Abstract
The zero-velocity surfaces in the three-dimensional ring problem of N + 1 bodies and their parametric evolution is the subject of this paper. These surfaces, which are also known as Hill's or Jacobian surfaces, provide us with valuable information concerning the regions of the permissible particle motion and the existence of equilibrium positions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banfi, V.: 1987, ‘On Gouda's surfaces in the magnetic binary problem’, Celest. Mech. & Dyn. Astr. 40, 67–76.
Bozis, G.: 1976, ‘Zero-velocity surfaces for the general planar three-body problem’, Astrop. Sp. Sci. 43, 355–368.
Bozis, G. and Michalodimitrakis, M.: 1982, ‘Bounded motion for a modified three-body problem’, Astrop. Sp. Sci. 86, 377–396.
Chapsiadis, A and Michalodimitrakis, M.: 1988, ‘Bounded motion in a two-body system consisting of a solid body and a material point’, Celest. Mech. & Dyn. Astr. 41, 53–63.
Desiniotis, C. D. and Kazantzis, P. G.: 1993, ‘The areas of motion in the three dipole problem’, Astrop. Sp. Sci. 205, 305–326.
Faintich, M.: 1973, ‘Three-dimensional zero-velocity contours’, Celest. Mech. & Dyn. Astr. 8, 291–296.
Goudas, C. L., Leftaki, M. and Petsagourakis, E. G.: 1990, ‘Motions in the fields of two rotating magnetic dipoles. III Zero-velocity curves and surfaces’, Celest. Mech. & Dyn. Astr. 47, 1–14.
Huang, Su-Shu: 1960, ‘Very restricted four-body problem’, NASA, Technical Note D-501, 1–13.
Huang, Su-Shu: 1961, ‘Some dynamical properties of natural and artificial satellites’, Astr. J. 66(4), 157–159.
Kalvouridis, T. J.: 1999, ‘A planar case of the n + 1 body problem: The ring problem’, Astrop. Sp. Sci. 260, (3), 309–325.
Kalvouridis, T. J., Mavraganis, A. G., Pangalos, C. and Zagouras, Ch.: 1985, ‘Areas of equatorial motion in the magnetic binary problem’, Celest. Mech. & Dyn. Astr. 35, 161–170.
Kalvouridis, T. J. and Mavraganis, A. G.: 1996, ‘The restricted 2+2 body problem: the Permissimble areas of motion of the minor bodies close to the collinear equilibria of their center of mass’, in: Docobo, J. A., Elipe, A. and McAlister, H. (eds.), Proceedings of the International Workshop on Multiple Stars and Celestial Mechanics (Santiago de Compostella, Spain, July 29-August 1, 1996). Astrophysics and Space Science Library, Kluwer Academic Publishers, Vol. 223, pp. 373–376.
Lundberg, J., Szebehely, V., Nerem, R. S. and Beal, B.: 1985, ‘Surfaces of zero-velocity in the restricted problem of three bodies’, Celest. Mech. & Dyn. Astr. 36, 191–205.
Maranhao, D. and Llibre, J.: 1999, ‘Ejection-collision orbits and invariant punctured tori in a restricted four-body problem’, Celest. Mech. & Dyn. Astr. 71, 1–14.
Marchal, C. and Saari, D.: 1975, ‘Hill regions for the general three-body problem’, Celest. Mech. & Dyn. Astr. 12, 115–129.
Michalodimitrakis, M. and Bozis, G.: 1985, ‘Bounded motion in a generalized two-body problem’, Astrop. Sp. Sci. 117, 217–225.
Moulton, R. F.: 1914, An introduction to Celestial Mechanics, 2nd, revised edn, McMillan, New York.
Ollongren, A.: 1988, ‘On a particular restricted five-body problem. An analysis with Computer Algebra’, J. Symb. Comput. 6, 117–126.
Ragos, O. and Zagouras, Ch.: 1988, ‘The zero-velocity surfaces in the photo-gravitational restricted three-body problem’, Earth, Moon Planets 41, 257–288.
Roy, A. E.: 1978, Orbital motion, Adam Hilger, Bristol.
Roy, A. E. and Steves, B. A.: 1998, ‘Some special restricted four-body problems-II From Caledonian to Copenhagen’, Planet. Space Sci. 46(11/12), 1475–1486.
Saari, D.: 1986, ‘Zero-velocity hypersurfaces for the general three-dimensional three-body problem’, Celest. Mech. & Dyn. Astr. 39, 341–343.
Salo, H. and Yoder, C. F.: 1988, ‘The dynamics of co-orbital satellite systems’, Astron. Astrophys. 205, 309–327.
Sergysels, R.: 1986, ‘Zero-velocity hypersurfaces for the general three-dimensional three-body problem’, 38, 207–214.
Sergysels, R. and Loks, A.: 1987, ‘Restrictions on the motion in the general four-body problem’, Astron. Astrop. 182, 163–166.
Steves, B. A. and Roy, A. E.: 1998, ‘Some special restricted four-body problems-I. Modelling the Caledonian problem’, Planet. Space Sci. 46(11/12), 1465–1474.
Subrahmanyam, P.: 1980, ‘Surfaces of zero relative velocity in the gravitational field of a pair of two interacting galaxies’, Astrop. Sp. Sci. 67, 503–510.
Subrahmanyam, P. and Narasimhan, K. S.: 1989, ‘Surfaces of zero relative velocity in the gravitational field of a pair of two interacting galaxies,II’, Astrop. Sp. Sci. 155, 319–326.
Szebehely, V.: 1967, Theory of orbits, Academic Press, New York.
Szebehely, V.: 1963, ‘Zero velocity curves and orbits in the restricted problem of three bodies’, Astron. J. 68(3), 147–151.
Rights and permissions
About this article
Cite this article
Kalvouridis, T. Zero-velocity surfaces in the three-dimensional ring problem of N + 1 bodies. Celestial Mechanics and Dynamical Astronomy 80, 133–144 (2001). https://doi.org/10.1023/A:1011919508410
Issue Date:
DOI: https://doi.org/10.1023/A:1011919508410