Abstract
The \(c^{2}E\) condition, for determining the Hill stability of a coplanar four-body system, is analyzed for the particular situation where the mass of a body is much greater than the other three. In this case, the criterion can be expressed in a closed form, which makes it easier to judge the stability of a system or predict the stable regions described by physical and orbital parameters. The criterion is applied to nine known four-body systems in our Solar System. Besides, the situations where a passing star moving on a parabolic or hyperbolic orbit encounters a triple system are considered. The Hill stability of all extrasolar triple systems are investigated during encounters of a passing star.
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Alvarez-Ramirez, M., Vidal, C.: Math. Probl. Eng. 2009(4), 266 (2014)
Arenstorf, R.F.: Celest. Mech. 28(1–2), 9 (1982)
Bottke, W.F., Vokrouhlický, D., Rubincam, D.P., Nesvorný, D.: Annu. Rev. Earth Planet. Sci. 34(1), 157 (2006). https://doi.org/10.1146/annurev.earth.34.031405.125154
Bozis, G.: Astrophys. Space Sci. 43(2), 355 (1976)
de la Fuente Marcos, C., de la Fuente Marcos, R., Aarseth, S.J.: Mon. Not. R. Astron. Soc. Lett. 476(1), 1 (2018)
Delbo’, M., dell’Oro, A., Harris, A.W., Mottola, S., Mueller, M.: Icarus 190(1) 236 (2007). https://doi.org/10.1016/j.icarus.2007.03.007
Donnison, J.R.: Mon. Not. R. Astron. Soc. 210, 915 (1984a)
Donnison, J.R.: Celest. Mech. 32(2), 145 (1984b)
Donnison, J.R.: Mon. Not. R. Astron. Soc. 369(3), 1267 (2006). https://doi.org/10.1111/j.1365-2966.2006.10372.x
Donnison, J.R.: Planet. Space Sci. 56(7), 927 (2008)
Donnison, J.R.: Planet. Space Sci. 57(7), 771 (2009). https://doi.org/10.1016/j.pss.2009.01.014
Donnison, J.R.: Mon. Not. R. Astron. Soc. 406, 1918 (2010). https://doi.org/10.1111/j.1365-2966.2010.16796.x
Donnison, J.R.: Mon. Not. R. Astron. Soc. 415(1), 470 (2011). https://doi.org/10.1111/j.1365-2966.2011.18720.x
Donnison, J.R., Williams, I.P.: Celest. Mech. 31(2), 123 (1983)
Érdi, B., Czirják, Z.: Celest. Mech. Dyn. Astron. 125(1), 33 (2016). https://doi.org/10.1007/s10569-016-9672-5
Georgakarakos, N.: Celest. Mech. Dyn. Astron. 100(2), 151 (2008). https://doi.org/10.1007/s10569-007-9109-2
Golubev, V.G.: Sov. Phys. Dokl. 12, 529 (1967)
Golubev, V.G.: Dokl. Akad. Nauk 180(308) (1968a)
Golubev, V.G.: Sov. Phys. Dokl. 13(13) (1968b)
Gong, S., Liu, C.: Mon. Not. R. Astron. Soc. 462(1), 547 (2016). https://doi.org/10.1093/mnras/stw1556
Hill, G.W.: Am. J. Math. 1(2), 129 (1878)
Johnston, W.R.: (2019). Last updated 3 February 2019 (http://www.johnstonsarchive.net/astro/asteroidmoons.html)
Leandro, E.S.G.: J. Differ. Equ. 226(1), 323 (2006)
Liu, C., Gong, S.: Mon. Not. R. Astron. Soc. 469(3), 3576 (2017a). https://doi.org/10.1093/mnras/stx1017
Liu, C., Gong, S.: Astrophys. Space Sci. 362(7) (2017b). https://doi.org/10.1007/s10509-017-3086-z
Liu, C., Gong, S.: Astrophys. Space Sci. 363(8), 162 (2018)
Liu, X., Baoyin, H., Georgakarakos, N., Donnison, J.R., Ma, X.: Mon. Not. R. Astron. Soc. 427(2), 1034 (2012). https://doi.org/10.1111/j.1365-2966.2012.22019.x
Liu, X., Baoyin, H., Marchis, F.: Astrophys. Space Sci. 349(2), 677 (2014). https://doi.org/10.1007/s10509-013-1663-3
Mamajek, E.E., Barenfeld, S.A., Ivanov, V.D., Kniazev, A.Y., Vaisanen, P., Beletsky, Y., Boffin, H.M.J.: Astrophys. J. Lett. 800(1) (2015)
Marchal, C., Saari, D.G.: Celest. Mech. 12(2), 115 (1975)
Qian, Y.-J., Yang, L.-Y., Yang, X.-D., Zhang, W.: Astrodynamics 2(2), 147 (2018). https://doi.org/10.1007/s42064-017-0017-2
Roy, A.E., Steves, B.A.: Celest. Mech. Dyn. Astron. 78(1–4), 299 (2000)
Scheeres, D.J.: Celest. Mech. Dyn. Astron. 70(2), 75 (1998)
Walker, I.W., Emslie, A.G., Roy, A.E.: Celest. Mech. 29(2), 149 (1980)
Yan, X., Li, J.: SCIENTIA SINICA, Phys., Mech. Astron. 49(8), 084511 (2019)
Zare, K.: Celest. Mech. 14(1), 73 (1976)
Zare, K.: Celest. Mech. 16(1), 35 (1977)
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No.11772167 and 11822205). The authors would like to thank the anonymous referees for their very careful reviews of the paper that include many important points and will improve significantly the clarity of this paper.
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Liu, C., Gong, S. Hill stability of coplanar four-body systems with application to the Solar System and extrasolar systems. Astrophys Space Sci 364, 155 (2019). https://doi.org/10.1007/s10509-019-3644-7
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DOI: https://doi.org/10.1007/s10509-019-3644-7