---
title: 'Sesquinary Catastrophe: Runaway Moon Erosion'
url: https://www.emergentmind.com/topics/sesquinary-catastrophe
type: topic
---

# Sesquinary Catastrophe: Runaway Moon Erosion

Sesquinary catastrophe is a runaway collisional erosion process in which ejecta launched from a small body escape that body, remain bound to the central gravitating primary, and later re-impact the source at velocities high enough to produce net mass loss rather than reaccretion. In planetary science, the mechanism was identified for small, close-in moons on dynamically excited orbits, where differential precession between the source moon and its ejecta drives high-speed returns; the expected end-state is destruction into debris followed by re-accretion onto a dynamically colder orbit [2309.02378]. Subsequent work has used the mechanism to reconcile Deimos’s plausibly excited past with its presently cool orbit and has extended the framework to minor planets orbiting white dwarfs, where analogous re-impact cascades can operate exterior to the Roche radius [2512.15976] [2507.05090].

## 1. Terminology and conceptual scope

In the relevant literature, “sesquinary” describes impact ejecta launched from a satellite that escape that satellite’s gravity, go onto their own planetocentric orbits, and later re-impact the same satellite. Ordinary cratering and secondary cratering are therefore distinct from sesquinary phenomena: primary craters are made by impactors unrelated to the target, whereas secondary craters are made by sub-orbital ejecta that fall back without escaping the target’s gravity [2309.02378].

A sesquinary catastrophe is the destabilized limit of this process. Instead of gentle reaccretion, returning ejecta collide at velocities sufficiently large that each impact excavates and liberates more mass than it reaccretes, producing a self-amplifying erosion sequence. The 2023 formulation describes a runaway erosion or disruption of a small, close-in moon on a dynamically excited orbit; the 2025 Deimos study reframes the same process as a runaway collisional cascade that converts the moon into a Roche-exterior debris disk and then into a dynamically cool, porous body through re-accretion [2309.02378] [2512.15976].

The mechanism is not restricted to Mars. The same logic has been applied to Saturnian resonant moonlets, Neptune’s Naiad, Jupiter’s Thebe, and minor planets orbiting white dwarfs. This suggests that sesquinary catastrophe is best understood as a generic dynamical-collisional instability of low-escape-speed bodies whose returning ejecta can be kinematically amplified by orbital excitation and secular misalignment [2507.05090].

## 2. Dynamical mechanism of runaway re-impact

The canonical sequence begins with an impact or mutual collision that produces ejecta. A fraction of the ejecta escape the source moon with speed at least the moon’s escape speed,
$$
v_{\rm esc,m} = \sqrt{\frac{2 G M_m}{R_m}},
$$
but remain bound to the central planet because $v_{\rm esc,m} \ll v_k$ for close-in moons, where
$$
v_k = \sqrt{\frac{G M_p}{a_m}}.
$$
The escaping fragments enter planet-bound orbits whose semimajor axis, eccentricity, and inclination differ only slightly from those of the source, so the fragments initially “inherit” the moon’s dynamical state [2309.02378].

The crucial amplification arises from differential precession. In the Solar System formulation, the longitude of ascending node and argument of periapse precess at rates controlled by the primary’s oblateness:
$$
n = \sqrt{\frac{G M_p}{a^3}},
$$
$$
\frac{d\Omega}{dt} = -\frac{3}{2} J_2 n \left(\frac{R_p}{a}\right)^2 \frac{\cos i}{(1 - e^2)^2},
$$
$$
\frac{d\omega}{dt} = \frac{3}{4} J_2 n \left(\frac{R_p}{a}\right)^2 \frac{(5\cos^2 i - 1)}{(1 - e^2)^2}.
$$
Even tiny differences in $a$, $e$, and $i$ between the source and the ejecta therefore cause their nodes and periapses to shear apart. Encounters then occur after apsidal and/or nodal misalignment, so the re-impact velocity is dominated not by the launch speed but by the orbital excitation of the source orbit [2309.02378].

The characteristic encounter speed scales as
$$
v_{\rm rel} \sim v_k \sqrt{e^2 + \sin^2 i},
$$
and the impact speed is
$$
v_{\rm imp} \approx \sqrt{v_{\rm rel}^2 + v_{\rm esc,m}^2}.
$$
For the smallest moons, $v_{\rm rel} \gg v_{\rm esc,m}$, so gravitational focusing is a minor correction. Once impacts are erosive rather than accretive, the moon loses mass, more ejecta are created, and the process accelerates into a runaway [2309.02378].

The Deimos-specific formulation adds an explicit positive-feedback channel through the mass dependence of the return time. Using
$$
\tau = \frac{P}{\Delta}\,\frac{a^{2}\,\delta e\,\delta i}{R^{2}}, \qquad \Delta \equiv \frac{v_{\rm escape}}{v_{\rm orbital}},
$$
the authors simplify the timescale to
$$
\tau = \frac{K}{M}, \qquad K = \sqrt{\frac{(2\pi)^{3}\,\rho}{3G}\,a^{3}\,\delta e\,\delta i}.
$$
As erosion lowers $M$, $\tau$ shortens and impact cadence accelerates, reinforcing the runaway [2512.15976].

## 3. Quantitative diagnostics and instability thresholds

The primary susceptibility metric in the 2023 and white-dwarf formulations is
$$
q_v \equiv \frac{v_{\rm imp}}{v_{\rm esc,m}} \approx \frac{v_k}{v_{\rm esc,m}} \sqrt{e^2 + \sin^2 i}.
$$
The paper adopts $q_{\rm crit} \sim 10$ for rubble piles and $q_{\rm crit} \gtrsim 20$ for cohesive or strong bodies. A necessary condition for susceptibility is therefore
$$
\frac{v_k}{v_{\rm esc,m}} \sqrt{e^2 + \sin^2 i} \gtrsim q_{\rm crit}.
$$
All else equal, small, low-density moons close to the planet are more vulnerable because $v_{\rm esc,m}$ is small while $v_k$ is large [2309.02378].

The Deimos study uses a related but differently named parameter,
$$
q = \sqrt{e^{2} + \sin^{2}i}\,\frac{v_{\rm orbital}}{v_{\rm escape}},
$$
with typical sesquinary re-impact speeds scaling as
$$
v_{\rm impact} \sim q\,v_{\rm escape}.
$$
Using N-body simulations with collisional fragmentation and a semi-analytical model calibrated to those simulations, the authors find a formal threshold near $q \approx 5.7$ for strengthless targets, but adopt $q \gtrsim 8$ as the conservative disruption threshold because of uncertainties in material strength, radiation forces on small debris, and numerical artifacts [2512.15976].

The Deimos work also gives an explicit angle-averaged cratering mass-loss law in terms of $\eta \equiv v_{\rm impact}/v_{\rm escape}$:
$$
\delta M(\eta) = \begin{cases}
0.02\,\eta^{2.2} + 0.071\,\eta^{0.88} - 0.85, & 0 \le \eta < 12,\\[6pt]
0.076\,\eta^{1.65} + 0.071\,\eta^{0.88} - 0.85, & 12 \le \eta < 16.79,\\[6pt]
0.076\,\eta^{1.65}, & \eta \ge 16.79.
\end{cases}
$$
From the N-body runs, the normalized distribution of $\eta/q$ is well fit by a log-normal probability density
$$
f(\eta;\,q) = \frac{1}{s\,\eta\,\sqrt{2\pi}} \exp\!\left[-\,\frac{\ln^{2}\!\left(\eta/(q\sigma_{\rm fit})\right)}{2s^{2}}\right],
$$
with shape $s \approx 1.002$ and scale $\sigma_{\rm fit} \approx 0.329$ [2512.15976].

The re-impact timescale in the 2023 study is
$$
\tau = \left(\frac{P}{\Delta}\right)\,\delta e\,\delta i \left(\frac{a}{R}\right)^2,
\qquad
\Delta \equiv \frac{v_{\rm esc}}{v_{\rm orb}},
$$
with
$$
\delta e = \sqrt{\Delta^2 + e^2}, \qquad \delta i = \sqrt{\Delta^2 + \sin^2 i}.
$$
Short $\tau$ values accelerate runaway onset; long $\tau$ can permit survival despite large nominal $q_v$ values [2309.02378].

## 4. Deimos and the Martian-moon problem

The Deimos application is motivated by a specific tension. In impact-generated circum-Martian disk scenarios, multiple inner moons form interior to Mars’s synchronous radius and subsequently interact via disk torques and tides. As inner moons migrate, they can encounter mean-motion resonances with Deimos, raising Deimos’s eccentricity and inclination well above modern values. Yet Deimos today is dynamically cool, with $e \approx 2.7\times10^{-4}$ and $i \approx 1.8^\circ$ relative to the local Laplace plane [2512.15976].

The sesquinary catastrophe is proposed as the mechanism that resolves this tension. If Deimos or its precursor were driven to sufficiently large $q$, it would undergo a runaway collisional cascade, break apart into a Roche-exterior debris disk, and later re-accrete into a dynamically cool body. Using N-body simulations with collisional fragmentation, the paper argues that breakup occurs for $q \gtrsim 8$ on timescales of $\sim 10^{3-4}$ years. In accelerated piecewise sequences at $q \approx 22$ with $e = 0.05$ and $i = 5^\circ$, Deimos rapidly loses mass and passes a tipping point near $95$–$99\%$ of its initial mass, beyond which the cascade rapidly completes [2512.15976].

The argument depends on Roche geometry. The standard fluid Roche limit is
$$
a_{\rm Roche} \simeq 2.44\,R_{p}\,\left(\frac{\rho_{p}}{\rho_{s}}\right)^{1/3}.
$$
For Mars and Deimos-like material, $a_{\rm Roche}$ is a few $R_M$ (approximately $3$–$4\,R_M$), well interior to Deimos’s orbit at $a \approx 6.91\,R_M$. Deimos fragments therefore form a Roche-exterior, planetocentric debris disk that is dynamically stable and can collisionally damp and re-accrete [2512.15976].

The paper further argues that tides cannot erase excitation quickly enough. The order-of-magnitude eccentricity damping rate due to tides raised on the satellite is
$$
\frac{de}{dt} \approx -\,\frac{21}{2}\,\frac{k_{2}}{Q}\,\frac{M_{p}}{m}\left(\frac{R}{a}\right)^{5} n\,e,
\qquad
n=\sqrt{\frac{G M_{p}}{a^{3}}}.
$$
For a small, porous Deimos, even optimistic choices of $k_2/Q \sim 10^{-3}$–$10^{-2}$ yield eccentricity damping times far exceeding $10^7$–$10^9$ years for $e \lesssim 10^{-2}$, which is orders of magnitude slower than the few $10^3$ years destruction times found for the sesquinary catastrophe [2512.15976].

The inferred end-state is a porous sand-pile moon assembled from fine debris. The paper explicitly connects this expectation to Deimos’s smooth surface and to recent ephemeris and moment-of-inertia fits consistent with near-uniform density bodies. This suggests that Deimos’s low eccentricity and inclination need not forbid strong past excitation; instead, strong excitation may have been self-limiting because it triggered destruction and re-accretion [2512.15976].

## 5. Other Solar System manifestations

The 2023 survey concludes that the large majority of small close-in moons in the Solar System have orbits that are immune to sesquinary catastrophe, but several notable exceptions illuminate the controlling physics. For Saturn’s resonant moonlets, large nominal $q_v$ values do not always imply destruction because resonances can keep ejecta co-aligned and re-impacts slow. Methone, for example, has $q_v \approx 46.6$ and $\tau \approx 5.4\times10^5$ yr, yet its $14{:}15$ corotation resonance with Mimas allows low-speed reaccretion; Anthe behaves similarly in $10{:}11$ corotation, while Pallene, with $q_v \approx 85.4$ and $\tau \approx 6.4\times10^6$ yr, may survive because of its long timescale and possible departure from resonance [2309.02378].

By contrast, Naiad and Thebe illustrate environments where resonance protection is weak or absent. Naiad has $q_v \approx 40.3$ and $\tau \approx 180$ yr; because its $73{:}69$ resonance with Thalassa affects inclination but does not confine debris, its survival implies substantial internal strength, consistent with independent Roche-limit arguments. Thebe has $q_v \approx 18.0$ and $\tau \approx 8.9\times10^3$ yr; its persistence and faint gossamer ring suggest ongoing but modest erosion, which the paper interprets as evidence that a higher effective threshold, around $q_{\rm crit} \approx 20$, may apply for Jupiter’s inner moons [2309.02378].

Mars’s moons also appear in the 2023 stability analysis. Present values from the paper’s table give Phobos $q_v \approx 4.6$ and $\tau \approx 38$ yr, and Deimos $q_v \approx 7.6$ and $\tau \approx 1600$ yr, both below the rubble-pile threshold. The derived constraints are
$$
\sqrt{e^2+\sin^2 i} \lesssim 0.053
$$
for Phobos and
$$
\sqrt{e^2+\sin^2 i} \lesssim 0.041
$$
for Deimos if long-lived rubble-pile stability is required. Sustained larger values over Myr would trigger sesquinary cascades, disfavoring prolonged high-excitation past orbits unless strength or other protection intervened [2309.02378].

## 6. Extension to minor planets orbiting white dwarfs

The white-dwarf generalization preserves the same core logic but shifts the dynamical scale dramatically. For a white dwarf of mass $M_{\rm WD}$,
$$
v_{\rm orb}(a) = \sqrt{\frac{G M_{\rm WD}}{a}},
\qquad
P(a) = 2\pi \sqrt{\frac{a^3}{G M_{\rm WD}}}.
$$
Near the white-dwarf Roche radius, orbital speeds are hundreds of km s$^{-1}$, so even modest orbital excitation yields $q_v \gg 10$ for kilometer-scale rubble piles. The key “danger zone” is $a \approx 1$–$4$ rubble-pile Roche radii, corresponding to periods of approximately $5$–$25$ hours for a fiducial $0.6\,M_\odot$ white dwarf [2507.05090].

The rubble-pile Roche radius used in the paper is
$$
r_{\rm Roche} = k \left(\frac{M_{\rm WD}}{\rho_{\rm mp}}\right)^{1/3},
$$
with $k = 0.78$ for a non-spinning body and $k = 0.89$ for synchronous spin. Inside roughly $1\,r_{R,{\rm RP}}$, classical tidal disruption dominates; outside roughly $4\,r_{R,{\rm RP}}$, orbital speed and differential precession slow enough that sesquinary timescales lengthen. Between these limits, returning ejecta impacts are fast enough and frequent enough to be erosive, implying destruction on $\sim 10^2$–$10^5$ yr timescales [2507.05090].

Apsidal and nodal misalignment are again essential. In white-dwarf systems, stellar oblateness and magnetic precession are usually too slow, whereas general relativity sets a robust floor on apsidal precession:
$$
\dot{\omega}_{\rm GR} =
\frac{3 G M_{\rm WD}}{a c^{2}(1-e^{2})}
\sqrt{\frac{G M_{\rm WD}}{a^{3}}}.
$$
Nearby massive planets can further accelerate both apsidal and nodal precession on $\sim 10^2$–$10^3$ yr timescales. The paper therefore argues that misalignment is effectively inevitable in the relevant orbital regime [2507.05090].

The destruction timescale is estimated as $t_{\rm cat} \approx$ a few $\times \tau$, where
$$
\tau \approx T \left(\frac{v_{\rm orb}}{v_{\rm esc}}\right)\left(\frac{a}{R_{\rm mp}}\right)^2
\sqrt{\left[\left(\frac{v_{\rm esc}}{v_{\rm orb}}\right)^2 + e^2\right]
\left[\left(\frac{v_{\rm esc}}{v_{\rm orb}}\right)^2 + \sin^2 i\right]}.
$$
The paper also recasts $\tau$ in terms of $q_v$ and emphasizes the steep scaling $\tau \propto T^3$. The astrophysical consequence is that debris discs around white dwarfs may be in a state of semi-continuous replenishment, because parent bodies exterior to the Roche radius can still be collisionally destroyed well inside typical disc-lifetime estimates [2507.05090].

## 7. Nonstandard catastrophe-theoretic extension and principal uncertainties

A separate 2025 paper, "Apocalypsis and Apocalyptic Events: The Morphogenetic Ontology of Synchronized Catastrophes" [2510.25431], does not explicitly define “sesquinary catastrophe.” It instead formalizes local catastrophes, synchronized apocalyptic events, and Apocalypsis as a topological meta-singularity generated by the coherent alignment of local singularities into a global structure of collapse. Consistent with that paper’s ontology, a sesquinary catastrophe can be rigorously defined as a partial, intermediate-order synchronized collapse: a multi-subsystem singular event in which at least two but not all subsystems synchronize and cross their catastrophe sets at the same control time, producing a connected cascade within a proper subset of the catastrophe graph [2510.25431].

In that proposed usage, the relevant objects are the coupled potential
$$
V_\epsilon(x;\alpha) = \sum_i V_i(x_i;\alpha_i) + \epsilon W(x;\alpha),
$$
the coupled critical set
$$
\mathcal{C}_\epsilon = \{(x,\alpha): \nabla_x V_\epsilon(x;\alpha)=0\},
$$
and the coupled discriminant
$$
\Sigma_\epsilon = \{(x,\alpha)\in\mathcal{C}_\epsilon : \det H_\epsilon(x,\alpha)=0\}.
$$
A sesquinary catastrophe occurs when the control trajectory hits a multi-singular stratum of codimension $m \ge 2$ in the coupled discriminant, while the triggered connected component $C$ of the catastrophe graph satisfies
$$
2 \le |C| < k
$$
or lies below a chosen global threshold $L_c$. This is explicitly presented as a proposed definition rather than established terminology [2510.25431].

The mainstream scientific meaning of the term remains the planetary-science and celestial-mechanics usage. The principal uncertainties there are material strength, fragmentation physics, and non-gravitational forces. The Deimos study notes that cohesive strength likely shifts the practical threshold upward from the formal $q \approx 5.7$ to $q \gtrsim 8$, that radiation and Lorentz forces can modulate the available impactor flux, and that a lower size cutoff of roughly $100$ m limits direct modeling of mm–cm dust. The white-dwarf application likewise assumes rubble-pile, low-cohesion bodies and treats ejecta as ballistic test particles; radiation pressure, sublimation forces, and magnetic drag are neglected in the re-impact calculation, although the authors argue that the very large orbital speeds make this conservative for impact timing and velocity [2512.15976] [2507.05090].

Across these domains, the central conclusion is stable: sesquinary catastrophe functions as a self-limiting mechanism for dynamical excitation. When orbital excitation raises returning-ejecta impacts above the erosive threshold, the source body is driven toward destruction, debris generation, dynamical cooling, and eventual re-accretion on a less excited orbit [2309.02378].

Source: https://www.emergentmind.com/topics/sesquinary-catastrophe