Abstract
The Hill stability of the nine known triple asteroid systems in the solar system has been investigated in a framework of the three body system. In this paper, the Sun and triple-asteroid system are treated as a four body system to analyze the influence of the Sun on the Hill stability of the triple subsystem. First, the relationship of the total energy and the angular momentum between the four body system and the triple subsystem is derived. It is found that the total energy of this 1–3 configuration four body system is the sum of the energy of the triple subsystem and the energy of a two-body system composed of the Sun and the mass center of the subsystem; so is the angular momentum. Then, the Hill stability of the triple subsystem is reinvestigated using a previous criterion in the four body problem (Gong and Liu in Mon. Not. R. Astron. Soc. 462:547–553, 2016) and the results are compared to those in the three body problem. Among the nine known triple-asteroid systems, 1995 CC and 1999 TC are Hill stable for both models; the others are stable in the three body model while not stable in the four body model. In addition, the exploration of Pluto by New Horizons has attracted great attention in recent years, the Sun-Pluto-Charon-Hydra four body system is investigated in the paper, and it is found that the system is Hill stable.
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References
Alvarez-Ramirez, M., Vidal, C.: Dynamical aspects of an equilateral restricted four-body problem. Math. Probl. Eng. 2009, 181360 (2009), 23 pp.
Arenstorff, R.F.: Central configurations of four bodies with one inferior mass. Celest. Mech. 28, 9–15 (1982)
Bozis, G.: Zero velocity surfaces for the general planar three-body problem. Astrophys. Space Sci. 43, 355–368 (1976)
Donnision, J.R.: The effects of eccentricity on the hierarchical stability of low-mass binaries in three body systems. Mon. Not. R. Astron. Soc. 231, 85–95 (1988)
Donnision, J.R., Williams, I.P.: The hierarchical stability of satellite systems. Mon. Not. R. Astron. Soc. 215, 567–573 (1985)
Donnison, J.R., Williams, I.P.: The stability of coplanar three-body systems with application to the solar system. Celest. Mech. 31, 123–128 (1983)
Gong, S., Liu, C.: Hill stability of the satellites in coplanar four-body problem. Mon. Not. R. Astron. Soc. 462, 547–553 (2016)
Hill, G.W.: Researches in the lunar theory. Am. J. Math. 1(2), 129–147 (1878)
Johnston, W.R.: Asteroids with satellites (2017). Last updated 29 January 2017 (http://www.johnstonsarchive.net/astro/asteroidmoons.html)
Leandro, E.S.G.: On the central configurations of the planar restricted four-body problem. J. Differ. Equ. 226, 323–351 (2006)
Liu, X.D., Baoyin, H.X., Georgakarakos, N., Donnison, J.R., Ma, X.R.: The Hill stability of triple minor planets in the solar system. Mon. Not. R. Astron. Soc. 427, 1034–1042 (2012)
Liu, X.D., Baoyin, H., Marchis, F.: The hierarchical stability of the seven known large size ratio triple asteroids using the empirical stability parameters. Astrophys. Space Sci. 349, 677–680 (2014)
Loks, A., Sergysels, R.: Zero velocity hypersurfaces for the general planar four body problem. Astron. Astrophys. 149, 462–464 (1985)
Mu, J.S., Gong, S.P., Li, J.F.: Coupled control of reflectivity modulated solar sail for GeoSail formation flying. J. Guid. Control Dyn. 38, 740–751 (2015)
Qi, Y., Xu, S.J.: Lunar capture in the planar restricted three-body problem. Celest. Mech. Dyn. Astron. 120(4), 401–422 (2014)
Roy, A.E., Steves, B.A.: The Caledonian symmetrical double binary four-body problem I: Surfaces of zero-velocity using the energy integral. Celest. Mech. Dyn. Astron. 78, 299–318 (2000)
Scheeres, D.J.: The restricted Hill four-body problem with applications to the Earth-Moon-Sun system. Celest. Mech. Dyn. Astron. 70, 75–78 (1998)
Szebehely, V.: Analytical determination of the measure of stability of triple stellar systems. Celest. Mech. 15, 107–110 (1977)
Szebehely, V., Zare, K.: Stability of classical triplets and of their hierarchy. Astron. Astrophys. 58, 145–152 (1977)
Walker, I.W., Roy, A.E.: Stability criteria in many-body systems I. An empirical stability criterion for co-rotational three-body systems. Celest. Mech. 22, 371–402 (1980)
Walker, I.W., Roy, A.E.: Stability criteria in many-body systems IV. Empirical stability parameters for general hierarchical dynamical systems. Celest. Mech. 29, 149–178 (1983a)
Walker, I.W., Roy, A.E.: Stability criteria in many-body systems V. On the totality of possible hierarchical general four-body systems. Celest. Mech. 29, 267–294 (1983b)
Zare, K.: The effects of integrals on the totality of solutions of dynamical systems. Celest. Mech. 14, 73–83 (1976)
Zare, K.: Bifurcation points in the planar problem of three bodies. Celest. Mech. 16, 35–38 (1977)
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This work was supported by the National Natural Science Foundation of China (Grant No. 11432001).
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Liu, C., Gong, S. The Hill stability of triple planets in the Solar system. Astrophys Space Sci 362, 127 (2017). https://doi.org/10.1007/s10509-017-3086-z
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DOI: https://doi.org/10.1007/s10509-017-3086-z