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arxiv: 2606.18050 · v1 · pith:TAHYF56Wnew · submitted 2026-06-16 · 🌌 astro-ph.HE · astro-ph.GA· gr-qc

A Semi-Analytical Loss Cone Theory for Tidal Disruption Event Rates Around Kerr Black Holes

Wenkang Xin This is my paper

Pith reviewed 2026-06-26 23:20 UTC · model grok-4.3

classification 🌌 astro-ph.HE astro-ph.GAgr-qc
keywords tidal disruption eventsKerr black holesloss cone theoryFokker-Planck equationsemi-analytical methodsblack hole spinretrograde orbitsTDE rates
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The pith

The global TDE rate around Kerr black holes stays nearly unchanged with spin even though retrograde stars are disrupted more readily.

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

This paper constructs a semi-analytical model that folds spin-dependent disruption thresholds into the classical loss-cone calculation for tidal disruption events. It treats stellar supply as Newtonian diffusion while fixing the angular-momentum cutoff with general-relativistic tidal tensors, then solves the resulting two-dimensional Fokker-Planck equation perturbatively. The calculation reveals a first-order excess of retrograde disruptions, yet the integrated rate across all inclinations shows almost no net dependence on black-hole spin. A reader cares because existing population estimates usually ignore spin; this result indicates those estimates remain usable for realistic Kerr holes.

Core claim

We develop the first semi-analytical framework that incorporates spin-dependent loss-cone boundaries into TDE rate theory. Using a novel tidal tensor formalism, we compute inclination-dependent thresholds for tidal disruption and direct capture by the event horizon. We then revisit the classical one-dimensional loss-cone problem with nested disruption and capture boundaries, deriving a closed-form capture fraction valid across all loss-cone regimes. Finally, we formulate a two-dimensional Fokker-Planck equation describing simultaneous diffusion in angular momentum magnitude and orientation. Through a perturbative treatment, we demonstrate that while the Kerr disruption boundary induces a fir

What carries the argument

The two-dimensional Fokker-Planck equation for diffusion in both angular-momentum magnitude and orientation, solved perturbatively with spin-dependent nested loss-cone boundaries.

If this is right

  • TDE rate estimates can continue to use the standard Newtonian loss-cone formalism at leading order even when black-hole spin is present.
  • Population-level studies need not treat black-hole spin as a dominant source of uncertainty in the total disruption rate.
  • The orientation bias may still appear in the statistics of individual events, such as light-curve shapes or preferred orbital planes.
  • The closed-form capture fraction gives an explicit spin- and inclination-dependent correction for stars swallowed whole rather than disrupted.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the global rate is truly spin-insensitive, observed TDE statistics could constrain the stellar cusp properties without requiring knowledge of the underlying black-hole spin distribution.
  • The first-order retrograde preference might produce a detectable signature in the average orbital inclinations of TDE hosts, accessible to future astrometric surveys.
  • Extending the same perturbative machinery to resonant relaxation or to second-order spin terms would test whether the insensitivity persists at extreme spins or in dense nuclei.

Load-bearing premise

The supply of stars to the disruption zone is governed by Newtonian stellar dynamics while the angular-momentum threshold for disruption is set by general-relativistic tidal dynamics.

What would settle it

A side-by-side comparison of the closed-form capture fraction against the fraction measured in full general-relativistic orbit integrations of stars around a rapidly spinning Kerr black hole.

Figures

Figures reproduced from arXiv: 2606.18050 by Wenkang Xin.

Figure 2.1
Figure 2.1. Figure 2.1: Trajectories showing the separatrix nature of the innermost bound spher [PITH_FULL_IMAGE:figures/full_fig_p015_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: The magnitude of the largest eigenvalue of the tidal tensor as a function [PITH_FULL_IMAGE:figures/full_fig_p020_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Loss cone boundaries as a function of inclination [PITH_FULL_IMAGE:figures/full_fig_p021_2_3.png] view at source ↗
Figure 2
Figure 2. Figure 2: demonstrates that the assumption of linearity is highly robust, failing only [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Heatmap showing failures of Ltde(x) ≈ L (0) tde(1 + ϵx) across the (M, a) parameter grid, with both linear-fit and small-slope conditions satisfied (grey solid), small-slope violated (green diagonal-hatching), and both conditions violated (orange cross-hatching). A similar analysis can be performed for the direct capture boundary. Recall that Lcap is defined by the IBSO conditions (Equations 2.13), which… view at source ↗
Figure 2
Figure 2. Figure 2: defines the region of valid angular momentum for tidal disruption. For [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: plots the TDE rates as a function of BH mass, comparing our semi [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: TDE rates as a function of BH mass for different BH spins. Left: Our [PITH_FULL_IMAGE:figures/full_fig_p028_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Capture fraction Ccap as a function of q for different values of φc, with φc = 0.3 (green solid), φc = 0.5 (orange dashed), φc = 0.9 (purple dash-dotted) [PITH_FULL_IMAGE:figures/full_fig_p036_3_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: a), permitting a perturbative treatment of the problem. [PITH_FULL_IMAGE:figures/full_fig_p042_2.png] view at source ↗
read the original abstract

A tidal disruption event (TDE) occurs when a star is scattered onto a near-radial orbit and is torn apart by a black hole (BH)'s tidal field. The angular momentum threshold for disruption is set by general relativistic tidal dynamics, while the supply of stars to the disruption zone is governed by Newtonian stellar dynamics. A spinning BH breaks the spherical symmetry of the disruption boundary, so a star's survival depends on both the magnitude and the orientation of its angular momentum. Existing treatments either assume a non-spinning BH or rely on numerical simulations of spinning BHs. We develop the first semi analytical framework that incorporates spin-dependent loss cone boundaries into TDE rate theory. Using a novel tidal tensor formalism, we compute inclination-dependent thresholds for tidal disruption and direct capture by the event horizon. We then revisit the classical one dimensional loss cone problem with nested disruption and capture boundaries, deriving a closed form capture fraction valid across all loss cone regimes. Finally, we formulate a two dimensional Fokker--Planck equation describing simultaneous diffusion in angular momentum magnitude and orientation. Through a perturbative treatment, we demonstrate that while the Kerr disruption boundary induces a first-order bias favouring the disruption of retrograde stars, the global TDE rate is remarkably insensitive to black hole spin. This approach offers a tractable route to including spin and orbital inclination in population-level TDE studies.

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

1 major / 0 minor

Summary. The manuscript develops the first semi-analytical framework for TDE rates around Kerr black holes. It introduces a tidal tensor formalism to compute inclination-dependent thresholds for tidal disruption and direct capture. It derives a closed-form capture fraction for the one-dimensional loss-cone problem with nested boundaries and formulates a two-dimensional Fokker-Planck equation for simultaneous diffusion in angular-momentum magnitude and orientation. A perturbative treatment of this equation is used to show that the Kerr boundary induces a first-order bias favoring retrograde disruptions, yet the global TDE rate remains remarkably insensitive to black hole spin.

Significance. If the derivations hold, the work is significant as the first tractable semi-analytical route to incorporating spin and inclination into population-level TDE calculations, replacing non-spinning approximations or full numerical simulations. The closed-form capture fraction valid across loss-cone regimes and the perturbative treatment of the 2D Fokker-Planck equation are explicit strengths that enable efficient, parameter-free estimates.

major comments (1)
  1. [Abstract, paragraph 2] Abstract, paragraph 2: The central claim of spin-insensitive global rates rests on separating Newtonian two-body relaxation (supply of stars) from GR tidal thresholds (disruption/capture boundary). For a ≈ 0.9 the relevant angular-momentum thresholds occur at r ∼ few r_g, where frame-dragging and geodesic precession alter diffusion coefficients and couple inclination evolution to radial motion; this directly undermines the 2D Fokker-Planck construction and the perturbative bias calculation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: The central claim of spin-insensitive global rates rests on separating Newtonian two-body relaxation (supply of stars) from GR tidal thresholds (disruption/capture boundary). For a ≈ 0.9 the relevant angular-momentum thresholds occur at r ∼ few r_g, where frame-dragging and geodesic precession alter diffusion coefficients and couple inclination evolution to radial motion; this directly undermines the 2D Fokker-Planck construction and the perturbative bias calculation.

    Authors: We agree that a fully relativistic treatment of orbital diffusion would be desirable at high spins. However, the framework follows the standard separation in loss-cone theory: two-body relaxation is accumulated through distant encounters at r ≫ r_g where Newtonian coefficients apply, while GR enters only via the inner boundary conditions. The 2D Fokker-Planck equation therefore retains Newtonian diffusion terms, with the Kerr tidal tensor supplying the inclination-dependent thresholds. The perturbative bias calculation isolates the first-order effect of those boundaries. We have added an explicit statement of this regime of validity and a note on the approximation's limitations in the revised text. revision: partial

Circularity Check

0 steps flagged

Derivation chain is self-contained; no reductions to inputs by construction

full rationale

The paper starts from classical loss-cone theory, introduces a novel tidal-tensor formalism to obtain inclination-dependent GR disruption/capture boundaries, derives a closed-form capture fraction for the 1D nested-boundary problem, and then performs a perturbative analysis on the 2D Fokker-Planck equation in angular-momentum magnitude and orientation. None of these steps are shown to reduce by definition or by self-citation to the final rate; the spin-insensitivity result is obtained from the perturbative expansion rather than being presupposed. No fitted parameters are relabeled as predictions, no uniqueness theorems are imported from the authors' prior work, and no ansatz is smuggled via citation. The separation of Newtonian supply dynamics from GR thresholds is an explicit modeling choice whose validity is external to the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on a clean separation between Newtonian scattering and GR tidal thresholds plus a perturbative treatment of the 2D diffusion equation; no free parameters or new entities are named in the abstract.

axioms (2)
  • domain assumption Supply of stars to the loss cone is governed by Newtonian stellar dynamics while disruption thresholds are set by general-relativistic tidal dynamics
    Stated directly in the second sentence of the abstract as the foundation for combining the two regimes.
  • ad hoc to paper A perturbative treatment of the two-dimensional Fokker-Planck equation suffices to capture the leading spin-induced bias
    The abstract invokes this treatment to demonstrate the retrograde bias and rate insensitivity.

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Reference graph

Works this paper leans on

54 extracted references · 1 linked inside Pith

  1. [1]

    M.et al.The process of stellar tidal disruption by supermassive black holes: the first pericenter passage.Space Science Reviews217(2021)

    Rossi, E. M.et al.The process of stellar tidal disruption by supermassive black holes: the first pericenter passage.Space Science Reviews217(2021)

    2021

  2. [2]

    Tidal disruption events.Annual Review of Astronomy and Astrophysics59, 21–58 (2021)

    Gezari, S. Tidal disruption events.Annual Review of Astronomy and Astrophysics59, 21–58 (2021)

    2021

  3. [3]

    van Velzen, S.et al.Seventeen tidal disruption events from the first half of ztf survey observa- tions: entering a new era of population studies.The Astrophysical Journal908, 4 (2021)

    2021

  4. [4]

    & Gomboc, A

    Bricman, K. & Gomboc, A. The prospects of observing tidal disruption events with the large synoptic survey telescope.The Astrophysical Journal890, 73 (2020)

    2020

  5. [5]

    & Gardner, M

    Mummery, A., Nathan, E., Ingram, A. & Gardner, M. Fitting transients with discs ( fitted ): a public light curve and spectral fitting package based on evolving relativistic discs.Monthly Notices of the Royal Astronomical Society544, 2225–2240 (2025)

    2025

  6. [6]

    Mummery, A.et al.Fundamental scaling relationships revealed in the optical light curves of tidal disruption events.Monthly Notices of the Royal Astronomical Society527, 2452–2489 (2023)

    2023

  7. [7]

    C.et al.Rates of stellar tidal disruption.Space Science Reviews216(2020)

    Stone, N. C.et al.Rates of stellar tidal disruption.Space Science Reviews216(2020)

    2020

  8. [8]

    E., Strader, J

    Greene, J. E., Strader, J. & Ho, L. C. Intermediate-mass black holes.Annual Review of Astronomy and Astrophysics58, 257–312 (2020)

    2020

  9. [9]

    Wang, Z.et al.The role of population iii star tidal disruption events in black hole growth at the cosmic dawn.The Astrophysical Journal990, 160 (2025)

    2025

  10. [10]

    W.-S., Onori, F., Hung, T

    van Velzen, S., Holoien, T. W.-S., Onori, F., Hung, T. & Arcavi, I. Optical-ultraviolet tidal disruption events.Space Science Reviews216(2020)

    2020

  11. [11]

    Hills, J. G. Possible power source of seyfert galaxies and qsos.Nature254, 295–298 (1975)

    1975

  12. [12]

    Reynolds, C. S. Observational constraints on black hole spin.Annual Review of Astronomy and Astrophysics59, 117–154 (2021)

    2021

  13. [13]

    C., Kesden, M., Cheng, R

    Stone, N. C., Kesden, M., Cheng, R. M. & van Velzen, S. Stellar tidal disruption events in general relativity.General Relativity and Gravitation51(2019)

    2019

  14. [14]

    & Loeb, A

    Hayasaki, K., Stone, N. & Loeb, A. Circularization of tidally disrupted stars around spinning supermassive black holes.Monthly Notices of the Royal Astronomical Society461, 3760–3780 (2016)

    2016

  15. [15]

    & Rees, M

    Frank, J. & Rees, M. J. Effects of massive central black holes on dense stellar systems.Monthly Notices of the Royal Astronomical Society176, 633–647 (1976)

    1976

  16. [16]

    Lightman, A. P. & Shapiro, S. L. The distribution and consumption rate of stars around a massive, collapsed object.The Astrophysical Journal211, 244 (1977)

    1977

  17. [17]

    & Kulsrud, R

    Cohn, H. & Kulsrud, R. M. The stellar distribution around a black hole: numerical integration of the fokker-planck equation.The Astrophysical Journal226, 1087–1108 (1978). URLhttps: //ui.adsabs.harvard.edu/abs/1978ApJ...226.1087C. 61

    1978

  18. [18]

    & Tremaine, S

    Magorrian, J. & Tremaine, S. Rates of tidal disruption of stars by massive central black holes. Monthly Notices of the Royal Astronomical Society309, 447–460 (1999).astro-ph/9902032

    Pith/arXiv arXiv 1999

  19. [19]

    & Merritt, D

    Wang, J. & Merritt, D. Revised rates of stellar disruption in galactic nuclei.The Astrophysical Journal600, 149–161 (2004)

    2004

  20. [20]

    Loss-cone dynamics.Classical and Quantum Gravity30, 244005 (2013)

    Merritt, D. Loss-cone dynamics.Classical and Quantum Gravity30, 244005 (2013)

    2013

  21. [21]

    Yao, Y.et al.Tidal disruption event demographics with the zwicky transient facility: volu- metric rates, luminosity function, and implications for the local black hole mass function.The Astrophysical Journal Letters955, L6 (2023)

    2023

  22. [22]

    Chang, J. N. Y., Dai, L., Pfister, H., Kar Chowdhury, R. & Natarajan, P. Rates of stellar tidal disruption events around intermediate-mass black holes.The Astrophysical Journal Letters 980, L22 (2025)

    2025

  23. [23]

    Stone, N. C. & Metzger, B. D. Rates of stellar tidal disruption as probes of the supermassive black hole mass function.Monthly Notices of the Royal Astronomical Society455, 859–883 (2015)

    2015

  24. [24]

    & Merritt, D

    Vasiliev, E. & Merritt, D. The loss-cone problem in axisymmetric nuclei.The Astrophysical Journal774, 87 (2013)

    2013

  25. [25]

    Tidal-disruption rate of stars by spinning supermassive black holes.Physical Review D85, 024037 (2012)

    Kesden, M. Tidal-disruption rate of stars by spinning supermassive black holes.Physical Review D85, 024037 (2012)

    2012

  26. [26]

    & Mummery, A

    Xin, W. & Mummery, A. Relativistic tidal tensor: general solutions for stationary axisymmetric spacetimes and the hills mass of naked singularities.Physical Review D(2025).arXiv:2511. 21499

    2025

  27. [27]

    & Kesden, M

    Singh, T. & Kesden, M. Distribution of orbital inclinations for tidal disruption events by kerr black holes.Physical Review D109, 043016 (2024)

    2024

  28. [28]

    Global structure of the kerr family of gravitational fields.Physical Review174, 1559–1571 (1968)

    Carter, B. Global structure of the kerr family of gravitational fields.Physical Review174, 1559–1571 (1968)

    1968

  29. [29]

    M., Press, W

    Bardeen, J. M., Press, W. H. & Teukolsky, S. A. Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation.The Astrophysical Journal178, 347 (1972)

    1972

  30. [30]

    Wilkins, D. C. Bound geodesics in the kerr metric.Physical Review D5, 814–822 (1972)

    1972

  31. [31]

    Solution to the equations of parallel transport in kerr geometry; tidal tensor

    Marck, J.-A. Solution to the equations of parallel transport in kerr geometry; tidal tensor. Proceedings of the Royal Society of London A385, 431–438 (1983)

    1983

  32. [32]

    P., Efstathiou, G

    Hobson, M. P., Efstathiou, G. P. & Lasenby, A. N.General relativity: an introduction for physicists(Cambridge University Press, 2006), 1. publ., 6. print. edn

    2006

  33. [33]

    Pirani, F. A. E. Republication of: on the physical significance of the riemann tensor.General Relativity and Gravitation41, 1215–1232 (2009)

    2009

  34. [34]

    Manasse, F. K. & Misner, C. W. Fermi normal coordinates and some basic concepts in differ- ential geometry.Journal of Mathematical Physics4, 735–745 (1963)

    1963

  35. [35]

    & Miller, J

    Tejeda, E., Gafton, E., Rosswog, S. & Miller, J. C. Tidal disruptions by rotating black holes: relativistic hydrodynamics with newtonian codes.Monthly Notices of the Royal Astronomical Society469, 4483–4503 (2017)

    2017

  36. [36]

    & Sesana, A

    Bortolas, E., Ryu, T., Broggi, L. & Sesana, A. Partial stellar tidal disruption events and their rates.Monthly Notices of the Royal Astronomical Society524, 3026–3038 (2023). 62

    2023

  37. [37]

    & Tremaine, S.Galactic dynamics

    Binney, J. & Tremaine, S.Galactic dynamics. No. v.13 in Princeton Series in Astrophysics Series (Princeton University Press, Princeton, 2008), 2nd ed. edn. URLhttps://ui.adsabs. harvard.edu/abs/2008gady.book.....B. Description based on publisher supplied metadata and other sources

    2008

  38. [38]

    & Gebhardt, K

    Schulze, A. & Gebhardt, K. Effect of a dark matter halo on the determination of black hole masses.The Astrophysical Journal729, 21 (2011)

    2011

  39. [39]

    Princeton Series in Astrophysics (Princeton University Press, New Jersey, 2013)

    Merritt, D.Dynamics and evolution of galactic nuclei. Princeton Series in Astrophysics (Princeton University Press, New Jersey, 2013). URLhttps://ui.adsabs.harvard.edu/abs/ 2013degn.book.....M

    2013

  40. [40]

    & Lacey, C

    Binney, J. & Lacey, C. The diffusion of stars through phase space.Monthly Notices of the Royal Astronomical Society230, 597–627 (1988)

    1988

  41. [41]

    & Kesden, M

    Servin, J. & Kesden, M. Unified treatment of tidal disruption by schwarzschild black holes. Physical Review D95, 083001 (2017)

    2017

  42. [42]

    Binney, J.The physics of quantum mechanics(Oxford University Press, Oxford, 2015), reprinted with corrections edn

    2015

  43. [43]

    J.Modern quantum mechanics(Cambridge University Press, Cambridge, United Kingdom, 2021), third edition edn

    Sakurai, J. J.Modern quantum mechanics(Cambridge University Press, Cambridge, United Kingdom, 2021), third edition edn. Comprend des r´ ef´ erences bibliographiques (pages 541-543) et un index

    2021

  44. [44]

    Rauch, K. P. & Tremaine, S. Resonant relaxation in stellar systems.New Astronomy1, 149–170 (1996)

    1996

  45. [45]

    & Alexander, T

    Bar-Or, B. & Alexander, T. Steady-state relativistic stellar dynamics around a massive black hole.The Astrophysical Journal820, 129 (2016)

    2016

  46. [46]

    & Will, C

    Merritt, D., Alexander, T., Mikkola, S. & Will, C. M. Stellar dynamics of extreme-mass-ratio inspirals.Physical Review D84, 044024 (2011)

    2011

  47. [47]

    & Tremaine, S

    Kocsis, B. & Tremaine, S. Resonant relaxation and the warp of the stellar disc in the galactic centre: Resonant relaxation and warp of the gc disc.Monthly Notices of the Royal Astronomical Society412, 187–207 (2011)

    2011

  48. [48]

    & Poon, M

    Merritt, D. & Poon, M. Y. Chaotic loss cones and black hole fueling.The Astrophysical Journal 606, 788–798 (2004)

    2004

  49. [49]

    & Lukes-Gerakopoulos, G

    Kerachian, M., Polcar, L., Skoup´ y, V., Efthymiopoulos, C. & Lukes-Gerakopoulos, G. Action- angle formalism for extreme mass ratio inspirals in kerr spacetime.Physical Review D108, 044004 (2023)

    2023

  50. [50]

    Celestial mechanics in kerr spacetime.Classical and Quantum Gravity19, 2743– 2764 (2002)

    Schmidt, W. Celestial mechanics in kerr spacetime.Classical and Quantum Gravity19, 2743– 2764 (2002)

    2002

  51. [51]

    & Loeb, A

    Stone, N. & Loeb, A. Observing lense-thirring precession in tidal disruption flares.Physical Review Letters108, 061302 (2012)

    2012

  52. [52]

    Collins, N. A. & Hughes, S. A. Towards a formalism for mapping the spacetimes of massive compact objects: bumpy black holes and their orbits.Physical Review D69, 124022 (2004)

    2004

  53. [53]

    & Psaltis, D

    Johannsen, T. & Psaltis, D. Metric for rapidly spinning black holes suitable for strong-field tests of the no-hair theorem.Physical Review D83, 124015 (2011)

    2011

  54. [54]

    Rotating traversable wormholes.Physical Review D58, 024014 (1998)

    Teo, E. Rotating traversable wormholes.Physical Review D58, 024014 (1998). 63

    1998