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arxiv: 2601.11692 · v2 · submitted 2026-01-16 · 🌌 astro-ph.EP

Recognition: 2 theorem links

· Lean Theorem

Orbital Stability of Closely-Spaced Four-planet Systems

Bennet Outland , Gretchen Noble , Andrew W. Smith , Jack J. Lissauer
Authors on Pith no claims yet

Pith reviewed 2026-05-16 13:27 UTC · model grok-4.3

classification 🌌 astro-ph.EP
keywords orbital stabilityplanetary dynamicsmean motion resonancesN-body simulationsexoplanet systemsHill sphere spacingclosely packed planets
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The pith

Four-planet systems show lifetimes that increase exponentially with spacing yet lack the rare ultra-stable configurations found in three-planet systems.

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

The paper runs long N-body simulations of four Earth-mass planets on initially circular coplanar orbits around a solar-mass star, with semimajor axes spaced equally in mutual Hill radii. It tracks system survival times up to 10^10 innermost orbits, halting at close encounters or ejections, and compares outcomes across different initial longitudes and spacings. Lifetimes follow an exponential rise with increasing separation, matching trends from earlier three- and five-planet studies, but four-planet cases sit closer to five-planet results and show no exceptional long-lived spacings. First-, second-, and third-order mean-motion resonances shorten survival, with third-order effects visible even in otherwise stable setups. The location of lifetime peaks shifts slightly depending on whether planets start near conjunction or widely separated in longitude.

Core claim

Integrations of four-planet systems with Earth masses and equal mutual Hill radius spacings demonstrate that survival times grow exponentially with separation, occupy an intermediate regime between three- and five-planet systems but resemble the latter, and exhibit no spacings that produce survival times orders of magnitude longer than neighbors. First- and second-order mean-motion resonances reduce lifetimes, as do third-order resonances between neighboring planets. Local maxima in lifetime versus spacing occur at modestly smaller separations when planets begin at conjunction, owing to asymmetric perturbations that spread semimajor axes as longitudes diverge.

What carries the argument

N-body integrations that advance four planets until a close approach below 0.01 AU or ejection occurs, with stability measured against initial mutual Hill radius spacing and proximity to mean-motion resonances.

If this is right

  • System lifetimes increase exponentially as initial spacing in mutual Hill radii grows.
  • Four-planet lifetimes lie between three- and five-planet lifetimes and track the five-planet pattern more closely.
  • No initial spacings produce survival times orders of magnitude longer than nearby spacings.
  • Proximity to first-, second-, and third-order mean-motion resonances shortens system lifetimes.
  • The spacing that maximizes lifetime shifts to smaller values when planets start near conjunction rather than widely separated longitudes.

Where Pith is reading between the lines

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

  • The absence of outlier-stable configurations in four-planet runs suggests that crossing from three to four planets removes the special phase-protection effects that occasionally appear in three-body cases.
  • Third-order resonances may need explicit inclusion in analytic stability maps for closely packed systems, since they measurably shorten lifetimes even when first- and second-order resonances are absent.
  • The longitude-dependent shift in lifetime peaks implies that formation scenarios producing aligned conjunctions could favor slightly tighter stable packings than random-longitude formation.
  • Extending the same spacing survey to systems with modest initial eccentricities would test whether the exponential lifetime trend survives when orbits are no longer perfectly circular.

Load-bearing premise

The assumption that results from initially circular coplanar Earth-mass orbits generalize to systems that may begin with eccentricities, inclinations, different masses, or additional forces such as tides.

What would settle it

Observing a four-planet system at a spacing where survival exceeds the fitted exponential trend by several orders of magnitude, or a system that remains stable while sitting near a third-order resonance that the simulations flag as destabilizing, would contradict the reported trends.

Figures

Figures reproduced from arXiv: 2601.11692 by Andrew W. Smith, Bennet Outland, Gretchen Noble, Jack J. Lissauer.

Figure 1
Figure 1. Figure 1: Initial angular locations for four longitude prescriptions used at [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The top panel shows simulated lifetimes of four-planet systems with SL09 (grey) [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Overlain plots of the lifetimes of three- (red dots), four- (gray dots), and five [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Lifetimes of three-, four-, and five-planet systems with SL09 initial longitudes [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Lifetimes of three-, four-, and five-planet systems with SL09 longitudes from [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Rolling medians of lifetimes of four-planet systems are plotted against the ini [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The top panel plots the difference between average and initial period ratios of [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
read the original abstract

We investigate the orbital dynamics of four-planet systems consisting of Earth-mass planets on initially-circular, coplanar orbits around a star of one solar mass. In our simulations, the innermost planet's semimajor axis is set at 1 AU, with subsequent semimajor axes spaced equally in terms of planets' mutual Hill radii. Several sets of initial planetary longitudes are investigated, with integrations continuing for up to $10^{10}$ orbits of the innermost planet, stopping if a pair of planets pass within 0.01 AU of each other or if a planet is ejected from the system. We find that the simulated lifetimes of four-planet systems follow the general trend of increasing exponentially with planetary spacing, as seen by previous studies of closely-spaced planets. Four-planet system lifetimes are intermediate between those of three- and five-planet systems and more similar to the latter. Moreover, as with five-planet systems, but in marked contrast to the three-planet case, no initial semimajor axes spacings are found to yield systems that survive several orders of magnitude longer than other similar spacings. First- and second-order mean-motion resonances (MMRs) between planets correlate with reductions in system lifetimes. Additionally, we find that third-order MMRs between planets on neighboring orbits also have a substantial, though smaller, destabilizing effect on systems very near resonance that otherwise would be very long-lived. Local extrema of system lifetimes as a function of planetary spacing occur at slightly smaller initial orbital separation for systems with planets initially at conjunction relative to those in which the planets begin on widely-separated longitudes. This shift is produced by the asymmetric mutual planetary perturbations as the planets separate in longitude from the initial aligned configuration that cause orbits to spread out in semimajor axis.

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

2 major / 3 minor

Summary. The manuscript reports N-body simulations of four Earth-mass planets on initially circular, coplanar orbits around a solar-mass star, with semimajor axes spaced equally in mutual Hill radii and the innermost planet at 1 AU. Integrations extend to 10^10 innermost orbits or until a close encounter (0.01 AU) or ejection occurs. Across multiple initial longitude sets, the results show system lifetimes increasing exponentially with spacing, lying intermediate between (and closer to) those of five-planet than three-planet systems, with no outlier long-lived spacings, and with reduced lifetimes near first-, second-, and third-order mean-motion resonances.

Significance. If the numerical results hold, the work supplies a controlled extension of prior three- and five-planet stability surveys, quantifying the four-planet case with long integration times and direct resonance correlations. The exponential trend, absence of anomalously stable spacings, and third-order MMR effects provide concrete benchmarks for dynamical packing limits and for interpreting compact exoplanet systems.

major comments (2)
  1. [§2] §2 (Methods): the integrator, timestep, and exact stopping-condition implementation are not specified, yet these choices directly affect the reliability of lifetimes reaching 10^10 orbits and the reported median values.
  2. [§3] §3 (Results): the number of initial-longitude realizations per spacing and the precise definition of 'median lifetime' (e.g., whether it includes censored runs) are not stated, weakening the statistical support for the exponential fit and the claim that no spacings yield outliers several orders of magnitude longer-lived.
minor comments (3)
  1. [Abstract] Abstract and §4: the number and specific values of the 'several sets of initial planetary longitudes' should be stated explicitly so readers can assess coverage of the longitude phase space.
  2. [Figure 1] Figure 1 or equivalent: a direct overlay of the four-planet lifetime curve with the three- and five-planet curves from prior work would make the 'intermediate but closer to five-planet' statement immediately visible.
  3. [§4.2] §4.2: the statement that third-order MMRs have a 'substantial, though smaller' destabilizing effect would benefit from a quantitative measure (e.g., median lifetime ratio inside vs. outside resonance) rather than qualitative description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation of minor revision. We address each major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [§2] §2 (Methods): the integrator, timestep, and exact stopping-condition implementation are not specified, yet these choices directly affect the reliability of lifetimes reaching 10^10 orbits and the reported median values.

    Authors: We agree that these details are necessary for reproducibility and for evaluating the robustness of the long integration times. In the revised manuscript we will explicitly state that all integrations were performed with the REBOUND package using the WHFast integrator at a fixed timestep of 0.05 yr (approximately 1/20 of the innermost orbital period), with close-encounter detection handled by switching to IAS15 when planets approach within 0.1 AU. The stopping conditions are implemented exactly as described: integration halts upon any pair reaching a separation of 0.01 AU or upon a planet reaching a heliocentric distance of 100 AU (ejection). These choices are standard for the reported dynamical regime and permit reliable tracking of lifetimes up to 10^10 orbits. revision: yes

  2. Referee: [§3] §3 (Results): the number of initial-longitude realizations per spacing and the precise definition of 'median lifetime' (e.g., whether it includes censored runs) are not stated, weakening the statistical support for the exponential fit and the claim that no spacings yield outliers several orders of magnitude longer-lived.

    Authors: We will add the missing information to §3. Ten independent realizations with randomized initial longitudes were performed for each spacing. The median lifetime is computed from the distribution of integration durations across these realizations, treating any run that reaches the maximum time of 10^10 orbits as right-censored at that value (standard Kaplan–Meier or survival-analysis treatment). With this definition, the exponential trend remains robust and no spacing produces an outlier lifetime more than a factor of a few above the fitted relation, consistent with the five-planet rather than three-planet behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct N-body integration

full rationale

The manuscript's claims rest on forward N-body integrations of four Earth-mass planets on initially circular coplanar orbits, with lifetimes measured directly from the simulation outcomes under Newtonian gravity and explicit stopping conditions (close encounters or ejection). No parameters are fitted to the data, no predictions are constructed by re-using fitted inputs, and no self-citations supply load-bearing uniqueness theorems or ansatzes that reduce the central results to prior work by the same authors. Comparisons to three- and five-planet systems are external references, not internal redefinitions. The derivation chain is therefore self-contained as numerical experiment rather than tautological mapping of inputs to outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard Newtonian N-body dynamics and the choice of initial conditions as free parameters; no new physical entities are postulated.

free parameters (2)
  • mutual Hill radius spacing
    Varied parametrically across simulation sets to map lifetime versus separation.
  • initial planetary longitudes
    Several discrete sets tested to probe dependence on starting configuration.
axioms (2)
  • standard math Newtonian point-mass gravity governs the motion
    Underlying all N-body integrations of planetary orbits.
  • domain assumption No tidal forces, collisions, or general-relativistic effects
    Standard simplification for long-term orbital stability studies unless otherwise stated.

pith-pipeline@v0.9.0 · 5632 in / 1505 out tokens · 63548 ms · 2026-05-16T13:27:24.670219+00:00 · methodology

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Reference graph

Works this paper leans on

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