Modelling the evolution of the two-planetary three-body system of variable masses

    Zhanar Imanova   Affiliation
    ; Alexander Prokopenya   Affiliation
    ; Mukhtar Minglibayev   Affiliation


A classical non-stationary three-body problem with two bodies of variable mass moving around the third body on quasi-periodic orbits is considered. In addition to the Newtonian gravitational attraction, the bodies are acted on by the reactive forces arising due to anisotropic variation of the masses. We show that Newtonian’s formalism may be generalized to the case of variable masses and equations of motion are derived in terms of the osculating elements of aperiodic motion on quasiconic sections. As equations of motion are not integrable the perturbative method is applied with the perturbing forces expanded into power series in terms of eccentricities and inclinations which are assumed to be small. Averaging these equations over the mean longitudes of the bodies in the absence of a mean-motion resonances, we obtain the differential equations describing the evolution of orbital parameters over long period of time. We solve the evolution equations numerically and demonstrate that the mass change modify essentially the system evolution.

Keyword : three-body problem, variable mass, equations of motion, reactive forces, evolution equations, Wolfram Mathematica

How to Cite
Imanova, Z., Prokopenya, A., & Minglibayev, M. (2023). Modelling the evolution of the two-planetary three-body system of variable masses. Mathematical Modelling and Analysis, 28(4), 636–652.
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Oct 20, 2023
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E.I. Abouelmagd. Solution of the two-body problem by KB averaging method within frame of the modified Newtonian potential. Journal of the Astronautical Sciences, 65(3):291–306, 2018.

A.A. Alshaery and E.I. Abouelmagd. Analysis of the spatial quantized three-body problem. Results in Physics, 17:103067, 2020.

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

L.M. Berkovič. Gylden-Meščerski problem. Celestial Mechanics, 24:407–429, 1981.

D. Boccaletti and G. Pucacco. Theory of Orbits. Vol. 2: Perturbative and Geometrical Methods. Springer, Berlin, Heidelberg, 1999.

D. Brouwer and G.M. Clemence. Methods of Celestial Mechanics. Academic Press, New York, 1961.

C.L. Charlier. Die Mechanik des Himmels. Walter de Gruyter: Berlin, Leipzig, 1927.

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

P. Eggleton. Evolutionary Processes in Binary and Multiple Stars. Cambridge University Press, New York, 2006.

J.D. Hadjidemetriou. Two-body problem with variable mass: A new approach. Icarus, 2:440–451, 1963.

L.G. Luk’yanov. Dynamical evolution of stellar orbits in close binary systems with conservative mass transfer. Astronomy Reports, 52:680–692, 2008.

I.V. Meshcherskii. Works on Mechanics of Bodies with Variable Masses. Gos. Izd-vo Tekhn.-Teor. Literatury, Moscow, 1952.

R. Mia, B.R. Prasadu and E.I. Abouelmagd. Analysis of stability of non-collinear equilibrium points: Application to Sun-Mars and Proxima Centauri systems. Acta Astronautica, 204:199–206, 2023.

E. Michaely and H.B. Perets. Secular dynamics in hierarchical three-body systems with mass loss and mass transfer. Astrophysical J., 794(2):122–133, 2014.

M. Minglibayev, A. Prokopenya and S. Shomshekova. Computing perturbations in the two-planetary three-body problem with masses varying nonisotropically at different rates. Mathematics in Computer Science, 14:241–251, 2020.

M.Zh. Minglibayev. Dynamics of gravitating bodies with variable masses and sizes [Dinamika gravitiruyushchikh tel s peremennymi massami i razmerami]. LAMBERT Academic, Saarbrucken, 2012. (in Russian)

M.Zh. Minglibayev and G.M. Mayemerova. Evolution of the orbital-plane orientations in the two-protoplanet three-body problem with variable masses. Astronomy Reports, 58(9):667–677, 2014.

M.Zh. Minglibayev, A.N. Prokopenya, G.M. Mayemerova and Zh.U. Imanova. Three-body problem with variable masses that change anisotropically at different rates. Mathematics in Computer Science, 11:383–391, 2017.

C.D. Murray and S.F. Dermott. Solar System Dynamics. Cambridge University Press, Cambridge, New York, 1999.

T.B. Omarov. Two-body problem with corpuscular radiation. Sov. Astron., 7:707–714, 1963.

A.N. Prokopenya, M.Zh. Minglibayev and G.M. Mayemerova. Symbolic computations in studying the problem of three bodies with variable masses. Programming and Computer Software, 40(2):79–85, 2014.

A.N. Prokopenya, M.Zh. Minglibayev, G.M. Mayemerova and Zh.U. Imanova. Investigation of the restricted problem of three bodies of variable masses using computer algebra. Programming and Computer Software, 43(5):289–293, 2017.

W.A. Rahoma, F.A. Abd El-Salam and M.K. Ahmed. Analytical treatment of the two-body problem with slowly varying mass. J. Astrophys. Astron., 30(34):187–205, 2009.

E.P. Razbitnaya. The problem of two bodies with variable masses: classification of different cases. Sov. Astronomy, 29:684–687, 1985.

N.S. Schulz. The Formation and Early Evolution of Stars. Springer-Verlag, Berlin, Heidelberg, 2012.

V. Szebehely. Theory of Orbits. The Restricted Problem of Three Bodies. Academic Press, New York/London, 1967.

D. Veras. Post-main-sequence planetary system evolution. Royal Soc. open sci., 3:150571, 2016.

D. Veras, J.D. Hadjidemetriou and C.A. Tout. An exoplanet’s response to anisotropic stellar mass-loss during birth and death. Monthly Notices Roy. Astron. Soc., 435(3):2416–2430, 2013.

S. Wolfram. An Elementary Introduction to the Wolfram Language. Wolfram Media, New York, 2016.