Advertisement

Astrophysics and Space Science

, Volume 338, Issue 1, pp 23–33 | Cite as

Computation of families of periodic orbits and bifurcations around a massive annulus

  • E. TresacoEmail author
  • A. Elipe
  • A. Riaguas
Original Article

Abstract

This paper studies the main features of the dynamics around a planar annular disk. It is addressed an appropriated closed expression of the gravitational potential of a massive disk, which overcomes the difficulties found in previous works in this matter concerning its numerical treatment. This allows us to define the differential equations of motion that describes the motion of a massless particle orbiting the annulus. We describe the computation methods proposed for the continuation of uni-parametric families of periodic orbits, these algorithms have been applied to analyze the dynamics around a massive annulus by means of a description of the main families of periodic orbits found, their bifurcations and linear stability.

Keywords

Periodic orbits Bifurcation of families Solid annulus disk potential 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abad, A., Elipe, E.: Evolution strategies for computing periodic orbits. In: AAS09-143 Proceedings, 19th AAS/AIAA Space Flight Mechanics Meeting, Savannah, Georgia (2009) Google Scholar
  2. Abad, A., Elipe, A., Tresaco, E.: Analytic model to find frozen orbits for a Lunar orbiter. J. Guid. Control Dyn. 32(3), 888–898 (2009) CrossRefGoogle Scholar
  3. Alberti, A., Vidal, C.: Dynamics of a particle in a gravitational field of a homogeneous annulus disk. Celest. Mech. Dyn. Astron. 98, 75–93 (2007) MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. Arribas, M., Elipe, A.: Bifurcations and equilibria in the extended N-body ring problem. Mech. Res. Commun. 31, 129–143 (2005) MathSciNetGoogle Scholar
  5. Arribas, M., Elipe, A., Kalvouridis, T.: Homographic solutions in the planar n+1 body problem with quasi-homogeneous potentials. Celest. Mech. Dyn. Astron. 99, 1–12 (2007) MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. Belbruno, E., Llibre, J., Ollé, M.: On the families of periodic orbits which bifurcate from the circular Sitnikov motions. Celest. Mech. Dyn. Astron. 60, 99–129 (1994) ADSzbMATHCrossRefGoogle Scholar
  7. Benet, L., Merlo, O.: Phase-space volume of regions of trapped motion: multiple ring components and arcs. Celest. Mech. Dyn. Astron. 103, 209–225 (2009) MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. Brack, M.: Bifurcation cascades and self-similarity of periodic orbits with analytical scaling constants in Hénon-Heiles type potentials. Found. Phys. 31, 209–229 (2001) MathSciNetCrossRefGoogle Scholar
  9. Breiter, S., Dybczynski, P.A., Elipe, A.: The action of the Galactic disk on the Oort cloud comets—qualitative study. Astron. Astrophys. 315, 618–624 (1996) ADSGoogle Scholar
  10. Broucke, R.A., Elipe, A.: The dynamics of orbits in a potential field of a solid circular ring. Regul. Chaotic Dyn. 10(2), 129–143 (2005) MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin (1945) Google Scholar
  12. Bulirsch, R.: Numerical calculation of elliptic integrals and elliptic functions. Numerische Mathematik, vol. 7, pp. 78–90. Springer, New York (1971) Google Scholar
  13. Carlson, B.C.: Computing elliptic integrals by duplication. Numer. Math. 33, 1–16 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  14. Deprit, A., Henrard, J.: Natural families of periodic orbits. Astron. J. 72, 158–172 (1967) ADSCrossRefGoogle Scholar
  15. Elipe, A., Arribas, M., Kalvouridis, T.: Periodic solutions and their parametric evolution in the planar case of the (n+1) ring problem with oblateness. J. Guid. Control Dyn. 30(6), 1640–1648 (2007) CrossRefGoogle Scholar
  16. Elipe, A., Lara, M.: Frozen orbits about the moon. J. Guid. Control Dyn. 26(2), 238–243 (2003) CrossRefGoogle Scholar
  17. Fukushima, T.: Fast computation of Jacobian elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic. Celest. Mech. Dyn. Astron. 105, 245–260 (2009) MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. Fukushima, T.: Precise computation of acceleration due to uniform ring or disk. Celest. Mech. Dyn. Astron. 108, 339–356 (2010) MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. Hénon, M.: Exploration numérique du problème restreint. II. Masses égales, stabilité des orbites périodiques. Ann. Astrophys. 28, 992–1007 (1965) ADSGoogle Scholar
  20. Kalvouridis, T.J.: Periodic solutions in the ring problem. Astrophys. Space Sci. 266(4), 467–494 (1999) ADSzbMATHCrossRefGoogle Scholar
  21. Kellogg, O.D.: Foundations of potential theory. Dover, New York (1929) Google Scholar
  22. Krough, F.T., Ng, E.W., Snyder, W.V.: The gravitational field of a disk. Celest. Mech. Dyn. Astron. 26, 395–405 (1982) Google Scholar
  23. Lara, M., Deprit, A., Elipe, E.: Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory. Celest. Mech. Dyn. Astron. 62, 167–181 (1995) MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. Lass, H., Blitzer, L.: The gravitational potential due to uniform disks and rings. Celest. Mech. Dyn. Astron. 30, 225–228 (1983) MathSciNetzbMATHGoogle Scholar
  25. Longaretti, P.Y.: Saturn’s main ring particle size distribution, An analytic approach. Icarus 81, 51–73 (1989) ADSCrossRefGoogle Scholar
  26. Mao, J.M., Delos, J.B.: Hamiltonian bifurcation theory of closed orbits in the diamagnetic Kepler problem. Phys. Rev. A 45(3), 1746–1761 (1992) ADSCrossRefGoogle Scholar
  27. Maxwell, J.: On the Stability of Motions of Saturn’s Rings. Macmillan & Co., Cambridge (1859) Google Scholar
  28. Riaguas, A., Elipe, A., Lara, M.: Periodic orbits around a massive straight segment. Celest. Mech. Dyn. Astron. 73, 169–178 (1999) MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. Scheeres, D.J.: On symmetric central configurations with application to the satellite motion about rings. Ph.D. thesis, University of Michigan (1992) Google Scholar
  30. Scheeres, D.J.: Satellite dynamics about asteroids: computing Poincaré maps for the general case. NATO Adv. Stud. Inst. Ser., Ser. C Math. Phys. Sci. 533, 554–557 (1999) MathSciNetGoogle Scholar
  31. Sicardy, B.: Numerical exploration of planetary arc dynamics. Icarus 89(2), 197–212 (1991) ADSCrossRefGoogle Scholar
  32. Stone, J.M., Balbus, S.A.: Angular momentum transport in accretion disks by convection. Astrophys. J. 464, 364 (1996) ADSCrossRefGoogle Scholar
  33. Tiscareno, M.S., Burns, J.A., Nicholson, P.D., Hedman, M., Porco, C.: Cassini imaging of Saturn’s rings II: A wavelet technique for analysis of density waves and other radial structure in the rings. Icarus 189, 14–34 (2007) ADSCrossRefGoogle Scholar
  34. Tresaco, E., Elipe, A., Riaguas, A.: Dynamics of a particle under the gravitational potential of a massive annulus: properties and equilibrium description. Celest. Mech. Dyn. Astron. (2011). doi: 10.1007/s10569-011-9371-1 Google Scholar
  35. Tresaco, E., Ferrer, S.: Some ring-shaped potentials as a generalized 4-D isotropic oscillator. Periodic orbits. Celest. Mech. Dyn. Astron. 107(3), 337–353 (2010) MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Centro Universitario de la DefensaZaragozaSpain
  2. 2.Grupo de Mecánica Espacial—IUMAUniversidad de ZaragozaZaragozaSpain
  3. 3.Dept. de Matemática AplicadaUniversidad de ValladolidSoriaSpain

Personalised recommendations