Abstract
In the current study, the existence of periodic orbits around a fixed homogeneous cube is investigated, and the results have powerful implications for examining periodic orbits around non-spherical celestial bodies. In the two different types of symmetry planes of the fixed cube, periodic orbits are obtained using the method of the Poincaré surface of section. While in general positions, periodic orbits are found by the homotopy method. The results show that periodic orbits exist extensively in symmetry planes of the fixed cube, and also exist near asymmetry planes that contain the regular Hex cross section. The stability of these periodic orbits is determined on the basis of the eigenvalues of the monodromy matrix. This paper proves that the homotopy method is effective to find periodic orbits in the gravity field of the cube, which provides a new thought of searching for periodic orbits around non-spherical celestial bodies. The investigation of orbits around the cube could be considered as the first step of the complicated cases, and helps to understand the dynamics of orbits around bodies with complicated shapes. The work is an extension of the previous research work about the dynamics of orbits around some simple shaped bodies, including a straight segment, a circular ring, an annulus disk, and simple planar plates.
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Alberti, A., Vidal, C.: Dynamics of a particle in a gravitational field of a homogeneous annulus disk. Celest. Mech. Dyn. Astron. 98(2), 75–93 (2007). doi:10.1007/s10569-007-9071-z
Arribas, M., Elipe, A.: Non-integrability of the motion of a particle around a massive straight segment. Phys. Lett. A 281(2/3), 142–148 (2001)
Azevêdo, C., Ontaneda, P.: On the existence of periodic orbits for the fixed homogeneous circle problem. J. Differ. Equ. 235(2), 341–365 (2007)
Azevêdo, C., Cabral, H.E., Ontaneda, P.: On the fixed homogeneous circle problem. Adv. Nonlinear Stud. 7(1), 47–75 (2007)
Blesa, F.: Periodic orbits around simple shaped bodies. Monogr. Semin. Mat. García de Galdeano 33, 67–74 (2006)
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)
Elipe, A., Riaguas, A.: Nonlinear stability under a logarithmic gravity field. Int. Math. J. 3, 435–453 (2003)
Fukushima, T.: Precise computation of acceleration due to uniform ring or disk. Celest. Mech. Dyn. Astron. 108(4), 339–356 (2010). doi:10.1007/s10569-010-9304-4
Geissler, P., Petit, J.-M., Durda, D., Greenberg, R., Bottke, W., Nolan, M., Moore, J.: Erosion and ejecta reaccretion on 243 Ida and its moon. Icarus 120(42), 140–157 (1996)
Gutiérrez–Romero, S., Palacián, J.F., Yanguas, P.: The invariant manifolds of a finite straight segment. Monogr. Real Acad. Ciencias Zaragoza 25, 137–148 (2004)
Liu, X., Baoyin, H., Ma, X.: Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube. Astrophys. Space Sci. (2011). doi:10.1007/s10509-011-0669-y
Michalodimitrakis, M., Bozis, G.: Bounded motion in a generalized two-body problem. Astrophys. Space Sci. 117(2), 217–225 (1985)
Palacián, J.F., Yanguas, P., Gutiérrez–Romero, S.: Approximating the invariant sets of a finite straight segment near its collinear equilibria. SIAM J. Appl. Dyn. Syst. 5(1), 12–29 (2006)
Riaguas, A., Elipe, A., Lara, M.: Periodic orbits around a massive straight segment. Celest. Mech. Dyn. Astron. 73(1/4), 169–178 (1999)
Riaguas, A., Elipe, A., López-Moratalla, T.: Non-linear stability of the equilibria in the gravity field of a finite straight segment. Celest. Mech. Dyn. Astron. 81(3), 235–248 (2001)
Scheeres, D.J., Ostro, S.J., Hudson, R.S., Werner, R.A.: Orbits close to asteroid 4769 Castalia. Icarus 121, 67–87 (1996). doi:10.1006/icar.1996.0072
Scheeres, D.J., Marzari, F., Tomasella, L., Vanzani, V.: ROSETTA mission: satellite orbits around a cometary nucleus. Planet. Space Sci. 46(6/7), 649–671 (1998a)
Scheeres, D.J., Ostro, S.J., Hudson, R.S., Dejong, E.M., Suzuki, S.: Dynamics of orbits close asteroid 4179 Toutatis. Icarus 132(1), 53–79 (1998b). doi:10.1006/icar.1997.5870
Scheeres, D.J., Williams, B.G., Miller, J.K.: Evaluation of the dynamic environment of an asteroid: applications to 433 Eros. J. Guid. Control Dyn. 23(3), 466–475 (2000)
Werner, R.A.: The gravitational potential of a homogeneous polyhedron or don’t cut corners. Celest. Mech. Dyn. Astron. 59(3), 253–278 (1994)
Werner, R.A., Scheeres, D.J.: Exterior gravitation of a polyhedron derived and compared with harmonic and mascon gravitation representations of asteroid 4769 Castalia. Celest. Mech. Dyn. Astron. 65(3), 313–344 (1997)
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Liu, X., Baoyin, H. & Ma, X. Periodic orbits in the gravity field of a fixed homogeneous cube. Astrophys Space Sci 334, 357–364 (2011). https://doi.org/10.1007/s10509-011-0732-8
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DOI: https://doi.org/10.1007/s10509-011-0732-8