Secondary resonances and the boundary of effective stability of Trojan motions

Original Article
  • 17 Downloads
Part of the following topical collections:
  1. Close Approaches and Collisions in Planetary Systems

Abstract

One of the most interesting features in the libration domain of co-orbital motions is the existence of secondary resonances. For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced ‘basic Hamiltonian model’ \(H_\mathrm{b}\) for Trojan dynamics (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez et al. in Celest Mech Dyn Astron 126:519, 2016): we show that the inner border of the secondary resonance of lowermost order, as defined by \(H_\mathrm{b}\), provides a good estimation of the region in phase space for which the orbits remain regular regardless of the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an ‘asymmetric expansion’ of the Hamiltonian around the libration points, which speeds up convergence. Applications to the determination of the effective stability domain for exoplanetary Trojans (planet-sized objects or asteroids) which may accompany giant exoplanets are discussed.

Keywords

Trojan dynamics Resonant dynamics Celestial Mechanics Perturbation theory Normal form 

Notes

Acknowledgements

Useful discussions with Prof. U. Locatelli are gratefully acknowledged. R.I.P. was supported by the Research Comittee of the Academy of Athens, under the Grant 200/854.

References

  1. Beaugé, C., Sándor, Z., Érdi, B., Süli, A.: Co-orbital terrestrial planets in exoplanetary systems: a formation scenario. Astron. Astrophys. 463, 359 (2007)ADSCrossRefGoogle Scholar
  2. Chirikov, B.V., Lieberman, M.A., Shepelyansky, D.L., Vivaldi, F.M.: A theory of modulational diffusion. Physica D 12, 289 (1985)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. Cresswell, P., Nelson, R.P.: On the growth and stability of Trojan planets. Astron. Astrophys. 493, 1141 (2009)ADSCrossRefMATHGoogle Scholar
  4. Dobrovolskis, A.: Effects of Trojan exoplanets on the reflex motions of their parent stars. Icarus 226, 1635 (2013)ADSCrossRefGoogle Scholar
  5. Dvorak, R., Bazsó, A., Zhou, L.Y.: Where are the Uranus Trojans? Celest. Mech. Dyn. Astron. 107, 51 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. Efthymiopoulos, C.: Canonical perturbation theory, stability and diffusion in Hamiltonian systems: applications in dynamical astronomy. In: Cincotta, P.M., Giordano, C.M., Efthymiopoulos, C. (eds.) Third La Plata International School on Astronomy and Geophysics: Chaos, Diffusion and Non-integrability in Hamiltonian Systems—Applications to Astronomy, p. 3 (2012)Google Scholar
  7. Efthymiopoulos, C.: High order normal form stability estimates for co-orbital motion. Celest. Mech. Dyn. Astron. 117, 101 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. Efthymiopoulos, C., Contopoulos, G., Voglis, N.: Cantori, islands and asymptotic curves in the stickiness region. Celest. Mech. Dyn. Astron. 73, 221 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. Efthymiopoulos, C., Giorgilli, A., Contopoulos, G.: Nonconvergence on formal integrals: II. Improved estimates for the optimal order of truncations. J. Phys. A 37, 10831 (2004)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. Érdi, B.: Long periodic perturbations of Trojan asteroids. Celest. Mech. Dyn. Astron. 43, 303 (1988)CrossRefMATHGoogle Scholar
  11. Érdi, B.: The Trojan problem. Celest. Mech. Dyn. Astron. 65, 149 (1997)MathSciNetCrossRefMATHGoogle Scholar
  12. Érdi, B., Sandor, Z.: Stability of co-orbital motion in exoplanetary systems. Celest. Mech. Dyn. Astron. 92, 113 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. Érdi, B., Nagy, I., Sándor, Z., Süli, A., Fröhlich, G.: Secondary resonances of co-orbital motions. MNRAS 381, 33 (2007)ADSCrossRefGoogle Scholar
  14. Froeschlé, C., Guzzo, M., Lega, E.: Graphical evolution of the Arnold web: from order to chaos. Science 289, 2108 (2000)ADSCrossRefGoogle Scholar
  15. Giorgilli, A., Skokos, C.: On the stability of the Trojan asteroids. Astron. Astrophys. 317, 254 (1997)ADSGoogle Scholar
  16. Giuppone, C., Benítez-Llambay, P., Beaugé, C.: Origin and detectability of co-orbital planets from radial velocity data. MNRAS 421, 356 (2012)ADSGoogle Scholar
  17. Haghighipour, N., Capen, S., Hinse, T.: Detection of Earth-mass and super-Earth Trojan planets using transit timing variation method. Celest. Mech. Dyn. Astron. 117, 75 (2013)ADSCrossRefGoogle Scholar
  18. Laughlin, G., Chambers, J.E.: Extrasolar Trojans: the viability and detectability of planets in the 1:1 resonance. Astron. J. 124, 592 (2002)ADSCrossRefGoogle Scholar
  19. Leleu, A.: Dynamics of co-orbital exoplanets. Ph.D. thesis, arXiv:1701.05585 (2016)
  20. Leleu, A., Robutel, P., Correia, A.C.M.: Detectability of quasi-circular co-orbital planets. Application to the radial velocity technique. Astron. Astrophys. 581, A128 (2015)ADSCrossRefGoogle Scholar
  21. Leleu, A., Robutel, P., Correia, A.C.M., Lillo-Box, J.: Detection of co-orbital planets by combining transit and radial-velocity measurements. Astron. Astrophys. 599, L7 (2017)ADSCrossRefGoogle Scholar
  22. Levison, H., Shoemaker, E., Shoemaker, C.: Dynamical evolution of Jupiter’s Trojan asteroids. Nature 385, 42 (1997)ADSCrossRefGoogle Scholar
  23. Lhotka, C., Efthymiopoulos, C., Dvorak, R.: Nekhoroshev stability at \(L4\) or \(L5\) in the elliptic restricted three-body problem—application to Trojan asteroids. MNRAS 384, 1165 (2008)ADSCrossRefGoogle Scholar
  24. Lykawka, P.S., Horner, J., Jones, B.W., Mukai, T.: Origin and dynamical evolution of Neptune Trojans: II. Long term evolution. MNRAS 412(1), 537 (2011)ADSCrossRefGoogle Scholar
  25. Lyra, W., Johansen, A., Klahr, H., Piskunov, N.: Standing on the shoulders of giants: Trojan Earths and vortex trapping in low mass self-gravitating protoplanetary disks of gas and solids. MNRAS 493, 1125 (2009)ADSGoogle Scholar
  26. Marzari, F., Scholl, H.: Dynamics of Jupiter Trojans during the 2:1 mean motion resonance crossing of Jupiter and Saturn. MNRAS 380, 479 (2007)ADSCrossRefGoogle Scholar
  27. Milani, A.: The Trojan asteroid belt: proper elements, stability, chaos and families. Celest. Mech. Dyn. Astron. 57, 59 (1993)ADSCrossRefGoogle Scholar
  28. Morais, M.H.M.: A secular theory for Trojan-type motion. Astron. Astrophys. 350, 318 (1999)ADSGoogle Scholar
  29. Morais, M.H.M.: Hamiltonian formulation on the secular theory for a Trojan-type motion. Astron. Astrophys. 369, 677 (2001)ADSCrossRefMATHGoogle Scholar
  30. Nauenberg, M.: Stability and eccentricity for two planets in a 1:1 resonances, and their possible occurrence in extrasolar planetary systems. Astron. J. 124, 2332 (2002)ADSCrossRefGoogle Scholar
  31. Neishtadt, A.I.: On the change in the adiabatic invariant on crossing a separatrix in systems with two degrees of freedom. Prikl. Matem. Mekhan. 51(5), 750 (1987); PMM USSR 51(5), 586Google Scholar
  32. Pierens, A., Raymond, S.N.: Disruption of co-orbital (1:1) planetary resonances during gas-driven orbital migration. MNRAS 442(2), 2296 (2014)ADSCrossRefGoogle Scholar
  33. Páez, R.I.: New normal form approaches adapted to the Trojan problem. Ph.D. thesis, arXiv:1703.08819 (2016)
  34. Páez, R.I., Efthymiopoulos, C.: Trojan resonant dynamics, stability, and chaotic diffusion, for parameters relevant to exoplanetary systems. Celest. Mech. Dyn. Astron. 121(2), 139 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. Páez, R.I., Locatelli, U.: Trojan dynamics well approximated by a new Hamiltonian normal form. MNRAS 453(2), 2177 (2015)ADSCrossRefGoogle Scholar
  36. Páez, R.I., Locatelli, U., Efthymiopoulos, C.: New Hamiltonian expansions adapted to the Trojan problem. Celest. Mech. Dyn. Astron. 126, 519 (2016)MathSciNetCrossRefMATHGoogle Scholar
  37. Robutel, P., Gabern, F.: The resonant structure of Jupiter’s Trojan asteroids: I. Long term stability and diffusion. MNRAS 372, 1463 (2006)ADSCrossRefGoogle Scholar
  38. Schwarz, R., Süli, Á., Dvorak, R., Pilat-Lohinger, E.: Stability of Trojan planets in multiplanetary systems. Celest. Mech. Dyn. Astron. 104, 69 (2009)ADSCrossRefMATHGoogle Scholar
  39. Tsiganis, K., Varvoglis, H., Dvorak, R.: Chaotic diffusion and effective stability of Jupiter Trojans. Celest. Mech. Dyn. Astron. 92, 71 (2005)ADSMathSciNetCrossRefMATHGoogle Scholar
  40. Voglis, N., Efthymiopoulos, C.: Angular dynamical spectra. A new method for determining frequencies, weak chaos and cantori. J. Phys. A 31, 2913 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Research Center for Astronomy and Applied MathematicsAcademy of AthensAthensGreece

Personalised recommendations