Skip to main content
SpringerLink
Log in

How does the oblateness coefficient influence the nature of orbits in the restricted three-body problem?

  • Original Article
  • Published:
Astrophysics and Space Science Aims and scope Submit manuscript

Abstract

We numerically investigate the case of the planar circular restricted three-body problem where the more massive primary is an oblate spheroid. A thorough numerical analysis takes place in the configuration \((x,y)\) and the \((x,E)\) space in which we classify initial conditions of orbits into three categories: (i) bounded, (ii) escaping and (iii) collisional. Our results reveal that the oblateness coefficient has a huge impact on the character of orbits. Interpreting the collisional motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape as well as the collisional basins and we managed to correlate them with the corresponding escape and collision times. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting version of the restricted three-body problem.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. We choose the \(\dot{\phi} < 0\) instead of the \(\dot{\phi} > 0\) part simply because in Zotos (2015b) we seen that it contains more interesting orbital content.

  2. An infinite number of regions of (stable) quasi-periodic (or small scale chaotic) motion is expected from classical chaos theory.

References

  • Abouelmagd, E.I.: Existence and stability of triangular points in the restricted three-body problem with numerical applications. Astrophys. Space Sci. 342, 45–53 (2012)

    Article  ADS  Google Scholar 

  • Altmann, E.G., Portela, J.S.E., Tél, T.: Leaking chaotic systems. Rev. Mod. Phys. 85, 869–918 (2013)

    Article  ADS  Google Scholar 

  • Barrio, R., Blesa, F., Serrano, S.: Is there chaos in Copenhagen problem? Monogr. Real Acad. Ci. Zaragoza 30, 43–50 (2006)

    MathSciNet  Google Scholar 

  • Barrio, R., Blesa, F., Serrano, S.: Fractal structures in the Hénon-Heiles Hamiltonian. Europhys. Lett. 82, 10003 (2008)

    Article  Google Scholar 

  • Barrio, R., Blesa, F., Serrano, S.: Bifurcations and safe regions in open Hamiltonians. New J. Phys. 11, 053004 (2009)

    Article  ADS  Google Scholar 

  • Beatty, J.K., Petersen, C.C., Chaikin, A.: The New Solar System, 4th edn. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  • Beevi, A.S., Sharma, R.K.: Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three-body problem. Astrophys. Space Sci. 340, 245–261 (2012)

    Article  ADS  Google Scholar 

  • Benet, L., Trautman, D., Seligman, T.: Chaotic scattering in the restricted three-body problem. I. The Copenhagen problem. Celest. Mech. Dyn. Astron. 66, 203–228 (1996)

    Article  ADS  Google Scholar 

  • Benet, L., Seligman, T., Trautman, D.: Chaotic scattering in the restricted three-body problem II. Small mass parameters. Celest. Mech. Dyn. Astron. 71, 167–189 (1998)

    Article  ADS  Google Scholar 

  • Bleher, S., Grebogi, C., Ott, E., Brown, R.: Fractal boundaries for exit in Hamiltonian dynamics. Phys. Rev. A 38, 930–938 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  • Bleher, S., Ott, E., Grebogi, C.: Routes to chaotic scattering. Phys. Rev. Lett. 63, 919–922 (1989)

    Article  ADS  Google Scholar 

  • Broucke, R.A.: Periodic orbits in the restricted three-body problem with Earth-Moon masses. Tech. Rep. 32–1168, Jet Propulsion Laboratory, California Institute of Technology (1968)

  • Contopoulos, G.: Asymptotic curves and escapes in Hamiltonian systems. Astron. Astrophys. 231, 41–55 (1990)

    MathSciNet  ADS  Google Scholar 

  • Contopoulos, G., Kaufmann, D.: Types of escapes in a simple Hamiltonian system. Astron. Astrophys. 253, 379–388 (1992)

    MathSciNet  ADS  MATH  Google Scholar 

  • Contopoulos, G., Kandrup, H.E., Kaufmann, D.: Fractal properties of escape from a two-dimensional potential. Physica D 64, 310–323 (1993)

    Article  ADS  MATH  Google Scholar 

  • de Assis, S.C., Terra, M.O.: Escape dynamics and fractal basin boundaries in the planar Earth–Moon system. Celest. Mech. Dyn. Astron. 120, 105–130 (2014)

    Article  ADS  Google Scholar 

  • de Moura, A.P.S., Grebogi, C.: Countable and uncountable boundaries in chaotic scattering. Phys. Rev. E 66, 046214 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  • de Moura, A.P.S., Letelier, P.S.: Fractal basins in Hénon-Heiles and other polynomial potentials. Phys. Lett. A 256, 362–368 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  • Ernst, A., Peters, T.: Fractal basins of escape and the formation of spiral arms in a galactic potential with a bar. Mon. Not. R. Astron. Soc. 443, 2579–2589 (2014)

    Article  ADS  Google Scholar 

  • Henon, M.: Numerical exploration of the restricted problem, V. Astron. Astrophys. 1, 223–238 (1969)

    ADS  MATH  Google Scholar 

  • Jung, C., Tél, T.: Dimension and escape rate of chaotic scattering from classical and semiclassical cross section data. J. Phys. A 24, 2793–2805 (1991)

    Article  ADS  Google Scholar 

  • Jung, C., Mejia-Monasterio, C., Seligman, T.H.: Scattering one step from chaos. Phys. Lett. A 198, 306–314 (1995)

    Article  ADS  Google Scholar 

  • Jung, C., Lipp, C., Seligman, T.H.: The inverse scattering problem for chaotic Hamiltonian systems. Ann. Phys. 275, 151–189 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Kalantonis, V.S., Markellos, V.V., Perdios, E.A.: Computing periodic orbits of the three-body problem: effective convergence of Newton’s method on the surface of section. Astrophys. Space Sci. 298, 441–451 (2005)

    Article  ADS  MATH  Google Scholar 

  • Kalantonis, V.S., Douskos, C.N., Perdios, E.A.: Numerical determination of homoclinic and heteroclinic orbits as collinear equilibria in the restricted three-body problem with oblateness. Celest. Mech. Dyn. Astron. 94, 135–153 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Kalantonis, V.S., Perdios, E.A., Perdiou, A.E.: The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness. Astrophys. Space Sci. 315, 323–334 (2008)

    Article  ADS  Google Scholar 

  • Kalvouridis, T., Gousidou-Koutita M.C.: Basins of attraction in the Copenhagen problem where the primaries are magnetic dipoles. Appl. Math. 3, 541–548 (2012)

    Article  Google Scholar 

  • Kandrup, H.E., Siopis, C., Contopoulos, G., Dvorak, R.: Diffusion and scaling in escapes from two-degrees-of-freedom Hamiltonian systems. Chaos 9, 381–392 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Markellos, V.V., Papadakis, K.E., Perdios, E.A.: Non-linear stability zones around triangular equilibria in the plane circular restricted three-body problem with oblateness. Astrophys. Space Sci. 245, 157–164 (1996)

    Article  ADS  MATH  Google Scholar 

  • Markellos, V.V., Roy, A.E., Velgakis, M.J., Kanavos, S.S.: A photogravitational Hill problem and radiation effects on Hill stability of orbits. Astrophys. Space Sci. 271, 293–301 (2000)

    Article  ADS  MATH  Google Scholar 

  • Millis, R.L., Wasserman, L.H., Franz, O.G., et al.: The size, shape, density, and albedo of Ceres from its occultation of BD+8 deg 471. Icarus 72, 507–518 (1987)

    Article  ADS  Google Scholar 

  • Milone, E.F., Wilson, W.J.F.: Solar System Astrophysics Background Science and the Inner Solar System, 2nd edn. Springer, New York (2014)

    Google Scholar 

  • Moulton, F.R.: An Introduction to Celestial Mechanics, 2nd edn. Dover, New York (1914)

    Google Scholar 

  • Nagler, J.: Crash test for the Copenhagen problem. Phys. Rev. E 69, 066218 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  • Nagler, J.: Crash test for the restricted three-body problem. Phys. Rev. E 71, 026227 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  • Nagler, J., Krieger, M., Linke, M., Schönke, J., Jan Wiersig, J.: Leaking billiards. Phys. Rev. E 75, 046204 (2007)

    Article  ADS  Google Scholar 

  • Navarro, J.F., Henrard, J.: Spiral windows for escaping stars. Astron. Astrophys. 369, 1112–1121 (2001)

    Article  ADS  Google Scholar 

  • Norton, O.R., Chitwood, L.A.: Field Guide to Meteors and Meteorites, 1st edn. Springer, London (2008)

    Book  Google Scholar 

  • Oberti, P., Vienne, A.: An upgraded theory for Helene, Telesto, and Calypso. Astron. Astrophys. 397, 353–359 (2003)

    Article  ADS  Google Scholar 

  • Perdios, A.E., Kalantonis, V.S.: Critical periodic orbits in the restricted three–body problem with oblateness. Astrophys. Space Sci. 305, 331–336 (2006)

    Article  ADS  MATH  Google Scholar 

  • Perdiou, A.E., Perdios, E.A., Kalantonis, V.S.: Periodic orbits of the Hill problem with radiation and oblateness. Astrophys. Space Sci. 342, 19–30 (2012)

    Article  ADS  Google Scholar 

  • Poincaré, H.: History of Modern Physics and Astronomy, vol. 13. AIP, New York (1993)

    Google Scholar 

  • Press, H.P., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in FORTRAN 77, 2nd edn. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  • Schneider, J., Tél, T.: Extracting flow structures from tracer data. Ocean Dyn. 53, 64–72 (2003)

    Article  ADS  Google Scholar 

  • Schneider, J., Tél, T., Neufeld, Z.: Dynamics of “leaking” Hamiltonian systems. Phys. Rev. E 66, 066218 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  • Seoane, J.M., Aguirre, J., Sanjuán, M.A.F., Lai, Y.C.: Basin topology in dissipative chaotic scattering. Chaos 16, 023101 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  • Seoane, J.M., Sanjuán, M.A.F., Lai, Y.C.: Fractal dimension in dissipative chaotic scattering. Phys. Rev. E 76, 016208 (2007)

    Article  ADS  Google Scholar 

  • Sharma, R.K.: Periodic orbits of the second kind in the restricted three-body problem when the more massive primary is an oblate spheroid. Astrophys. Space Sci. 76, 255–258 (1981)

    Article  ADS  MATH  Google Scholar 

  • Sharma, R.K.: The linear stability of libration points of the photogravitational restricted three-body problem when the smaller primary is an oblate spheroid. Astrophys. Space Sci. 135, 271–281 (1987)

    Article  ADS  MATH  Google Scholar 

  • Sharma, R.K.: The periodic orbits of the second kind in terms of Giacaglia’s variables with oblateness. Earth Moon Planets 45, 213–218 (1989)

    Article  ADS  MATH  Google Scholar 

  • Sharma, R.K.: Periodic orbits of the third kind in the restricted three-body problem with oblateness. Astrophys. Space Sci. 166, 211–218 (1990)

    Article  ADS  MATH  Google Scholar 

  • Sharma, R.K., Subba Rao, P.V.: Stationary solutions and their characteristic exponents in the restricted three-body-problem when the more massive primary is an oblate spheroid. Celest. Mech. 13, 137–149 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Sharma, R.K., Subba Rao, P.V.: Effect of oblateness on triangular solutions at critical mass. Astrophys. Space Sci. 60, 247–250 (1979)

    Article  ADS  MATH  Google Scholar 

  • Sharma, R.K., Subba Rao, P.V.: On finite periodic orbits around the equilibrium solutions of the planar restricted three-body problem. In: Bhatnagar, K.B. (ed.) Space Dynamics and Celestial Mechanics, pp. 71–85. Reidel, Dordrecht (1986)

    Chapter  Google Scholar 

  • Simó, C., Stuchi, T.: Central stable/unstable manifolds and the destruction of KAM tori in the planar Hill problem. Physica D 140, 1–32 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Singh, J., Leke, O.: Equilibrium points and stability in the restricted three-body problem with oblateness and variable masses. Astrophys. Space Sci. 340, 27–41 (2012)

    Article  ADS  MATH  Google Scholar 

  • Singh, J., Leke, O.: Effect of oblateness, perturbations, radiation and varying masses on the stability of equilibrium points in the restricted three-body problem. Astrophys. Space Sci. 344, 51–61 (2013)

    Article  ADS  MATH  Google Scholar 

  • Singh, J., Leke, O.: Motion in a modified Chermnykh’s restricted three-body problem with oblateness. Astrophys. Space Sci. 350, 143–154 (2014)

    Article  ADS  Google Scholar 

  • Stuchi, T.J., Yokohama, A.A., et al.: Dynamics of a spacecraft and normalization around Lagrangian points in the Neptune–Triton system. Adv. Space Res. 42, 1715–1722 (2008)

    Article  ADS  Google Scholar 

  • Subba Rao, P.V., Sharma, R.K.: Oblateness effect on finite periodic orbits at L4. In: 39th Congress of the International Astronautical Federation (IAF-88-300), pp. 8–15 (1988). 6 pages

    Google Scholar 

  • Subba Rao, P.V., Sharma, R.K.: Effect of oblateness on the non-linear stability of \(L_{4}\) in the restricted three-body problem. Celest. Mech. Dyn. Astron. 65, 291–312 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Szebehely, V.: Theory of Orbits. Academic Press, New York (1967)

    Google Scholar 

  • Tuval, I., Schneider, J., Piro, O., Tél, T.: Opening up fractal structures of three-dimensional flows via leaking. Europhys. Lett. 65, 633–639 (2004)

    Article  ADS  Google Scholar 

  • Zotos, E.E.: A Hamiltonian system of three degrees of freedom with eight channels of escape: the great escape. Nonlinear Dyn. 76, 1301–1326 (2014a)

    Article  MathSciNet  Google Scholar 

  • Zotos, E.E.: Revealing the escape mechanism of three-dimensional orbits in a tidally limited star cluster. Mon. Not. R. Astron. Soc. 446, 770–792 (2014a)

    Article  ADS  Google Scholar 

  • Zotos, E.E.: Escapes in Hamiltonian systems with multiple exit channels: part I. Nonlinear Dyn. 78, 1389–1420 (2014b)

    Article  MathSciNet  Google Scholar 

  • Zotos, E.E.: Crash test for the Copenhagen problem with oblateness. Celest. Mech. Dyn. Astron. 122, 75–99 (2015b)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Acknowledgements

I would like to express my warmest thanks to the anonymous referee for the careful reading of the manuscript and for all the apt suggestions and comments which allowed us to improve both the quality and the clarity of the paper.

Author information

Authors and Affiliations

  1. Department of Physics, School of Science, Aristotle University of Thessaloniki, 541 24, Thessaloniki, Greece

    Euaggelos E. Zotos

Authors
  1. Euaggelos E. Zotos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Euaggelos E. Zotos.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zotos, E.E. How does the oblateness coefficient influence the nature of orbits in the restricted three-body problem?. Astrophys Space Sci 358, 33 (2015). https://doi.org/10.1007/s10509-015-2435-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10509-015-2435-z

Keywords

Search

Navigation