# Evolution equations of multi-planet systems with variable masses

### Abstract

In celestial mechanics and astrodynamics, the study of the dynamical evolution of exoplanetary systems is the relevant topics. For today more than 3,000 exoplanetary systems are known. In this paper, we study the dynamic evolution of extrasolar systems, when the leading factor of evolution is the variability of the masses of gravitating bodies. The problem of spherically symmetric bodies with variable masses is considered in a relative coordinate system, this bodies inter-gravitating according to Newton's law. The quasi-elliptical motions of planets whose orbits do not intersect during evolution are investigated. It is believed that the mass of bodies under consideration varies isotropically by various known laws with different velocities. The mass of the parent star is considered to be the most massive than its planets and the origin of the relative coordinate system is in the center of the parent star.. Due to the variability of the masses, the differential equations of motion become non-autonomous and the task is difficult. The problem is investigated by methods of perturbation theory. The canonical perturbation theory based on a periodic motion over a quasi-canonical section is used. Canonical equations of motion are obtained in analogues of the second Poincare system, which are effective in the case when the analogues of eccentricities and the analogues of the inclination of the orbital plane of planets are sufficiently small. The secular perturbations of the planets, which determine the behavior of the orbital parameters over long time intervals, are studied.The evolutionary equations of many planetary systems with isotropically varying masses in analogues of the second system of Poincare variables are derived in an analytical form which are obtained using the Wolfram Mathematica computer algebra system. This takes into account the effects of the decreasing mass of the parent star and the growth of the masses of the planets due to the accretion of matter from the remnants of the protoplanetary disk. For the three-planetary problem of four bodies with variable masses, the evolutionary equations in dimensionless variables are obtained explicitly. In the future, these results will be used to study the dynamics of the three-planet system K2-3 in the non-stationary stage of its evolution.

### References

[1] Sannikova T.N., Kholshevnikov K.V., "Motion in a Central Field in the Presence of a Constant Perturbing Acceleration in a Co-moving Coordinate System" , Astronomy Reports, 59(8) (2015): 806–817.

[2] Sokolov L.L., Kholshevnikov K.V., "Ob integpipyemocti zadachi N tel [On the integrability of the N-body problem]" , Astronomy Letters, 12(7) (1986): 557–561.

[3] Perminov A.S., Kuznetsov E.D., "The Implementation of Hori–Deprit Method to the Construction Averaged Planetary Motion Theory by Means of Computer Algebra System Piranha" , Mathematics in Computer Science, 14(2) (2020): 305–316. DOI: 10.1007/s11786-019-00441-4.

[4] Kuznetsov E.D., Kholshevnikov K.V., "Orbital’naya evolyuciya dvuplanetnoj sistemy Solnce – YUpiter – Saturn [Orbital evolution of the Sun – Jupiter – Saturn bi - planetary system]" , Vestniks of Saint Petersburg University, 1(1) (2009): 139–150.

[5] Kuznetsov E.D., "Orbital evolution of phaethon cluster" , Meteoritics & Planetary Science, 56 (2021): 1.

[6] Belkina S.O., Kuznetsov E.D., "Orbital flips due to solar radiation pressure for space debris in near-circular orbits", Acta Astronautica, 178 (2021): 360–369. https://doi.org/10.1016/j.actaastro.2020.09.025.

[7] Kuznetsov E.D., Rosaev A.E., Plavalova E., Safronova V.S., Vasileva M.A., "A Search for Young Asteroid Pairs with Close Orbits" , Solar System Research, 54(3) (2020): 236–252. DOI: 10.1134/s0038094620030077.

[8] Perminov A., Kuznetsov E., "The orbital evolution of the Sun–Jupiter–Saturn–Uranus–Neptune system on long time scales" , Astrophysics and Space Science, 365(8) (2020): 144. DOI: 10.1007/s10509-020-03855-w.

[9] Kholshevnikov K.V., Mullari A.A., Tolumbaeva D.A., Vavilov D.E., "Opredelenie pervonachal’nyh orbit vnesolnechnyh planet metodom luchevyh skorostej: zamknutye formuly [Determination of the initial orbits of extrasolar planets by the method of radial velocities: closed formulas]" , Vestniks of Saint Petersburg University, 1(3) (2011): 143–152.

[10] Kholshevnikov K.V., Tolumbaeva D.A., Mullari A.A., "Opredelenie pervonachal’nyh orbit vnesolnechnyh planet metodom luchevyh skorostej: stepennye ryady [Determination of the initial orbits of extrasolar planets by the method of radial velocities: power series]" , Vestniks of Saint Petersburg University, 1(1) (2011): 166–172.

[11] Perminov A.S., Kuznetsov E.D., "Orbital’naya evolyuciya vnesolnechnyh planetnyh sistem HD 39194, HD 141399 I HD 160691 [Orbital evolution of extrasolar planetary systems HD 39194, HD 141399 and HD160691]" , The Astronomical Journal, 96(10) (2019): 795–813. DOI:10.1134/S1063772919090075.

[12] https://exoplanets.nasa.gov/.

[13] Prokopenya A., Minglibayev Ì., Shomshekova S., "Computing Perturbations in the Two-Planetary Three-Body Problem with Masses Varying Non-Isotropically at Different Rates" , Mathematics in Computer Science, 14(2) (2020): 241–251. https://doi.org/10.1007/s11786-019-00437-0.

[14] Minglibayev M.Zh., Dinamika gravitiruyushchikh tel s peremennymi massami i razmerami [Dynamics of gravitating bodies with variable masses and sizes] (LAP LAMBERT Academic Publishing, 2012): 224. Germany. ISBN:978-3-659-29945-2.

[15] Minglibayev M.Zh., Kosherbayeva A.B., "Differential equations of planetary systems" , Reports of the National Academy of Sciences of the Republic of Kazakhstan, 2(330) (2020): 14–20. https://doi.org/10.32014/2020.2518-1483.26.

[16] Minglibayev M.Zh., Kosherbayeva A.B., "Equations of planetary systems motion" , News of NAS RK. Physicalmathematical series, 6 (2020): 53–60.

[17] Minglibayev M. Zh., Mayemerova G.M., "Evolution of the orbital-plane orientations in the two-protoplanet three-body problem with variable masses" , Astronomy Reports, 58(9) (2014): 667–677. DOI: 10.1134/S1063772914090066.

[18] Prokopenya A.N., Minglibayev M.Zh., Kosherbayeva A.B., "Derivation of evolutionary equations in the multi-body problem with isotropically varying masses using computer algebra" , Programming and Computer Software, 48(2) (2022): 1–11.

[19] Charlier K., Nebesnaya mekhanika [Celestial mechanics] [Òåêñò] / Perevod s nem. V.G. Demina; Pod red. prof. B.M. Shchigoleva [Translated from German by V.G. Demin; Edited by prof. B.M. Shchigolev] (Moscow: Nauka, 1966): 627.

[20] Wolfram S., An elementary introduction to the Wolfram Language (New York: Wolffram Media, Inc., 2017): 324. ISBN: 978-1-944183-05-9.

[21] Prokopenya A.N., Reshenie fizicheskih zadach c ispolzovaniem sistemy Mathematica [Solving physical problems using the Mathematica system] (BSTU Publishing, Brest., 2005): 260.

[2] Sokolov L.L., Kholshevnikov K.V., "Ob integpipyemocti zadachi N tel [On the integrability of the N-body problem]" , Astronomy Letters, 12(7) (1986): 557–561.

[3] Perminov A.S., Kuznetsov E.D., "The Implementation of Hori–Deprit Method to the Construction Averaged Planetary Motion Theory by Means of Computer Algebra System Piranha" , Mathematics in Computer Science, 14(2) (2020): 305–316. DOI: 10.1007/s11786-019-00441-4.

[4] Kuznetsov E.D., Kholshevnikov K.V., "Orbital’naya evolyuciya dvuplanetnoj sistemy Solnce – YUpiter – Saturn [Orbital evolution of the Sun – Jupiter – Saturn bi - planetary system]" , Vestniks of Saint Petersburg University, 1(1) (2009): 139–150.

[5] Kuznetsov E.D., "Orbital evolution of phaethon cluster" , Meteoritics & Planetary Science, 56 (2021): 1.

[6] Belkina S.O., Kuznetsov E.D., "Orbital flips due to solar radiation pressure for space debris in near-circular orbits", Acta Astronautica, 178 (2021): 360–369. https://doi.org/10.1016/j.actaastro.2020.09.025.

[7] Kuznetsov E.D., Rosaev A.E., Plavalova E., Safronova V.S., Vasileva M.A., "A Search for Young Asteroid Pairs with Close Orbits" , Solar System Research, 54(3) (2020): 236–252. DOI: 10.1134/s0038094620030077.

[8] Perminov A., Kuznetsov E., "The orbital evolution of the Sun–Jupiter–Saturn–Uranus–Neptune system on long time scales" , Astrophysics and Space Science, 365(8) (2020): 144. DOI: 10.1007/s10509-020-03855-w.

[9] Kholshevnikov K.V., Mullari A.A., Tolumbaeva D.A., Vavilov D.E., "Opredelenie pervonachal’nyh orbit vnesolnechnyh planet metodom luchevyh skorostej: zamknutye formuly [Determination of the initial orbits of extrasolar planets by the method of radial velocities: closed formulas]" , Vestniks of Saint Petersburg University, 1(3) (2011): 143–152.

[10] Kholshevnikov K.V., Tolumbaeva D.A., Mullari A.A., "Opredelenie pervonachal’nyh orbit vnesolnechnyh planet metodom luchevyh skorostej: stepennye ryady [Determination of the initial orbits of extrasolar planets by the method of radial velocities: power series]" , Vestniks of Saint Petersburg University, 1(1) (2011): 166–172.

[11] Perminov A.S., Kuznetsov E.D., "Orbital’naya evolyuciya vnesolnechnyh planetnyh sistem HD 39194, HD 141399 I HD 160691 [Orbital evolution of extrasolar planetary systems HD 39194, HD 141399 and HD160691]" , The Astronomical Journal, 96(10) (2019): 795–813. DOI:10.1134/S1063772919090075.

[12] https://exoplanets.nasa.gov/.

[13] Prokopenya A., Minglibayev Ì., Shomshekova S., "Computing Perturbations in the Two-Planetary Three-Body Problem with Masses Varying Non-Isotropically at Different Rates" , Mathematics in Computer Science, 14(2) (2020): 241–251. https://doi.org/10.1007/s11786-019-00437-0.

[14] Minglibayev M.Zh., Dinamika gravitiruyushchikh tel s peremennymi massami i razmerami [Dynamics of gravitating bodies with variable masses and sizes] (LAP LAMBERT Academic Publishing, 2012): 224. Germany. ISBN:978-3-659-29945-2.

[15] Minglibayev M.Zh., Kosherbayeva A.B., "Differential equations of planetary systems" , Reports of the National Academy of Sciences of the Republic of Kazakhstan, 2(330) (2020): 14–20. https://doi.org/10.32014/2020.2518-1483.26.

[16] Minglibayev M.Zh., Kosherbayeva A.B., "Equations of planetary systems motion" , News of NAS RK. Physicalmathematical series, 6 (2020): 53–60.

[17] Minglibayev M. Zh., Mayemerova G.M., "Evolution of the orbital-plane orientations in the two-protoplanet three-body problem with variable masses" , Astronomy Reports, 58(9) (2014): 667–677. DOI: 10.1134/S1063772914090066.

[18] Prokopenya A.N., Minglibayev M.Zh., Kosherbayeva A.B., "Derivation of evolutionary equations in the multi-body problem with isotropically varying masses using computer algebra" , Programming and Computer Software, 48(2) (2022): 1–11.

[19] Charlier K., Nebesnaya mekhanika [Celestial mechanics] [Òåêñò] / Perevod s nem. V.G. Demina; Pod red. prof. B.M. Shchigoleva [Translated from German by V.G. Demin; Edited by prof. B.M. Shchigolev] (Moscow: Nauka, 1966): 627.

[20] Wolfram S., An elementary introduction to the Wolfram Language (New York: Wolffram Media, Inc., 2017): 324. ISBN: 978-1-944183-05-9.

[21] Prokopenya A.N., Reshenie fizicheskih zadach c ispolzovaniem sistemy Mathematica [Solving physical problems using the Mathematica system] (BSTU Publishing, Brest., 2005): 260.

How to Cite

MINGLIBAYEV, M.; KOSHERBAYEVA, A..
Evolution equations of multi-planet systems with variable masses.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 116, n. 4, dec. 2022. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/1030>. Date accessed: 28 jan. 2023. doi: https://doi.org/10.26577/JMMCS.2022.v116.i4.04.
Section

Mechanics

Keywords
variable mass;, perturbation theory;, evolutionary equations;, exoplanetary systems;, Poincare elements