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.
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References
Abad, A., Elipe, E.: Evolution strategies for computing periodic orbits. In: AAS09-143 Proceedings, 19th AAS/AIAA Space Flight Mechanics Meeting, Savannah, Georgia (2009)
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)
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)
Arribas, M., Elipe, A.: Bifurcations and equilibria in the extended N-body ring problem. Mech. Res. Commun. 31, 129–143 (2005)
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)
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)
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)
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)
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)
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)
Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin (1945)
Bulirsch, R.: Numerical calculation of elliptic integrals and elliptic functions. Numerische Mathematik, vol. 7, pp. 78–90. Springer, New York (1971)
Carlson, B.C.: Computing elliptic integrals by duplication. Numer. Math. 33, 1–16 (1979)
Deprit, A., Henrard, J.: Natural families of periodic orbits. Astron. J. 72, 158–172 (1967)
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)
Elipe, A., Lara, M.: Frozen orbits about the moon. J. Guid. Control Dyn. 26(2), 238–243 (2003)
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)
Fukushima, T.: Precise computation of acceleration due to uniform ring or disk. Celest. Mech. Dyn. Astron. 108, 339–356 (2010)
Hénon, M.: Exploration numérique du problème restreint. II. Masses égales, stabilité des orbites périodiques. Ann. Astrophys. 28, 992–1007 (1965)
Kalvouridis, T.J.: Periodic solutions in the ring problem. Astrophys. Space Sci. 266(4), 467–494 (1999)
Kellogg, O.D.: Foundations of potential theory. Dover, New York (1929)
Krough, F.T., Ng, E.W., Snyder, W.V.: The gravitational field of a disk. Celest. Mech. Dyn. Astron. 26, 395–405 (1982)
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)
Lass, H., Blitzer, L.: The gravitational potential due to uniform disks and rings. Celest. Mech. Dyn. Astron. 30, 225–228 (1983)
Longaretti, P.Y.: Saturn’s main ring particle size distribution, An analytic approach. Icarus 81, 51–73 (1989)
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)
Maxwell, J.: On the Stability of Motions of Saturn’s Rings. Macmillan & Co., Cambridge (1859)
Riaguas, A., Elipe, A., Lara, M.: Periodic orbits around a massive straight segment. Celest. Mech. Dyn. Astron. 73, 169–178 (1999)
Scheeres, D.J.: On symmetric central configurations with application to the satellite motion about rings. Ph.D. thesis, University of Michigan (1992)
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)
Sicardy, B.: Numerical exploration of planetary arc dynamics. Icarus 89(2), 197–212 (1991)
Stone, J.M., Balbus, S.A.: Angular momentum transport in accretion disks by convection. Astrophys. J. 464, 364 (1996)
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)
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
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)
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Tresaco, E., Elipe, A. & Riaguas, A. Computation of families of periodic orbits and bifurcations around a massive annulus. Astrophys Space Sci 338, 23–33 (2012). https://doi.org/10.1007/s10509-011-0925-1
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DOI: https://doi.org/10.1007/s10509-011-0925-1