Abstract
Equilibrium points and periodic orbits in irregular gravitational fields are significant for an understanding of dynamical behaviors around asteroids as well as deep space exploring missions. The dipole segment is a good alternative model to study qualitative dynamical properties near dumbbell-shaped asteroids. In this paper, the dipole segment model and its equilibrium points are simply introduced. The stability of the two triangular equilibrium points of the system is numerically examined. Next, periodic orbits are presented around the dipole segment model in two different cases, in which triangular equilibria are linearly stable and unstable, respectively. New types of periodic orbits are illustrated in detail, including their orbital shapes, periods and the Jacobi integral. The orbital stability, topological classification and bifurcations of these orbits are also analyzed with numerical continuations.
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Werner R A. The gravitational potential of a homogeneous polyhedron or don’t cut corners. Celestial Mech Dyn Astr, 1994, 59: 253–278
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, 1997, 65: 313–344
Lan L, Yang M, Baoyin H, et al. The periodic dynamics of the irregular heterogeneous celestial bodies. Astrophys Space Sci, 2017, 362: 38
Kaula W M. Theory of Satellite Geodesy: Application of Satellite to Geodesy. Mineola: Dover Publications, 2000
Geissler P, Petit J M, Durda D D, et al. Erosion and ejecta reaccretion on 243 Ida and its moon. Icarus, 1996, 120: 140–157
Bartczak P, Breiter S, Jusiel P. Ellipsoids, material points and material segments. Celestial Mech Dyn Astr, 2006, 96: 31–48
Broucke R A, Elipe A. The dynamics of orbits in a potential field of a solid circular ring. Regul Chaotic Dyn, 2005, 10: 129–143
Azevêdo C, Ontaneda P, Cabral H E. On the fixed homogeneous circle problem. Adv NOnlinear Stud, 2007, 7: 47–75
Azevêdo C, Ontaneda P. On the existence of periodic orbits for the fixed homogeneous circle problem. J Differ Equ, 2007, 235: 341–365
Alberti A, Vidal C. Dynamics of a particle in a gravitational field of a homogeneous annulus disk. Celestial Mech Dyn Astr, 2007, 98: 75–93
Fukushima T. Precise computation of acceleration due to uniform ring or disk. Celest Mech Dyn Astr, 2010, 108: 339–356
Blesa F. Periodic orbits around simple shaped bodies. Monografías del Seminario Matemático García de Galdeano, 2006, 33: 67–74
Guibout V, Scheeres D J. Stability of surface motion on a rotating ellipsoid. Celest Mech Dyn Astron, 2003, 87: 263–290
Liu X, Baoyin H, Ma X. Periodic orbits in the gravity field of a fixed homogeneous cube. Astrophys Space Sci, 2011, 334: 357–364
Riaguas A, Elipe A, Lara M. Periodic orbits around a massive straight segment. Celest Mech Dyn Astron, 1999, 73: 169–178
Arribas M, Elipe A. Non-integrability of the motion of a particle around a massive straight segment. Phys Lett A, 2001, 281: 142–148
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, 2001, 81: 235–248
Elipe A, Riaguas A. Nonlinear stability under a logarithmic gravity field. Int Math J, 2003, 3: 435–453
Gutiérrez-Romero S, Palacián J F, Yanguas P. The invariant manifolds of a finite straight segment. Monografías de la Real Academia de Ciencias de Zaragoza, 2004, 25: 137–148
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, 2006, 5: 12–29
Romanov V A, Doedel E J. Periodic orbits associated with the libration points of the massive rotating straight segment. Int J Bifurcat Chaos, 2014, 24: 1430012
Bartczak P. Double material segment as the model of irregular bodies. Celest Mech Dyn Astron, 2003, 86: 131–141
Najid N E, Haj Elourabi E, Zegoumou M. Potential generated by a massive inhomogeneous straight segment. Res Astron Astrophys, 2011, 11: 345–352
Chermnykh S V. On the stability of libration points in a certain gravitational field. Vestn Leningr Univ, 1987, 2: 73–77
Gozdziewski K. Nonlinear stability of the lagrangian libration points in the chermnykh problem. Celest Mech Dyn Astron, 1998, 70: 41–58
Li X, Qiao D, Cui P. The equilibria and periodic orbits around a dumbbell-shaped body. Astrophys Space Sci, 2013, 348: 417–426
Zeng X, Jiang F, Li J, et al. Study on the connection between the rotating mass dipole and natural elongated bodies. Astrophys Space Sci, 2015, 356: 29–42
Zeng X Y, Liu X D, Li J F. Extension of the rotating dipole model with oblateness of both primaries. Res Astron Astrophys, 2017, 17: 11–20
Shang H, Wu X, Cui P. Periodic orbits in the doubly synchronous binary asteroid systems and their applications in space missions. Astrophys Space Sci, 2015, 355: 69–87
Yu Y, Richardson D C, Michel P. Structural analysis of rubble-pile asteroids applied to collisional evolution. Astrodynamics, 2017, 1: 57–69
Yang HW, Zeng XY, et al. Feasible region and stability analysis for hovering around elongated asteroids with low thrust. Res Astron Astrophys, 2015, 15: 1571–1586
Feng F, Tang L N, Xu J F, et al. A review of the end-effector of large space manipulator with capabilities of misalignment tolerance and soft capture. Sci China Tech Sci, 2016, 59: 1621–1638
Yu Y, Baoyin H. Orbital dynamics in the vicinity of asteroid 216 Kleopatra. Astron J, 2012, 143: 62–70
Jiang Y, Baoyin H. Periodic orbit families in the gravitational field of irregular-shaped bodies. Astron J, 2016, 152: 137
Zeng X, Liu X. Searching for time optimal periodic orbits near irregularly shaped asteroids by using an indirect method. IEEE Trans Aerosp Electron Syst, 2017, 53: 1221–1229
Shang H, Wu X, Ren Y, et al. An efficient algorithm for global periodic orbits generation near irregular-shaped asteroids. Commun NOnlinear Sci Numer Simul, 2017, 48: 550–568
Zeng X Y, Alfriend K T. Periodic orbits in the Chermnykh problem. Astrodynamics, 2017, 1: 41–55
Yu Y, Baoyin H. Generating families of 3D periodic orbits about asteroids. Mon Not R Astron Soc, 2012, 427: 872–881
Zeng X Y, Fang, B D. Li J F, et al. Generalized flyby trajectories around elongated minor celestial bodies as a rotating mass dipole. Acta Mechanica Sinica, 2016, 32: 535–545
Jiang F, Baoyin H, Li J. Practical techniques for low-thrust trajectory optimization with homotopic approach. J Guidance Control Dyn, 2012, 35: 245–258
Jiang Y, Yu Y, Baoyin H. Topological classifications and bifurcations of periodic orbits in the potential field of highly irregular-shaped celestial bodies. NOnlinear Dyn, 2015, 81: 119–140
Ni Y S, Jiang Y, Baoyin H X. Multiple bifurcations in the periodic orbit around Eros. Astrophys Space Sci, 2016, 361: 1–15
Jiang Y, Baoyin H, Li H. Periodic motion near the surface of asteroids. Astrophys Space Sci, 2015, 360: 63
Yu Y, Baoyin H, Jiang Y. Constructing the natural families of periodic orbits near irregular bodies. Mon Not R Astron Soc, 2015, 453: 3270–3278
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Zhang, Y., Zeng, X. & Liu, X. Study on periodic orbits around the dipole segment model for dumbbell-shaped asteroids. Sci. China Technol. Sci. 61, 819–829 (2018). https://doi.org/10.1007/s11431-017-9099-y
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DOI: https://doi.org/10.1007/s11431-017-9099-y