Hayes (2007) showed that within the current observational uncertainty of the Jovian planets' positions - a few parts in ten million - the Newtonian outer Solar System (the Sun plus Jupiter, Saturn, Uranus and Neptune) admits both chaotic solutions with Lyapunov times ranging from about 2 to 230 million years and solutions showing no evidence of chaos over billion-year integrations, interleaved on phase-space scales at least an order of magnitude finer than those addressed by existing three-body mean-motion resonance theory. The unresolved problem, following the source paper's statement of future work, is linked and twofold: determine the dynamical state of the actual outer Solar System and its true Lyapunov time, which will require a more accurate observational determination of the orbits of the jovian planets; and achieve a full understanding of the nature and consequences of the chaos, including the dynamical mechanism, possibly a web of four-body resonances, that generates the fine-scale interleaving of chaotic and regular trajectories.

Background

The term outer Solar System is used here for the system of the Sun and the four giant planets Jupiter, Saturn, Uranus and Neptune, which is often studied numerically as an isolated five-body system. The Solar System is known to be practically stable - no planet is likely to collide with another or be ejected over the next several billion years - but practical stability is compatible with chaos. A trajectory is chaotic if the distance between two solutions whose initial conditions differ by an infinitesimal amount d(0)d(0) grows approximately as d(0)eλtd(0)e^{\lambda t}, where λ\lambda is the Lyapunov exponent; its inverse, the Lyapunov time 1/λ1/\lambda, is the timescale beyond which the system's state cannot be accurately predicted. A regular (non-chaotic) trajectory instead shows only polynomial divergence of nearby solutions.

The existence of chaos among the Jovian planets was contested for about fifteen years before the source paper. Averaged secular integrations by Laskar (1989), which smooth over the orbital motion, found the outer planets to be non-chaotic, while the direct nine-planet integration of Sussman and Wisdom (1992) found chaos with a Lyapunov time that varied puzzlingly with integration parameters. The puzzle sharpened when Murray and Holman (1999) proposed that chaos in the outer Solar System is driven by three-body mean-motion resonances: resonances defined not by a near-integer relation between the mean motions (average orbital angular frequencies) of two planets, but by a relation among the mean motions of three planets, such as n1−2n2+n3≈0n_1-2n_2+n_3\approx 0. Because no pair of giant planets is near a two-body resonance, only these much narrower three-body resonances can trap the system. Guzzo (2005) numerically detected a dense web of such three-planet resonances in the outer Solar System precisely where Murray and Holman's theory predicts them.

Against this background, Hayes (2007) asked which regime the real outer Solar System occupies. The positions of the giant planets are known from planetary ephemerides - numerical tables fitted to observations, in his case the Jet Propulsion Laboratory's DE405 - only to a few parts in ten million: the semi-major axes of Saturn, Uranus and Neptune were then known to roughly 3, 5 and 7 parts in 10710^7. Integrating many initial conditions drawn from within this observational error box with three independent high-precision schemes, he found chaotic solutions with Lyapunov times from 2 million years (Myr) to 230 Myr coexisting with solutions showing no evidence of chaos over 200 Myr to 1 Gyr (a billion years); about 70% of sampled initial conditions were chaotic over 200 Myr, while fewer than 10% remained regular over 1 Gyr, and the most authoritative initial condition fitted to all DE405 observations showed no evidence of chaos after 1 Gyr. He further observed that this fine-scale interleaving of chaotic and regular trajectories occurs at relative scales at least an order of magnitude below the roughly one-part-in-10610^6 scale about which existing three-body resonance theory makes statements, and that it is unclear whether extensions of that theory - for example to four-body resonances, of which there can be at most a limited hierarchy in a four-planet system - can explain it. His model also replaced the inner planets by an augmented central mass and ignored a list of physical effects, judging them unlikely to alter the qualitative presence of chaos but leaving their quantitative influence on the Lyapunov-time distribution unestablished. A companion review of chaotic diffusion in the Solar System is Laskar (2008).

The author's own follow-up, Hayes (2008), strengthened the picture: perturbing the initial conditions by as little as one part in 10710^7 can switch the system between chaotic and regular behavior over 200 Myr, and the canonical DE405 initial condition, when integrated to 5 Gyr, is weakly chaotic with a Lyapunov time of 200-400 Myr. Whether the actual Solar System is chaotic, and on what timescale, therefore remains observationally undecided, and the mechanism behind the delicate phase-space structure remains theoretically unexplained.

Problem Statement

Decide the dynamical character of the actual outer Solar System - the Sun plus the four giant planets - and account for the structure of its phase space within the current observational error box. This comprises two linked objectives, following the source paper's statement of future work: "Determining the true Lyapunov time of the outer Solar System will require a more accurate observational determination of the orbits of the jovian planets. A full understanding of the nature and consequences of the chaos may require further theoretical development."

  1. Determine whether the trajectory realized by the real Jovian planets is chaotic, and its true Lyapunov time, given that the observational uncertainty of their positions (semi-major axes of Saturn, Uranus and Neptune known to roughly 3, 5 and 7 parts in 10710^7 at the time of the source paper) admits both chaotic solutions with Lyapunov times from about 2 Myr to 230 Myr and solutions showing no evidence of chaos over 200 Myr to 1 Gyr. As the source states, settling this question will require a more accurate observational determination of the orbits of the jovian planets; within the current observational error box the objective is to establish the strongest defensible statement of the actual system's dynamical character, and to state what improvement in those observations would be needed to settle the true Lyapunov time, which the present data may leave undetermined.

  2. Explain the dynamical origin of the observed fine-scale interleaving of chaotic and regular trajectories at relative scales of order one part in 10710^7 - at least an order of magnitude below the roughly one-part-in-10610^6 scale about which existing three-body mean-motion resonance theory makes statements. Candidate hypotheses include a web of four-body mean-motion resonances; the source notes that if the fine structure persists at still smaller scales, even four-body resonance theory may not suffice (there are no five-body resonances in a four-planet system), so it is unclear whether extensions of existing resonance theory can fully explain the structure.

These objectives are linked: objective 1 is the target fact, and objective 2 explains the phase-space delicacy that leaves objective 1 observationally undecided and is needed for the full understanding of the nature and consequences of the chaos that the source calls for. As background rather than as a required goal, the source notes that its model ignored many physical effects that may be important to the detailed motion of the planets, while expressing the belief that none of these effects will alter the chaotic nature of solutions.

References

  1. Wayne B. Hayes (2007), 'Is the outer Solar System chaotic?', Nature Physics 3(10), 689-691, doi:10.1038/nphys728, arXiv:astro-ph/0702179, https://doi.org/10.1038/nphys728
  2. Wayne B. Hayes (2008), 'Surfing on the edge: chaos versus near-integrability in the system of Jovian planets', Monthly Notices of the Royal Astronomical Society 386(1), 295-306, doi:10.1111/j.1365-2966.2008.13024.x, arXiv:0708.2875, https://doi.org/10.1111/j.1365-2966.2008.13024.x
  3. Wayne B. Hayes, Anton V. Malykh, Christopher M. Danforth (2010), 'The interplay of chaos between the terrestrial and giant planets', Monthly Notices of the Royal Astronomical Society 407(3), 1859-1865, doi:10.1111/j.1365-2966.2010.17027.x, https://doi.org/10.1111/j.1365-2966.2010.17027.x
  4. N. Murray, M. Holman (1999), 'The Origin of Chaos in the Outer Solar System', Science 283(5409), 1877-1881, doi:10.1126/science.283.5409.1877, https://doi.org/10.1126/science.283.5409.1877
  5. Massimiliano Guzzo (2005), 'The web of three-planet resonances in the outer Solar System', Icarus 174(1), 273-284, doi:10.1016/j.icarus.2004.10.015, https://doi.org/10.1016/j.icarus.2004.10.015
  6. Massimiliano Guzzo (2006), 'The web of three-planet resonances in the outer Solar System: II. A source of orbital instability for Uranus and Neptune', Icarus 181(2), 475-485, doi:10.1016/j.icarus.2005.11.019, https://doi.org/10.1016/j.icarus.2005.11.019
  7. Gerald Jay Sussman, Jack Wisdom (1992), 'Chaotic Evolution of the Solar System', Science 257(5066), 56-62, doi:10.1126/science.257.5066.56, https://doi.org/10.1126/science.257.5066.56
  8. J. Laskar (1989), 'A numerical experiment on the chaotic behaviour of the Solar System', Nature 338, 237-238, doi:10.1038/338237a0, https://doi.org/10.1038/338237a0
  9. J. Laskar (2008), 'Chaotic diffusion in the Solar System', Icarus 196(1), 1-15, doi:10.1016/j.icarus.2008.02.017, https://doi.org/10.1016/j.icarus.2008.02.017
  10. Alice C. Quillen (2011), 'Three-body resonance overlap in closely spaced multiple-planet systems', Monthly Notices of the Royal Astronomical Society 418(2), 1043-1054, doi:10.1111/j.1365-2966.2011.19555.x, https://doi.org/10.1111/j.1365-2966.2011.19555.x
  11. Tabare Gallardo (2014), 'Atlas of three body mean motion resonances in the Solar System', Icarus 231, 273-286, doi:10.1016/j.icarus.2013.12.020, https://doi.org/10.1016/j.icarus.2013.12.020
  12. Tabare Gallardo, Leonardo Coito, Luciana Badano (2016), 'Planetary and satellite three body mean motion resonances', Icarus 274, 83-98, doi:10.1016/j.icarus.2016.03.018, https://doi.org/10.1016/j.icarus.2016.03.018
  13. Antoine C. Petit (2021), 'An integrable model for first-order three-planet mean motion resonances', Celestial Mechanics and Dynamical Astronomy 133, 39, doi:10.1007/s10569-021-10035-7, https://doi.org/10.1007/s10569-021-10035-7
  14. Jeremy Rath, Sam Hadden, Yoram Lithwick (2022), 'The Criterion for Chaos in Three-planet Systems', The Astrophysical Journal 932(1), 61, doi:10.3847/1538-4357/ac5f57, arXiv:2110.02956, https://doi.org/10.3847/1538-4357/ac5f57
  15. Cezary Migaszewski, Mariusz Slonina, Krzysztof Gozdziewski (2012), 'A dynamical analysis of the Kepler-11 planetary system', Monthly Notices of the Royal Astronomical Society 427(1), 770-789, doi:10.1111/j.1365-2966.2012.21976.x, arXiv:1205.0822, https://doi.org/10.1111/j.1365-2966.2012.21976.x
  16. E. Woillez, F. Bouchet (2017), 'Long-term influence of asteroids on planet longitudes and chaotic dynamics of the solar system', Astronomy & Astrophysics 607, A62, doi:10.1051/0004-6361/201731576, arXiv:1707.03577, https://doi.org/10.1051/0004-6361/201731576
  17. Richard E. Zeeve (2015), 'Dynamic Stability of the Solar System: Statistically Inconclusive Results from Ensemble Integrations', The Astrophysical Journal 798(1), 8, doi:10.1088/0004-637X/798/1/8, arXiv:1506.07602, https://doi.org/10.1088/0004-637X/798/1/8
  18. Alexander Perminov, Eduard Kuznetsov (2020), 'The orbital evolution of the Sun-Jupiter-Saturn-Uranus-Neptune system on long time scales', Astrophysics and Space Science 365, 144, doi:10.1007/s10509-020-03855-w, https://doi.org/10.1007/s10509-020-03855-w
  19. Ryan S. Park, William M. Folkner, James G. Williams, Dale H. Boggs (2021), 'The JPL Planetary and Lunar Ephemerides DE440 and DE441', The Astronomical Journal 161(3), 105, doi:10.3847/1538-3881/abd414, https://doi.org/10.3847/1538-3881/abd414
  20. Federico Mogavero, Nam H. Hoang, Jacques Laskar (2023), 'Timescales of Chaos in the Inner Solar System: Lyapunov Spectrum and Quasi-integrals of Motion', Physical Review X 13, 021018, doi:10.1103/PhysRevX.13.021018, arXiv:2305.01683, https://doi.org/10.1103/PhysRevX.13.021018