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Excitation of Tumbling in Phobos and Deimos

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Impact-triggered tumbling can explain the low orbital eccentricities of Phobos and Deimos without an early high-eccentricity phase.

desk verdict A useful, honest numerical study that turns Wisdom's tumbling-damping speculation into a concrete mechanism, but the quantitative Deimos claim rests on a clearly flagged rheological extrapolation. read the letter →

arxiv 1908.05720 v2 pith:A7KNDR5H submitted 2019-08-15 astro-ph.EP

Alice C. Quillen , Mckenzie Lane , Miki Nakajima , Esteban Wright This is my paper · ORCID
classification astro-ph.EP
keywords PhobosDeimostumblingtidaldissipationspin-orbitresonanceorbitaleccentricitychaoticrotationmass-springmodel
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

Using mass-spring simulations of viscoelastic Phobos and Deimos, this paper tries to establish that the moons' current low orbital eccentricities can be explained by episodes of chaotic tumbling excited by rare, nearly catastrophic impacts. It first shows that crossing the strongest spin-orbit and mean-motion resonances does not knock either moon out of tidal lock, so those resonances cannot be the spin-excitation mechanism. Once a moon is tumbling, however, simulations show it stays tumbling for thousands of orbits while tidal dissipation runs one to six orders of magnitude above the locked-state rate, strongly damping eccentricity, even down to the very small values Phobos and Deimos have today. If true, the old problem that a tidally locked Deimos could not damp its eccentricity within the age of the Solar system disappears.

What carries the argument

The paper's central object is the mass-spring model: a moon is represented by a few dozen point masses connected by damped springs, so tidal deformation, torque, heat, and eccentricity damping arise self-consistently from the viscoelastic response of the resolved body. The analytical idea that carries the argument is the chaotic-zone volume estimate from the classic spin-orbit problem. Near the spin-synchronous separatrix, the 3:2 resonance perturbation strength is proportional to eccentricity, giving a chaotic zone width in energy proportional to $e$ and a phase-space volume proportional to $\sqrt{e}$; because this volume shrinks so weakly, a dissipating tumbling body is almost as likely to leave the chaotic zone at one decade of eccentricity as at the next, which explains why simulations lock at very low eccentricity. The simulations also calibrate the dissipation-rate contrast: while tumbling the mean rate is approximately $\dot E_{\rm wob} = A_w C_e n e$ with $A_w \approx 7$–14, one to six orders of magnitude above the locked-state rate $7 C_e n e^2$ at low eccentricity.

What would settle it

Measure the frequency-dependent tidal quality factor of plausible Phobos and Deimos material, or run a high-frequency Maxwell or Andrade simulation of the same mass-spring configurations: if at the real moons' tidal frequencies the tumbling-state dissipation rate is less than roughly a thousand times the locked-state rate, Deimos's eccentricity could not have been damped in the age of the Solar system, and the impact-tumbling explanation would fail.

Watch

Extended reading notes

Core claim

The central claim is that the low eccentricities of Phobos and Deimos could in part be due to spin excitation by nearly catastrophic impacts rather than tidal evolution following orbital resonance excitation. In the simulations, an initially tidally locked Phobos crossing the 2:1 spin-orbit resonance with Mars, and an initially locked Deimos crossing the 2:1 mean-motion resonance with Phobos, both remain locked; the resonances raise eccentricity by only a small amount and do not excite tumbling. Simulations started in a tumbling state, by contrast, remain tumbling for thousands of orbital periods, and the elevated tidal dissipation during that time substantially reduces the orbital eccentricity. The paper attributes the tendency to drop into spin-synchronous rotation at very low eccentricity to the insensitivity of the tumbling chaotic zone volume to eccentricity: in the classic spin-orbit problem the chaotic zone volume scales roughly as the square root of eccentricity, so a body is only about three times more likely to escape the chaotic zone per decade of eccentricity decrease. After entering spin-synchronous rotation, long-lived non-principal axis rotation can keep dissipation elevated further. With estimates of impactor fluxes, the paper concludes that sub-catastrophic impacts of roughly 0.7 km (Phobos) and 0.2 km (Deimos) projectiles were likely frequent enough to have excited such tumbling.

Load-bearing premise

The load-bearing assumption is that the tumbling-versus-locked boost in tidal dissipation measured at the simulations' low tidal frequencies still holds at the much higher tidal frequencies of the real moons, so that Deimos's eccentricity can actually be damped within the age of the Solar system.

Editorial extensions

If this is right

  • Crossing the 2:1 spin-orbit resonance with Mars or the 2:1 mean-motion resonance between Phobos and Deimos does not excite tumbling, so these resonance crossings cannot be the mechanism that set the moons' spins or eccentricities.
  • A single tumbling episode can damp a moon's eccentricity to values below the current ones within tens of millions of years, even for Deimos, whose locked-state eccentricity damping time is about $10^{12}$ years.
  • Sub-catastrophic impacts of the size that formed Stickney crater on Phobos are frequent enough on billion-year timescales to have excited tumbling in both moons.
  • When a tumbling body falls into spin-synchronous rotation, it can remain in long-lived non-principal axis rotation, keeping tidal dissipation elevated and making standard locked-state tidal formulas underestimates of the eccentricity damping.
  • The broad, low range of eccentricities at which simulated moons lock is consistent with a tumbling chaotic-zone volume that shrinks only as $\sqrt{e}$, implying that tumbling bodies often reach very low eccentricity before becoming locked.

Reading between the lines

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

  • If impact-triggered tumbling is a general satellite mechanism, other small, heavily cratered moons on nearly circular orbits should show a 'reset' eccentricity set by their last tumbling episode, not by their formation; a survey correlating cratering ages with eccentricities across small satellites would test this.
  • Because the chaotic-zone volume shrinks as $\sqrt{e}$, tumbling should persist even at extremely low eccentricity, so small moons observed currently tumbling may be in this long-lived state rather than freshly impacted.
  • The high-frequency rheology assumption could be checked directly with laboratory measurements of frequency-dependent dissipation in rocky and icy regolith; if the tumbling-to-locked dissipation ratio drops at high frequency, the mechanism would apply only to moons with different internal dissipation.
  • If such tumbling episodes reset eccentricity after the last big impact, Phobos need not have recently crossed Deimos's orbit, which would make the moons' current near-circular, low-inclination orbits a natural outcome of formation in a circumplanetary disk rather than capture.
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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 investigates whether past episodes of chaotic tumbling could explain the present low orbital eccentricities of Phobos and Deimos. Using mass-spring viscoelastic simulations of the spinning bodies, the authors first show that adiabatic crossing of the 2:1 resonance with Mars' rotating figure and of the 2:1 Phobos–Deimos mean-motion resonance does not excite tumbling in a tidally locked moon. They then simulate initially tumbling Phobos- and Deimos-like bodies with exaggerated tidal forcing and find that tumbling persists for thousands of orbital periods and can damp eccentricity to very low values before the body falls into the spin-synchronous state. The tendency to reach low eccentricity is attributed to the weak (square-root) dependence of the chaotic zone volume on eccentricity. The authors also report long-lived non-principal-axis rotation in the spin-synchronous state and conclude that sub-catastrophic impacts could have excited tumbling episodes, so that impact-driven tumbling, rather than tidal evolution following resonance excitation, may in part explain the low eccentricities.

Significance. If the central scenario is correct, the paper offers a physically self-consistent mechanism that resolves a long-standing problem: Deimos's eccentricity damping timescale in the tidally locked state is longer than the Solar-system age, yet its current eccentricity is extremely small. The work extends Wisdom (1987)'s speculation with direct numerical experiments in which tidal dissipation, torque, and orbital evolution arise from the internal spring damping rather than being imposed through the standard constant-Q or constant-phase-lag formulas. The strengths of the paper are its explicit calibration of the simulations against analytic tidal formulas (Eqs. 19–23), its reproduction of the chaotic-zone behavior predicted by the conservative spin-orbit problem (Eq. 17–18), and its honest delineation of several caveats, including the statement in Section 6 that sensitivity to rheology was neglected. The principal weakness is that the quantitative claim for Deimos depends on an untested rheological extrapolation and on simulation parameters far from the real moons' values, so the result is plausible but not yet established.

major comments (3)
  1. [§4.1 (text after Eq. 23)] The load-bearing step for the Deimos eccentricity-damping claim is the assertion that 'the ratio of dissipation rates, between spin-synchronous and tumbling, would be similar for the high frequency limit as for the low frequency limit.' The simulations are in the low-frequency Kelvin–Voigt regime, with Kd~10^-5 (Tables 2 and 4, §4.3), whereas Table 1 gives Kd~10^-9 for Deimos and the text notes that the moons may be in the opposite limit (chi>>1). Section 6 explicitly states that 'sensitivity to rheology was neglected.' The Deimos timescale estimate in §4.1 rests on a factor of ~5 orders of magnitude in the dissipation-rate ratio at e=0.003; if the true high-frequency ratio is two or more orders of magnitude smaller, the required tumbling duration would exceed the age of the Solar system, and the abstract's closing explanation for Deimos's low eccentricity would collapse. This is not a mere uncertainty but an unsupported internal step: the authors need either an analytic derivation of the high-frequency ratio or simulations or measurements in that regime.
  2. [§4, Table 4 and §4.3 (Eq. 26)] The simulations use a significantly inflated radius (a/R=48 versus 830 for Phobos and 3778 for Deimos) and a correspondingly reduced central mass, with the statement that this 'should not significantly affect the dynamics... primarily sensitive to the body axis ratios.' However, the relative rate of eccentricity damping to spin evolution is governed by Ke (Eq. 26), and the paper itself states that the PhS simulations have Ke about 10^3 times that of Phobos and the DeS simulations about 10^6 times that of Deimos. The observed outcomes—including the duration of tumbling and the distribution of eccentricities at lock-in—could depend on this mismatch. Because the central scenario is an extrapolation from a regime where eccentricity damping is artificially fast, the authors should either run simulations with Ke closer to realistic values or provide a scaling argument showing that the qualitative behavior is unchanged when the ratio of the eccentricity-damping rate to the spin-dissipation rate is reduced by orders of magnitude.
  3. [§3, DeP 2:1 simulation (Table 3)] The negative conclusion that crossing the 2:1 Phobos–Deimos mean-motion resonance does not excite Deimos's spin is based on a simulation in which Phobos's mass is enhanced by a factor of 100, and the drift rate imposed is not demonstrated to satisfy the adiabatic criterion for the enhanced resonance. Because a stronger resonance requires a slower drift rate for adiabatic passage, the simulation may be testing a non-adiabatic crossing, where the eccentricity kick is suppressed. The later analytic estimate of adiabaticity uses real masses and drift rates, not the simulation's values. To support the abstract's claim that such resonance crossings 'do[es] not excite tumbling,' the simulation should be run at a drift rate that is adiabatic for the enhanced-mass system, or the conclusion should be explicitly qualified as applying to the simulated non-adiabatic crossing only.
minor comments (5)
  1. [§4.1 (Figs. 6 and 7)] The brown lines in Figures 6 and 7 are fits with Aw=14 and Aw=7, but the paper does not describe how these coefficients were obtained or what their scatter is; a brief fitting procedure or an estimate of the uncertainty would be helpful.
  2. [§3, text near Eqs. (12)–(14)] The symbol tau_an in the eccentricity-jump estimate is not defined explicitly; it should be written as the product tau_a n to avoid confusion with a single parameter.
  3. [Abstract and §3] The abstract states that 'crossing of a spin-orbit resonance with Mars or a mean motion resonance with each other does not excite tumbling,' but the paper tests only two specific resonances (the 2:1 Mars-figure resonance and the 2:1 Phobos–Deimos resonance). The statement should be qualified to 'the tested resonances' to avoid over-generalization.
  4. [§5] The estimate of impact rates relies on scaling from asteroid 433 Eros using cross-sectional areas, but the authors do not discuss the uncertainty in the cratering scaling or the effect of Mars's gravity focusing on impactor flux; a brief caveat would be appropriate.
  5. [Introduction] The reference 'Efroimsky, private communication' is not a citable source; it should either be replaced with a published reference or removed from the formal text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claim is produced by the mass-spring simulations and checked against independent analytic tidal formulas; the main caveat is an acknowledged rheological extrapolation, not a circular reduction.

full rationale

The paper's derivation chain is self-contained rather than circular. The central result, that tumbling bodies can remain tumbling for thousands of orbits and substantially damp their orbital eccentricity, is a measured outcome of the mass-spring simulations: dissipation comes from damping in the springs, and the enhanced dissipation while tumbling emerges from the dynamics rather than being imposed as an input. The simulations are calibrated against independent analytic expressions: Equation 19 is the standard Kaula/Cassen tidally locked dissipation formula, Equations 21 and 22 are standard spin-down estimates, and Wisdom (1987) provides the prior prediction that tumbling enhances dissipation by orders of magnitude. The brown fit lines Ẏwob(e) = AwCene are summaries of the simulation data, not fitted parameters that are then renamed as predictions; the qualitative conclusions do not depend on the particular Aw values. The paper's self-citations (Quillen et al. 2016a,b, 2017, 2019a,b; Frouard et al. 2016) support the code and earlier validation, but the novel claim is demonstrated in this paper and compared with an independent external prediction by Wisdom (1987). No uniqueness theorem is imported from the authors' prior work, and no known result is merely renamed. The main legitimate concern is the extrapolation in Section 4.1 that the tumbling-to-synchronous dissipation ratio is similar in the high-frequency limit, and the Section 6 admission that 'sensitivity to rheology was neglected.' That is an acknowledged, untested assumption about real Phobos and Deimos, and it is a correctness/robustness risk, but it is not a circular reduction: the paper does not define tumbling dissipation in terms of the high-frequency ratio, nor does it fit that ratio from the data it later 'predicts.' Therefore the appropriate finding is no significant circularity.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claim depends on model parameters that are chosen for computational tractability (inflated radius, enhanced mass, fast drift rates) rather than fit to the real moons, and on the fitted tumbling-dissipation coefficients Aw. The key untested premise is that dissipation-rate ratios measured in the simulated low-frequency Kelvin-Voigt regime transfer to the high-frequency regime of the real moons. No new physical entities are proposed.

free parameters (6)
  • Central mass ratio M*/M = 10^4
    Adopted in tumbling simulations (Table 4) to reduce simulation time by effectively inflating the body radius to a/R=48, versus real Phobos a/R=830 and Deimos 3778. The authors assert this does not significantly affect spin-orbit dynamics, which is primarily sensitive to body axis ratios, but this is an unverified modeling assumption.
  • Spring damping parameter gamma_s = 0.1 (F series), 0.01 (S series)
    Sets the viscoelastic dissipation strength. The resulting dimensionless dissipation parameter Kd is about 10^-5 for the slow series, which is 2 to 4 orders of magnitude larger than the Kd values for Phobos (10^-7) and Deimos (10^-9) listed in Table 1. Tumbling persistence and eccentricity damping timescales depend on this choice.
  • Tumbling-state dissipation coefficient Aw = 14 (Deimos simulations), 7 (Phobos simulations)
    Fitted to the box-averaged simulation dissipation rates in Eq. 23 (brown lines in Figures 6 and 7). The 5-orders-of-magnitude enhancement over tidally locked dissipation used to estimate the Deimos damping time is read from this fitted line.
  • Phobos mass enhancement = 100x
    In the DeP 2:1 resonance simulation (Table 3), Phobos's mass is increased by a factor of 100 so the resonance can be crossed in feasible simulation time. Adiabaticity is checked with analytic scaling, but the direct test of spin excitation uses the enhanced mass.
  • Initial eccentricities = 0.2 and 0.05
    Chosen initial conditions for the four tumbling simulation series (Table 4). The claim that tumbling can reduce eccentricity to very low values depends on these starting eccentricities and the distribution of initial spin and obliquity draws.
  • Migration timescale tau_a = |tau_a|^-1 tg = 2e-6 (Phobos), 1e-7 (Deimos)
    Forced inward drift rates in resonance-crossing simulations (Table 3). The Phobos simulation crosses the Mars 2:1 resonance faster than adiabatic, so the eccentricity jump (0.008) is smaller than Yoder's 0.03; the no-tumbling conclusion is drawn from a non-adiabatic crossing. A slower crossing could produce a larger eccentricity jump and was not simulated.
assumptions (6)
  • domain assumption The mass-spring network with N ~ 40 nodes and random springs adequately represents the viscoelastic continuum response of Phobos and Deimos for spin-orbit and tumbling dynamics.
    Invoked throughout; the base parameters in Table 2 (N=40, ks=0.22, dI=0.5) are used in all simulations. Prior papers by the same group validate the code for tidal spin-down and wobble damping, but the specific coarse network used here is not validated against an independent benchmark for tumbling dissipation rates.
  • domain assumption The ratio of energy dissipation rates between tumbling and spin-synchronous states is similar in the low-frequency (simulated) and high-frequency (real Phobos and Deimos) tidal regimes.
    Stated in section 4.1: 'We expect that the ratio of dissipation rates, between spin-synchronous and tumbling, would be similar for the high frequency limit as for the low frequency limit.' This assumption is load-bearing for using the simulation dissipation ratios to estimate real Deimos eccentricity damping.
  • domain assumption Mars's rotating quadrupole gravity field (C20, C22, S22) and Phobos/Deimos mutual gravity are the only perturbations needed to test the two strongest resonance crossings; secular spin resonances, the Sun, and other planets are negligible.
    The PhM 2:1 and DeP 2:1 simulations model the Mars quadrupole terms (Eq. 10-11) and include the other moon only in the Deimos case. Section 6 notes secular phenomena were not simulated and argues that body-axis precession frequencies are fast enough to make strong secular spin resonance coupling unexpected.
  • ad hoc to paper The chaotic zone volume of the tumbling state scales with the square root of eccentricity, as estimated from the classic single-angle spin-orbit problem.
    The attribution in section 4.3 and the abstract that bodies drop into spin-synchronous at very low eccentricity due to insensitivity of chaotic zone volume to eccentricity is based on the 2D classic spin-orbit problem's separatrix width analysis, assuming it extends to the full 3D problem. The authors call these 'rough arguments' and note they could be tested in the full 3D problem.
  • domain assumption Impactor flux scaling from 433 Eros (Richardson et al. 2005) to Phobos and Deimos by cross-section ratio gives realistic impact rates for sub-catastrophic spin excitation.
    Section 5 estimates mean times of a few times 10^9 years for 0.7 km impactors on Phobos and 10^9 years for 0.2 km impactors on Deimos by multiplying Eros rates by the cross-section ratio. This neglects secondary impacts from Mars and assumes the impact rate was not much lower; the authors note the bombardment rate was higher in the past.
  • standard math Spin-orbit resonance overlap criteria (Wisdom et al. 1984; Wisdom 1987) and the dissipationless spin-orbit Hamiltonian (Eq. 24) describe the conservative dynamics underlying the tumbling chaotic zone.
    Used in section 4.3 to estimate chaotic zone volume scaling and to interpret simulation behavior. These are standard results in celestial mechanics.

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Pith. "Pith review of Excitation of Tumbling in Phobos and Deimos." pith.science (2026). https://pith.science/paper/A7KNDR5H

@misc{pith2026190805720,
  author       = {Pith},
  title        = {Pith review of: Excitation of Tumbling in Phobos and Deimos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7KNDR5H}},
  note         = {Machine review of arXiv:1908.05720}
}
read the original abstract

Mass-spring model simulations are used to investigate past spin states of a viscoelastic Phobos and Deimos. From an initially tidally locked state, we find crossing of a spin-orbit resonance with Mars or a mean motion resonance with each other does not excite tumbling in Phobos or Deimos. However, once tumbling our simulations show that these moons can remain so for an extended period and during this time their orbital eccentricity can be substantially reduced. We attribute the tendency for simulations of an initially tumbling viscoelastic body to drop into spin-synchronous state at very low eccentricity to the insensitivity of the tumbling chaotic zone volume to eccentricity. After a tumbling body enters the spin synchronous resonance, it can exhibit long lived non-principal axis rotation and this too can prolong the period of time with enhanced tidally generated energy dissipation. The low orbital eccentricities of Phobos and Deimos could in part be due to spin excitation by nearly catastrophic impacts rather than tidal evolution following orbital resonance excitation.

Figures

Figures reproduced from arXiv: 1908.05720 by the authors.

Figure 1
Figure 1. A snapshot of a simulation of Phobos showing the mass [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A simulation of a spinning Phobos drifting inward and [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. A simulation of a spinning Deimos crossing the 2:1 mean [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Spin and eccentricity evolution for series of Deimos simulations with parameters listed in Table 4. a) (left panels) Each panel shows [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Spin and eccentricity evolution for series of Phobos simulations with parameters listed in Table 4. Similar to Figure 4. a) (left [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Energy dissipation rate vs eccentricity for the simulations shown in Figure 4. a) (on the left) The De [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Energy dissipation rate vs eccentricity for the simulations shown in Figure 5. Similar to Figure 6 a) (on the left) The Ph [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 2
Figure 2. Figure 2: In Figure 8, the second panel from the bottom [PITH_FULL_IMAGE:figures/full_fig_p016_2.png]
Figure 8
Figure 8. Figure 8: A simulation that displayed long lived non-principal axis [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.