Letter | Published:

An example of stable chaos in the Solar System

Naturevolume 357pages569571 (1992) | Download Citation



MANY planets have been shown to have chaotic instabilities in their orbital motions, but the long-term significance of this is not fully understood1. The eccentricity of Mercury, for example, changes by about 25% of its value over 40 times the Lyapunov time2,3(the e-folding time for divergence of nearby orbits), but the orbit of Pluto, in an integration lasting 50 Lyapunov times4, shows no significant change. Here we show that the orbit of the near-Jupiter asteroid 522 Helga is chaotic, with an unusually short Lyapunov time of 6,900 yr. We integrate its motion, including perturbations from the outer giant planets, over a period 1,000 times longer than this, and find no significant instability. Chaos in the orbit of 522 Helga is caused by a 7:12 resonance with the orbit of Jupiter, but the size of the chaotic region in phase space is small; stability is ensured because the eccentricity and precession of the orbit are such that it avoids close encounters with Jupiter. Asteroid orbits with larger proper eccentricity would, we suggest, be genuinely unstable, consistent with the sparse asteroid population near Helga. Although Helga is the first clear-cut example of a stable chaotic orbit, we argue that 'stable chaos' may be a rather common feature of Solar System dynamics.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.


  1. 1

    Nobili, A. M. & Burns, J. A. Science 244, 1425 (1989).

  2. 2

    Laskar, J. Icarus 88, 266–291 (1990).

  3. 3

    Laskar, J. in Chaos et Déterminisme (eds Dahan. A., Chabert, J. L. & Chemla, K.) (Seuil, Paris, in the press).

  4. 4

    Wisdom, J. & Holman, M. Astr. J. 102, 1528–1538 (1991).

  5. 5

    Milani, A. & Nobili, A. M. Celest. Mech. 34, 343–355 (1984).

  6. 6

    Brouwer, D. & Clemence, G. M. Methods of Celestial Mechanics Ch. XVI (Academic, New York, 1961).

  7. 7

    Schubart, J. Celest. Mech. 43, 309–317 (1988).

  8. 8

    Milani, A. & Knez̆ević, Z. Celest. Mech. 49, 347–411 (1990).

  9. 9

    Nobili, A. M. in Asteroids II (eds Binzel, R. P., Geherels, T. & Matthews, M. S.) 862–879 (University of Arizona Press. 1989).

  10. 10

    Bevilacqua, R., Menchi, O., Milani, A., Nobili, A. M. & Farinella, P. Moon Planets 22, 141–152 (1980).

  11. 11

    Wisdom, J. Icarus 56, 51–74 (1983).

  12. 12

    Wisdom, J. Nature 315, 731–733 (1985).

  13. 13

    Milani, A. & Nobili, A. M. Astr. Astrophys. 144, 261–274 (1985).

  14. 14

    Sussman, G. J. & Wisdom, J. Science 241, 433–437 (1988).

  15. 15

    Nobili, A. M., Milani, A. & Carpino, M. Astr. Astrophys. 210, 313–336 (1989).

  16. 16

    Laskar, J. Nature 338, 237–238 (1989).

  17. 17

    Milani, A., Nobili, A. M. & Carpino, M. Icarus 82, 200–217 (1989).

Download references

Author information


  1. Gruppo di Meccanica Spaziale, Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 1-56127, Pisa, Italia

    • Andrea Milani
    •  & Anna M. Nobili
  2. Observatoire de Paris, Meudon, Groupe EUROPA-DAEC, 5 place Jules Janssen, F-92195, Meudon, France

    • Andrea Milani
    •  & Anna M. Nobili


  1. Search for Andrea Milani in:

  2. Search for Anna M. Nobili in:

About this article

Publication history



Issue Date



Further reading


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.