Abstract
The bifurcations of relative equilibrium points in the gravitational field of asteroids during homotopy deformation are investigated in this paper. A theorem concerning the continuity of non-degenerate equilibrium points under small variations of parameters and degeneration of equilibrium points after collisions is presented. This can contribute to understanding the bifurcations of the number of equilibrium points. We concentrate on four types of homotopy deformation: from a rotating ellipsoid to a rotating sphere; from 216 Kleopatra to a rotating ellipsoid, a cube and a rectangular parallelepiped. The results show that during the process of deforming a rotating ellipsoid to a sphere, the number of relative equilibria may vary from 1 to 3 to 5. Infinitely many relative equilibria may occur in some critical cases. When deforming 216 Kleopatra to a rotating ellipsoid or a rectangular parallelepiped, the number of relative equilibria changes from 7 to 5 and collisional annihilation of two relative equilibria inside the body occurs, corresponding to a saddle-node bifurcation. The number of relative equilibria changes from 7 to 5 to 9 when 216 Kleopatra is deformed into a rotating cube. Both the saddle-node bifurcation and Hopf bifurcation occur during deformation. Moreover, the positions, eigenvalues, topological types and stability of equilibrium points are studied here.
Similar content being viewed by others
References
Delsate, N.: Analytical and numerical study of the ground-track resonances of dawn orbiting Vesta. Planet Space Sci. 59, 1372–83 (2011)
Descamps, P., Marchis, F., Berthier, J.: Triplicity and physical characteristics of asteroid (216) Kleopatra. Icarus 211(2), 1022–1033 (2011)
Doedel, E., Romanov, V.A.: Periodic orbits associated with the libration points of the massive rotating straight segment. Int J Bifurcation Chaos 22 (2012)
Fukushima, T.: Precise computation of acceleration due to uniform ring or disk. Celest. Mech. Dyn. Astron. 108(4), 339–356 (2010)
Gong, S., Li, J.: Asteroid capture using lunar flyby. Adv. Space Re. 56(5), 3848–58 (2015a)
Gong, S., Li, J.: Equilibria near asteroids for solar sails with reflection control devices. Astrophys. Space Sci. 355, 213–223 (2015b)
Hartmann, W.K.: The shape of Kleopatra. Science 288, 820 (2000)
Hirabayashi, M., Scheeres, D.J.: Analysis of asteroid (216) Kleopatra using dynamical and structural constraints. Astrophys. J. 780(2), 158 (2014)
Holsapple, K.A., Michel, P.: Tidal disruptions: A continuum theory for solid bodies. Icarus 183(2), 331–348 (2006)
Holsapple, K.A., Michel, P.: Tidal disruptions. II. A continuum theory for solid bodies with strength, with applications to the solar system. Icarus 193(1), 283–301 (2008)
Jewitt, D., Weaver, H., Agarwal, J., Mutchler, M., Drahus, M.: A recent disruption of the main-belt asteroid p/2010 a2. Nature 467(4), 817–819 (2010)
Jewitt, D., Agarwal, J., Li, J., Weaver, H., Mutchler, M., Larson, S.: Disintegrating asteroid p/2013 r3. Astrophys. J. Lett. 784(4), L8 (2014)
Jiang, Y.: Equilibrium points and orbits around asteroid with the full gravitational potential caused by the 3d irregular shape. Astrodynamics 2(4), 361–373 (2018)
Jiang, Y., Baoyin, H.: Parameters and bifurcations of equilibrium points in the gravitational potential of irregular-shaped bodies subjected to a varying external shape. Adv. Space Res. 62, 3199–3213 (2019)
Jiang, Y., Baoyin, H., Li, J., Li, H.: Orbits and manifolds near the equilibrium points around a rotating asteroid. Astrophys. Space Sci. 349(1), 83–106 (2014)
Jiang, Y., Baoyin, H., Li, H.: Collision and annihilation of relative equilibrium points around asteroids with a changing parameter. Month. Notices R. Astronom. Soc. 4, 3924–3931 (2015)
Jiang, Y., Baoyin, H., Wang, X., Yu, Y., Li, H., Peng, C.: Order and chaos near equilibrium points in the potential of rotating highly irregular-shaped celestial bodies. Nonlinear Dyn. 83(1), 231–252 (2016)
Li, X., Dong, Q., Cui, P.: The equilibria and periodic orbits around a dumbbell-shaped body. Astrophys. Space Sci. 348(2), 417–426 (2013)
Liu, X., Baoyin, H., Ma, X.: Periodic orbits in the gravity field of a fixed homogeneous cube. Astrophys. Space Sci. 333(2), 409–418 (2011)
Liu, X., Baoyin, H., Ma, X.: Dynamics of surface motion on a rotating massive homogeneous body. Sci. China Phys. Mech. Astron. 56(4), 818 (2013)
Macmillan, W.: The Theory of the Potential. McGraw-Hill, New York (1930)
Michel, P., Cheng, A., Kuppers, M., Pravec, P., Blum, J., Delbo, M., et al.: Science case for the asteroid impact mission (aim): a component of the asteroid impact & deflection assessment (AIDA) mission. Adv. Space Res. 57(12), 2529–2547 (2016)
Neese, C.: Small body radar shape models v2.0.ear-a 5-ddr-radarshape-models-v2.0. NASA Planetary Data System (2004)
Nesvorny, D., Bottke, W.F., Jr., Levison HF, D.L.: The recent breakup of an asteroid in the main-belt region. Nature 417(6890), 720–771 (2002)
Ostro, S.J., Hudson, R.S., Nolan, M.C., Margot, J.L., Scheeres, D.J., Campbell, D.B., et al.: Radar observations of asteroid 216 Kleopatra. Science 288(5467), 836–839 (2000)
Scheeres, D.J.: Dynamics about uniformly rotating triaxial ellipsoids: Applications to asteroids. Icarus 110(2), 225–238 (1994)
Scheeres, D.J.: Orbit mechanics about asteroids and comets. J. Guidance Control Dyn. 35(3), 987–997 (2012)
Scheeres, D.J., Ostro, S.J., Hudson, R.S., Werner, R.A.: Orbits close to asteroid 4769 Castalia. Icarus 121(4), 67–87 (1996)
Scheeres, D.J., Ostro, S.J., Hudson, R.S., Dejong, E.M., Suzuki, S.: Dynamics of orbits close to asteroid 4179 Toutatis. Icarus 132(1), 53–79 (1998)
Scheeres, D.J., Williams, B.G., Miller, J.K.: Evaluation of the dynamic environment of an asteroid: Applications to 433 Eros. J. Guidance Control Dyn. 23(3), 466–475 (2000)
Shepard, M.K., Timerson, B., Scheeres, D.J., Benner, L.A.M., Giorgini, J.D., Howell, E.S., et al.: A revised shape model of asteroid (216) Kleopatra. Icarus 311, 197–209 (2018)
Takahashi, Y., Busch, M.W., Scheeres, D.J.: Spin state and moment of inertia characterization of 4179 Toutatis. Astronom. J. 146(4), 95 (2013)
Walsh, K.J., Richardson, D.C., Michel, P.: Rotational breakup as the origin of small binary asteroids. Nature 454(4), 188–191 (2008)
Wang, X., Gong, S., Li, J.: A method for classifying orbits near asteroids. Acta Mech. Sinica 30, 316–325 (2014a)
Wang, X., Jiang, Y., Gong, S.: Analysis of the potential field and equilibrium points of irregular-shaped minor celestial bodies. Astrophys. Space Sci. 353(1), 105–121 (2014)
Wang, Y., Xu, S.: Non-equatorial equilibrium points around an asteroid with gravitational orbit-attitude coupling perturbation. Astrodynamics 4(1), 1–16 (2020)
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 (1996)
Yu, Y., Baoyin, H.: Routing the asteroid surface vehicle with detailed mechanics. Acta Mech. Sinica 30(3), 301–9 (2014)
Zhang, Y., Michel, P.: Tidal distortion and disruption of rubble-pile bodies revisited. soft-sphere discrete element analyses. Astron. Astrophys. 640:A102 (2020)
Acknowledgements
We gratefully acknowledge the reviewers for their helpful and constructive suggestions that helped us to improve the paper substantially.
Funding
This work was supported by the National Natural Science Foundation of China (Grant No. 11772356).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Liu, Y., Jiang, Y. & Li, H. Bifurcations of relative equilibrium points during homotopy deformation of asteroids. Celest Mech Dyn Astr 133, 42 (2021). https://doi.org/10.1007/s10569-021-10040-w
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10569-021-10040-w