole, the MG model gives peak fallback rates and peak times that match SPH simulations across the whole mass range.
  • For evolved (MAMS and TAMS) stars the simulated fallback peaks are later and lower than predicted, with late-time $t^{-9/4}$ scaling indicating partial disruptions that reform a core; the model's $\beta_c$ still marks the transition.
  • The Paczynski-Wiita generalization predicts smaller critical impact parameters for more massive black holes, in line with relativistic simulations, and shows $\beta_c$ peaks near $1.4\,M_\odot$ because that star has the smallest core-to-stellar-radius ratio.
  • Because $t_{\rm peak}$ depends mainly on black hole mass, the model translates an observed TDE peak time into a black hole mass estimate, and the observed roughly 30 to 50 day rise times of the optically selected TDE sample are consistent with $\sim 10^6\,M_\odot$ black holes.

Reading between the lines

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

  • Editorial inference: the tidal-compression explanation implies a simple upgrade path: replace the fixed $R_c$ with an effective core radius that depends on the pericenter depth, and calibrate that prescription against the same simulations to close most of the 35 to 50 percent gap for evolved stars.
  • Editorial inference: if the MG criterion is the right threshold, then for sufficiently massive black holes even very deep encounters may fail to destroy evolved stars in a Newtonian treatment, making repeating partial TDEs a natural outcome; the relativistic PW extension sharpens that boundary and could be used to predict which observed repeating transients involve massive progenitors.
  • Editorial inference: the model's success suggests that the peak time of a TDE lightcurve is largely set before accretion begins, so fits that use both $t_{\rm peak}$ and the shape around the peak could break degeneracies between black hole mass and stellar structure.
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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 tests the maximum gravity (MG) model of Coughlin & Nixon (2022) against smoothed-particle hydrodynamics simulations of tidal disruption events. The MG model predicts that a star is completely destroyed when the black hole's tidal field exceeds the maximum self-gravitational field g_max of the unperturbed star, located at the core radius R_c, and uses this to predict the peak fallback rate Mdot_peak and the time to peak t_peak. The authors simulate a grid of 24 MESA stellar models with masses 0.2–5.0 M_sun at three main-sequence ages (ZAMS, MAMS, TAMS), disrupted by a 10^6 M_sun SMBH in phantom, and compare the numerical peaks with the MG predictions. They report excellent agreement for ZAMS stars and systematically later/lower peaks for evolved stars, attributing the evolved-star discrepancy to a smaller effective R_c caused by tidal compression. They additionally generalize the MG model to the Paczyński–Wiita potential and compare the resulting critical impact parameter with relativistic simulations from the literature, finding good agreement except in the most relativistic cases.

Significance. If the conclusions hold, the MG model provides a simple, parameter-free analytic predictor for the peak of TDE fallback lightcurves that is much more accurate than the frozen-in approximation, with direct relevance to inferring black hole mass from observed rise times. The study is careful in several respects: the MG predictions are evaluated from the same MESA profiles used to initialize the simulations, no constants are fitted to the simulation peaks, and Appendix A demonstrates convergence of the peak of the fallback rate from 10^5 to 10^7 particles. The honest reporting of the evolved-star discrepancies is a strength. The main weakness is that the paper's proposed explanation for those discrepancies is not directly tested, and one of the supporting experiments has a logical gap. The quantitative claim of accuracy is also stated inconsistently in different parts of the manuscript.

major comments (3)
  1. [Section 2.2] The inference that the β = β_c + 1 simulations show that 'the core radius is smaller than that predicted by the MG model' and that 'the β_c predicted by the model is still accurate' is not supported. Going from β_c to β_c + 1 increases the tidal compression of the star, which is exactly the mechanism the paper invokes to shrink R_c, so the experiment cannot separate a smaller effective R_c from an incorrect MG threshold or from residual partial-disruption effects. A direct measurement of the compression-modified density profile or core radius at pericenter in the simulations, or a scan over β bracketing β_c, is needed to sustain this load-bearing claim about the evolved-star branch of the validation.
  2. [Abstract, Section 1, Section 4] The paper gives inconsistent quantitative statements of the evolved-star accuracy. The abstract states the predictions are 'within ∼ 35−50%' of the simulations, the introduction states the peaks are 'still generally within ∼ few × 10% of the MG values', and the discussion says the model is 'at most 30-50% inaccurate'. These statements span nearly an order of magnitude in the claimed accuracy and directly affect the central quantitative claim. The authors should tabulate the actual ratios t_peak,sim/t_peak,MG and Mdot_peak,sim/Mdot_peak,MG for each evolved star and use one consistent error measure throughout.
  3. [Sections 2.1 and 3] The MG threshold and its Paczyński–Wiita generalization in Eq. (10) evaluate g(R_c) from the unperturbed MESA density profile, but the paper itself argues in Section 2.2 that tidal compression augments the central density and self-gravity, making R_c smaller during the deep encounters needed to destroy evolved stars. If this compression is important, the threshold condition is not being evaluated at the self-gravity of the disrupted star, and the model's applicability to highly evolved, centrally concentrated stars rests on an untested assumption. The authors should either include a compression-aware estimate of g_max/R_c or explicitly frame the evolved-star predictions as an empirical approximation rather than a first-principles criterion.
minor comments (5)
  1. [Section 2.2] The statement that the authors 'run three sets of simulations of 24 stars' is inaccurate for the MAMS and TAMS sets, which exclude stars below 0.9 M_sun; the number of stars in each set should be stated explicitly.
  2. [Throughout] The Paczyński–Wiita name is misspelled in several places, including 'Packzyński-Wiita' after Eq. (9) and 'Paczyński-Witta' in Section 4; these should be corrected to Paczyński–Wiita.
  3. [Section 2.1, Eq. (5)] The values A = −1.8 and n = 3.6 are quoted for the 1 M_sun ZAMS star, but the text does not state how the fit parameters are determined for the other masses and ages used in the next-to-leading-order predictions shown in Figure 2.
  4. [Figure 9] The caption refers to the Gafton et al. result as a 'green dashed curve' while the text at the end of Section 3 calls it a 'green curve'; the figure and text should agree on the line style designation.
  5. [Section 4] The statement that the PW predictions are in 'good agreement' with Jankovič & Gomboc (2023) and Ryu et al. (2020c) is qualitative; given that the TAMS comparison is acknowledged to be systematically different, it would be helpful to include a quantitative comparison or a scatter plot for these cases.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MG-model predictions are computed from unperturbed stellar profiles and tested against independent hydrodynamical simulations without any parameters fitted to the simulation outcomes.

full rationale

The paper's central claim is that the maximum-gravity model, originally proposed by the same group in Coughlin & Nixon (2022), predicts t_peak and Mdot_peak for tidal disruption events. The predictions are derived from MESA stellar structure profiles (R_c, rho_c, g_max) and the analytical formulae of Coughlin & Nixon (2022). These predictions are then compared to SPH (phantom) simulations in which the pericenter is set using the model's beta_c, but the fallback rate itself is computed self-consistently by the hydrodynamics code. There is no parameter fitted to the simulation results; the only shared input is the stellar model. The agreement for ZAMS stars and the 35-50% deviations for evolved stars are therefore genuine tests of the model rather than reductions to its inputs. The paper's explanation for evolved-star discrepancies (tidal compression reducing the effective R_c) is a physical hypothesis, not a circular argument, and it is explicitly tested with beta_c+1 simulations. Self-citations to Coughlin & Nixon (2022), Bandopadhyay et al. (2024b), and Nixon & Coughlin (2022) provide the model and supporting numerical results, but the present paper independently validates the model against its own simulations and against external works such as Gafton et al. (2015), Law-Smith et al. (2020), and Jankovic & Gomboc (2023). No equation in the paper is equal to its input by construction, and no prediction is statistically forced by a fitted parameter. The derivation chain is self-contained and the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The model's predictions rest on the physical postulate that full disruption occurs at the g_max core radius, on the use of unperturbed stellar profiles, and on standard SPH and PW-potential assumptions. The unperturbed-profile assumption is the one most likely to fail for evolved stars, and the paper itself argues this is the source of the 35-50% offset. No new particles, forces, or dimensions are introduced.

free parameters (2)
  • A (mass profile fit coefficient) = -1.8
    In Section 2.1, Equation (5), A and n are fit to the mesa-generated mass profile of a 1 Msun ZAMS star for the next-to-leading-order approximation of the self-gravitational field. This approximation is illustrative; the primary comparison uses the exact core radius from the mesa profile.
  • n (mass profile fit exponent) = 3.6
    Same fit as A, used only for the NLO demonstration in Figure 2, not for the main validation.
assumptions (5)
  • domain assumption A star is completely destroyed when the tidal field of the SMBH exceeds the maximum self-gravitational field g_max at the core radius Rc.
    This is the MG postulate from Coughlin & Nixon 2022, restated in Section 2.1. It is the physical criterion the paper tests; if false, the model's predictions do not follow.
  • domain assumption The unperturbed mesa profile's g(R) remains representative during the encounter; tidal compression does not significantly alter g_max at the destruction threshold.
    Invoked implicitly throughout Section 2.1 and discussed in Section 2.2, where tidal compression is suggested to reduce the effective core radius for evolved stars. This assumption is load-bearing exactly where the model is least accurate.
  • domain assumption The phantom SPH simulations with 10^6 particles and the described accretion-radius and sink-particle prescriptions produce converged fallback rates near the peak.
    The convergence appendix (Appendix A) checks four cases, and the peak is stable across resolutions. This assumption justifies using the simulation peak as ground truth.
  • domain assumption The Paczynski-Wiita potential adequately approximates relativistic tidal fields for the SMBH masses and impact parameters considered.
    Section 3 replaces the Newtonian potential with Phi = -GM/(r-2rg). The paper itself notes this approximation fails near pericenters less than about 6rg, so the assumption holds only in the moderate-relativistic regime.
  • domain assumption The fallback rate computed by counting particles crossing the accretion radius equals the mass return rate that powers the observed lightcurve.
    Standard practice in TDE simulation papers, used in Section 2.2; not directly verified against radiative transfer or disk formation.

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Cite this review

Pith. "Pith review of Predicting the Properties of the Fallback Rate from Tidal Disruption Events: Investigating the Maximum Gravity Model." pith.science (2026). https://pith.science/paper/SJDIZFV7

@misc{pith2026250716912,
  author       = {Pith},
  title        = {Pith review of: Predicting the Properties of the Fallback Rate from Tidal Disruption Events: Investigating the Maximum Gravity Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SJDIZFV7}},
  note         = {Machine review of arXiv:2507.16912}
}
abstract

A star destroyed by the tidal field of a supermassive black hole (SMBH) in a tidal disruption event (TDE) gives rise to a luminous flare. TDEs are being detected at an ever-increasing rate, motivating the need for accurate models of their lightcurves. The ``maximum gravity'' (MG) model posits that a star is completely destroyed when the tidal field of the SMBH exceeds the maximum self-gravitational field within the star, $g_{\rm max}$, and predicts the peak fallback rate $\dot{M}_{\rm peak}$ and the time to peak $t_{\rm peak}$. Here we perform hydrodynamical simulations of the complete disruption of 24 stars with masses ranging from $0.2-5.0 M_\odot$, at different stages of their main sequence evolution, to test the predictions of this model. We find excellent agreement between the MG model predictions and our simulations for stars near the zero-age main sequence, while the predictions are less accurate (but still within $\sim 35-50\%$ of the simulation results) for highly evolved stars. We also generalize the MG model to incorporate the Paczy{\'n}ski-Wiita potential to assess the impact of strong-gravity effects -- which are especially important for deep encounters that are required to completely destroy evolved and centrally concentrated stars -- and find good agreement with recent works that include relativistic gravity. Our results demonstrate that this model provides accurate constraints on the peak timescale of TDE lightcurves and their correlation with black hole mass.

Figures

Figures reproduced from arXiv: 2507.16912 by the authors.

Figure 2
Figure 2. The peak fallback times tpeak and rates M˙ peak for 0.2 − 5M⊙ ZAMS stars disrupted by a 106M⊙ SMBH. The 5-pointed stars represent the exact MG prediction, obtained by using the true location of the core radius Rc. The cir￾cles and diamonds represent the predicted peaks using the leading order (LO) and next to leading order (NLO) approx￾imations for the self-gravitational field, and the correspond￾ing values of Rc. T… view at source ↗
Figure 3
Figure 3. The fallback curves following the disruption of a 0.2M⊙ ZAMS star (left) and a 2.0M⊙ ZAMS star (right) by a 106M⊙ SMBH. The insets zoom in on the peak, where the inverted triangle shows the numerical peak value and the star marks the MG model prediction. The horizontal line shows 95% of the peak value, and the vertical lines show the time at which the fallback rate reaches 95% of the peak value, represented in [PIT… view at source ↗
Figure 4
Figure 4. The predictions of the MG model for the disruption of ZAMS stars at varying masses (stars) compared to the numerical value of the peak fallback rate (triangles). The horizontal lines provide the amount of time for which the fallback rate is above 95% of its peak value; see the insets in Figures 3 and 6 for examples. ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽ ▽… view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: Same as [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: Same as [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Left: The critical impact parameter β PWc obtained using the Packzy´nski-Wiita potential, as a function of stellar mass, for zero-age main sequence stars disrupted by SMBHs of masses 105 , 106 and 107M⊙ (represented by the the blue, yellow and green lines respectively)…
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: The critical impact parameter predicted using the Packzy´nski-Wiita potential, required for the complete disruption of a sun-like star modeled as a 5/3−polytrope (blue curve), as a function of SMBH mass, M•. Also shown are the results of numerical simulations performed…
Figure 10
Figure 10. Figure 10: shows the fallback rates obtained using the three different resolutions for a 0.3M⊙ ZAMS star (top-left), 1.0M⊙ ZAMS star (top-right), 1.0M⊙ MAMS star (bottom-left) and 1.0M⊙ TAMS star (bottom-right). The simulation setup is identical to the description provided in Se…

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