Celestial Mechanics and Dynamical Astronomy

, Volume 120, Issue 1, pp 77–104 | Cite as

On the stability of dust orbits in mean-motion resonances perturbed by from an interstellar wind

  • Pavol Pástor
Original Article


Circumstellar dust particles can be captured in a mean-motion resonance (MMR) with a planet and simultaneously be affected by non-gravitational effects. It is possible to describe the secular variations of a particle orbit in the MMR analytically using averaged resonant equations. We derive the averaged resonant equations from the equations of motion in near-canonical form. The secular variations of the particle orbit depending on the orientation of the orbit in space are taken into account. The averaged resonant equations can be derived/confirmed also from Lagrange’s planetary equations. We apply the derived theory to the case when the non-gravitational effects are the Poynting–Robertson effect, the radial stellar wind, and an interstellar wind. The analytical and numerical results obtained are in excellent agreement. We found that the types of orbits correspond to libration centers of the conservative problem. The averaged resonant equations can lead to a system of equations which holds for stationary points in a subset of resonant variables. Using this system we show analytically that for the considered non-gravitational effects, all stationary points should correspond to orbits which are stationary in interplanetary space after an averaging over a synodic period. In an exact resonance, the stationary orbits are stable. The stability is achieved by a periodic repetition of the evolution during the synodic period. Numerical solutions of this system show that there are no stationary orbits for either the exact or non-exact resonances.


Interplanetary dust Mean-motion resonances Averaged resonant equations Poynting–Robertson effect Stellar wind Interstellar gas flow 



I would like to thank the referees of this paper for their useful comments and suggestions.


  1. Alouani-Bibi, F., Opher, M., Alexashov, D., Izmodenov, V., Toth, G.: Kinetic versus multi-fluid approach for interstellar neutrals in the heliosphere: exploration of the interstellar magnetic field effects. Astrophys. J. 734, 45 (2011)CrossRefADSGoogle Scholar
  2. Baines, M.J., Williams, I.P., Asebiomo, A.S.: Resistance to the motion of a small sphere moving through a gas. Mon. Not. R. Astron. Soc. 130, 63–74 (1965)CrossRefADSGoogle Scholar
  3. Belyaev, M.A., Rafikov, R.R.: The dynamics of dust grains in the outer solar system. Astrophys. J. 723, 1718–1735 (2010)CrossRefADSGoogle Scholar
  4. Bate, R.R., Mueller, D.D., White, J.E.: Fundamentals of Astrodynamics. Dover Publications, New York (1971)Google Scholar
  5. Beaugé, C.: Asymmetric librations in exterior resonances. Celest. Mech. Dyn. Astron. 60, 225–248 (1994)CrossRefzbMATHADSGoogle Scholar
  6. Beaugé, C., Ferraz-Mello, S.: Resonance trapping in the primordial solar nebula: the case of a Stokes drag dissipation. Icarus 103, 301–318 (1993)CrossRefADSGoogle Scholar
  7. Beaugé, C., Ferraz-Mello, S.: Capture in exterior mean-motion resonances due to Poynting–Robertson drag. Icarus 110, 239–260 (1994)CrossRefADSGoogle Scholar
  8. Brouwer, D., Hori, G.I.: Theoretical evaluation of atmospheric drag effects in the motion of an artificial satellite. Astron. J. 66, 193–225 (1961)CrossRefMathSciNetADSGoogle Scholar
  9. Brownlee, D.E.: The ring around us. Nature 369, 706 (1994)CrossRefADSGoogle Scholar
  10. Buenzli, E., Thalmann, C., Vigan, A., Boccaletti, A., Chauvin, G., Augereau, J.C., Meyer, M.R., Ménard, F., Desidera, S., Messina, S., Henning, T., Carson, J., Montagnier, G., Beuzit, J.L., Bonavita, M., Eggenberger, A., Lagrange, A.M., Mesa, D., Mouillet, D., Quanz, S.P.: Dissecting the Moth: discovery of an off-centered ring in the HD 61005 debris disk with high-resolution imaging. Astron. Astrophys. 524, L1 (2010)CrossRefADSGoogle Scholar
  11. Danby, J.M.A.: Fundamentals of Celestial Mechanics, 2nd edn. Willmann-Bell, Richmond (1988)Google Scholar
  12. Debes, J.H., Weinberger, A.J., Kuchner, M.J.: Interstellar medium sculpting of the HD 32297 debris disk. Astrophys. J. 702, 318–326 (2009)CrossRefADSGoogle Scholar
  13. Dermott, S.F., Jayaraman, S., Xu, Y.L., Gustafson, B.A.S., Liou, J.-C.: A circumsolar ring of asteroidal dust in resonant lock with the Earth. Nature 369, 719–723 (1994)CrossRefADSGoogle Scholar
  14. Dermott, S.F., Grogan, K., Durda, D.D., Jayaraman, S., Kehoe, T.J.J., Kortenkamp, S.J., Wyatt, M.C.: Orbital evolution of interplanetary dust. In: Grün, E., Gustafson, B.A.S., Dermott, S.F., Fechtig, H. (eds.) Interplanetary Dust, pp. 569–639. Springer, Berlin (2001)CrossRefGoogle Scholar
  15. Dohnanyi, J.S.: Particle dynamics. In: McDonnell, J.A.M. (ed.) Cosmic Dust, pp. 527–605. Wiley-Interscience, Chichester (1978)Google Scholar
  16. Frisch, P.C., Bzowski, M., Grün, E., Izmodenov, V., Krüger, H., Linsky, J.L., McComas, D.J., Möbius, E., Redfield, S., Schwadron, N., Shelton, R., Slavin, J.D., Wood, B.E.: The galactic environment of the sun: interstellar material inside and outside of the heliosphere. Space Sci. Rev. 146, 235–273 (2009)CrossRefADSGoogle Scholar
  17. Golimowski, D.A., Krist, J.E., Stapelfeldt, K.R., Chen, C.H., Ardila, D.R., Bryden, G., Clampin, M., Ford, H.C., Illingworth, G.D., Plavchan, P., Rieke, G.H., Su, K.Y.L.: Hubble and Spitzer Space Telescope observations of the debris disk around the nearby K dwarf HD 92945. Astrophys. J. 142, 30 (2011)ADSGoogle Scholar
  18. Gomes, R.S.: The effect of nonconservative forces on resonance lock: stability and instability. Icarus 115, 47–59 (1995)CrossRefADSGoogle Scholar
  19. Gomes, R.S.: Orbital evolution in resonance lock. I. The restricted 3-body problem. Astron. J. 114, 2166–2176 (1997)CrossRefADSGoogle Scholar
  20. Gustafson, B.A.S.: Physics of zodiacal dust. Annu. Rev. Earth Planet. Sci. 22, 553–595 (1994)CrossRefADSGoogle Scholar
  21. Hines, D.C., Schneider, G., Hollenbach, D., Mamajek, E.E., Hillenbrand, L.A., Metchev, S.A., Meyer, M.R., Carpenter, J.M., Moro-Martín, A., Silverstone, M.D., Kim, J.S., Henning, T., Bouwman, J., Wolf, S.: The Moth: an unusual circumstellar structure associated with HD 61005. Astrophys. J. 671, L165–L168 (2007)CrossRefADSGoogle Scholar
  22. Kirkwood, D.: Meteoric Astronomy: A Treatise on Shooting-Stars, Fireballs, and Aerolites. Lippincott, Philadelphia (1867)Google Scholar
  23. Klačka, J., Kómar, L., Pástor, P., Petržala, J.: The non-radial component of the solar wind and motion of dust near mean motion resonances with planets. Astron. Astrophys. 489, 787–793 (2008)CrossRefADSGoogle Scholar
  24. Klačka, J., Petržala, J., Pástor, P., Kómar, L.: Solar wind and motion of dust grains. Mon. Not. R. Astron. Soc. 421, 943–959 (2012)CrossRefADSGoogle Scholar
  25. Klačka, J., Petržala, J., Pástor, P., Kómar, L.: The Poynting–Robertson effect: a critical perspective. Icarus 232, 249–262 (2014)CrossRefADSGoogle Scholar
  26. Lallement, R., Quémerais, E., Bertaux, J.L., Ferron, S., Koutroumpa, D., Pellinen, R.: Deflection of the interstellar neutral hydrogen flow across the heliospheric interface. Science 307, 1447–1449 (2005)CrossRefADSGoogle Scholar
  27. Leinert, C., Grün, E.: Interplanetary dust. In: Schwen, R., Marsch, E. (eds.) Physics of the Inner Heliosphere I, pp. 207–275. Springer, Berlin (1990)CrossRefGoogle Scholar
  28. Liou, J.-C., Zook, H.A.: Evolution of interplanetary dust particles in mean motion resonances with planets. Icarus 128, 354–367 (1997)CrossRefADSGoogle Scholar
  29. Maness, H.L., Kalas, P., Peek, K.M.G., Chiang, E.I., Scherer, K., Fitzgerald, M.P., Graham, J.R., Hines, D.C., Schneider, G., Metchev, S.A.: Hubble Space Telescope optical imaging of the eroding debris disk HD 61005. Astrophys. J. 707, 1098–1114 (2009)CrossRefADSGoogle Scholar
  30. Margheri, A., Ortega, R., Rebelo, C.: Some analytical results about periodic orbits in the restricted three body problem with dissipation. Celest. Mech. Dyn. Astron. 113, 279–290 (2012)CrossRefzbMATHMathSciNetADSGoogle Scholar
  31. Marzari, F.: Interstellar medium perturbations on transport-dominated debris discs in binary star systems. Mon. Not. R. Astron. Soc. 421, 3431–3442 (2012)CrossRefADSGoogle Scholar
  32. Marzari, F., Thébault, P.: On how optical depth tunes the effects of the interstellar medium on debris discs. Mon. Not. R. Astron. Soc. 416, 1890–1899 (2011)CrossRefADSGoogle Scholar
  33. Murray, C.D., Dermott, S.F.: Solar System Dynamics. Cambridge University Press, New York (1999)zbMATHGoogle Scholar
  34. Pástor, P.: Influence of fast interstellar gas flow on the dynamics of dust grains. Celest. Mech. Dyn. Astron. 112, 23–45 (2012a)CrossRefADSGoogle Scholar
  35. Pástor, P.: Orbital evolution under the action of fast interstellar gas flow with a non-constant drag coefficient. Mon. Not. R. Astron. Soc. 426, 1050–1060 (2012b)CrossRefADSGoogle Scholar
  36. Pástor, P.: Dust particles in mean motion resonances influenced by an interstellar gas flow. Mon. Not. R. Astron. Soc. 431, 3139–3149 (2013)CrossRefADSGoogle Scholar
  37. Pástor, P., Klačka, J., Kómar, L.: Motion of dust in mean motion resonances with planets. Celest. Mech. Dyn. Astron. 103, 343–364 (2009)CrossRefzbMATHGoogle Scholar
  38. Pástor, P., Klačka, J., Kómar, L.: Orbital evolution under the action of fast interstellar gas flow. Mon. Not. R. Astron. Soc. 415, 2637–2651 (2011)CrossRefGoogle Scholar
  39. Pástor, P., Klačka, J., Petržala, J., Kómar, L.: Eccentricity evolution in mean motion resonance and non-radial solar wind. Astron. Astrophys. 501, 367–374 (2009)CrossRefzbMATHGoogle Scholar
  40. Poynting, J.M.: Radiation in the solar system: its effect on temperature and its pressure on small bodies. Philos. Trans. R. Soc. Lond. Ser. A 202, 525–552 (1904)CrossRefzbMATHADSGoogle Scholar
  41. Reach, W.T., Franz, B.A., Welland, J.L., Hauser, M.G., Kelsall, T.N., Wright, E.L., Rawley, G., Stemwedel, S.W., Splesman, W.J.: Observational confirmation of a circumsolar dust ring by the COBE satellite. Nature 374, 521–523 (1995)CrossRefADSGoogle Scholar
  42. Robertson, H.P.: Dynamical effects of radiation in the solar system. Mon. Not. R. Astron. Soc. 97, 423–438 (1937)CrossRefzbMATHADSGoogle Scholar
  43. Rodigas, T.J., Hinz, P.M., Leisenring, J., Vaitheeswaran, V., Skemer, A.J., Skrutskie, M., Su, K.Y.L., Bailey, V., Schneider, G., Close, L., Mannucci, F., Esposito, S., Arcidiacono, C., Pinna, E., Argomedo, J., Agapito, G., Apai, D., Bono, G., Boutsia, K., Briguglio, R., Brusa, G., Busoni, L., Cresci, G., Currie, T., Desidera, S., Eisner, J., Falomo, R., Fini, L., Follette, K., Fontana, A., Garnavich, P., Gratton, R., Green, R., Guerra, J.C., Hill, J.M., Hoffmann, W.F., Jones, T.J., Krejny, M., Kulesa, C., Males, J., Masciadri, E., Mesa, D., McCarthy, D., Meyer, M., Miller, D., Nelson, M.J., Puglisi, A., Quiros-Pacheco, F., Riccardi, A., Sani, E., Stefanini, P., Testa, V., Wilson, J., Woodward, C.E., Xompero, M.: The gray needle: large grains in the HD 15115 debris disk from LBT/PISCES/ Ks and LBTI/LMIRcam/ L’ adaptive optics imaging. Astrophys. J. 752, 57 (2012)CrossRefADSGoogle Scholar
  44. Whipple, F.L.: A comet model III. The zodiacal light. Astrophys. J. 121, 750–770 (1955)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Tekov ObservatoryLeviceSlovak Republic

Personalised recommendations