Pith. sign in

REVIEW 3 major objections 5 minor 69 references

Numerical approach to second-order canonical perturbation theory in the planetary 3-body problem: Application to exoplanets

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Second-order secular theory reproduces exoplanet precession to 0.05-1%.

desk verdict Second-order secular theory via FFT is genuinely new and the SJS/WASP-148 tests are convincing, but the resonant exoplanet benchmarks rest on an unverified truncation that needs a convergence check before the headline claim is fully secure. read the letter →

arxiv 2506.13745 v2 pith:UARNXQFS submitted 2025-06-16 astro-ph.EP physics.class-phphysics.comp-ph

Aya Alnajjarine , Federico Mogavero , Jacques Laskar This is my paper · ORCID
classification astro-ph.EPphysics.class-phphysics.comp-ph MSC 70F1570H15
keywords second-ordersecularperturbationtheoryLietransformmean-motionresonanceFFTaveragingthree-bodyplanetaryproblemexoplanetlong-termdynamicsorbitalprecessionfrequenciessmalldivisors
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

The paper develops a numerical route to second-order canonical perturbation theory for two planets orbiting a star. Standard secular theories stop at first order in the planetary mass ratio, which distorts the slow precession frequencies of orbits near mean-motion resonances. The authors use a Lie transform to remove the fast orbital angles from the Hamiltonian and compute the required Fourier coefficients with a fast Fourier transform, avoiding expansions in eccentricity, inclination, or semi-major axis ratio. The second-order Hamiltonian reduces relative errors in the dominant precession frequencies from roughly 14-20% to 0.05-1% for the Sun-Jupiter-Saturn system, and from 112% to 2.6% for the resonant system GJ 876. If the results hold, the method extends accurate long-term secular modeling to compact, massive, and resonant exoplanetary systems.

What carries the argument

The load-bearing object is the second-order secular Hamiltonian of Eqs. (20) and (23): sums over non-zero Fourier harmonics $h^k_1$ of the disturbing function, divided by the small-divisor denominators $k\cdot n'_0$ (or $k\cdot \tilde n_0$ in the linearized resonant version) and combined through Poisson brackets over the slow variables. It is built by a Lie transform that eliminates the fast mean longitudes; the Fourier coefficients and their first derivatives are computed by evaluating the disturbing function on a two-dimensional grid in the mean longitudes and applying a 2D FFT, which removes any need for expansions in eccentricities, inclinations, or semi-major axis ratios. The second derivatives needed for the equations of motion are evaluated with a five-point finite-difference stencil.

What would settle it

For GJ 876 or TIC 279401253, raise the resonant truncation from $K'=20$ to $K'=30$ or $40$, recompute the second-order precession frequencies $g_1$ and $g_2$, and compare with n-body values; if $g_2$ shifts by more than the reported few-percent accuracy, the claimed agreement is controlled by the unverified truncation rather than by the second-order theory.

Watch

Extended reading notes

Core claim

The central claim is that a second-order secular Hamiltonian, obtained by Lie-transforming the three-body Hamiltonian and evaluating its Fourier sums numerically, reproduces the long-term secular dynamics of planetary systems near or inside mean-motion resonances much better than first-order theory. The second-order terms contain the small-divisor denominators $k\cdot n'_0$ that encode near-resonant coupling, so they correct exactly what first-order averaging misses. In the validations, the relative error in the Saturn perihelion frequency $g_6$ drops from about 20% with the first-order model to 1% with the second-order secular model and to 0.05% when the 5:2 resonant harmonics are included; the error in the GJ 876 frequency $g_2$ drops from 112% to 2.6%; and the WASP-148 perihelion frequencies $g_1$ and $g_2$ improve from 12% and 6.5% error to below 0.1%. The paper concludes that second-order terms significantly improve the accuracy of orbital precession frequencies, most notably for systems with strong mean-motion resonances or large planetary masses.

Load-bearing premise

The load-bearing premise is that the Fourier coefficients of the perturbation decay exponentially with harmonic order, so truncating the series at $K$ (and at $K'$ for resonant systems) leaves the second-order secular dynamics essentially unchanged, even though convergence is demonstrated numerically only for the Sun-Jupiter-Saturn system.

Editorial extensions

If this is right

  • Second-order secular models reproduce n-body precession frequencies with relative errors near or below 1% in systems where first-order models err by 10-100%.
  • Adding the resonant harmonics of a mean-motion resonance, as done for the 5:2 Jupiter-Saturn resonance, removes most of the remaining frequency error, showing that those residuals come from higher-order mass terms associated with the resonance.
  • Because the method avoids expansions in orbital elements, it can be applied to highly eccentric, inclined, or tightly packed planets where classical secular expansions fail to converge.
  • For systems locked in a 2:1 resonance, the non-resonant secular model is inadequate, and the second-order resonant secular model is required to track the evolution accurately.
  • The method isolates the main planetary interactions driving secular dynamics, making it a tool for studying long-term stability of exoplanetary architectures.

Reading between the lines

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

  • A natural extension is to push the resonant truncation limit $K'$ upward for GJ 876 and TIC 279401253: a convergence scan would show whether their residual discrepancies come from the unverified truncation or from neglected third-order mass terms.
  • The same FFT-plus-Lie-transform construction should extend to more than two planets, since neither the Poisson-bracket sums nor the exponential-decay argument is specific to the three-body case.
  • For high-eccentricity systems, the mean-motion correction currently absorbs only the zero-eccentricity part of the perturbation; a version that folds in higher-degree terms could further improve systems like TIC.
  • If the method scales, it could serve as a fast screening tool for the long-term stability of newly discovered resonant exoplanet systems, replacing expensive n-body integrations for population-level studies.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 manuscript presents a numerical implementation of second-order canonical perturbation theory for the planetary three-body problem. Using Lie transforms, the authors construct secular Hamiltonians to second order in the planet-to-star mass ratio for both non-resonant and resonant configurations, including a mean-motion frequency correction. The Fourier coefficients of the disturbing function and their derivatives are computed on a grid via FFT, and the secular equations are integrated with a high-order Adams method. The method is benchmarked against direct n-body integrations for the Sun-Jupiter-Saturn system, WASP-148, TIC 279401253, and GJ 876. For SJS, second-order terms reduce precession frequency errors from 14--20% to below 1%, and the resonant second-order model reduces the g6 error to 0.05%. For WASP-148, errors drop below 0.1% in the non-resonant second-order model. For the resonant systems TIC and GJ 876, the resonant second-order model improves the g2 frequency substantially (for GJ 876 from a large first-order error of reported 112% down to 2.6%), although for TIC the g1 error slightly increases from 0.2% to 1%.

Significance. If the results hold, the paper offers a useful and comparatively general numerical route to second-order secular dynamics, avoiding expansions in eccentricities, inclinations, and semi-major axis ratios. The derivation is self-contained and the SJS and WASP-148 validations are convincing, with clear algorithmic detail and quantified improvements. The main significance is the potential applicability to high-eccentricity, resonant exoplanetary systems where classical analytical expansions are problematic. However, the support for the central claim in exactly those regimes is incomplete: for TIC and GJ 876 the resonant second-order sum is truncated at K'=20 without any convergence check, and the benchmark initial conditions are taken from low-pass-filtered n-body solutions of the same systems. These issues weaken, but do not by themselves invalidate, the otherwise sound SJS/WASP-148 results.

major comments (3)
  1. [§IV.a, §V (TIC/GJ 876)] The truncation of the resonant second-order sum in Eq. (23b) to K'=20 for TIC 279401253 and GJ 876 is introduced solely to avoid small-divisor instabilities, and no convergence check is provided for these systems. The exponential-decay justification in Eq. (29) is demonstrated only for SJS (Figs. 1 and 2), and the text itself states that convergence is slower for TIC because of higher eccentricities and masses. Since the claimed improvements for these systems (e.g., the g2 error reduction from 112% to 2.6% for GJ 876) are comparable to the expected magnitude of the truncated terms, the reported accuracy could be an artifact of the cutoff. I request a convergence study over K' (for example, 12, 16, 20, 24, 32) for both systems, reporting the secular frequencies g1 and g2 and, if possible, the eccentricity curves. If larger K' produces small-divisor instabilities, then the method is not validated for these high-eccentricity resonant configurations and the conclusions should be restricted accordingly.
  2. [§IV.e and §V] The initial conditions for the secular integrations are obtained by low-pass filtering the very same n-body solution against which the secular models are compared (Eq. (33) and the text of Sect. IV.e). This makes the benchmark a test of internal consistency between the secular equations and filtered initial conditions, not a test of the method's ability to evolve a system from published osculating elements. I recommend adding at least one case (for example, SJS or WASP-148) in which the secular model is initialized directly from osculating elements, with the first- or second-order Lie-transform corrections applied, and compared to an n-body integration from the same osculating elements. Such a test would substantially strengthen the practical applicability of the method.
  3. [§V (TIC) and Table I] For TIC, the second-order resonant model improves g2 (from 5.1% to 0.3%) but degrades g1 (from 0.2% to 1%). The paper's explanation in terms of high masses, high eccentricities, and the limited validity of the mean-motion correction is plausible, but the net improvement is less clear than for SJS and WASP-148. I suggest an error-budget analysis for the TIC case, separating the contributions of the K' truncation, the mean-motion correction, and the initial-condition offset, and a more guarded statement of the method's performance for strongly resonant, high-eccentricity systems.
minor comments (5)
  1. [Title/running header] The title and running header contain a typo: 'perturbatio n theory' with a space.
  2. [§V (GJ 876) and Table I] The text reports a 112% relative error for g2 at first order, but the values in Table I (n-body -0.0202, resonant first order -0.0025) give a relative error of about 88%. Please correct this inconsistency.
  3. [Abstract] The statement in the abstract that the method 'avoids the need for expansions in orbital elements' is too strong; the method still requires a finite truncation of the Fourier series (K and K') and relies on the exponential decay of the coefficients. Please qualify the claim.
  4. [Eq. (29)] The statement in Eq. (29) that the neglected remainder of the Fourier series is 'of order ε²' should be clarified; the truncation error of H1 depends on the analyticity domain and is not automatically O(ε²), especially at high eccentricity.
  5. [§VI] The paper mentions an implementation in C but does not state whether the code will be made publicly available. Given the numerical nature of the work, a code availability statement would be useful for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the second-order secular Hamiltonian is derived from first principles and validated against direct n-body integrations.

full rationale

The central derivation (Eqs. 7, 20, 23) constructs the second-order secular Hamiltonian from the canonical 3-body Hamiltonian via Lie-transform perturbation theory, with no fitted constants. The Fourier coefficients h^k_1 are computed by FFT from H1, and the second-order terms are algebraic functions of these coefficients and the mean-motion denominators; the precession frequencies compared in Table I are dynamical outputs of the secular equations, not inputs. The validations are benchmarked against independent symplectic n-body integrations, which is a standard and non-circular test. The use of low-pass-filtered n-body solutions to set initial conditions (Sec. IV.e, Eq. 33) does not reduce to circularity: initial conditions do not determine the frequencies, and the test is whether the secular equations reproduce the n-body secular evolution from a consistent initial state. The K'=20 truncation for TIC and GJ876 is an unverified convergence assumption, but that is a correctness risk, not a circular reduction. Self-citations, such as the mean-motion correction following Laskar 1985, are standard external methodological results and are not used to force the paper's conclusions.

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

The central claim rests on no fitted physical constants. All listed parameters are numerical discretization or initialization choices, with the resonant truncation K' being the most load-bearing because its convergence is not tested. The main domain assumptions are short-term regularity and exponential decay of Fourier coefficients.

free parameters (5)
  • Fourier truncation limit K = 32 for SJS/WASP-148, 64 for TIC/GJ876
    Chosen per system; convergence shown for SJS only (Figs. 1-2).
  • Resonant second-order truncation K' = 20 for TIC and GJ876
    Chosen to avoid small-divisor instabilities; no convergence test provided for these systems.
  • Butterworth filter cutoff period = e.g., 5000 yr (SJS secular), 200 yr (SJS resonant)
    Set by identifying the frequency gap in the n-body spectrum; affects initialization.
  • FFT grid size N = 64 for SJS/WASP-148, 128 for TIC/GJ876
    Grid size chosen as trade-off between accuracy and cost; Table II explores 16, 32, 64 for SJS.
  • Integration time-step = 250 yr (SJS secular), 18 yr (SJS resonant), 1 yr (WASP-148), 3.6 d (TIC), 1.8 d (GJ876)
    Chosen to resolve fastest relevant secular/resonant timescale.
assumptions (5)
  • domain assumption Short-term dynamics of the planetary system is regular, with no overlap of mean-motion resonances causing significant diffusion of mean-motion frequencies.
    Invoked in Sect. II to justify the averaging; the method is not valid for strongly chaotic short-term motion.
  • domain assumption Fourier coefficients of H1 decay exponentially with harmonic order |k|.
    Invoked in Sect. IV.a (Eq. 29) to justify truncation; references Arnold (1963). Applicability is questionable for high eccentricities and masses, acknowledged in the TIC/GJ876 setup.
  • domain assumption The mean-motion corrected integrable Hamiltonian H'_0 = H0 + epsilon h0_1(0,0;Lambda) remains integrable and provides a better unperturbed frequency n'_0.
    Sect. III.c, Eq. (18); assumes low eccentricity/inclination dominance of the averaged perturbation.
  • ad hoc to paper The epsilon^2 corrections in the Lie-inverted initial conditions are negligible.
    Sect. IV.e, Eq. (33) drops the order-epsilon^2 terms in the averaging of the Lie transform; stated as typically small, but not tested for all systems.
  • domain assumption For resonant systems, delta-Lambda = O(sqrt(epsilon) Lambda_0), so the linearization of H'_0 introduces an O(epsilon) error in the Hamiltonian.
    Sect. III.b, Eqs. (25)-(27), used in the variant with fixed mean motions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Numerical approach to second-order canonical perturbation theory in the planetary 3-body problem: Application to exoplanets." pith.science (2026). https://pith.science/paper/UARNXQFS

@misc{pith2026250613745,
  author       = {Pith},
  title        = {Pith review of: Numerical approach to second-order canonical perturbation theory in the planetary 3-body problem: Application to exoplanets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UARNXQFS}},
  note         = {Machine review of arXiv:2506.13745}
}
read the original abstract

Extrasolar planetary systems commonly exhibit planets on eccentric orbits, with many systems located near or within mean-motion resonances, showcasing a wide diversity of orbital architectures. Such complex systems challenge traditional secular theories, which are limited to first-order approximations in planetary masses or rely on expansions in orbital elements--eccentricities, inclinations, and semi-major axis ratios--that are subject to convergence issues, especially in highly eccentric, inclined, or tightly-packed systems. To overcome these limitations, we develop a numerical approach to second-order perturbation theory based on the Lie transform formalism. Our method avoids the need for expansions in orbital elements, ensuring broader applicability and more robust convergence. We first outline the Hamiltonian framework for the 3-body planetary problem, and apply a canonical transformation to eliminate fast angle dependencies, deriving the secular Hamiltonian up to second order in the mass ratio. We then use the fast Fourier transform algorithm to numerically simulate, in an accurate way, the long-term evolution of planetary systems near or away from mean-motion resonances. Finally, we validate our methods against well-known planetary configurations, such as the Sun-Jupiter-Saturn system, as well as to exoplanetary systems like WASP-148, TIC 279401253 and GJ 876, demonstrating the applicability of our models across a wide range of planetary configurations.

Figures

Figures reproduced from arXiv: 2506.13745 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Amplitude of the Fourier transform of the variable [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Eccentricity evolution of Jupiter ( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Eccentricity evolution of the inner planet ( [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Eccentricity evolution of the inner planet ( [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Eccentricity evolution of the inner planet ( [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Eccentricity evolution of planet c ( [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

69 extracted references · 64 canonical work pages

  1. [1]

    (20) defines the (non-resonant) secular models considered in Sects

    Eq. (20) defines the (non-resonant) secular models considered in Sects. IV and V. d. Secular equations of motion. The long-term evo- lution of the planetary orbits is governed by Hamilton’s equations corresponding to the secular Hamiltonian given in Eq. (20). The momenta ˆΛ are integrals of motion for its flow, and the time evolution of the orbits is determ...

    work page

  2. [2]

    (30) We then compute the normalized value |ǫˆh0,κ 2 /h0 1| as a function of κ, at the initial conditions of the non-resonant model detailed in Sect

    We define the contribution to this term from a given harmonic order κ ∈ N+ as ˆh0,κ 2 = − 1 2 ∑ k∈Z2 |k|=κ [ ι{hk 1 , hk 1 }⋆ k · n′ 0 + k · ∂ ∂ ˆΛ ( |hk 1 |2 k · n′ 0 ) ] . (30) We then compute the normalized value |ǫˆh0,κ 2 /h0 1| as a function of κ, at the initial conditions of the non-resonant model detailed in Sect. V. The results are shown in Fig. 1,...

    work page

  3. [3]

    P. S. Laplace, M´ emoires de l’Acad´ emie Royale des Sci- ences de Paris 8, 201 (1773)

    work page

  4. [4]

    Lagrange, Nouveaux m´ emoires de l’Acad´ emie des Sci- ences et Belles-Lettres de Berlin 5, 125 (1781)

    J. Lagrange, Nouveaux m´ emoires de l’Acad´ emie des Sci- ences et Belles-Lettres de Berlin 5, 125 (1781)

    work page

  5. [5]

    Lagrange, Nouveaux m´ emoires de l’Acad´ emie des Sci- ences et Belles-Lettres de Berlin 5, 211 (1782)

    J. Lagrange, Nouveaux m´ emoires de l’Acad´ emie des Sci- ences et Belles-Lettres de Berlin 5, 211 (1782)

    work page

  6. [6]

    G. W. Hill, Astronomical Papers of the American Ephemeris 1, 315 (1882)

    work page

  7. [7]

    U. J. Le Verrier, Annales de l’Observatoire de Paris 1, 258 (1855)

    work page

  8. [8]

    Boquet, Annales de l’Observatoire de Paris 19, B.1 (1889)

    F. Boquet, Annales de l’Observatoire de Paris 19, B.1 (1889)

    work page

Show all 69 references
  1. [9]

    Laskar, Astronomy and Astrophysics 198, 341 (1988)

    J. Laskar, Astronomy and Astrophysics 198, 341 (1988)

    1988

  2. [10]

    Brouwer and A

    D. Brouwer and A. J. J. van Woerkom, Astronomical papers prepared for the use of the American ephemeris and nautical almanac 13, 81 (1950)

    1950

  3. [11]

    Laskar, Astronomy and Astrophysics 144, 133 (1985)

    J. Laskar, Astronomy and Astrophysics 144, 133 (1985)

    1985

  4. [12]

    Seager, Exoplanets, The University of Arizona Space Science Series (University of Arizona Press, Tucson, 2011)

    S. Seager, Exoplanets, The University of Arizona Space Science Series (University of Arizona Press, Tucson, 2011)

    2011

  5. [13]

    Laskar, Nature (London) 338, 237 (1989)

    J. Laskar, Nature (London) 338, 237 (1989)

    1989

  6. [14]

    Laskar, Icarus 88, 266 (1990)

    J. Laskar, Icarus 88, 266 (1990)

    1990

  7. [15]

    J. J. e. a. Lissauer, Astrophysical Journal, Supplemen t 197, 8 (2011), arXiv:1102.0543 [astro-ph.EP]

    2011 arXiv

  8. [16]

    J. N. Winn and D. C. Fabrycky, Annual Review of As- tronomy and Astrophysics 53, 409–447 (2015)

    2015

  9. [17]

    Kley, ISSI Scientific Reports Series 6, 39 (2006)

    W. Kley, ISSI Scientific Reports Series 6, 39 (2006)

    2006

  10. [18]

    Kozai, Astronomical Journal 67, 591 (1962)

    Y. Kozai, Astronomical Journal 67, 591 (1962)

    1962

  11. [19]

    P. A. Hansen, Abhandl. d. K. S. Ges. d. Wissensch (1852)

  12. [20]

    Tisserand, Trait´ e de m´ ecanique c´ eleste, Vol

    F. Tisserand, Trait´ e de m´ ecanique c´ eleste, Vol. 3 (Gauthier-Villars, Paris, 1894)

  13. [21]

    M. H. Lee and S. J. Peale, The Astrophysical Journal 592, 1201–1216 (2003)

    2003

  14. [22]

    M. L. Lidov, Planetary and Space Science 9, 719 (1962)

    1962

  15. [23]

    E. B. Ford, B. Kozinsky, and F. A. Rasio, Astrophys. J. 535, 385 (2000)

    2000

  16. [24]

    A. S. Libert and J. Henrard, Icarus 183, 186 (2006)

    2006

  17. [25]

    Laskar and G

    J. Laskar and G. Bou´ e, Astronomy and Astrophysics522, A60 (2010)

    2010

  18. [26]

    A. S. Libert and J. Henrard, Celestial Mechanics and 17 Dynamical Astronomy 93, 187 (2005)

    2005

  19. [27]

    Beaug´ e and T

    C. Beaug´ e and T. A. Michtchenko, Monthly Notices of the RAS 341, 760 (2003)

    2003

  20. [28]

    A. S. Libert and M. Sansottera, Celestial Mechanics and Dynamical Astronomy 117, 149 (2013)

    2013

  21. [29]

    Sansottera and A

    M. Sansottera and A. S. Libert, Celestial Mechanics and Dynamical Astronomy 131, 38 (2019)

    2019

  22. [30]

    J. B. Delisle, J. Laskar, A. C. M. Correia, and G. Bou´ e, Astronomy and Astrophysics 546, A71 (2012)

    2012

  23. [31]

    Callegari, S

    N. Callegari, S. Ferraz-Mello, and T. A. Michtchenko, Celestial Mechanics and Dynamical Astronomy 94, 381 (2006)

    2006

  24. [32]

    Batygin and A

    K. Batygin and A. Morbidelli, Astronomy and Astro- physics 556, A28 (2013)

    2013

  25. [33]

    Schubart, SAO Special Report 149 (1964)

    J. Schubart, SAO Special Report 149 (1964)

    1964

  26. [34]

    Gauss, Werke 3, 331 (1818)

    C. Gauss, Werke 3, 331 (1818)

  27. [35]

    Musen, Celestial Mechanics 2, 41 (1970)

    P. Musen, Celestial Mechanics 2, 41 (1970)

    1970

  28. [36]

    Laskar, Analytical framework in poincare variables for the motion of the solar system, in Predictability, Stability, and Chaos in N-Body Dynamical Systems (Springer US, 1991)

    J. Laskar, Analytical framework in poincare variables for the motion of the solar system, in Predictability, Stability, and Chaos in N-Body Dynamical Systems (Springer US, 1991)

    1991

  29. [37]

    Moons and A

    M. Moons and A. Morbidelli, Celestial Mechanics and Dynamical Astronomy 57, 99 (1993)

    1993

  30. [38]

    Moons, Celestial Mechanics and Dynamical Astron- omy 60, 173 (1994)

    M. Moons, Celestial Mechanics and Dynamical Astron- omy 60, 173 (1994)

    1994

  31. [39]

    Laskar, in Chaos, Resonance, and Collective Dynam- ical Phenomena in the Solar System , IAU Symposium, Vol

    J. Laskar, in Chaos, Resonance, and Collective Dynam- ical Phenomena in the Solar System , IAU Symposium, Vol. 152, edited by S. Ferraz-Mello (Kluwer Academic Publishers, Dordrecht, 1992) p. 1

    1992

  32. [40]

    Hori, Publications of the Astronomical Society of Japan 18, 287 (1966)

    G. Hori, Publications of the Astronomical Society of Japan 18, 287 (1966)

    1966

  33. [41]

    Deprit, Celestial Mechanics 1, 12 (1969)

    A. Deprit, Celestial Mechanics 1, 12 (1969)

    1969

  34. [42]

    Locatelli and A

    U. Locatelli and A. Giorgilli, Celestial Mechanics and Dynamical Astronomy 78, 47 (2000)

    2000

  35. [43]

    Mogavero and J

    F. Mogavero and J. Laskar, Astronomy and Astrophysics 662, L3 (2022), arXiv:2205.03298 [astro-ph.EP]

    2022 arXiv

  36. [44]

    The computational cost involves N 2 evaluations per each function, followed by a 2D FFT, with complex- ity 2 N 2 log(N ) per function

    on the evaluated data to obtain the Fourier coef- ficients. The computational cost involves N 2 evaluations per each function, followed by a 2D FFT, with complex- ity 2 N 2 log(N ) per function. Note that performing an FFT with N × N points yields N/2 positive and N/2 negative ...

    2023

  37. [45]

    Morbidelli, Modern celestial mechanics: aspects of so- lar system dynamics (Taylor & Francis, 2002)

    A. Morbidelli, Modern celestial mechanics: aspects of so- lar system dynamics (Taylor & Francis, 2002)

    2002

  38. [46]

    V. I. Arnold, Russian Mathematical Surveys 18, 9 (1963)

    1963

  39. [47]

    J. W. Cooley and J. W. Tukey, Mathematics of Compu- tation 19, 297 (1965)

    1965

  40. [48]

    Abramowitz and I

    M. Abramowitz and I. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965)

    1965

  41. [49]

    Sauer, Numerical Analysis (Pearson, Boston, 2012)

    T. Sauer, Numerical Analysis (Pearson, Boston, 2012)

    2012

  42. [50]

    Laskar, Astronomy and Astrophysics 287, L9 (1994)

    J. Laskar, Astronomy and Astrophysics 287, L9 (1994)

    1994

  43. [51]

    Laskar and J

    J. Laskar and J. Simon, Celestial mechanics 43, 37 (1987)

    1987

  44. [52]

    Virtanen, R

    P. Virtanen, R. Gommers, Oliphant, and et al. , Nature Methods 17, 261 (2020)

    2020

  45. [53]

    Farr´ es, J

    A. Farr´ es, J. Laskar, S. Blanes, F. Casas, J. Makazaga, and A. Murua, Celestial Mechanics and Dynamical As- tronomy 116, 141 (2013)

    2013

  46. [54]

    G. W. Hill, Astronomical Journal 17, 81 (1897)

  47. [55]

    Duriez, Astronomy and Astrophysics 54, 93 (1977)

    L. Duriez, Astronomy and Astrophysics 54, 93 (1977)

    1977

  48. [56]

    Varadi, M

    F. Varadi, M. Ghil, and W. M. Kaula, Icarus 139, 286 (1999)

    1999

  49. [57]

    Michtchenko and S

    T. Michtchenko and S. Ferraz-Mello, Icarus 149, 357 (2001b)

    2001

  50. [58]

    Locatelli and A

    U. Locatelli and A. Giorgilli, Discrete and Continuous Dynamical Systems - B 7, 377 (2007)

    2007

  51. [59]

    Mogavero and J

    F. Mogavero and J. Laskar, Astronomy and Astrophysics 655, A1 (2021), arXiv:2105.14976 [astro-ph.EP]

    2021 arXiv

  52. [60]

    Carpino, A

    M. Carpino, A. Milani, and A. M. Nobili, Astronomy and Astrophysics 181, 182 (1987)

    1987

  53. [61]

    Laskar, in Hamiltonian Systems and Fourier Analy- sis: New Prospects For Gravitational Dynamics , edited by D

    J. Laskar, in Hamiltonian Systems and Fourier Analy- sis: New Prospects For Gravitational Dynamics , edited by D. Benest, C. Froeschl´ e, and E. Lega (Cambridge Scientific Publishers Ltd, Cambridge, 2005) pp. 93–114, arXiv:math/0305364

    2005 arXiv

  54. [62]

    H´ ebrard andet al

    G. H´ ebrard andet al. , Astronomy and Astrophysics 640, A32 (2020)

    2020

  55. [63]

    J. M. Almenara and et al. , Astronomy and Astrophysics 663, A134 (2022)

    2022

  56. [64]

    , Astrophysical Journal, Letters 946, L36 (2023)

    Bozhilov and et al. , Astrophysical Journal, Letters 946, L36 (2023)

    2023

  57. [65]

    Laskar, Physica D Nonlinear Phenomena 67, 257 (1993)

    J. Laskar, Physica D Nonlinear Phenomena 67, 257 (1993)

    1993

  58. [66]

    Couetdic, J

    J. Couetdic, J. Laskar, A. C. M. Correia, M. Mayor, and S. Udry, Astronomy and Astrophysics 10.1051/0004-6361/200913635 (2010)

    2010 doi

  59. [67]

    G. W. Marcy, R. P. Butler, D. Fischer, S. S. Vogt, J. J. Lissauer, and E. J. Rivera, apj 556, 296 (2001)

    2001

  60. [68]

    Laughlin and J

    G. Laughlin and J. E. Chambers, apjl 551, L109 (2001), arXiv:astro-ph/0101423 [astro-ph]

    2001 arXiv

  61. [69]

    Rein and D

    H. Rein and D. S. Spiegel, Monthly Notices of the RAS 446, 1424 (2015), arXiv:1409.4779 [astro-ph.EP]

    2015 arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.