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Three-Body Problem with Variable Masses that Change Anisotropically at Different Rates

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Abstract

In this paper we consider a general case of the three-body problem with variable masses that change anisotropically at different rates. Due to the change of masses reactive forces appear which significantly complicate the problem. Equations of motion of the system have been derived in Jacobi coordinates for the first time. Using these equations of motion and applying the methods of perturbation theory in modified Jacobi and Delaunay elements, we have obtained canonical equations of perturbed motion of the system in the presence of reactive forces. Canonical system of equations for secular perturbations in the three-body problem with variable masses changing anisotropically was derived in explicit form in terms of the analogues of the second system of Poincaré elements. An approximate analytical solution of the differential equations for secular perturbations was obtained by Picard’s method.

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References

  1. Omarov, T.B. (ed.): Non-Stationary Dynamical Problems in Astronomy. Nova Science Publishers. Inc., New York (2002)

    Google Scholar 

  2. Bekov, A.A., Omarov, T.B.: The theory of orbits in non-stationary stellar systems. Astron. Astrophys. Trans. 22, 145–153 (2003)

    Article  Google Scholar 

  3. Eggleton, P.: Evolutionary Processes in Binary and Multiple Stars. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  4. Luk’yanov, L.G.: Dynamical evolution of stellar orbits in close binary systems with conservative mass transfer. Astron. Rep. 52, 680–692 (2008)

    Article  Google Scholar 

  5. Minglibayev, M.Zh.: Dinamika gravitiruyushchikh tel s peremennymi massami i razmerami. Postupatel’noye i postupatel’no-vrashchatel’noye dvizheniye. LAP LAMBERT Academic Publishing, Saarbrucken (2012)

  6. Minglibayev, M.Zh., Mayemerova, G.M.: Investigation of the evolution equations of the three-body problem with variable masses. Appl. Math. Sci. 7, 4439–4454 (2013)

  7. Prokopenya, A.N., Minglibayev, M.Zh., Mayemerova, G.M.: Symbolic calculations in studying the problem of three bodies with variable masses. Program. Comput. Softw. 40, 79–85 (2014)

  8. Minglibayev, M.Zh., Mayemerova, G.M.: Evolution of the orbital-plane orientations in the two-protoplanet three-body problem with variable masses. Astron. Rep. 58, 762–772 (2014)

  9. Meshchersky, I.V.: Rabotji po mehanike tel peremennoj massji. Gos. Izd. tehniko-teoret. lit, Moscow (1952)

    Google Scholar 

  10. Robe, H.A.: A new kind of three body problem. Celest. Mech. Dyn. Astron. 16, 343–351 (1977)

    Article  MATH  Google Scholar 

  11. Prokopenya, A.N.: Solving Physical Problems with Mathematica. Brest State Technical University Publications, Brest (2005)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Al-Farabi Kazakh National University, Al-Farabi av. 71, Almaty, Kazakhstan, 050040

    M. Zh. Minglibayev, G. M. Mayemerova & Zh. U. Imanova

  2. Fesenkov Astrophysical Institute, Observatory 23, Almaty, Kazakhstan, 050020

    M. Zh. Minglibayev

  3. Warsaw University of Life Sciences, Nowoursynowska str. 166, 02-787, Warsaw, Poland

    A. N. Prokopenya

Authors
  1. M. Zh. Minglibayev
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  2. A. N. Prokopenya
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  3. G. M. Mayemerova
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  4. Zh. U. Imanova
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Correspondence to G. M. Mayemerova.

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Minglibayev, M.Z., Prokopenya, A.N., Mayemerova, G.M. et al. Three-Body Problem with Variable Masses that Change Anisotropically at Different Rates. Math.Comput.Sci. 11, 383–391 (2017). https://doi.org/10.1007/s11786-017-0306-4

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  • DOI: https://doi.org/10.1007/s11786-017-0306-4

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