Skip to main content
SpringerLink
Log in

Review of the dynamics in the Kirkwood gaps

  • Published:
Celestial Mechanics and Dynamical Astronomy Aims and scope Submit manuscript

Abstract

The study of mean motion resonance dynamics was motivated by the search for an explanation for the puzzling problem of the Kirkwood gaps. The most important contributions in this field within the last 32 years are reviewed here. At the beginning of that period, which coincides with the first long-term numerical investigations of resonant motion, different hypotheses (collisional, gravitational, statistical and cosmological) to explain the origin of the gaps were still competing with each other. At present, a general theory, based on gravitational mechanisms only, is capable of explaining in a uniform way all the Kirkwood gaps except the 2/1 one. Indeed, in the 4/1, 3/1, 5/2 and 7/3 mean motion commensurabilities, the overlap of secular resonances leads to almost overall chaos where asteroids undergo large and wild variations in their orbital elements. Such asteroids, if not thrown directly into the Sun, are sooner or later subject to strong close encounters with the largest inner planets, the typical time scale of the whole process being of the order of a few million years. Unfortunately, this mechanism is not capable of explaining the 2/1 gap where the strong chaos produced by the overlapping secular resonances does not attain orbits with moderate eccentricity, of low inclination and with low to moderate amplitude of libration. In the light of the most recent studies, it appears that the 2/1 gap is the global consequence of slow diffusive processes. At present, the origin of these processes remains under study.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Andoyer, H.: 1903, “Contribution à la théorie des petites planètes dont le moyen mouvement est sensiblement double de celui de Jupiter”, Bull. Astron., 20, 321–356.

    Google Scholar 

  • Brouwer, D.: 1963, “The Problem of the Kirkwood Gaps in the Asteroidal Belt”, Astron. J., 68, 152–159.

    Google Scholar 

  • Carpino, M., Milani, A. and Nobili, A.M.: 1987, “Long-Term numerical integrations and synthetic theories for the motion of the outer planets”, Astron. Astrophys., 181, 182–194.

    Google Scholar 

  • Chirikov, B.V.: 1979, “A universal instability of many-dimensional oscillator systems”, Phys. Rept., 52, 263–379.

    Google Scholar 

  • Farinella, P., Froeschlé, Ch. and Gonczi, R.: 1993, “Meteorites from the asteroid 6 Hebe”, Celest. Mech., 56, 287–305.

    Google Scholar 

  • Farinella, P., Froeschlé, Ch., Froeschlé, C., Gonczi, R., Hahn, G., Morbidelli, A. and Valsecchi, G.B.: 1994, “Asteroids falling into the Sun”, Nature, 371, 314–317.

    Google Scholar 

  • Ferraz-Mello, S.: 1987, “Expansion of the disturbing force-function for the study of higheccentricity librations”, Astron. Astrophys., 183, 397–402.

    Google Scholar 

  • Ferraz-Mello, S.: 1988, “The high-eccentricity libration of the Hildas”, Astron. J., 96, 400–408.

    Google Scholar 

  • Ferraz-Mello, S.: 1994a, “The convergence domain of the Laplacian expansion of the disturbing function”, Celest. Mech., 58, 37–52.

    Google Scholar 

  • Ferraz-Mello, S.: 1994b, “Kirkwood gaps and resonant groups”, in Asteroids, Comets, Meteors 1993 (A. Milani, M. Di Martino and A. Cellino, eds.), Kluwer, Dordrecht, 175–188.

    Google Scholar 

  • Ferraz-Mello, S.: 1994c, “Dynamics of the asteroidal 2/1 resonance”, Astron. J., 108, 2330–2337.

    Google Scholar 

  • Ferraz-Mello, S.: 1996, “On the Hecuba gap”, in Dynamics, Ephemerides and Astrometry of the Solar System (S. Ferraz-Mello, B. Morando and J.-E. Arlot, eds.), Kluwer, Dordrecht, 177–182.

    Google Scholar 

  • Ferraz-Mello, S. and Klafke, J.C.: 1991, “A model for the study of very-high-eccentricity asteroidal motion: the 3:1 resonance”, in Predictability, Stability, and Chaos in N-body Dynamical Systems (A.E. Roy, ed.), Plenum Press, New-York, 177–184.

    Google Scholar 

  • Ferraz-Mello, S. and Sato, M.: 1989, “The very-high-eccentricity asymmetric expansion of the disturbing function near resonanes of any order”, Astron. Astrophys., 225, 541–547.

    Google Scholar 

  • Ferraz-Mello, S., Dvorak, R. and Michtchenko, T.A.: 1994, “Depletion of the asteroidal belt at resonanes”, in From Newton to Chaos (A.E. Roy and B.A. Steves, eds.), Plenum Press, New-York, 157–169.

    Google Scholar 

  • Franklin, F.: 1994, “An examination of the relation between chaotic orbits and the Kirkwood gap at the 2:1 resonance. I., Astron, J. 107, 1890–1899.

    Google Scholar 

  • Franklin, F., Lecar, M. and Murison, M.: 1993, “Chaotic orbits and long term stability: an example from asteroids of the Hilda group”, Astron. J., 105, 2336–2343.

    Google Scholar 

  • Froeschlé, C. and Greenberg, R.: 1989, “Mean motion resonanes”, in Asteroids II (R. P. Binzel, T. Gehrels and M. S. Matthews, eds.), University of Arizona Press, Tucson. 827–844.

    Google Scholar 

  • Froeschlé, C. and Scholl, H.: 1976, “On the Dynamical Topology of the Kirkwood Gaps”, Astron. Astrophys., 48, 389–393.

    Google Scholar 

  • Froeschld, C. and Scholl, H.: 1982, “A Systematic Exploration of Three-dimensional Asteroidal Motion at the 2/1 Resonance”, Astron. Astrophys., 111, 346–356.

    Google Scholar 

  • Giffen, R.: 1973, “A study of Commensurable Motion in the Asteroid Belt”, Astron. Astrophys., 23, 387–403.

    Google Scholar 

  • Gladman B.J., Migliorini, F., Morbidelli, A., Zappalà, V., Michel, P. Cellino, A., Levison, H., Froeschlé, Ch. and Bailey, M.E.: 1997, to be submitted to Science.

  • Greenberg, R. and Scholl, H.: 1979, “Resonances in the asteroidal belt”, in Asteroids (T. Gehrels, ed.), The University of Arizona Press, Tucson, 310–333.

    Google Scholar 

  • Hadjidemetriou, J.D.: 1995, “Mechanisms of Generation of Chaos in the Solar System”, in From Newton to Chaos (A.E. Roy and B.A. Steves, eds.), Plenum Press, New-York, 79–96.

    Google Scholar 

  • Hadjidemetriou, J.D.: 1996, “Symplectic mappings”, in Dynamics, Ephemerides and Astrometry of the Solar System (S. Ferraz-Mello, B. Morando and J.-E. Arlot, eds.), Kluwer, Dordrecht, 255–266.

    Google Scholar 

  • Hagihara, Y.: 1961, “Gaps in the distribution of asteroids”, Smithsonian Contr to astrophysics, 5,6, 59–67.

    Google Scholar 

  • Henrard, J.: 1988, “Resonances in the Planar Elliptic Restricted Problem”, in Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems (A.E. Roy, ed.), Kluwer, Dordrecht, 405–425.

    Google Scholar 

  • Henrard, J.: 1990, “A semi-numerical perturbation method for separable Hamiltonian systems”, Celest. Mech., 49, 43–67.

    Google Scholar 

  • Henrard, J.: 1996, “A note concerning the 2:1 and the 3:2 resonanancs in the asteroid belt”, Celest. Mech., 64, 107–114.

    Google Scholar 

  • Henrard, J. and Caranicolas, N.D.: 1990, “Motion near the 3/1 resonance of the planar elliptic restricted three body problem”, Celest. Mech., 47, 99–121.

    Google Scholar 

  • Henrard, J. and Lemaitre, A.: 1986, “A perturbation method for problems with two critical arguments”, Celest. Mech., 39, 213–238.

    Google Scholar 

  • Henrard, J. and Lemaitre, A.: 1987, “A Perturbadve Treatment of the 2/1 Jovian Resonance”, Icarus, 69, 266–279.

    Google Scholar 

  • Henrard, J., Watanabe, N. and Moons, M.: 1995, “A Bridge between Secondary and Secular Resonances inside the Hecuba Gap”, Icarus, 115, 336–346.

    Google Scholar 

  • Henrard, J., Lemaitre, A., Milani, A. and Murray, C.D.: 1986, “The Reducing Transformation and Apocentric Librators”, Celest. Mech., 38, 335–344.

    Google Scholar 

  • Hill, G.W: 1902a, “Illustration of periodic solutions in the problem of three bodies. I”, Astron. J., 22, 93–97.

    Google Scholar 

  • Hill, G.W.: 1902b, “Illustration of periodic solutions in the problem of three bodies. II”, Astron. J., 22, 117–121.

    Google Scholar 

  • Jefferys, W.H. and Standish, E.M.: 1972, “Further Periodic Solutions of the Three-Dimensional Restricted Problem. II”, Astron. J., 77, 394–1300.

    Google Scholar 

  • Kirkwood, D.: 1867, Meteoric Astronomy, Lippincott, Philadelphia, ch. 13.

    Google Scholar 

  • Klafke, J.C., Ferraz-Mello, S. and Michtchenko, T.A.: 1992, “Very-high-eccentricity librations at some higher order resonances”, in Chaos, Resonance and Collective Dynamical Phenomena in the Solar System (S. Ferraz-Mello, ed.), Kluwer, Dordrecht, 153–158.

    Google Scholar 

  • Kozai, Y.: 1962, “Secular Perturbations of Asteroids with High Inclination and Eccentricity”, Astron. J., 67, 591–598.

    Google Scholar 

  • Lemaitre, A. and Henrard, J.: 1988, “The 3/2 resonance”, Celest. Mech., 43, 91–98.

    Google Scholar 

  • Lemaitre, A. and Henrard, J.: 1990, “On the Origin of Chaotic Behavior in the 2/1 Kirkwood Gap”, Icarus, 83, 391–409.

    Google Scholar 

  • Message, P.J.: 1966, “On nearly-commensurable periods in the restricted problem of three bodies, with calculation of the long-period variations in the interior 2:1 case”, in The Theory of Orbits in the Solar System and in Stellar Systems (G. Contopoulos, ed.), Academic, London, 197–222.

    Google Scholar 

  • Michtchenko, T.A. and Ferraz-Mello, S.: 1995, “Comparative study of the asteroidal motion in the 3:2 and 2:1 resonances with Jupiter. 1. Planar model”, Astron. Astrophys., 303, 945–963.

    Google Scholar 

  • Michtchenko, T.A. and Ferraz-Mello, S.: 1996, “Comparative study of the asteroidal motion in the 3:2 and 2:1 resonances with Jupiter. II. Three-dimensional model”, Astron. Astrophys., 310, 1021–1035.

    Google Scholar 

  • Migliorini F., Morbidelli, A., Zappalà, V., Gladman, B.J., Bailey, M.E. and Cellino, A.: 1997, “ Vesta Fragments for ν6 and 3:1 resonances: implications for V-type NEAs and HED meteorites ”, submitted to Meteoritics.

  • Moons, M.: 1994, “Extended Schubart averaging”, Celest. Mech., 60, 173–186.

    Google Scholar 

  • Moons, M. and Morbidelli, A.: 1993, “The main mean motion commensurabilities in the planar circular and elliptic problem”, Celest. Mech., 57, 99–108.

    Google Scholar 

  • Moons, M. and Morbidelli, A.: 1995, “Secular Resonances in Mean Motion Commensurabilities: The 4/1, 3/1, 5/2 and 7/3 Cases”, Icarus, 114, 33–50.

    Google Scholar 

  • Morbidelli, A.: 1993, “On the Successive Elimination of Perturbation Harmonics”, Celest. Mech., 55,101–130.

    Google Scholar 

  • Morbidelli, A.: 1996, “The Kirkwood gap at the 2/1 commensurability with Jupiter: new numerical results”, Astron. J., 111, 2453–2461.

    Google Scholar 

  • Morbidelli, A. and Moons, M.: 1993, “Secular resonances in mean motion commensurabilities: The 2/1 and 3/2 Cases”, Icarus, 102, 316–332.

    Google Scholar 

  • Murray, C.D.: 1986, “Structure of the 2/1 and 3/2 Jovian Resonances”, Icarus, 65, 70–82.

    Google Scholar 

  • Murray, C.D. and Fox, K.: 1984, “Structure of the 3:1 Jovian Resonance: A Comparison of Numerical Methods”, Icarus, 59, 221–233.

    Google Scholar 

  • Nakai, H. and Kinoshita, H.: 1985, “Secular Perturbations of Asteroids in Secular Resonance”, Celest. Mech., 36, 391–407.

    Google Scholar 

  • Nobili, A., Milani, A. and Carpino, M.: 1989, “Fundamental frequencies and small divisors in the orbits of the outer planets”, Astron. Astrophys., 210, 313–336.

    Google Scholar 

  • Poincaré, M.H.: 1902a, “Les solutions périodiques et les planètes du type d' Hécub”, Bull. Astron., 19, 177–198.

    Google Scholar 

  • Poincaré, M.H.: 1902b, “Sur les plan\'tes du type d'Hécube”, Bull. Astron., 19, 289–310.

    Google Scholar 

  • Šidlichovský, M.: 1987a, “Application of the method of Wisdom to resonances, 5/2 resonance of asteroids”, in Figure and Dynamics of the Earth, Moon, and Planets (M. Šidlichovský, ed.), Praha, Czechoslovakia, 571–580.

    Google Scholar 

  • Šidlichovský, M.: 1987b, “Comparison of perturbation and Wisdom methods for 5/2 resonance” inn Dynamics of the Solar System, (M. Šidlichovský, ed.), Praha, Czechoslovakia, 33–37.

    Google Scholar 

  • Šidlichovský, M. and Melendo, B.: 1986, “Mapping for 5/2 asteroidal commensurability”, Bull. Astron. Inst. Czechosl., 37, 65–80.

    Google Scholar 

  • Scholl, H.: 1979, “Recent work on the origin of the Kirkwood gaps”, in Dynamics of the Solar System (R.L. Duncombe, ed.), 217–222.

  • Scholl, H. and Froeschld, C.: 1974, “Asteroidal motion at the 3/1 commensurability”, Astron. Astrophys., 33,455–458.

    Google Scholar 

  • Scholl, H. and Froeschlé, C.: 1975, “Asteroidal motion at the 5/2, 7/3 and 2/1 resonances”, Astron. Astrophys., 42,457–463.

    Google Scholar 

  • Scholl, H. and Froeschlé, C.: 1977, “The Kirkwood gaps as an asteroidal source of meteorites”, in Comets, asteroids, meteorites (E.H. Delsemme, ed.), 293–295.

  • Schubart, J.: 1964, “Long-Period effects in nearly commensurable cases of the restricted three-body problem”, Smithsonian Astrophys. Obs. Spec. Rept. 149.

  • Schubart, J.: 1968, “Long-Period effects in the motion of Hilda-type planets”, Astron. J., 73, 99–103.

    Google Scholar 

  • Schubart, J.: 1978, “New results on the commensurability cases of the problem SunJupiter-Asteroid”, in Dynamics of Planets and Satellites and Theories of Their Motion (V. Szebehely, ed.), D. Reidel, Dordrecht, 137–143.

    Google Scholar 

  • Schwarzschild, K.: 1903, “Über die periodischen Bahnen vom Hecubatypus”, Astron. Nachr., 160, 385–400.

    Google Scholar 

  • Sessin, W. and Ferraz-Mello, S.: 1984, “Motion of two planets with periods commensurable in the ratio 2:1. Solutions of the Hod auxiliary system”, Celest. Mech., 32, 307–332.

    Google Scholar 

  • Sundman, K.F.: 1916, “Sur les conditions nécessaires et suffisantes pour la convergence du développement de la fonction perturbarice dans le mouvement plan”, Öfversigt Finska Vetenskaps-Soc. Föhr., 58 A, (24).

  • Wetherill, G.W.: 1975, “Late heavy bombardment of the Moon and terrestrial planets”, in Proceedings of the Sixth Lunar Science Conference, Pergamon, Oxford, 1539–1561.

    Google Scholar 

  • Williams, J.G.: 1969, “Secular Perturbations in the Solar System”, Ph. D. Dissertation, Univ. of California, Los Angeles.

    Google Scholar 

  • Wisdom, J.: 1982, “The origin of the Kirkwood gaps: a mapping for asteroidal motion near the 3/1 commensurability”, Astron. J., 87, 577–593.

    Google Scholar 

  • Wisdom, J.: 1983, “Chaotic behavior and the origin of the 3/1 Kirkwood gap”, Icarus, 63, 272–289.

    Google Scholar 

  • Wisdom, J.: 1985a, “Meteorites may follow a chaotic route to Earth”, Nature, 315, 731–733.

    Google Scholar 

  • Wisdom, J.: 1985b, “A perturbative treatment of the motion near the 3/1 commensurability”, Icarus, 63, 279–282.

    Google Scholar 

  • Wisdom, J.: 1986, “Canonical Solution of the Twp Critical Argument Problem”, Celest. Mech., 38, 175–180.

    Google Scholar 

  • Wisdom, J.: 1987, “Urey Price Lecture: Chaotic Dynamics in the Solar System”, Icarus, 72, 241–275.

    Google Scholar 

  • Woltjer, J.: 1928, “The motion of Hyperion”, Ann. Sterrew. Leiden, 16,3.

  • Yoshikawa, M.: 1989, “A survey on the motion of asteroids in commensurabilities with Jupiter”, Astron. Astrophys., 213, 436–458.

    Google Scholar 

  • Yoshikawa, M.: 1990, “Motions of asteroids at the Kirkwood gaps. l. On the 3:1 resonance with Jupiter”, Icarus, 87, 78–102.

    Google Scholar 

  • Yoshikawa, M.: 1991, “Motions of asteroids at the Kirkwood gaps. 11. On the 5:2, 7:3 and 2:1 resonances with Jupiter”, Icarus, 92, 94–117.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de mathématique FUNDP, 8, Rempart de la Vierge, B-5000, Namur, Belgique

    Michèle Moons

Authors
  1. Michèle Moons
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Moons, M. Review of the dynamics in the Kirkwood gaps. Celestial Mech Dyn Astr 65, 175–204 (1996). https://doi.org/10.1007/BF00048446

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00048446

Key words

Search

Navigation