Hot Jupiters from secular planet–planet interactions

Journal name:
Nature
Volume:
473,
Pages:
187–189
Date published:
DOI:
doi:10.1038/nature10076
Received
Accepted
Published online

About 25 per cent of ‘hot Jupiters’ (extrasolar Jovian-mass planets with close-in orbits) are actually orbiting counter to the spin direction of the star1. Perturbations from a distant binary star companion2, 3 can produce high inclinations, but cannot explain orbits that are retrograde with respect to the total angular momentum of the system. Such orbits in a stellar context can be produced through secular (that is, long term) perturbations in hierarchical triple-star systems. Here we report a similar analysis of planetary bodies, including both octupole-order effects and tidal friction, and find that we can produce hot Jupiters in orbits that are retrograde with respect to the total angular momentum. With distant stellar mass perturbers, such an outcome is not possible2, 3. With planetary perturbers, the inner orbit's angular momentum component parallel to the total angular momentum need not be constant4. In fact, as we show here, it can even change sign, leading to a retrograde orbit. A brief excursion to very high eccentricity during the chaotic evolution of the inner orbit allows planet–star tidal interactions to rapidly circularize that orbit, decoupling the planets and forming a retrograde hot Jupiter.

At a glance

Figures

  1. Dynamical evolution of a representative planet and brown dwarf system.
    Figure 1: Dynamical evolution of a representative planet and brown dwarf system.

    Here we ignore tidal dissipation, but we do include the lowest-order post-Newtonian precession rate for the inner orbit. The star has mass 1M, the planet has mass 1MJ and the outer brown dwarf has mass 40MJ. The inner orbit has a1 = 6au and the outer orbit has a2 = 100au. The initial eccentricities are e1 = 0.001 and e2 = 0.6, and the initial relative inclination is i = 65°. Red curves show the following: a, the inner orbit's inclination, i1; b, the eccentricity of the inner orbit (as 1e1); and c, d, the z-component of the angular momentum of the inner (Lz,1; c) and outer (Lz,2; d) orbit, normalized to the total angular momentum (where the z-axis is defined to be along the total angular momentum). The dotted line in a marks the 90° boundary, separating prograde and retrograde orbits. The initial mutual inclination of 65° corresponds to an inner and outer inclination with respect to the total angular momentum (parallel to z) of 64.7° and 0.3°, respectively. During the evolution, the eccentricity and inclination of the inner orbit oscillate, but, in contrast to what would be predicted from evolution equations truncated to quadrupole order (shown by green curves in a and b), the eccentricity of the inner orbit can occasionally reach extremely high values and its inclination can become higher then 90°. The outer orbit's inclination always remains near its initial value. We note that more compact systems usually do not exhibit the same kind of regular oscillations between retrograde and prograde orbits illustrated here, as chaotic effects become more important and are revealed at octupole order (see Fig. 2). We find that for ~50% of the time, the inner orbit is retrograde.

  2. Dynamical evolution of a representative two-planet system with tidal dissipation included.
    Figure 2: Dynamical evolution of a representative two-planet system with tidal dissipation included.

    The inner planet becomes retrograde at 112Myr, and remains retrograde after circularizing into a hot Jupiter. Here the star has mass 1M, the inner planet has mass 1MJ and the outer planet has mass 3MJ. The inner orbit has a1 = 6au and the outer orbit has a2 = 61au. The initial eccentricities are e1 = 0.01 and e2 = 0.6, the initial relative inclination i = 71.5°, and the argument of periapsis is 45°. Left panels, complete simulation; right panels, zoomed-in view around time t110Myr. Red curves show: a, the inner orbit's inclination (i1); b, the eccentricity of the inner orbit (as 1e1); c, the semi-major axis for the inner orbit and the outer orbit; d, the magnitude of the angular momentum of the inner orbit; and in e and f, the z-component of the angular momentum of the inner (Lz,1; e) and outer (Lz,2; f) orbit, normalized to the total angular momentum. The black curves in c are the pericentre and apocentre distances of the inner and outer orbits, respectively. The initial mutual inclination of 71.5° corresponds to inner- and outer-orbit inclinations of 64.7° and 6.8°, respectively. During each excursion to very high eccentricity for the inner orbit (marked with vertical lines in b and c right panels), tidal dissipation becomes significant. Eventually the inner planet is tidally captured by the star and its orbit becomes decoupled from the outer body. After this point, the orbital angular momenta remain nearly constant. The final semi-major axis for the inner planet is 0.022au, typical of a hot Jupiter. The green curves in a, b, d, e and f show the evolution in the quadrupole approximation (but including tidal friction), demonstrating that the octupole-order effects lead to a qualitatively different behaviour. For the tidal evolution in this example, we assume tidal quality factors Q* = 5.5×106 for the star and QJ = 5.8×106 for the hot Jupiter (see Supplementary Information). We monitor the pericentre distance of the inner planet to ensure that it always remains outside the Roche limit29. Here, as in Fig. 1, we also include the lowest-order post-Newtonian precession rate for the inner orbit.

References

  1. Triaud, A. H. M. J. et al. Spin-orbit angle measurements for six southern transiting planets. New insights into the dynamical origins of hot Jupiters. Astron. Astrophys. 524, A25 (2010)
  2. Fabrycky, D. & Tremaine, S. Shrinking binary and planetary orbits by Kozai cycles with tidal friction. Astrophys. J. 669, 12981315 (2007)
  3. Wu, Y., Murray, N. W. & Ramsahai, J. M. Hot Jupiters in binary star systems. Astrophys. J. 670, 820825 (2007)
  4. Ford, E. B., Kozinsky, B. & Rasio, F. A. Secular evolution of hierarchical triple star systems. Astrophys. J. 535, 385401 (2000)
  5. Chatterjee, S., Matsumura, S., Ford, E. B. & Rasio, F. A. Dynamical outcomes of planet-planet scattering. Astrophys. J. 686, 580602 (2008)
  6. Lai, D., Foucart, F. & Lin, D. N. C. Evolution of spin direction of accreting magnetic protostars and spin-orbit misalignment in exoplanetary systems. Mon. Not. R. Astron. Soc. (submitted); preprint at left fencehttp://arxiv.org/abs/1008.3148right fence (2011)
  7. Nagasawa, M., Ida, S. & Bessho, T. Formation of hot planets by a combination of planet scattering, tidal circularization, and the Kozai mechanism. Astrophys. J. 678, 498508 (2008)
  8. Schlaufman, K. C. Evidence of possible spin-orbit misalignment along the line of sight in transiting exoplanet systems. Astrophys. J. 719, 602611 (2010)
  9. Takeda, G., Kita, R. & Rasio, F. A. Planetary systems in binaries. I. Dynamical classification. Astrophys. J. 683, 10631075 (2008)
  10. Winn, J. N., Fabrycky, D., Albrecht, S. & Johnson, J. A. Hot stars with hot Jupiters have high obliquities. Astrophys. J. 718, L145L149 (2010)
  11. Wu, Y. & Lithwick, Y. Secular chaos and the production of hot Jupiters. Preprint at left fencehttp://arxiv.org/abs/1012.3475right fence (2010)
  12. Lin, D. N. C. & Papaloizou, J. On the tidal interaction between protoplanets and the proto-planetary disk. III — Orbital migration of protoplanets. Astrophys. J. 309, 846857 (1986)
  13. Masset, F. S. & Papaloizou, J. Runaway migration and the formation of hot Jupiters. Astrophys. J. 588, 494508 (2003)
  14. Gaudi, B. S. & Winn, J. N. Prospects for the characterization and confirmation of transiting exoplanets via the Rossiter-McLaughlin effect. Astrophys. J. 655, 550563 (2007)
  15. Holman, M., Touma, J. & Tremaine, S. Chaotic variations in the eccentricity of the planet orbiting 16 Cygni B. Nature 386, 254256 (1997)
  16. Eggleton, P. P., Kiseleva, L. G. & Hut, P. The equilibrium tide model for tidal friction. Astrophys. J. 499, 853870 (1998)
  17. Kozai, Y. Secular perturbations of asteroids with high inclination and eccentricity. Astron. J. 67, 591598 (1962)
  18. Lidov, M. L. The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies. Planet. Space Sci. 9, 719759 (1962)
  19. Mazeh, T. & Shaham, J. The orbital evolution of close triple systems — the binary eccentricity. Astron. Astrophys. 77, 145151 (1979)
  20. Harrington, R. S. The stellar three-body problem. Celest. Mech. 1, 200209 (1969)
  21. Krymolowski, Y. & Mazeh, T. Studies of multiple stellar systems — II. Second-order averaged Hamiltonian to follow long-term orbital modulations of hierarchical triple systems. Mon. Not. R. Astron. Soc. 304, 720732 (1999)
  22. Kiseleva, L. G., Eggleton, P. P. & Mikkola, S. Tidal friction in triple stars. Mon. Not. R. Astron. Soc. 300, 292302 (1998)
  23. Zdziarski, A. A., Wen, L. & Gierlin´ski, M. The superorbital variability and triple nature of the X-ray source 4U 1820–303. Mon. Not. R. Astron. Soc. 377, 10061016 (2007)
  24. Mikkola, S. & Tanikawa, K. Does Kozai resonance drive CH Cygni? Astron. J. 116, 444450 (1998)
  25. Ford, E. B. & Rasio, F. A. On the relation between hot Jupiters and the Roche limit. Astron. J. 638, L45L48 (2006)
  26. Kalas, P. et al. Optical images of an exosolar planet 25 light-years from Earth. Science 322, 13451348 (2008)
  27. Marois, C. et al. Direct imaging of multiple planets orbiting the star HR 8799. Science 322, 13481352 (2008)
  28. Pollack, J. B. et al. Formation of the giant planets by concurrent accretion of solids and gas. Icarus 124, 6285 (1996)
  29. Matsumura, S., Peale, S. J. & Rasio, F. A. Formation and evolution of close-in planets. Astrophys. J. 725, 19952016 (2010)

Download references

Author information

Affiliations

  1. Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA), Northwestern University, Evanston, Illinois 60208, USA

    • Smadar Naoz,
    • Will M. Farr,
    • Yoram Lithwick,
    • Frederic A. Rasio &
    • Jean Teyssandier

Contributions

S.N. performed numerical calculations with help from J.T. All authors developed the mathematical model, discussed the physical interpretation of the results and jointly wrote the manuscript.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary Information (112K)

    This file contains Supplementary Notes and Data , Supplementary Figure 1 and legend and additional references.

Additional data