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Orbital Stability of Hierarchical 3 and 4-Body Systems with Inclination: Results for Kepler-1625, 1708, and HD 23079

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper maps where moons and submoons around exoplanets stay bound over 100,000 years, and finds that the proposed exomoon orbits in Kepler-1625 and Kepler-1708 fall mostly in stable regions.

desk verdict Useful inclined stability maps for the leading exomoon candidates; the 13:2 secular resonance claim is a fit, not a demonstrated resonance. read the letter →

arxiv 2501.12258 v1 pith:7J2ZCUEN submitted 2025-01-21 astro-ph.EP

Shaan D. Patel , Billy Quarles , Manfred Cuntz This is my paper · ORCID
classification astro-ph.EP
keywords exomoonstabilitysubmoonsecularresonancevonZeipel-Lidov-KozaieffectN-bodysimulationsKepler-1625Kepler-1708HD23079
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 uses 100,000-year N-body simulations to map where a moon around an exoplanet, and a moon-of-a-moon (a submoon) around that moon, can remain bound, treating orbital size and inclination as free parameters. Applying the maps to the two strongest exomoon candidates, it finds that the observationally proposed orbits of Kepler-1625b-i and Kepler-1708b-i lie mostly in stable, low-eccentricity regions. For a hypothetical submoon around Kepler-1625b-i, it identifies stable islands alongside ridges of elevated eccentricity, which it attributes to a secular resonance where the moon's and submoon's orbits precess together at a 13:2 rate ratio. The result is a general framework for screening exomoon and submoon candidates against dynamical stability before investing in follow-up observations.

What carries the argument

The central machinery is a grid of stability maps built from 100,000-year N-body integrations that vary the satellite's semimajor axis and inclination, using maximum eccentricity as a stability proxy and checking orbit crossing beyond the Hill radius. The resonance claim rests on the resonant-angle derivative $\dot{\phi} = 13\dot{\omega}_{\rm m} - 2\dot{\omega}_{\rm sm}$, computed by linear fits to the moon's and submoon's arguments of pericenter; secular perturbation theory (the classical disturbing-function expansion) reproduces the amplitude of the submoon's inclination oscillation, which anchors the identification of the ridges as secular rather than mean-motion resonances. A double coordinate rotation—tilting the submoon relative to the moon, then the moon relative to the planet—is needed to initialize the inclined hierarchical 4-body problem correctly.

What would settle it

Re-run the 4-body stability maps with tidal dissipation included, using tidal quality factors comparable to those in Kollmeier & Raymond (2018), and check whether any submoon in the 20-33% moon-Hill-radius range remains bound for 100,000 years; if none do, the claimed stable regions are not physically real.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that the published parameter estimates for the exomoon candidate Kepler-1625b-i are dynamically viable: most of the observationally allowed semimajor-axis and inclination range falls in regions of low maximum eccentricity in 100,000-year N-body integrations, with instability only where the von Zeipel-Lidov-Kozai mechanism excites eccentricity at inclinations above roughly 40 degrees. The same holds even more strongly for Kepler-1708b-i, whose proposed orbit sits so close to its planet that maximum eccentricity stays below about 0.005 for inclinations under 40 degrees. Extending to a fourth body, the paper finds that a submoon can remain bound around Kepler-1625b-i in portions of the 20-33% moon-Hill-radius range for inclinations below about 40 degrees, and it identifies the curved ridges of elevated eccentricity in that stability map as secular resonances, confirmed by a 13:2 ratio of apsidal precession rates and by secular perturbation theory reproducing the amplitude of the submoon's inclination oscillation.

Load-bearing premise

The load-bearing premise is that 100,000 years of gravity-only motion is a fair test of stability: the simulations omit tidal dissipation, which the paper's own cited sources say would drag submoons inward and preferentially remove them, and they assume the contested exomoon detections with their adopted masses and orbits.

Editorial extensions

If this is right

  • Kepler-1625b-i remains dynamically viable: most of its observed parameter range falls in low-maximum-eccentricity, bound regions, with only high-inclination parts susceptible to von Zeipel-Lidov-Kozai excitation.
  • Kepler-1708b-i is even more robust: its proposed semimajor axis sits at 6-11% of the planet's Hill radius, where maximum eccentricity stays below about 0.005 for inclinations under roughly 40 degrees.
  • Stable submoon orbits exist around Kepler-1625b-i in the 20-33% moon-Hill-radius range for inclinations below about 40 degrees, so a moon of a moon is dynamically possible.
  • The elevated-eccentricity ridges in the submoon stability map are secular resonances, identified by a 13:2 ratio of apsidal precession rates and confirmed by secular perturbation theory matching the inclination amplitude.
  • The same framework applied to HD 23079 (which has no known exomoon) predicts a stable region similar to Kepler-1625's, which can inform future exomoon searches there.

Reading between the lines

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

  • Because the integrations are purely Newtonian, the stable submoon regions are upper limits: tidal dissipation, as the paper itself notes citing Kollmeier & Raymond (2018), would likely remove submoons on shorter timescales, so a submoon detection would require either weak tidal dissipation or a young system.
  • The 13:2 apsidal precession resonance is a high-order secular resonance; extending the maps to longer integration times (10^6 years) would test whether the resonant ridges eventually drive the submoon to ejection, and whether other precession ratios (e.g., 11:2, 15:2) produce similar ridges.
  • The coordinate-rotation fix for inclined hierarchical initial conditions suggests that earlier coplanar-only stability studies may have missed inclination-driven resonances of this kind; future exomoon stability surveys should scan inclination as a matter of course.
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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 uses the rebound N-body package to map orbital stability for hierarchical 3-body (star-planet-moon) systems for Kepler-1625, Kepler-1708, and HD 23079, and for 4-body (star-planet-moon-submoon) systems for Kepler-1625 and HD 23079. The authors validate their integration setup against known von Zeipel-Lidov-Kozai behavior, identify and patch a coordinate-frame issue in rebound's hierarchical-element initialization, and produce log maximum-eccentricity maps over satellite semimajor-axis and inclination grids. They interpret elevated-eccentricity ridges in the submoon stability map as secular resonances between the host moon and submoon precession, supported by measured apsidal precession rates, a claimed 13:2 commensurability, MEGNO maps, and a secular perturbation theory comparison. The main conclusions are that the proposed Kepler-1625 exomoon parameters are mostly orbitally stable and that certain submoon configurations remain gravitationally bound over 10^5 yr.

Significance. If confirmed, the paper would provide a useful framework for evaluating exomoon and submoon stability with inclination, and it would identify a specific secular-resonance mechanism for submoon eccentricity excitation. The manuscript has several genuine strengths: the authors compare multiple integrators, explicitly document and patch a coordinate-frame bug in rebound, randomize mean anomalies to reduce phase bias, verify their secular-theory implementation against Murray and Dermott's Jupiter-Saturn test case, and use MEGNO as an independent chaos indicator. These elements make the 3-body stability maps and the general 4-body instability structure credible and reproducible. The main weakness is that the central novel claim, the identification of the Fig. 5(a) ridges as secular resonances, is not demonstrated with a librating resonant angle or a matched secular frequency; it rests on an approximate commensurability of fitted precession rates.

major comments (3)
  1. [Sec. 2.3, Eq. (3); Sec. 3.2.2] The secular-resonance identification is not demonstrated. The coefficients 13 and 2 in Eq. (3) are chosen because the measured precession-rate ratio omega_dot_sm/omega_dot_m ~ 6.58 is close to 13/2, but the paper does not show that the resonant angle phi librates or remains bounded on any timescale. With the quoted rates omega_dot_m = 1.22 deg/yr and omega_dot_sm = 8.026 deg/yr, Eq. (3) gives phi_dot ~ -0.19 deg/yr, corresponding to a circulation period of roughly 1900 years; the 100-year simulations in Fig. 7 cover only a small fraction of that cycle, so slow circulation and trapped libration are indistinguishable. The authors should show phi(t) over at least a full cycle, report the libration width or a stroboscopic map, and test whether the 13:2 choice is dynamically preferred over nearby integer combinations.
  2. [Sec. 2.4; Fig. 9] The secular perturbation theory comparison does not confirm the resonance because it matches only the amplitude of the submoon inclination variation and not its frequency or phase, as the text and figure caption admit. Since a secular resonance is defined by a commensurability of precession frequencies, an amplitude match is insufficient evidence. The authors should compare the dominant periods or eigenfrequencies of the secular theory with the N-body result, or otherwise show that the frequency mismatch arises from a benign non-secular effect, before concluding that the ridges in Fig. 5(a) are secular resonances.
  3. [Sec. 4, final discussion of submoon viability] The paper's own cited literature limits the physical interpretation of the 4-body stability maps. Section 4 cites Kollmeier and Raymond (2018) and Rosario-Franco et al. (2020) showing that tidal migration would preferentially remove submoons, but the 10^5-year simulations in Sec. 2.1 are pure Newtonian and include no tides. The stable regions in Fig. 5 are therefore gravitational-stability regions, not necessarily regions where a submoon could survive in the real Kepler-1625 or HD 23079 systems. The conclusion that a submoon 'could be orbitally stable' should be explicitly qualified as stability in the absence of tides, with a discussion of whether the tidal migration timescale is longer than the 10^5-year simulation time or the age of the system.
minor comments (5)
  1. [Abstract] The phrase 'pursing orbital stability analyses' contains a typo; it should be 'pursuing.'
  2. [Table 1] In the HD 23079 4-body row, the entry '0.0298 9' is visually ambiguous; the moon semimajor axis and inclination should be separated or labeled so that the reader can identify a_m = 0.0298 au and i_m = 9 deg without guessing.
  3. [Fig. 6 caption] The caption says 'Kepler-162' where 'Kepler-1625' is meant.
  4. [Sec. 3.2.2] The statement that the precession-rate ratio is '~13:2' is imprecise; the quoted values give 8.026/1.22 = 6.58, which is close to but measurably different from 13/2 = 6.50, and the paper should quantify this difference and discuss whether it is within the fitting uncertainty.
  5. [Sec. 2.1] The description of the coordinate-frame bug says the second transformation was not applied, but the precise failure mode in rebound's hierarchical initialization would be clearer if the authors stated whether the bug affected the submoon's inclination relative to the moon or its longitude of ascending node, since the correction in Appendix A addresses only inclination rotations.

Circularity Check

1 steps flagged · score 6.0 of 10

Submoon 'secular resonance' identification reduces to a 13:2 commensurability chosen from the same simulation; the stability maps themselves are self-contained.

  1. self definitional [Sec. 3.2.2, Eq. (3) and surrounding text]
    "From these calculations, we find \dot{\omega}_m = 1.22° yr−1 and \dot{\omega}_sm = 8.026° yr−1. The ratio between the apsidal precession rates for the submoon and moon is∼13:2, which points to a potential resonance. We find the time derivative of the resonant angle as given in Eq. 3. This points to a secular resonance between the precession of these two elements causing an increase in the eccentricity and inclination of the submoon."

    The integers 13 and 2 in Eq. (3) are not derived independently; they are selected because the measured precession ratio from the same 100-year simulation is ≈6.58 ≈ 13/2. Consequently \dot{\phi} = 13\dot{\omega}_m − 2\dot{\omega}_sm ≈ 13(1.22) − 2(8.026) ≈ −0.19°/yr, which is near zero by construction. The paper then presents this near-zero value as evidence of a secular resonance, without showing that \phi(t) librates; over the 100-yr run the circulation period is roughly 1900 yr, so slow drift and libration are not distinguished. The secular-theory test in Fig. 9 matches only the inclination amplitude and explicitly admits a frequency/phase difference.

full rationale

The 3-body and 4-body stability maps (Figs. 3, 5, 6) are self-contained N-body experiments: the conclusion that quoted exomoon parameters fall in stable regions is a forward simulation result, not a fit to the output. The adopted exomoon masses/orbits and neglect of tides are modeling assumptions, not circular inputs. The self-citations (Rosario-Franco et al. 2020; Jagtap et al. 2021; Quarles et al. 2024) supply stability ranges and secular equations; they are not the load-bearing justification for the resonance claim, and the secular implementation is checked against Murray & Dermott (2000), so they do not raise the score. The only identified circular step is the 13:2 secular-resonance identification: Eq. (3) defines the resonant angle using precession rates measured from the same simulation, making the near-zero \dot{\phi} a restatement of the observed ratio rather than an independent prediction. Since this resonance identification is one of the paper's central novel conclusions, the score is 6 rather than lower.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central maps are essentially numerical scans, so the main free parameters are the chosen initial-condition ranges and the resonance coefficients that are fitted to the simulation output. The load-bearing axioms are that the contested exomoons exist, that a tide-free 10^5-yr integration captures physical stability, and that the ad hoc 13:2 precession combination is a genuine resonance.

free parameters (3)
  • Secular resonance coefficients (13, 2) in phi = 13*omega_m - 2*omega_sm = 13 and 2
    The coefficients are selected to match the measured precession ratio omega_dot_sm/omega_dot_m ~ 6.58 computed from numpy.polyfit on the same simulation (Sec. 3.2.2, Eq. 3), so the resonant angle is a fit to the data, not a parameter-free prediction.
  • HD 23079 exomoon initial condition (a_m, i_m) for 4-body runs = a_m = 0.0298 au, i_m = 9 deg
    No exomoon parameters exist for HD 23079, so the paper states the choice is made somewhat arbitrarily from Fig. 3b. The HD 23079 submoon stability map depends directly on this chosen orbit (Table 1, Sec. 2.2).
  • Submoon semimajor-axis range 0.20-0.33 R_H,m = 0.20-0.33 R_H,m
    This scan window is adopted from Rosario-Franco et al. (2020) rather than derived; the claimed resonance ridges in Fig. 5 are located inside this chosen range.
assumptions (5)
  • domain assumption The Kepler-1625b-i and Kepler-1708b-i exomoon candidates are real with the adopted masses and orbital parameters.
    Table 1 and Sec. 2.2 use Teachey & Kipping (2018) and Kipping et al. (2022). Sec. 1 acknowledges these candidates are contested, so a large part of the stability maps is conditional on the detections being correct.
  • domain assumption Orbital stability over 10^5 yr without tidal dissipation, collisions, or additional bodies is a sufficient proxy for physical viability of submoons.
    Sec. 2.1 sets the 10^5 yr timescale; Sec. 4 notes prior tidal studies show tidal migration would preferentially remove such submoons. This assumption is load-bearing for the practical interpretation of the stable regions.
  • domain assumption SABA(10,6,4) integration faithfully represents the dynamics after the manual coordinate rotations.
    Sec. 2.1 reports WHFast failed to reproduce von Zeipel-Lidov-Kozai expectations; the authors fixed coordinate-frame issues and checked SABA(10,6,4) against IAS15, but the corrected pipeline is not independently verified in this paper.
  • domain assumption MEGNO from 100-yr runs identifies the same resonant and chaotic structures as the 10^5-yr eccentricity maps.
    Sec. 2.3 and Fig. 6; the paper notes one curve from Fig. 5a does not appear in the MEGNO map, possibly due to short runtime, so the correspondence is incomplete.
  • ad hoc to paper The 13:2 combination of precession rates defines a meaningful secular resonance.
    Eq. 3 is introduced after the simulation yields omega_dot_sm/omega_dot_m ~ 13/2; no first-principles derivation of this commensurability is given, and the secular theory comparison matches amplitude but not frequency or phase (Fig. 9).
invented entities (1)
  • Hypothetical submoon of mass 1.67e-4 M_earth orbiting Kepler-1625b-i or a HD 23079 exomoon
    purpose: Test particle for 4-body stability maps and secular resonance analysis
    Submoons are not observed. The paper adopts a mass near limits suggested by Kollmeier & Raymond (2018) and provides no falsifiable handle outside the simulation maps.

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Cite this review

Pith. "Pith review of Orbital Stability of Hierarchical 3 and 4-Body Systems with Inclination: Results for Kepler-1625, 1708, and HD 23079." pith.science (2026). https://pith.science/paper/7J2ZCUEN

@misc{pith2026250112258,
  author       = {Pith},
  title        = {Pith review of: Orbital Stability of Hierarchical 3 and 4-Body Systems with Inclination: Results for Kepler-1625, 1708, and HD 23079},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7J2ZCUEN}},
  note         = {Machine review of arXiv:2501.12258}
}
read the original abstract

As the number of potential exomoon candidates grows, there is a heightened motivation of pursing orbital stability analyses. In this work, we provide an in-depth investigation into 4-body systems, consisting of a star, planet, moon, and submoon by using the N-body simulator rebound. Particularly, we focus on the system of Kepler-1625, where evidence of a possible exomoon has been obtained. We investigate the 3-body star--planet--moon system for the proposed exomoon parameters allowing us to identify stable regions associated with most of the space parameters. Thereafter, we consider a 4-body system including a potential submoon. We find that there are both stable and unstable regions, as expected, as well as resonance patterns that are further explored using numerical and analytical methods including secular perturbation theory. We are able to identify these resonances as secular in nature. In addition, we investigate 3-body versions of two other systems, Kepler-1708 and HD 23079, while also studying a 4-body version of HD 23079. Our work may serve as a generalized framework for exploring other planet--moon cases in the future while noting that the current 4-body study may be an incentive for studying further exomoon and submoon systems.

Figures

Figures reproduced from arXiv: 2501.12258 by the authors.

Figure 1
Figure 1. Side view of the 4-body system setup for Kepler-1625 in the rebound simulations: (a) Kepler-1625b (red) orbits Kepler-1625 (yellow) with the proposed exomoon (blue) orbiting Kepler-1625b, (b) the exomoon orbits Kepler-1625b zoomed in showing the inclination of the moon’s orbit, and (c) a theoretical submoon (purple) in orbit about the exomoon. The gray areas in panel c represent the parameter space for the submoon a… view at source ↗
Figure 2
Figure 2. Time series of the moon’s argument of pericenter from years 23 through 26 overlaid with the median values taken every 0.08 years. The line of best fit (using numpy.polyfit) for the median points is overlaid [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Logarithm of maximum eccentricity (log10 (max em); color coded) from stability simulations that vary the exomoon’s initial inclination and semi-major axis in (a) Kepler-1625, (b) HD 23079, and (c) Kepler-1708. The black lines denote the derived parameters for the exomoon given in the literature by (a) Teachey & Kipping (2018) and (c) Kipping et al. (2022). The white dot represents the exomoon parameters chosen for t… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Time evolution of the exomoon’s parameters in the 𝑒 − 𝜔 plane (black dots) through our Kepler-1625 3-body numerical simulations (with 𝑎m = 0.0218 au, 𝑖m = 54◦ , MAm = 0 ◦ ). The overlaid contours represent the trajectories within the test particle quadrupole approximat…
Figure 5
Figure 5. Figure 5: Logarithm of maximum eccentricity (log10 (max esm); color coded) from stability simulations that vary a putative submoon’s initial inclination and semi-major axis in (a) Kepler-1625 and (b) HD 23079. The gray cells denote those initial conditions where the maximum ecce…
Figure 6
Figure 6. Figure 6: Map of MEGNO values centered around 0 (log10 | 〈Y 〉 − 2|) from 4-body stability simulations for Kepler-162 that vary the submoon’s initial inclination and semi-major axis. Herman M. K., Zhu W., Wu Y., 2019, AJ, 157, 248 Jagtap O., Quarles B., Cuntz M., 2021, Publ. Astr…
Figure 7
Figure 7. Figure 7: Time series evolution of the moon (black) and submoon (blue) orbital parameters for the 4-body Kepler-1625 simulation with the initial conditions 𝑎 = 0.00084 au, 𝑖 = 3.6 ◦ , and MA = 0 ◦ for the submoon. The star, planet, and moon’s initial conditions can be found in …
Figure 8
Figure 8. Figure 8: Comparison of the time evolution for the inclination and eccentricity of the submoon in Kepler-1625 using a secular approach (blue) and N-body simulations (black). Each panel shows the short-term (< 5 yr) variations on the left, while the longer-term variations are on …
Figure 9
Figure 9. Figure 9: Comparison of the time evolution for the submoon’s inclination in Kepler-1625 using a secular approach (blue) and N-body simulations (black). There is good agreement in the magnitude of the inclination variations, although there are differences in phase. Note that the …

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.