Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance

Abstract

Extrasolar systems with planets on eccentric orbits close to or in mean-motion resonances are common. The classical low-order resonant Hamiltonian expansion is unfit to describe the long-term evolution of these systems. We extend the Lagrange-Laplace secular approximation for coplanar systems with two planets by including (near-)resonant harmonics and realize an expansion at high order in the eccentricities of the resonant Hamiltonian both at orders one and two in the masses. We show that the expansion at first order in the masses gives a qualitative good approximation of the dynamics of resonant extrasolar systems with moderate eccentricities, while the second order is needed to reproduce more accurately their orbital evolutions. The resonant approach is also required to correct the secular frequencies of the motion given by the Laplace-Lagrange secular theory in the vicinity of a mean-motion resonance. The dynamical evolutions of four (near-)resonant extrasolar systems are discussed, namely GJ 876 (2:1 resonance), HD 60532 (3:1), HD 108874 and GJ 3293 (close to 4:1).

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Notes

  1. 1.

    Here, we expand around the initial values, but the average values over a long-term numerical integration could also be considered (see, e.g., Sansottera et al. (2013)).

  2. 2.

    Let us note that Rivera et al. (2010) have revealed the presence of an additional planet in a three-body Laplace resonance with the previously two known giant planets.

  3. 3.

    Let us stress that, to better visualize the evolution of \(\sigma _1\), we plot the evolution on a much smaller timescale.

References

  1. Alves, A., Michtchenko, T., Tadeu dos Santos, M.: Dynamics of the 3/1 planetary mean-motion resonance. An application to the HD60532 b-c planetary system. CeMDA 124, 311–334 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  2. Astudillo-Defru, N., Bonfils, X., Delfosse, X., et al.: The HARPS search for southern extra-solar planets XXXV. Planetary systems and stellar activity of the M dwarfs GJ 3293, GJ 3341, and GJ 3543. A&A 575, A119 (2015)

    ADS  Article  Google Scholar 

  3. Astudillo-Defru, N., Forveille, T., Bonfils, X., et al.: The HARPS search for southern extra-solar planets XLI. A dozen planets around the M dwarfs GJ 3138, GJ 3323, GJ 273, GJ 628, and GJ 3293. A&A 602, A88 (2017)

    ADS  Article  Google Scholar 

  4. Batygin, K., Morbidelli, A.: Analytical treatment of planetary resonances. A&A 556, A28 (2013)

    ADS  Article  Google Scholar 

  5. Beaugé, C., Michtchenko, T.: Modelling the high-eccentricity planetary three-body problem. Application to the GJ876 planetary system. MNRAS 341, 760 (2003)

    ADS  Article  Google Scholar 

  6. Butler, R.P., Marcy, G.W., Vogt, S.S., Fischer, D.A., Henry, G.W., Laughlin, G., et al.: Seven new Keck planets orbiting G and K dwarfs. Astrophys. J. 582, 455–466 (2003)

    ADS  Article  Google Scholar 

  7. Callegari Jr., N., Michtchenko, T.A., Ferraz-Mello, S.: Dynamics of two planets in the 2/1 mean-motion resonance. CeMDA 556(89), 201–234 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  8. Callegari Jr., N., Ferraz-Mello, S., Michtchenko, T.A.: Dynamics of two planets in the 3/2 mean-motion resonance: application to the planetary system of the pulsar PSR B1257+12. CeMDA 94, 381–397 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  9. Celletti, A., Chierchia, L.: KAM stability and celestial mechanics. Mem. Am. Math. Soc. 187, 1–134 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Correia, A.C.M., Udry, S., Mayor, M., et al.: The HARPS search for southern extra-solar planets—XVI. HD45364, a pair of planets in a 3:2 mean motion resonance. A&A 496, 521–526 (2009)

    ADS  Article  Google Scholar 

  11. Desort, M., Lagrange, A.-M., Galland, F., Beust, H., Udry, S., Mayor, M., et al.: Extrasolar planets and brown dwarfs around A-F type stars. V. A planetary system found with HARPS around the F6IV-V star HD 60532. Astron. Astrophys. 491, 883–888 (2008)

  12. Duriez, L.: Le développement de la fonction perturbatrice, Les méthodes modernes de la mécanique céleste: théorie des perturbations et chaos intrinsèque / comptes rendus de la 13e Ecole de printemps d’astrophysique de Goutelas, France, 24–29 avril 1989 ; éd. par Daniel Benest et Claude Froeschlé. ISBN 2-86332-091-2. http://adsabs.harvard.edu/abs/1990mmcm.conf (1989a)

  13. Duriez, L.: Le problème des deux corps revisité, Les méthodes modernes de la mécanique céleste: théorie des perturbations et chaos intrinsèque / comptes rendus de la 13e Ecole de printemps d’astrophysique de Goutelas, France, 24–29 avril 1989 ; éd. par Daniel Benest et Claude Froeschlé. ISBN 2-86332-091-2. http://adsabs.harvard.edu/abs/1990mmcm.conf (1989b)

  14. Ferraz-Mello, S.: The convergence domain of the Laplacian expansion of the disturbing function. CeMDA 58, 37–52 (1994)

    ADS  MathSciNet  Article  Google Scholar 

  15. Gabern, F., Jorba, A., Locatelli, U.: On the construction of the Kolmogorov normal form for the Trojan asteroids. Nonlinearity 18(4), 1705–1734 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  16. Giorgilli A (1995) Quantitative methods in classical perturbation theory. In: Roy AE, Steves BA (eds) From newton to chaos. NATO ASI series (Series B: Physics), vol 336. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1085-1_3

    Google Scholar 

  17. Giorgilli, A., Sansottera, M.: Methods of algebraic manipulation in perturbation theory. Workshop Ser. Asoc. Argent. Astron. 3, 147–183 (2011)

    ADS  Google Scholar 

  18. Giorgilli, A., Locatelli, U., Sansottera, M.: Kolmogorov and Nekhoroshev theory for the problem of three bodies. CeMDA 104, 159–173 (2009)

    ADS  MathSciNet  Article  Google Scholar 

  19. Giorgilli, A., Locatelli, U., Sansottera, M.: Secular dynamics of a planar model of the Sun–Jupiter–Saturn–Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories. Regular Chaotic Dyn. 22, 54–77 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  20. Hadjidemetriou, J.: Resonant periodic motion and the stability of extrasolar planetary systems. CeMDA 83, 141 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  21. Henrard, J.: The algorithm of the inverse for lie transform, recent advances in dynamical astronomy. Astrophys. Space Sci. Libr. 39, 248–257 (1973)

    ADS  Google Scholar 

  22. Laskar, J.: Systèmes de variables et éléments, Les méthodes modernes de la mécanique céleste: théorie des perturbations et chaos intrinsèque / comptes rendus de la 13e Ecole de printemps d’astrophysique de Goutelas, France, 24-29 avril 1989 ; éd. par Daniel Benest et Claude Froeschlé. ISBN 2-86332-091-2. http://adsabs.harvard.edu/abs/1990mmcm.conf (1989)

  23. Laskar, J.: Secular evolution over 10 million years. A&A 198, 341–362 (1988)

    ADS  Google Scholar 

  24. Laskar, J., Correia, A.C.M.: HD60532, a planetary system in a 3:1 mean motion resonance. A&A 496, L5 (2009)

    ADS  Article  Google Scholar 

  25. Laskar, J., Robutel, P.: Stability of the planetary three-body problem—I. Expans. Planet. Hamilt. CeMDA 62, 193–217 (1995)

    MATH  Google Scholar 

  26. Laskar, J., Robutel, P.: High order symplectic integrators for perturbed Hamiltonian systems. CeMDA 80, 39–62 (2001)

    ADS  MathSciNet  Article  Google Scholar 

  27. Laughlin, G., Chambers, J.: Short-term dynamical interactions among extrasolar planets. ApJ 551, L109–L113 (2001)

    ADS  Article  Google Scholar 

  28. Libert, A.-S., Henrard, J.: Analytical approach to the secular behaviour of exoplanetary systems. CeMDA 93, 187–200 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  29. Libert, A.-S., Henrard, J.: Analytical study of the proximity of exoplanetary systems to mean-motion resonances. A&A 461, 759–763 (2007)

    ADS  Article  Google Scholar 

  30. Libert, A.-S., Sansottera, M.: On the extension of the Laplace–Lagrange secular theory to order two in the masses for extrasolar systems. CeMDA 117, 149–168 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  31. Locatelli, U., Giorgilli, A.: Invariant tori in the Sun–Jupiter–Saturn system. DCDS-B 7, 377–398 (2007)

    MathSciNet  Article  Google Scholar 

  32. Marcy, G., Butler, P., Fisher, D., et al.: A pair of resonant planets orbiting GJ 876. ApJ 556, 296 (2001)

    ADS  Article  Google Scholar 

  33. Poincaré, H.: Les méthodes nouvelles de la Mécanique Céleste. Gauthier-Villars, Paris (1893)

    Google Scholar 

  34. Rivera, E., Laughlin, G., Butler, P., et al.: The Lick-Carnegie exoplanet survey: a uranus-mass fourth planet for GJ 876 in an extrasolar Laplace configuration. ApJ 719, 890 (2010)

    ADS  Article  Google Scholar 

  35. Robutel, P.: Stability of the planetary three-body problem—II. KAM theory existence quasiperiodic motions. CeMDA 62, 219–261 (1995)

    ADS  MathSciNet  Article  Google Scholar 

  36. Sansottera, M., Locatelli, U., Giorgilli, A.: A semi-analytic algorithm for constructing lower dimensional elliptic Tori in planetary systems. CeMDA 111, 337–361 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  37. Sansottera, M., Locatelli, U., Giorgilli, A.: On the stability of the secular evolution of the planar Sun–Jupiter–Saturn–Uranus system. Math. Comput. Simulat. 88, 1–14 (2013)

    MathSciNet  Article  Google Scholar 

  38. Sansottera, M., Grassi, L., Giorgilli, A.: On the relativistic Lagrange–Laplace secular dynamics for extrasolar systems. Proc. IAU Symp. S310, 74–77 (2015)

    ADS  Google Scholar 

  39. Sundman, K.F.: Sur les conditions nécessaires et suffisantes pour la convergence du développement de la fonction perturbatrice dans le mouvement plan. Öfvers. Fin. Vetensk. Soc. Förh 58A, 24 (1916)

    Google Scholar 

  40. Tan, X., Payme, M., Lee, M.H. et al.: Characterizing the orbital and dynamical state of the HD 82943 planetary system with Keck radial velocity data. Astrophys. J. 777, id. 101, pp. 21 (2013)

    ADS  Article  Google Scholar 

  41. Veras, D.: A resonant-term-based model including a nascent disk, precession, and oblateness: application to GJ 876. CeMDA 99, 197–243 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  42. Vogt, S.S., Butler, R.P., Marcy, G.W., Fischer, D.A., Henry, G.W., Laughlin, G., Wright, J.T., Johnson, J.A.: Five new multicomponent planetary systems. Astrophys. J. 632, 638–658 (2005)

    ADS  Article  Google Scholar 

  43. Wittenmyer, R.A., Tan, X., Lee, M.H., et al.: A detailed analysis of the HD 73526 2:1 resonant planetary system. ApJ 780, id. 140, pp. 9 (2014)

    ADS  Article  Google Scholar 

  44. Wright, J.T., Upadhyay, S., Marcy, G.W., Fisher, D.A., et al.: Ten new and updated multiplanet systems and a survey of exoplanetary systems. Astrophys. J. 693, 1084–1099 (2009)

    ADS  Article  Google Scholar 

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Acknowledgements

The authors seize the opportunity of the Topical Collection for the 50th birthday of CM&DA to dedicate this paper to the memory of Jacques Henrard. This work follows the path traced in his two contributions published in the first volume of Celestial Mechanics. The work of M. S. has been partially supported by the National Group of Mathematical Physics (GNFM-INdAM). Computational resources have been provided by the PTCI (Consortium des Équipements de Calcul Intensif CECI), funded by the FNRS-FRFC, the Walloon Region, and the University of Namur (Conventions No. 2.5020.11, GEQ U.G006.15, 1610468 et RW/GEQ2016).

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Appendix Low-order expansions of \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}1)}\) and \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}2)}\) for GJ 876

Appendix Low-order expansions of \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}1)}\) and \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}2)}\) for GJ 876

We report here the low-order expansion of \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}1)} = {\overline{{\mathcal {H}}}}^{({\mathcal {T}})} + {\widetilde{{\mathcal {H}}}}^{({\mathcal {T}})}\) and \({\mathcal {H}}_{\mathrm{res}}^{({\mathcal {O}}2)} = {\overline{{\mathcal {H}}}}^{({\mathcal {O}}2)} + {\widetilde{{\mathcal {H}}}}^{({\mathcal {O}}2)}\) (see (5) and  (10), respectively) for GJ 876. We refer to Table 1 for the physical and orbital parameters of the system and Subsect. 4.2 for a complete description of the system (Tables 2, 3).

Table 2 Low-order secular contributions of the resonant Hamiltonians at order one and two in the masses, namely \({\overline{{\mathcal {H}}}}^{({\mathcal {T}})}\) and \({\overline{{\mathcal {H}}}}^{({\mathcal {O}}2)}\)
Table 3 Low-order resonant contributions of the resonant Hamiltonians at orders one and two in the masses, namely \({\widetilde{{\mathcal {H}}}}^{({\mathcal {T}})}\) and \({\widetilde{{\mathcal {H}}}}^{({\mathcal {O}}2)}\)

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Sansottera, M., Libert, AS. Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance. Celest Mech Dyn Astr 131, 38 (2019). https://doi.org/10.1007/s10569-019-9913-5

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Keywords

  • Extrasolar systems
  • n-Body problem
  • Mean-motion resonances
  • Perturbation theory