A crater is a negative mass anomaly
The paper models a central body as a sphere with constant mass distribution plus an equatorial crater dug into its surface, orbited by a test particle in the body's equatorial plane. The crater is treated as a localized deficit of mass, so the mass-ratio parameter is strictly negative in the simulation codes; for clarity the authors report results using its absolute magnitude, μ = |m/M|, where m is the mass removed to form the crater and M is the total mass of the system. A larger μ therefore means a deeper, more massive crater.
This sign convention matters because it separates the crater case from the mass-anomaly case studied earlier in the same series. A positive μ corresponds to an added mass anomaly; the crater configuration instead removes mass, displacing the centre of mass away from the geometric centre of the spherical portion. With the adopted normalization (G = 1, M = 1, Rc = 1), the two component centres sit at x1 = −μ and x2 = 1 − μ, and the potential in Equation 2 omits the indirect term proportional to λ² because the frame is already barycentric.
Two parameters control the dynamics. The rotation rate is λ = ω/ωk, the ratio of the body's angular velocity to the critical Keplerian frequency at the surface; the crater mass ratio is μ. The nominal case uses Máni's physical parameters, λ = 0.52 and μ = 0.002, with a crater depth exceeding 10% of the body's radius. Trajectories are integrated with an explicit Runge–Kutta RK45 solver at relative and absolute tolerances of 10⁻⁸.
Three techniques map stability, chaos and resonance width
Poincaré surfaces of section (PSS) record where trajectories cross a chosen plane in phase space, revealing islands of quasi-periodic motion around central periodic orbits and the chaotic separators between them. Survival maps instead distribute test particles on a semi-major axis versus eccentricity grid and record whether each particle survives the integration interval, collides with the body (r ≤ Rc or pericentre inside the physical radius) or is ejected (r ≥ 100 Rc or e ≥ 1). The finite-time Lyapunov exponent (FTLE) measures the exponential growth rate γ(t) = (1/t) ln[d(t)/d0] of an initial phase-space separation, with the Lyapunov time TL = 1/γ giving the timescale over which that separation grows by a factor of e.
The paper is explicit that these metrics answer different questions and should not be substituted for one another. Survival time delimits the boundaries of instability and gives a preliminary identification of where resonances might persist at high eccentricity, but it is insufficient to evaluate long-term dynamical resilience: chaotic trajectories with slow diffusion can survive the entire simulation without macroscopic instability, masking the underlying chaos. The FTLE exposes those internal nuances, resolving stability gradients inside apparently stable zones and highlighting chaos along the resonance separators produced by resonance overlap.
The authors also note a practical asymmetry. Precisely delimiting resonance boundaries and widths from PSS requires exhaustive manual inspection of multiple graphs generated at different Jacobi constant values, whereas the FTLE-based technique characterizes the morphology and width variations automatically. For regular quasi-periodic motion the Lyapunov estimator decreases roughly as ln(t)/t, so a finite-time calculation can return a small non-zero γ even for a regular trajectory; TL should then be read comparatively rather than as an absolute long-term stability timescale.
The nominal case recovers four spin-orbit resonances
A spin-orbit resonance (SOR) occurs when the orbital frequency of a particle and the rotation frequency of the central body are commensurable, following m n = (m − j) ωp + j ϖ̇, where j sets the resonance order. In the nominal case, the 1:1, 2:1, 3:1 and 4:1 SORs are consistently identified by all three techniques. In the PSS at μ = 2 × 10⁻³ and λ = 0.52, the innermost pink island corresponds to the 1:1 SOR, the blue structure to the 2:1 SOR, and the green three-island structure to the third-order 4:1 SOR; black structures correspond to fourth-order or higher resonances.
The comparison between numerically obtained resonance positions and analytical models gives a maximum relative deviation of 2.82% for the 1:1 SOR, falling progressively to 1.34% for the 4:1 SOR. The authors interpret this trend as evidence that the analytical correction, based on an expansion in μ, is more affected by higher-order terms closer to the primary.
The 3:1 SOR is the most prominent structure and shows a bifurcated shape at high eccentricities. Within the interval CJ ∈ (1.9300, 1.9328) Rc²ωk², a period-doubling bifurcation splits the central fixed point into a pair of periodic and quasi-periodic orbits, so the resonance effectively transitions into a 6:2 state. The resulting periodic orbits in the rotating frame are perfectly symmetric about y = 0, unlike the asymmetric trajectories found in the mass-anomaly model of the same series, and the particle's velocity drops significantly near the turning points and loops.
Faster rotation and a deeper crater widen resonances
Increasing either the rotation rate λ or the crater mass ratio μ enlarges resonance widths and promotes overlap, making the 2:1 and 4:1 SORs progressively dominant. In the stability maps at fixed λ = 0.523, the μ = 0.01 case shows the 2:1 and 4:1 structures visibly expanded relative to μ = 0.0001 and μ = 0.005, consistent with the established proportionality between resonance width and the magnitude of the gravitational perturbation set by μ and λ.
The 3:1 SOR behaves differently under this growth. Despite the overall expansion at μ = 0.01, it does not undergo a significant structural change and keeps a comparatively less regular core than the 2:1 and 4:1 resonances. Between roughly 1.0 and 1.75 Rc, only a few dispersed particles survive to the end of the simulation, spanning a broad range of behaviour that includes regions of faster chaotic diffusion. Fine structures associated with the 5:3, 7:3, 11:3, 25:6 and 13:3 SORs remain discernible, though less pronounced than at lower μ.
Rotation rate changes the eccentricity ceiling for survival. As λ approaches the critical velocity (λ → 1), the maximum eccentricity of surviving particles decreases significantly, restricting stability to lower limits. A tenfold reduction to λ = 0.052 shifts the resonances further apart, minimizing overlap and allowing survival at considerably higher eccentricities. The 1:1 SOR is highly sensitive to stronger perturbations and disappears in the most extreme cases; even in the nominal survival map it contains comparatively more regular particles than its surroundings but is not clearly characterized as a resonant structure, consistent with its proximity to the central body.
The crater preserves stable zones a mass anomaly would clear
The paper's central comparison is between the crater configuration and mass-anomaly models motivated by ringed trans-Neptunian objects. Those models suggest that torques from a non-spherical body can sweep the surrounding disc and render the 2:1 SOR unstable on short timescales, clearing the region interior to it. In the crater simulations, stable particles exist in that inner region over 10⁴ rotations, indicating that this specific mass distribution allows stable zones to persist where a mass anomaly would typically deplete them.
The authors frame this as a broader statement about geometry rather than about craters alone. Because the model represents the crater as a point of removed mass that shifts the centre of mass relative to the spin axis, it more generally explores the dynamical consequences of a non-coincident spin axis and centre of mass, whether that offset arises from topographical depressions or from internal density variations.
The stability gradient identified by the FTLE is interpreted as going beyond the numerical value of the Lyapunov exponent: a region characterized as more stable implies greater dynamical robustness against external perturbations that the model does not include, such as non-modelled spherical harmonics (J2 and C22) or non-conservative mechanisms including solar radiation pressure and viscous dissipation. On that reading, the low-chaos zones are the most likely environments where material could be preserved over long timescales despite unmodelled physical complexity.
Grid maps probe one set of initial orbital elements
The survival and FTLE grids are constructed under a single set of initial orbital elements: inclination I = 0°, ϖ0 = 180°, f0 = 180° and Ω0 = 0°. For the PSS to reflect that same set, all considered resonance structures were evaluated precisely at ẋ0 = 0. Each individual stability grid is therefore restricted to probing the evolution of one unique set of initial orbital elements, even though the PSS itself can characterize resonance structures across the whole phase space for each value of CJ.
This restriction shapes what the grid maps show. The bifurcation pattern appears at the top of the structures at the highest eccentricities, a split-path geometry that directly manifests the period-doubling bifurcation (3:1 → 6:2) taking place at this apocentre configuration. The visible architecture of even-order resonances at elevated eccentricities therefore depends heavily on the initial orbital phase and orbital orientation.
The authors also flag the limits of the survival metric itself, noting that in applications to Centaur dynamics such as Chiron, and again here, survival time alone cannot fully evaluate long-term dynamical resilience because slowly diffusing chaotic trajectories can persist for the whole simulation. The FTLE is presented as the technique that resolves this ambiguity, at the cost of requiring a finite integration interval whose length affects the estimate.
Máni emerges as a candidate for rings or satellites
Applying the PSS results to the reference body, the authors conclude that the 1:1, 2:1, 3:1 and 4:1 SORs are the most prominent resonant structures around Máni. Given recent discoveries of ring systems near the 3:1 resonance in other trans-Neptunian objects and Centaurs, they identify Máni as a strong candidate for hosting rings or satellites within these stable regions, particularly around the 2:1, 3:1 and 4:1 SORs.
The priority this paper establishes is a comparison between two ways of producing an offset between the spin axis and the centre of mass. A mass anomaly can clear the region interior to the 2:1 SOR; the crater configuration preserves stable particles there over 10⁴ rotations. Which scenario applies therefore depends on whether the observed non-sphericity is a mass deficit or a mass excess, and the paper's own variables — μ, λ and the resulting resonance widths — are the quantities that decide whether the 2:1 and 4:1 resonances dominate or whether the 1:1 resonance survives at all.
The reference parameters themselves carry observational uncertainty that bounds how far the nominal result transfers. Máni's rotation period is reported as either 7.44 h or 10.44 h, its global density as either 0.72 g cm⁻³ or 0.36 g cm⁻³, and its estimated polar radius as a minimum of 234 km and a maximum of 387 km, with a topography including a 322 km feature and a 45 km deep crater. The nominal λ = 0.52 and μ = 0.002 are one point inside that range, and the paper's own parameter sweeps show that moving λ or μ changes which resonances dominate.
Sources used in this answer
Dynamics around non-spherical symmetric bodies - III. The case of a spherical body with a crater
Using Máni as a reference body with a crater deeper than 10% of its radius, the 1:1, 2:1, 3:1 and 4:1 spin-orbit resonances are recovered by Poincaré surfaces of section, survival maps and finite-time Lyapunov exponents; the 3:1 resonance undergoes a period-doubling bifurcation into a 6:2 state within CJ ∈ (1.9300, 1.9328) Rc²ωk², and unlike mass-anomaly models the crater configuration preserves stable particles interior to the 2:1 SOR over 10⁴ rotations.