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A novel type of ER3BP introducing Milankovitch cycles or seasonal irradiation processes influencing onto orbit of planet

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Abstract

A developing approach for solving equations of a trapped motion of small satellite near the secondary planet mplanet (Earth) in case of the elliptic restricted problem of three bodies, ER3BP, is presented hereby. In ordinary way, this problem includes consideration of two primaries, MSun and mplanet (mplanet <  < MSun), both are orbiting around their center of mass on Keplerian orbits. Eccentricity of orbit for the aforementioned mplanet is considered to be quasiperiodically variable depending on long-term Milankovitch cycles or various types of seasonal irradiation processes influencing onto orbit of planet (hereafter, Earth). Our aim is first to establish and second to investigate a novel type of ER3BP with variable eccentricity of secondary planet stemming from long-term Milankovitch cycles where in the formulation of above problem, small satellite will always maintain its orbit located near the secondary planet, mplanet. Indeed, Milankovitch cycles govern the dynamics of slow changing the eccentricity of the secondary planet orbit quasiperiodically during a long-time period of secondary planet’s motion around the primary. This planet moves on quasi-stable elliptic orbit with negligible deviations from purely elliptical motion. Semi-analytical solutions and numerical findings with graphical plots are presented accordingly.

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References

  1. Ershkov, S., Aboeulmagd, E., Rachinskaya, A.: A novel type of ER3BP introduced for hierarchical configuration with variable angular momentum of secondary planet. Arch. Appl. Mech. 91(11), 4599–4607 (2021)

    Article  Google Scholar 

  2. Arnold, V.: Mathematical Methods of Classical Mechanics. Springer, New York (1978)

    Book  Google Scholar 

  3. Lagrange J.: 'OEuvres' (M.J.A. Serret, Ed.). Vol. 6, published by Gautier-Villars, Paris. (1873)

  4. Duboshin G.N.: Nebesnaja mehanika. Osnovnye zadachi i metody. Moscow: “Nauka” (handbook for Celestial Mechanics, in russian). (1968)

  5. Szebehely V.: Theory of Orbits. The Restricted Problem of Three Bodies. Yale University, New Haven, Connecticut. Academic Press New-York and London. (1967)

  6. Ferrari, F., Lavagna, M.: Periodic motion around libration points in the Elliptic restricted three-body problem. Nonlinear Dyn. 93(2), 453–462 (2018)

    Article  Google Scholar 

  7. Llibre, J., Conxita, P.: On the elliptic restricted three-body problem. Celest. Mech. Dyn. Astron. 48(4), 319–345 (1990)

    Article  MATH  Google Scholar 

  8. Ershkov, S., Rachinskaya, A.: Semi-analytical solution for the trapped orbits of satellite near the planet in ER3BP. Arch. Appl. Mech. 91(4), 1407–1422 (2021)

    Article  Google Scholar 

  9. Ershkov, S., Leshchenko, D., Rachinskaya, A.: Note on the trapped motion in ER3BP at the vicinity of barycenter. Arch. Appl. Mech. 91(3), 997–1005 (2021)

    Article  Google Scholar 

  10. Abouelmagd, E.I.: Stability of the triangular points under combined effects of radiation and oblateness in the restricted three-body problem. Earth Moon Planet 110(3), 143–155 (2013)

    Article  MATH  Google Scholar 

  11. Ershkov, S., Leshchenko, D., Aboeulmagd, E.: About influence of differential rotation in convection zone of gaseous or fluid giant planet (Uranus) onto the parameters of orbits of satellites. Eur. Phys. J. Plus 136, 387 (2021)

    Article  Google Scholar 

  12. Ershkov, S.V.: About tidal evolution of quasi-periodic orbits of satellites. Earth Moon Planet. 120(1), 15–30 (2017)

    Article  Google Scholar 

  13. Milankovitch M.: Mathematische Klimalehre und Astronomische Theorie der Klimaschwankungen. Handbuch der Klimatologie. Publisher: Borntraeger, Berlin. (1930)

  14. Ershkov, S.V., Leshchenko, D.: Revisiting Apophis 2029 approach to Earth (staying on shoulders of NASA’s experts) or Can we be sure in almost ricocheting fly-by of Apophis on 13 of April 2029 near the Earth? J. Space Saf. Eng. 9(3), 363–1374 (2022)

    Article  Google Scholar 

  15. Ershkov, S., Leshchenko, D., Rachinskaya, A.: Revisiting the dynamics of finite-sized satellite near the planet in ER3BP. Arch. Appl. Mech. 92(8), 2397–2407 (2022)

    Article  Google Scholar 

  16. Ershkov, S., Leshchenko, D., Rachinskaya, A.: Capture in regime of a trapped motion with further inelastic collision for finite-sized asteroid in ER3BP. Symmetry 14(8), 1548 (2022)

    Article  Google Scholar 

  17. Ferraz-Mello, S., Rodríguez, A., Hussmann, H.: Tidal friction in close-in satellites and exoplanets: The Darwin theory re-visited. Celest. Mech. Dyn. Astron. 101, 171–201 (2008). https://doi.org/10.1007/s10569-008-9133-x

    Article  MATH  Google Scholar 

  18. Sidorenko, V.V.: The eccentric Kozai-Lidov effect as a resonance phenomenon. Celest. Mech. Dyn. Astron. 130(1), 4 (2018)

    Article  MATH  Google Scholar 

  19. Singh, J., Umar, A.: On motion around the collinear libration points in the elliptic R3BP with a bigger triaxial primary. New Astron. 29, 36–41 (2014)

    Article  Google Scholar 

  20. Zotos, E.E.: Crash test for the Copenhagen problem with oblateness. Celest. Mech. Dyn. Astron. 122(1), 75–99 (2015)

    Article  Google Scholar 

  21. Abouelmagd, E.I., Sharaf, M.A.: The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness. Astrophys. Space Sci. 344(2), 321–332 (2013)

    Article  MATH  Google Scholar 

  22. Kushvah, B.S., Sharma, J.P., Ishwar, B.: Nonlinear stability in the generalised photogravitational restricted three body problem with Poynting-Robertson drag. Astrophys. Space Sci. 312(3–4), 279–293 (2007)

    Article  MATH  Google Scholar 

  23. Nekhoroshev, N.N.: Exponential estimate on the stability time of near integrable Hamiltonian systems. Russ. Math. Surv. 32(6), 1–65 (1977)

    Article  MATH  Google Scholar 

  24. Lidov, M.L., Vashkov’yak, M.A.: Theory of perturbations and analysis of the evolution of quasi-satellite orbits in the restricted three-body problem. Kosmicheskie Issledovaniia 31, 75–99 (1993)

    Google Scholar 

  25. Peale, S.J.: Orbital resonances in the solar system. Ann. Rev. Astron. Astrophys. 14, 215–246 (1976)

    Article  Google Scholar 

  26. Wiegert, P., Innanen, K., Mikkola, S.: The stability of quasi satellites in the outer solar system. Astron. J. 119, 1978–1984 (2000). https://doi.org/10.1086/301291

    Article  Google Scholar 

  27. Lhotka C.: Nekhoroshev stability in the elliptic restricted three body problem. Thesis for: Doktor rerum naturalium. DOI: https://doi.org/10.13140/RG.2.1.2101.3848. (2008)

  28. Berger, A., Loutre, M.-F., Tricot, Ch.: Insolation and Earth’s orbital periods. J. Geophys. Res. 98(D6), 10341–10362 (1993)

    Article  Google Scholar 

  29. Ershkov, S., Leshchenko, D., Rachinskaya, A.: Semi-analytical findings for rotational trapped motion of satellite in the vicinity of collinear points L1, L2 in planar ER3BP. Arch. Appl. Mech. 92(10), 3005–3012 (2022)

    Article  Google Scholar 

  30. Melnikov, A.V.: Rotational dynamics of asteroids approaching planets. Sol. Syst. Res. 56, 241–251 (2022)

    Article  Google Scholar 

  31. Ershkov, S., Leshchenko, D.: Analysis of the size of Solar system close to the state with zero total angular momentum via Sundman’s inequality. Anais da Academia Brasileira de Ciências 93(Suppl. 3), e20200269 (2021)

    Article  Google Scholar 

  32. Ershkov, S., Leshchenko, D.: Estimation of the size of the solar system and its spatial dynamics using Sundman inequality. Pramana J. Phys. 96, 158 (2022)

    Article  Google Scholar 

  33. Ershkov, S., Leshchenko, D.: On the stability of Laplace resonance for Galilean moons (Io, Europa, Ganymede). Ann. Braz. Acad. Sci. 93(4), e20201016 (2021)

    Article  Google Scholar 

  34. Abouelmagd, E.I., Ansari, A.A.: The motion properties of the infinitesimal body in the framework of bicircular Sun perturbed Earth-Moon system. New Astron. 73, 101282 (2019)

    Article  Google Scholar 

  35. Zotos, E.E., Chen, W., Abouelmagd, E.I., Han, H.: Basins of convergence of equilibrium points in the restricted three-body problem with modified gravitational potential. Chaos Solitons Fractals 134, 109704 (2020)

    Article  MATH  Google Scholar 

  36. Alshaery, A.A., Abouelmagd, E.I.: Analysis of the spatial quantized three-body problem. Results Phys. 17, 103067 (2020)

    Article  Google Scholar 

  37. Abozaid, A.A., Selim, H.H., Gadallah, K.A., Hassan, I.A., Abouelmagd, E.I.: Periodic orbit in the frame work of restricted three bodies under the asteroids belt effect. Appl. Math. Nonlinear Sci. 5(2), 157–176 (2020)

    Article  Google Scholar 

  38. Abouelmagd, E.I., Mostafa, A., Guirao, J.L.G.: A first order automated Lie transform. Int. J. Bifurc. Chaos 25(14), 1540026 (2015)

    Article  MATH  Google Scholar 

  39. Abouelmagd, E.I., Pal, A.K., Guirao, J.L.: Analysis of nominal halo orbits in the Sun-Earth system. Arch. Appl. Mech. 91(12), 4751–4763 (2021)

    Article  Google Scholar 

  40. Ershkov, S.V., Leshchenko, D.: Revisiting dynamics of Sun center relative to barycenter of Solar system or Can we move towards stars using Solar self-resulting photo-gravitational force? J. Space Saf. Eng. 9(2), 160–164 (2022)

    Article  Google Scholar 

  41. Ershkov, S.V.: Forbidden zones for circular regular orbits of the moons in solar system, R3BP. J. Astrophys. Astron. 38(1), 1–4 (2017)

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Authors and Affiliations

  1. Plekhanov Russian University of Economics, Scopus Number 60030998, Moscow, Russia

    Sergey Ershkov

  2. Sternberg Astronomical Institute, M.V. Lomonosov’s Moscow State University, 13 Universitetskij Prospect, Moscow, Russia, 119992

    Sergey Ershkov

  3. Odessa State Academy of Civil Engineering and Architecture, Odessa, Ukraine

    Dmytro Leshchenko

  4. Institute of Engineering Science UB RAS, Ekaterinburg, Russia

    E. Yu. Prosviryakov

  5. B.N. Yeltsin Ural Federal University, Ekaterinburg, Russia

    E. Yu. Prosviryakov

Authors
  1. Sergey Ershkov
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  2. Dmytro Leshchenko
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  3. E. Yu. Prosviryakov
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Contributions

In this research, Dr. S.E. is responsible for the general ansatz and the solving procedure, simple algebra manipulations, calculations, results of the article and also is responsible for the search for analytical and semi-analytical solutions. Dr. D.L. is responsible for theoretical investigations as well as for the deep survey of the literature on the problem under consideration. Dr. E.P. is responsible for obtaining numerical solutions related to approximated ones (including their graphical plots). All authors agreed with results and conclusions of each other in Sects. 1–4.

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Correspondence to Sergey Ershkov.

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Ershkov, S., Leshchenko, D. & Prosviryakov, E.Y. A novel type of ER3BP introducing Milankovitch cycles or seasonal irradiation processes influencing onto orbit of planet. Arch Appl Mech 93, 813–822 (2023). https://doi.org/10.1007/s00419-022-02300-4

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