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Dynamical Instability of Multi-planet Systems and Free-floating Planets

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read In planetary systems with super-Earths plus a cold Jupiter, close encounters with the Jupiter eject about 38% of the super-Earths, mostly at speeds below 6 km/s, making dynamical instability a promising source of free-floating super-Earths.

desk verdict A plausible new channel for free-floating super-Earths, but the 38% ejection fraction needs an energy-based check at the 1000 au removal radius. read the letter →

arxiv 2507.21216 v1 pith:5EWZE3AU submitted 2025-07-28 astro-ph.EP astro-ph.GA

Ruocheng Zhai , Man Hoi Lee , Tianjun Gan , Shude Mao This is my paper · ORCID
classification astro-ph.EPastro-ph.GA MSC 70F1585A04 PACS 95.10.Ce
keywords free-floatingplanetssuper-EarthscoldJupitersplanetarydynamicsorbitalinstabilityN-bodysimulationsSafronovnumbermicrolensing
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

This paper asks whether the dynamical instability of multi-planet systems can produce the free-floating super-Earths suggested by recent microlensing surveys. Using N-body simulations of five super-Earths, with and without an outer cold Jupiter, it argues that the cold Jupiter is the decisive agent: the ejection fraction jumps from under 2% to about 38% when the Jupiter is present, and nearly all ejected super-Earths leave with relative speeds below 6 km/s. The simulations also show that first close encounters and first planet losses follow different time scalings, and that the Safronov number, rather than planet mass alone, controls whether encounters end in collision or ejection. If these results hold, they give concrete predictions for the kinematics of free-floating super-Earths and for the orbital architecture of the systems left behind.

What carries the argument

The argument runs on dimensionless quantities. The orbital spacing $K$ divides the separation between neighboring planets by their mutual Hill radius and sets how quickly interactions begin. Times are measured in units of the innermost planet's orbital period $P_1$, which makes the differing scalings of first-encounter and first-loss times visible. The Safronov number $\Theta = (M_p/M_*)(a_p/r_p)$ — the ratio of a planet's surface escape speed squared to its orbital speed squared — is the switch that decides whether a close encounter ends in collision ($\Theta<1$) or ejection ($\Theta>1$); the cold Jupiter's large Safronov number is why it scatters super-Earths out of the system. The machinery also includes the simulation's event rules, especially the definition of ejection at 1000 au from the system center of mass.

What would settle it

Rerun the super-Earth+cold-Jupiter suite with a collision model that allows hit-and-run or partial mass loss; if the ejected fraction drops well below 38% or the surviving two-planet systems no longer favor inward, eccentric super-Earths, the central numerical claim fails. Observational check: measure the space velocities of microlensing free-floating super-Earth candidates; if most move much faster than the local stellar population, the predicted low ejection speeds are wrong.

Watch

Extended reading notes

Core claim

The paper's central discovery is that a cold Jupiter can act as an ejection engine for inner super-Earths. In the reference sample with a $0.5$ au innermost super-Earth, orbital spacing $K=5$, and one Jupiter-mass planet at the outer edge, 38% of all super-Earths end up escaping the system, with 91% of those ejections happening during the early chaotic phase; in the identical super-Earth-only sample the ejection fraction is below 2%. The ejected planets have low velocities relative to their host stars, most below 6 km/s, so their Galactic motions should match the stellar population rather than form a fast-moving halo. Systems that survive typically contain one super-Earth and the cold Jupiter; the surviving super-Earth has migrated inward and gained eccentricity (median 0.21), and more than 86% of these two-planet systems are judged long-term stable by the empirical criterion used. A supporting discovery is that the time to first planet loss, measured in units of the innermost orbital period, grows roughly as $a_1^{2.1}$ with the inner semimajor axis, while the first-close-encounter time does not, so encounter times alone do not describe instability.

Load-bearing premise

The simulations assume that every planet-planet collision results in a perfect merger of the two bodies, conserving total mass and momentum; if collisions instead bounce, fragment, or lose mass, the 38% ejection fraction and the final orbit statistics could change.

Editorial extensions

If this is right

  • Super-Earth+cold-Jupiter systems become a quantitatively significant source of free-floating super-Earths: about 38% of the super-Earths in such systems are ejected during instability.
  • The ejected planets' low relative speeds mean microlensing surveys should detect them with timescales and kinematics similar to those of the ambient stellar population, not as a high-velocity population.
  • Surviving systems are typically one super-Earth plus the cold Jupiter; their inner super-Earths should be on inward-migrated, eccentric orbits, and over 86% of the survivors should be dynamically stable on long timescales.
  • In super-Earth-only systems, the first-close-encounter time in units of $P_1$ is independent of orbital radius, while the first-planet-loss time grows as $a_1^{2.1}$, so instability should be described by planet-loss times rather than encounter times.
  • Ejection fraction is organized by the Safronov number: systems with the same Safronov number but different masses eject planets at similar rates, which explains why earlier low-Safronov-number simulations saw no ejections.

Reading between the lines

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

  • If the 38% ejection rate applies to real super-Earth+cold-Jupiter systems, then the observed coexistence of super-Earths with cold Jupiters implies a substantial steady production of free-floating super-Earths; the paper notes this possibility but does not quantify a galactic rate.
  • A testable extension: because the Safronov number grows with distance from the star and shrinks with stellar mass, super-Earths around low-mass stars with a cold Jupiter should be ejected even more readily; a scaled simulation suite could check this.
  • The low relative ejection speeds imply that the free-floating planets from this channel should be kinematically indistinguishable from their birth population; any future detection of fast-moving free-floating super-Earths would point to other channels such as stellar flybys or binary disruption.
  • The predicted inward migration and eccentricity growth of surviving super-Earths give a demographic signature: observed systems with a cold Jupiter and a tight, eccentric inner super-Earth may be the aftermath of this instability, which transit and radial-velocity surveys can search for.
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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 / 5 minor

Summary. The paper uses N-body simulations (REBOUND with IAS15 and Mercurius, plus a GR correction) to study instability, collisions, and ejections in systems of five super-Earths and in systems of five super-Earths plus one cold Jupiter. It reports that the first close encounter time in units of the innermost period is nearly independent of the innermost semimajor axis, whereas the half-loss time increases with it; that a cold Jupiter ejects about 38% of super-Earths at low relative velocities; and that surviving two-planet systems are mostly stable and contain inward-migrated, eccentric super-Earths. The paper connects these results to free-floating planet observations and to the Safronov number.

Significance. If the quantitative ejection statistics are correct, the paper makes a useful contribution: it identifies a plausible dynamical channel for free-floating super-Earths, gives a predictor for their low ejection velocities, and quantifies the role of cold Jupiters. The modeling uses standard, well-tested integrators with GR corrections, and the main statistical results are derived directly from simulations rather than from circular fitting. The paper also honestly states the perfect-merger limitation and gives a conservative stability estimate. However, the headline 38% ejection fraction and the velocity histogram are directly tied to the distance-based ejection criterion, and the lack of an energy check means the quantitative central claim is not yet established.

major comments (2)
  1. [§2.2, §3.2, Fig. 7] The ejection criterion is purely distance-based: a planet is removed when its barycentric distance exceeds 1000 au (§2.2). For a 1 M☉ host, the two-body escape speed at 1000 au is about 1.33 km/s, not the 40 km/s quoted in §3.2 as the escape speed at 1 au. Figure 7 reports most ejected super-Earths at relative velocities of 0–6 km/s, so any object in that histogram with relative speed below about 1.33 km/s is formally bound and will return on an orbit with semimajor axis greater than 1000 au unless it is subsequently scattered again. Because the manuscript does not check the sign of the total energy at removal, the reported 38% ejection fraction and the “mostly below 6 km/s” velocity distribution may significantly overstate the true production of free-floating super-Earths. I request an energy-based ejection criterion (or a tracking test for returning planets), a report of how many removals are actually unbound, and recomputed fej,SE and velocity statistics.
  2. [§5, §2.2] The perfect-merger assumption is acknowledged in §5 but not tested. In the super-Earth–cold-Jupiter systems, a super-Earth can receive velocity kicks near the surface escape velocity of Jupiter during encounters with the cold Jupiter, so subsequent super-Earth–super-Earth collisions may occur at relative velocities above their mutual escape speeds; hit-and-run or partial accretion outcomes are then plausible. Such outcomes would change the number of surviving planets, the fraction of super-Earths removed by collision versus ejection, and the final two-planet fraction. Since the 38% ejection fraction and the 94% two-planet fraction are quantitative central claims, a sensitivity test with a simple alternative collision prescription, or at least a quantitative estimate of the affected encounters, is needed before these numbers can be taken as robust.
minor comments (5)
  1. [Abstract, §3.1, Fig. 3] The abstract says the first close encounter time “is identical” regardless of the innermost semimajor axis, but the text and Figure 3 show no clear dependence, not exact equality; I suggest “nearly independent of a1,0.”
  2. [Table 3] The table reports fej,SE = 38% for all three K values and gives eccentricity percentiles without uncertainties; with 180 systems per sample, Poisson errors on the ejection counts are nontrivial, and the exact equality across K should be discussed with error bars.
  3. [Fig. 7] The histogram should state the number of ejected planets included and the bin width; without these, the shape of the low-velocity peak cannot be assessed.
  4. [§4.3] The observational sample is selected by requiring eccentricity measurements, which may bias the comparison; this selection effect should be acknowledged, and the small sample size (16 single-super-Earth systems and 8 multi-super-Earth systems) should be stated as a limitation.
  5. [§4.1, Eq. (5)] The parameter ℓ ≈ 3 in Equation (5) is introduced as an approximate value, but no fit or derivation is shown; please clarify whether it is an empirical constant or an adjusted parameter.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all central results are direct outputs of N-body integrations with no target fitting.

full rationale

All quantitative claims in this paper (half-loss times, first-close-encounter times, ejection fractions, velocity histograms, surviving-system eccentricities) are direct outputs of the REBOUND N-body integrations described in Sections 2.2 and 2.3. No parameter is fitted to the target outcome and then renamed a prediction; the scaling laws in Equations (3) through (5) are post-hoc regressions of simulated data, which is standard data analysis rather than circular derivation. The cold-Jupiter ejection enhancement is a comparison of two independent simulation ensembles (Section 3.2) with identical integrator settings and no feedback from the claimed result into the physics. The stability estimate in Section 4.4 uses an external empirical criterion (Petrovich 2015), not a self-citation. The only self-citation, to Yuan and Lee 2024 for the two-phase integration scheme, is a numerical methodology choice and is not load-bearing for the scientific conclusion. The 1000 au ejection radius in Section 2.2 is an operational definition; whether an object at 1000 au is truly unbound is a physical correctness and sensitivity question, not a case of the conclusion being equivalent to the input by construction. Likewise, the low-velocity histogram in Figure 7 is a measured simulation output; the paper's comparison to the 40 km/s escape speed at 1 au is loose, but no equation reduces the histogram to the ejection definition. The acknowledged perfect-merger assumption in Section 5 affects quantitative accuracy but is also not circular.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The paper's conclusions rest on the chosen initial conditions and on several modeling approximations. The listed free parameters are the hand-picked initial conditions of the simulations. The axioms are the background physical assumptions and the empirical stability criterion used to interpret the final systems. No new entities are postulated.

free parameters (6)
  • Orbital spacing K = 5 (also 4 and 6)
    Sets the initial separations between planets in Hill radii; the main super-Earth-cold-Jupiter runs use K=5.
  • Innermost semimajor axis a1,0 = 0.5 au for main SE+Jupiter sample
    Sets the scale of the system; also varied across 0.1-2 au for parameter study.
  • Super-Earth mass = 5 M_Earth
    Chosen as representative super-Earth mass; not varied in main runs.
  • Cold Jupiter mass = 1 M_Jupiter
    Chosen as representative cold Jupiter.
  • Ejection distance threshold = 1000 au
    Defines ejection; a planet beyond 1000 au is removed.
  • Number of super-Earths = 5
    Chosen initial condition.
assumptions (5)
  • standard math Newtonian gravity with GR apsidal precession governs the dynamics
    Used throughout via REBOUND and gr-potential; not proved in the paper.
  • domain assumption The chosen initial conditions are representative of super-Earth and cold Jupiter systems after disk dispersal
    The paper assumes this to draw conclusions about real systems; Section 2.3.
  • domain assumption Perfect merger approximation for collisions
    Stated in Section 5; if hit-and-run occurs, results may change.
  • domain assumption Petrovich (2015) empirical stability criterion is valid for the surviving two-planet systems
    Used in Section 4.4 to estimate stability fraction.
  • domain assumption A planet is ejected when its distance exceeds 1000 au
    Modeling choice; highly eccentric bound orbits with apocenter beyond 1000 au could be misclassified as ejected. Section 2.2.

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Cite this review

Pith. "Pith review of Dynamical Instability of Multi-planet Systems and Free-floating Planets." pith.science (2026). https://pith.science/paper/5EWZE3AU

@misc{pith2026250721216,
  author       = {Pith},
  title        = {Pith review of: Dynamical Instability of Multi-planet Systems and Free-floating Planets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5EWZE3AU}},
  note         = {Machine review of arXiv:2507.21216}
}
abstract

The ejection of planets by the instability of planetary systems is a potential source of free-floating planets. We numerically simulate multi-planet systems to study the evolution process, the properties of surviving systems, and the statistics of ejected planets. For systems with only super-Earth planets, we find that the time (in units of the orbital period $P_{1}$ of the innermost planet) for the system to lose the first planet by collision or ejection increases with the semimajor axis of the innermost planet. In contrast, the time (in units of $P_{1}$) for the first close encounter between two planets is identical. These two timescales also depend differently on the orbital spacing between the planets. Most systems with only super-Earths do not have planets ejected. In systems with super-Earths and a cold Jupiter, we discover that a cold Jupiter significantly increases the probability of ejection of the super-Earths by close encounters. Of 38\% of ejected super-Earths, most velocities relative to their parent stars are smaller than $6\ \mathrm{km\ s^{-1}}$. We conservatively estimate that more than 86\% of the surviving two-planet systems in the super-Earths plus cold Jupiter sample are long-term stable by using empirical criteria. Most super-Earths in the remaining two-planet systems are on highly elliptical but stable orbits and have migrated inwards compared with their initial states.

Figures

Figures reproduced from arXiv: 2507.21216 by the authors.

Figure 1
Figure 1. The fraction of five-planet systems in the su￾per-Earths only simulations with K = 5 as a function of time, which is normalized by the period P1 of the initial innermost orbit of each system. Five curves are shown for a1,0 = 0.2, 0.4, 0.6, 0.8, and 1 au, where a1,0 defines the ini￾tial distribution of the semimajor axis of the innermost orbit (see [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Half-loss time vs. the semimajor axis of the in￾nermost orbit for the super-Earths only samples with K = 5. The red dashed line shows the linear regression result. t˜ is the time normalized by P1: t˜= t/P1 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The first close encounter time (t˜FCE) vs. the initial semimajor axis of the innermost orbit a1,0 for su￾per-Earths only samples with K = 5. The first close en￾counter time depends little on the distance between the plan￾ets and the host. there is no clear dependence of the first close encounter time on the distance between the planets and the host. So even though many previous works (J. Chambers et al. 1996; A. W. … view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: Evolution of the fractions of planetary systems with different numbers of planets in the super-Earth-cold-Jupiter sample with a1,0 = 0.5 au and K = 5. Phases 1 and 2 are in the left and right panels, respectively. Curves labeled fnp are the fractions of systems with n …
Figure 6
Figure 6. Figure 6: Evolution of the semimajor axes (upper left panel), eccentricities (lower left panel), apastron distances (upper right panel), and periastron distances (lower right panel) in a super-Earth-cold-Jupiter system. When a planet-planet collision happens, the new planet will…
Figure 7
Figure 7. Figure 7: Distribution of the velocities of ejected plan￾ets relative to the host stars in the super-Earth-cold-Jupiter sample. Most ejected planets have relative velocities lower than 6 km s−1 . with increasing K (as found in previous studies). For all values of K, the half-los…
Figure 8
Figure 8. Figure 8: Orbital properties of the super-Earths in the final stable two-planet systems from the super-Earths-cold-Jupiter simulations. Top left panel: Final eccentricities vs. final semimajor axes. Top right panel: Final semimajor axes vs. initial semimajor axes. In this panel,…
Figure 9
Figure 9. Figure 9: The first close encounter time t˜FCE vs. the initial semimajor axis of the innermost orbit a1,0 for super-Earths only simulations with orbital spacing K = 4, 5, and 6, de￾fined in Equation (2) [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: The half-loss time t˜1/2 vs. the initial semimajor axis of the innermost orbit a1,0 for super-Earths only simu￾lations with orbital spacing K = 4, 5, and 6. The vertical dash-dotted line is log a1,0 = −0.45, which roughly divides the dependence of t˜1/2 on K into two …
Figure 12
Figure 12. Figure 12: Histogram of the number of ejections per planet vs. Safronov number in the equal-mass planet sample with randomly chosen masses and semimajor axis of the innermost orbit. the Safronov number Θ ≈ 0.11(a1,0/au), which would be ≈ 1.1 and 11 for the simulations with a1,0 …
Figure 14
Figure 14. Figure 14: aout/ain vs. rap−Y for the remaining two-planet systems from the super-Earths-cold-Jupiter sample in Sec￾tion 3.2 with a1,0 = 0.5 au and K = 5 for all planets . The left and right vertical dashed lines show the 95% instability and stability boundaries, respectively. T…

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