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Collisional damping in debris discs: Only significant if collision velocities are low

T0 review · 2 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Debris disc collisions damp vertical structure only when collision velocities are low.

desk verdict The inefficient-damping branch (Yc << 1) is robust and worth citing; the efficient branch and the 40 m/s threshold lean on maximal-damping and should be tested against partial restitution. read the letter →

arxiv 2411.13991 v1 pith:ALXDUHA4 submitted 2024-11-21 astro-ph.EP

Marija R. Jankovic , Mark C. Wyatt , Torsten Löhne This is my paper · ORCID
classification astro-ph.EP
keywords debrisdiscscollisionaldampingverticalscaleheightinclinationdistributioncascadecriticaldisruptionenergykineticmodelimpactvelocities
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 asks when inelastic collisions actually flatten the vertical structure of a debris disc, and answers: only when the typical collision velocity is low enough. Damping is governed by the critical projectile-to-target mass ratio $Y_c$, roughly twice the specific disruption energy divided by impact velocity squared. If $Y_c$ is much smaller than one, a particle can be destroyed by a projectile far less massive than itself; such collisions damp little, so eccentricities and inclinations are reshaped by destruction rates rather than by damping, and the disc keeps the inclination distribution it was born with for much longer than the collisional timescale of its largest bodies. If $Y_c$ is of order one or larger, collisions mostly leave targets intact and damping acts at the classical rate. For millimetre-sized grains with common material strength assumptions, this efficient-damping regime requires impact velocities below roughly $40\ \mathrm{m\,s^{-1}}$.

What carries the argument

The governing object is the critical projectile-to-target mass ratio $Y_c$, which measures whether a projectile much smaller than the target can still destroy it. The argument compares the damping rate, whose integrand is proportional to $m_t m_p/(m_t+m_p)^2$ so that equal-size collisions dominate, with the fragmentation rate, dominated by projectiles just above the critical size; when the fragmentation rate exceeds the damping rate, collisional damping becomes inefficient. The numerical machinery is a kinetic model that evolves particle numbers over a phase space of mass, eccentricity, and inclination, with collision rates and impact velocities supplied by Monte Carlo sampling of orbit-overlap regions and with collisional outcomes including catastrophic fragmentation, cratering, and growth. A key simplifying assumption is that every collision is completely inelastic, so post-collision remnants and fragments move at the centre-of-mass velocity of the colliders, which maximizes the damping efficiency of each collision.

What would settle it

Measure the vertical scale-height of an edge-on debris disc at both millimetre and near-infrared wavelengths: the paper predicts that a disc with typical collision velocities well above about $40\ \mathrm{m\,s^{-1}}$ should show a wavelength-independent scale-height even without viscous stirring, whereas a lower-velocity disc should appear thinner at shorter wavelengths; a high-velocity disc showing thinner small grains would contradict the central claim.

Watch

Extended reading notes

Core claim

The central claim is that collisional damping in a debris disc is controlled not by the classical damping rate alone but by the ratio of the critical disruption energy to the square of the impact velocity, expressed as the critical projectile-to-target mass ratio $Y_c = 2Q_D^*/v_{\mathrm{imp}}^2$. When $Y_c \ll 1$, projectiles far smaller than the target can destroy it, and because collisions with similar-mass bodies are the ones that damp efficiently, the damping rate falls below the fragmentation rate. In that regime the eccentricity and inclination distributions are shaped by the destruction probability being slightly different for particles on different orbits; the average eccentricity and inclination evolve slowly, at the same rate for all particle sizes, and the disc retains its birth vertical thickness. When $Y_c$ is of order unity or larger, collisions mostly leave targets intact, damping proceeds at roughly the classical rate, and particles of different sizes can be damped at different rates, producing a scale-height that varies with wavelength.

Load-bearing premise

The paper assumes every collision, including cratering and catastrophic disruption, is completely inelastic, so all post-collision material moves at the centre-of-mass velocity of the two colliders, which maximizes the damping each collision can do; if real collisions retain more kinetic energy, the efficient-damping branch for $Y_c$ of order unity or larger is weakened, while the inefficient-damping branch for $Y_c \ll 1$ is unaffected or strengthened.

Editorial extensions

If this is right

  • A vertically thick debris disc with a scale-height that does not change with observing wavelength does not have to be viscously stirred; it can simply be in the fragmentation-dominated regime where $Y_c \ll 1$.
  • When the critical projectile-to-target mass ratio is of order unity or larger, small particles are damped faster than large ones, so the disc scale-height should increase with observing wavelength during the damping phase.
  • In a wide disc, collisional damping can be a non-monotonic function of radius: inefficient at small radii because collision velocities are high, slow at large radii because collision rates are low, and efficient only in an intermediate zone.
  • For an exo-Kuiper belt with aspect ratio above roughly $0.01$ around a $2\ M_\odot$ star at $100$ au, collision velocities exceed the critical value and collisional damping is inefficient, whereas white-dwarf discs and planetary rings may still thin significantly within their lifetimes even in the high-velocity regime.
  • In the fragmentation-dominated regime the average eccentricity falls faster than the average inclination, driving the ratio $\langle e\rangle/\langle i\rangle$ below unity, opposite to the value near two expected from gravitational equipartition.

Reading between the lines

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

  • The paper's logic suggests that the observed diversity of debris disc aspect ratios could largely reflect where each disc sits relative to the $Y_c \sim 1$ boundary, with high-velocity discs remaining thick and wavelength-independent and low-velocity discs collapsing vertically.
  • A testable extension is to use multi-wavelength imaging of edge-on discs to map whether the scale-height is wavelength-independent in high-velocity systems and wavelength-dependent in low-velocity systems, directly testing the $Y_c$ criterion.
  • Because $Y_c$ depends on the material-strength law $Q_D^*(s,v_{\mathrm{imp}})$, laboratory impact experiments at speeds of order $1$ to $100\ \mathrm{m\,s^{-1}}$ could sharpen or revise the quoted $40\ \mathrm{m\,s^{-1}}$ threshold for millimetre grains.
  • For viscously stirred discs, the paper's results imply that the relevant balance may be between viscous stirring and destruction rather than between stirring and damping, because fragments inherit the velocities of their parent bodies and small grains may track the velocities of the largest bodies.
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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

2 major / 6 minor

Summary. The paper extends an existing kinetic model of debris-disc collisional evolution (van Lieshout et al. 2014) by adding orbital inclination to the phase space and by computing collision probabilities and impact velocities via Monte Carlo simulations (Wyatt et al. 2010). The model evolves mass, eccentricity, and inclination on a grid of particle sizes from 1 mm to 100 m in a single semi-major-axis bin, with a treatment of catastrophic disruption, cratering, and growth. The central question is when collisional damping reduces the eccentricity and inclination dispersion of a pre-stirred disc. Using the critical projectile-to-target mass ratio Yc ~ 2Q*_D/v_imp^2, the paper shows that for Yc << 1 collisional damping is inefficient: destructive collisions remove particles faster than damping can act, and fragments inherit the velocity distribution of the massive target, so the average inclination evolves slowly and mostly independently of particle size. For Yc of order unity or larger, damping proceeds at roughly the classical rate. The authors apply the criterion to observations, predicting a wavelength-independent scale-height for high-excitation discs and deriving a threshold of ~40 m/s for mm grains. The model is tested against analytic collision rates and steady-state size distributions, and the main qualitative claims are robust, although the efficient-damping branch rests on the maximal-damping assumption.

Significance. If correct, the paper supplies a simple and falsifiable criterion for when collisional damping matters for debris-disc vertical structure, replacing the classical damping rate by a criterion based on Yc = 2Q*_D/v^2. It also makes a nontrivial prediction: discs with Yc << 1 should show a wavelength-independent scale-height that changes slowly even without viscous stirring, contrary to a common interpretation of such observations. The numerical implementation is carefully tested: Appendix A validates the Monte Carlo collision rates against analytical 2D results and the Bottke et al. velocity distribution, and Appendix B compares the code with analytic '2D' models, the expected steady-state size distribution, and the M(t) = M0/(1+ct) mass evolution. Grid-resolution tests show that the qualitative distinction between efficient and inefficient damping is convergent. The inefficient-damping branch is robust to the main physical assumption (complete inelasticity), because fragments inherit the target velocity in that regime.

major comments (2)
  1. [Sect. 3.1.3, Eq. (18); Sect. 5.4] The conclusion that collisional damping is efficient for Yc of order unity or larger (Abstract; Conclusions item 2) rests on the maximal-damping prescription: Eq. (18) assigns the pre-collision centre-of-mass velocity to every remnant and fragment in every collision, including cratering and bouncing. This maximizes the damping rate for each collision and is the opposite extreme from the partial restitution expected for non-catastrophic impacts. Section 5.4 acknowledges the limitation but gives no sensitivity estimate. A partial-restitution sensitivity test (for example, retaining a fraction epsilon of the relative kinetic energy in non-catastrophic outcomes, or a lower dissipation efficiency for cratering) is needed to establish whether the efficient branch and the boundary velocity of ~40 m/s are robust or shift to lower velocities. The inefficient-damping branch for Yc << 1 is not affected because fragments inherit the massive target's velocity, so the central qualitative result stands; nevertheless, the abstract's symmetric claim is not yet quantitatively supported.
  2. [Sect. 5.1, Table 1 (model 3)] The headline threshold of ~40 m/s for mm-sized grains is obtained by solving 1 = 2Q*_D(s,v)/v^2 with Q*_D model 3, whose velocity dependence (vimp/3 km/s)^0.5 is fitted to impact experiments at km/s velocities. At 40 m/s this is an extrapolation by roughly two orders of magnitude in impact velocity, and low-velocity disruption physics (different energy partitioning, possible effects of porosity or van der Waals forces) could change Q*_D substantially. The manuscript mentions this caveat in Sect. 5.1, but the abstract states the number without qualification. Please either add a low-velocity Q*_D prescription, or explicitly demote the number to an illustrative estimate in the abstract and conclusions.
minor comments (6)
  1. [Key words] The key word 'cellestial mechanics' contains a typo and should read 'celestial mechanics'.
  2. [References] The reference 'Löhne. 2008' is malformed; it should include the author's initial (e.g., Löhne, T. 2008) to be consistent with the rest of the reference list.
  3. [Appendix B, Fig. B.2 caption] The analytic mass evolution is written as '1/(1+ct)' in the caption; writing M(t)/M0 = 1/(1+ct) would be clearer.
  4. [Sect. 4.1] The statement that damping is limited by the lowest grid bins is important, but the main text could clarify that the apparent slowdown at late times in Fig. 2 is partly numerical; the caption of Fig. 2 already shows the floor, yet a sentence in the text noting which portion of the curves is affected would help the reader.
  5. [Sect. 3.2] The text 'we adopt n(m)dm ∝ m^{-11/6} dm' contains a redundant differential; this should be written as n(m) ∝ m^{-11/6}.
  6. [Abstract] The claim of 'vertical thickness independent of wavelength' should be tied to the model's assumptions (pre-stirred, single-annulus, no stirring); consider adding 'in the framework of this model' to avoid overgeneralization.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Yc is a diagnostic parameter, not a fitted input, and the inefficient-damping branch is an emergent result of the rate comparison and numerical evolution.

full rationale

The derivation chain is self-contained. The paper's central parameter Yc = 2Q*_D/v^2_imp is computed from the adopted material-strength prescriptions and initial excitation levels (Fig. 1) and is explicitly not used inside the kinetic model's collisional-outcome logic, which instead applies the standard Q*_tp criteria described in Sect. 3.1.3. No quantity is fitted to a subset of data and then re-predicted. The inefficient-damping branch follows from a rate comparison in Sect. 2 (R_frag > R_damp when the critical projectile size is below the target size) and is confirmed by the numerical models of Sect. 4; it does not depend on the maximal-damping assumption, as the authors state in Sect. 5.4. The efficient-damping branch does rely on Eq. (18)'s complete-inelasticity prescription, but this is an openly stated physical assumption, explicitly flagged in Sect. 5.4 ('This assumption maximises the rate at which the average eccentricity and inclination decrease with time'), rather than a hidden circular step. Citations to the authors' earlier code (Krivov et al. 2006; van Lieshout et al. 2014; Wyatt et al. 2010) are methodological and are validated against independent benchmarks (Bottke et al. 1994; Krivov et al. 2006) in Appendices A and B. No load-bearing self-citation, uniqueness theorem, or definitional equivalence forces the headline conclusions.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The model introduces no new physical entities. Its input parameters (Q*_D, v_stick, disc mass, size cutoffs, initial excitation) come from prior literature or are chosen to probe the desired regimes, and the central claim is an emergent property of the simulation rather than a fitted quantity.

free parameters (4)
  • Critical sticking velocity v_stick = 1 m/s
    Chosen by hand as the threshold below which collisions lead to net growth; affects the low-velocity outcome (Sect. 3.1.3).
  • Material strength parameters in Q*_D model 3 = A = 5e6 erg/g, exponents -0.37 and 1.38, velocity exponent 0.5
    Taken from Benz & Asphaug (1999) impact simulations; the ~40 m/s critical velocity for mm grains scales with the square root of Q*_D and would shift if a low-velocity strength law were used (Sect. 5.1).
  • Lower size cutoff smin = 1 mm
    Computational cutoff for the mass grid; creates an artificial over-abundance of the smallest grains, acknowledged in Sect. 4.2.
  • Initial excitation level emax = 0.01 and 0.2
    Chosen to probe Yc below and above unity; the paper notes results in the slow-evolution regime may depend on this choice (Sects. 4.2, 5.4).
assumptions (7)
  • domain assumption Collision timescales are much longer than orbital timescales, so the distribution function evolves according to the continuity equation (Eq. 10).
    Standard kinetic approach assumption stated in Sect. 3; fails for very dense discs.
  • domain assumption Semi-major axis evolution due to collisional damping is negligible over the simulated time.
    Justified in Sect. 2 by timescale arguments from equal-size N-body simulations (Brahic 1977; Lithwick & Chiang 2007); the paper acknowledges it may not hold for small bodies in a real cascade (Sect. 5.4).
  • domain assumption All collisions are completely inelastic; post-collision velocity is the center-of-mass velocity of the colliders (Eq. 18).
    Maximal-damping assumption used for all remnants and fragments; flagged as a limitation in Sect. 5.4.
  • domain assumption The disc is axisymmetric, allowing averaging over the remaining orbital angles.
    Standard assumption of the kinetic model, stated in Sect. 3.
  • domain assumption Fragments are distributed in mass following n(m) proportional to m^{-11/6}.
    Adopted from van Lieshout et al. (2014); standard power-law for collision cascades.
  • domain assumption Particles over the full size range 1 mm to 100 m are in the material strength regime; gravitational aggregates are not included.
    Grid extends only to 100 m; discussed in Sect. 5.1.
  • ad hoc to paper Initial eccentricity and inclination distributions are uniform.
    Chosen for simplicity; paper notes possible sensitivity in slowly evolving regimes (Sect. 5.4).

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Pith. "Pith review of Collisional damping in debris discs: Only significant if collision velocities are low." pith.science (2026). https://pith.science/paper/ALXDUHA4

@misc{pith2026241113991,
  author       = {Pith},
  title        = {Pith review of: Collisional damping in debris discs: Only significant if collision velocities are low},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALXDUHA4}},
  note         = {Machine review of arXiv:2411.13991}
}
read the original abstract

Context. Dusty debris discs around main sequence stars are observed to vary widely in terms of their vertical thickness. Their vertical structure may be affected by damping in inelastic collisions. Although kinetic models have often been used to study the collisional evolution of debris discs, these models have not yet been used to study the evolution of their vertical structure. Aims. We extend an existing implementation of a kinetic model of collisional evolution to include the evolution of orbital inclinations and we use this model to study the effects of collisional damping in pre-stirred discs. Methods. We evolved the number of particles of different masses, eccentricities, and inclinations using the kinetic model and used Monte Carlo simulations to calculate collision rates between particles in the disc. We considered all relevant collisional outcomes including fragmentation, cratering, and growth. Results. Collisional damping is inefficient if particles can be destroyed by projectiles that are of much lower mass. If that is the case, catastrophic disruptions shape the distributions of eccentricities and inclinations, and their average values evolve slowly and at the same rate for all particle sizes. Conclusions. The critical projectile-to-target mass ratio (Yc) and the collisional timescale jointly determine the level of collisional damping in debris discs. If Yc is much smaller than unity, a debris disc retains the inclination distribution that it is born with for much longer than the collisional timescale of the largest bodies in the disc. Such a disc should exhibit a vertical thickness that is independent of wavelength even in the absence of other physical processes. Collisional damping is efficient if Yc is of order unity or larger. For millimetre-sized dust grains and common material strength assumptions, this requires collision velocities of lower than ~40 m/s. Abridged

Figures

Figures reproduced from arXiv: 2411.13991 by the authors.

Figure 1
Figure 1. Approximate initial critical projectile-to-target mass ratio (Yc) for the models considered in Sects. 4.2 and 4.3. Blue and orange lines are for low-excitation (emax = 0.01) and high-excitation discs (emax = 0.2), respectively, and the different line styles are for differ￾ent Q ∗ D models shown in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Evolution of average eccentricity (solid lines) and inclination (in radians; dashed lines) in a disc of indestructible particles, inelastically bouncing off each other in collisions. The left- and right-hand-side panels show a low-excitation disc and high-excitation disc, respectively. Horizontal blue lines show the lowest grid bin values, i.e. the lowest values to which eccentricity and inclination can be damped in… view at source ↗
Figure 3
Figure 3. Cross-section area per base-10 logarithmic unit of size at different times (see plot legend) in a model of a collisional cascade with constant Q ∗ D (model 1 in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Evolution of average eccentricity (solid lines) and inclination (in radians; dashed lines) in a model of a collisional cascade with constant Q ∗ D (model 1 in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Evolution of eccentricity (top panels) and inclination (bottom panels) distributions for 1 m bodies in the models shown in Figs. 3 and 4. The left- and right-hand-side panels show low-excitation disc and high-excitation disc, respectively. See Sect. 4.2. collisions in …
Figure 6
Figure 6. Figure 6: Geometric collision probability (top), average impact velocity (middle), and the geometric part of the collision rate (bottom) in the high-excitation simulation discussed in Sect. 4.2, as a function of the eccentricity et and the inclination it of target particles coll…
Figure 7
Figure 7. Figure 7: Rate of change of average eccentricity (top panels) and inclination (bottom panels) for 1 m bodies in the models shown in Figs. 3 and 4. The left- and right-hand-side panels show low-excitation disc and high-excitation disc, respectively. The total rate of change is sh…
Figure 8
Figure 8. Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: Example calculation of critical projectile-to-target mass ratio (in blue) and the classical damping rate given by eq. (5) for 1 mm grains as functions of radius in a wide disc around a solar-mass star, with particle Q ∗ D given by model 3 in [PITH_FULL_IMAGE:figures/…

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Works this paper leans on

56 extracted references · 49 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archiveprefix author booktitle chapter edition editor howpublished institution eprint journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

  3. [3]

    1997, Annual Review of Earth and Planetary Sciences, 25, 175

    Artymowicz , P. 1997, Annual Review of Earth and Planetary Sciences, 25, 175

    work page 1997

  4. [4]

    & Asphaug , E

    Benz , W. & Asphaug , E. 1999, , 142, 5

    work page 1999

  5. [5]

    F., Nolan , M

    Bottke , W. F., Nolan , M. C., Greenberg , R., & Kolvoord , R. A. 1994, , 107, 255

    work page 1994

  6. [6]

    1977, , 54, 895

    Brahic , A. 1977, , 54, 895

    work page 1977

  7. [7]

    & Henon , M

    Brahic , A. & Henon , M. 1977, , 59, 1

    work page 1977

  8. [8]

    R., Farinella , P., & Paolicchi , P

    Campo Bagatin , A., Cellino , A., Davis , D. R., Farinella , P., & Paolicchi , P. 1994, , 42, 1079

    work page 1994

Show all 56 references
  1. [9]

    M., Carter , E

    Daley , C., Hughes , A. M., Carter , E. S., et al. 2019, , 875, 87

    2019

  2. [10]

    Dohnanyi , J. S. 1969, , 74, 2531

    1969

  3. [11]

    & Decin , G

    Dominik , C. & Decin , G. 2003, , 598, 626

    2003

  4. [12]

    1986, , 57, 47

    Fujiwara , A. 1986, , 57, 47

    1986

  5. [13]

    S., Marino , S., Sheehan , P

    Hales , A. S., Marino , S., Sheehan , P. D., et al. 2022, , 940, 161

    2022

  6. [14]

    C., & Matr \`a , L

    Han , Y., Wyatt , M. C., & Matr \`a , L. 2022, , 511, 4921

    2022

  7. [15]

    1985, , 64, 295

    Hornung , P., Pellat , R., & Barge , P. 1985, , 64, 295

    1985

  8. [16]

    M., Duch \^e ne , G., & Matthews , B

    Hughes , A. M., Duch \^e ne , G., & Matthews , B. C. 2018, , 56, 541

    2018

  9. [17]

    & Makino , J

    Ida , S. & Makino , J. 1992, , 96, 107

    1992

  10. [18]

    M., Marino , S., Matr \`a , L., et al

    Kennedy , G. M., Marino , S., Matr \`a , L., et al. 2018, , 475, 4924

    2018

  11. [19]

    Kennedy , G. M. & Wyatt , M. C. 2010, , 405, 1253

    2010

  12. [20]

    Kenyon , S. J. & Bromley , B. C. 2002 a , , 123, 1757

    2002

  13. [21]

    Kenyon , S. J. & Bromley , B. C. 2002 b , , 577, L35

    2002

  14. [22]

    Kenyon , S. J. & Bromley , B. C. 2004, , 127, 513

    2004

  15. [23]

    Kenyon , S. J. & Bromley , B. C. 2008, , 179, 451

    2008

  16. [24]

    Kenyon , S. J. & Bromley , B. C. 2010, , 188, 242

    2010

  17. [25]

    Kenyon , S. J. & Luu , J. X. 1999, , 118, 1101

    1999

  18. [26]

    & Tanaka , H

    Kobayashi , H. & Tanaka , H. 2010, , 206, 735

    2010

  19. [27]

    Krivov , A. V. & Booth , M. 2018, , 479, 3300

    2018

  20. [28]

    V., L \"o hne , T., & Srem c evi \'c , M

    Krivov , A. V., L \"o hne , T., & Srem c evi \'c , M. 2006, , 455, 509

    2006

  21. [29]

    V., Srem c evi \'c , M., & Spahn , F

    Krivov , A. V., Srem c evi \'c , M., & Spahn , F. 2005, , 174, 105

    2005

  22. [30]

    Leinhardt , Z. M. & Stewart , S. T. 2012, , 745, 79

    2012

  23. [31]

    Lissauer , J. J. & Stewart , G. R. 1993, in Protostars and Planets III, ed. E. H. Levy & J. I. Lunine , 1061

    1993

  24. [32]

    & Chiang , E

    Lithwick , Y. & Chiang , E. 2007, , 656, 524

    2007

  25. [33]

    V., Kirchschlager , F., Sende , J

    L \"o hne , T., Krivov , A. V., Kirchschlager , F., Sende , J. A., & Wolf , S. 2017, , 605, A7

    2017

  26. [34]

    2008, PhD thesis, Friedrich-Schiller-Universität Jena, Germany

    Löhne. 2008, PhD thesis, Friedrich-Schiller-Universität Jena, Germany

    2008

  27. [35]

    P., Milli , J., Choquet , E., et al

    Marshall , J. P., Milli , J., Choquet , E., et al. 2023, , 521, 5940

    2023

  28. [36]

    C., Wilner , D

    Matr \`a , L., Wyatt , M. C., Wilner , D. J., et al. 2019, , 157, 135

    2019

  29. [37]

    Murray, C. D. & Dermott, S. F. 2000, Solar System Dynamics (Cambridge University Press)

    2000

  30. [38]

    R., Kenyon , S

    Najita , J. R., Kenyon , S. J., & Bromley , B. C. 2022, , 925, 45

    2022

  31. [39]

    1988, , 230, 551

    Nakano , T. 1988, , 230, 551

    1988

  32. [40]

    2022, , 513, 713

    Olofsson , J., Th \'e bault , P., Kral , Q., et al. 2022, , 513, 713

    2022

  33. [41]

    & Schlichting , H

    Pan , M. & Schlichting , H. E. 2012, , 747, 113

    2012

  34. [42]

    1996, , 121, 126

    Paolicchi , P., Verlicchi , A., & Cellino , A. 1996, , 121, 126

    1996

  35. [43]

    C., Morbidelli , A., & Moore , A

    Quillen , A. C., Morbidelli , A., & Moore , A. 2007, , 380, 1642

    2007

  36. [44]

    Rigley , J. K. & Wyatt , M. C. 2020, , 497, 1143

    2020

  37. [45]

    Stewart , S. T. & Leinhardt , Z. M. 2009, , 691, L133

    2009

  38. [46]

    A., et al

    Terrill , J., Marino , S., Booth , R. A., et al. 2023, , 524, 1229

    2023

  39. [47]

    2009, , 505, 1269

    Th \'e bault , P. 2009, , 505, 1269

    2009

  40. [48]

    & Augereau , J

    Th \'e bault , P. & Augereau , J. C. 2007, , 472, 169

    2007

  41. [49]

    C., & Beust , H

    Th \'e bault , P., Augereau , J. C., & Beust , H. 2003, , 408, 775

    2003

  42. [50]

    & Wu , Y

    Th \'e bault , P. & Wu , Y. 2008, , 481, 713

    2008

  43. [51]

    1971, , 12, 329

    Trulsen , J. 1971, , 12, 329

    1971

  44. [52]

    2014, , 571, A51

    van Lieshout , R., Dominik , C., Kama , M., & Min , M. 2014, , 571, A51

    2014

  45. [53]

    M., Carter , E

    Vizgan , D., Hughes , A. M., Carter , E. S., et al. 2022, , 935, 131

    2022

  46. [54]

    Wyatt , M. C. 2008, , 46, 339

    2008

  47. [55]

    C., Booth , M., Payne , M

    Wyatt , M. C., Booth , M., Payne , M. J., & Churcher , L. J. 2010, , 402, 657

    2010

  48. [56]

    C., Smith , R., Greaves , J

    Wyatt , M. C., Smith , R., Greaves , J. S., et al. 2007, , 658, 569

    2007

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.