Simulation of the Interaction of Oppositely Directed Particle Flows


A mathematical model of the early stage of the formation of interstellar clouds resulting from the interaction of oppositely directed cosmic particle flows and evolution of the clouds into planetary systems is proposed. An important role in the evolution is played by mechanical energy dissipation caused mainly by particle collisions. The simulation is based on the classical n-body problem and the Newtonian theory of instantaneous collisions of rigid bodies with a relative velocity recovery factor smaller than unity. The performed computations suggest that this model is applicable in cosmogonic theories of planetary system formation.

This is a preview of subscription content, log in to check access.


  1. 1

    O. K. Sil’chenko, Origin and Evolution of Galaxies, Ed. by V. G. Surdin (Vek 2, Fryazino, 2017) [in Russian].

  2. 2

    E. M. Galimov, A. M. Krivtsov, A. V. Zabrodin, M. S. Legkostupov, and T. M. Eneev, “Dynamic model of the formation of the Earth–Moon system,” Geokhimiya 11, (2005).

  3. 3

    T. Eneev and N. Kozlov, The Dynamics of Planet Formation: Theory and Computer Simulation (Lap Lambert Academic, Saarbrucken, Germany, 2016).

    Google Scholar 

  4. 4

    T. M. Eneev, N. N. Kozlov, E. I. Kugushev, and D. A. Chechevatov, Preprint No. 072, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2006).

  5. 5

    R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles (McGraw-Hill, New York, 1987).

    Google Scholar 

  6. 6

    V. A. Dorofeeva and A. B. Makalkin, Evolution of the Early Solar System: Cosmochemical and Physical Aspects (Editorial URSS, Moscow, 2004) [in Russian].

    Google Scholar 

  7. 7

    Ya. B. Zeldovich and I. D. Novikov, The Structure and Evolution of the Universe (Nauka, Moscow, 1975) [in Russian].

    Google Scholar 

  8. 8

    T. Matsumoto, K. Dobashi, and T. Shimoikura, “Star formation in turbulent molecular clouds with colliding flow,” Astrophys. J. 801 (2) (2015).

  9. 9

    A. P. Markeev, Libration Points in Celestial Mechanics and Astrodynamics (Nauka, Moscow, 1978) [in Russian].

    Google Scholar 

  10. 10

    K. Kordylewski, “Photographische Untersuchungen des Librationspunktes im System Erde-Mond,” Acta Astron. 11 (3), 165–169 (1961).

    MathSciNet  Google Scholar 

  11. 11

    V. V. Vedenyapin, T. V. Salnikova, and S. Ya. Stepanov, “Vlasov–Poisson–Poisson equations, critical mass, and Kordylewski clouds,” Dokl. Math. 99 (2), 221–224 (2019).

    Article  MATH  Google Scholar 

  12. 12

    T. V. Salnikova, S. Ya. Stepanov, and A. I. Shuvalova, “Probabilistic model of Kordylewski clouds,” Dokl. Phys. 61 (7), 243–246 (2016).

    Article  Google Scholar 

  13. 13

    T. V. Salnikova and S. Ya. Stepanov, “Mathematical model of formation of Kordylewski cosmic dust clouds,” Dokl. Phys. 60 (2), 323–326 (2015).

    Article  Google Scholar 

  14. 14

    T. Salnikova and S. Stepanov, “Existence of elusive Kordylewsky cosmic dust clouds,” Acta Astron. (2019).

  15. 15

    T. Salnikova, S. Stepanov, and A. Shuvalova, “Probabilistic model of the Kordylewski dust clouds formation,” Acta Astron. 150, 85–91 (2018).

    Article  Google Scholar 

  16. 16

    T. Salnikova and S. Stepanov, “Dust charged particles motion in vicinity of the Lagrange libration points,” Adv. Astron. Sci. 170, 91–96 (2020).

    Google Scholar 

  17. 17

    T. Salnikova, S. Stepanov, and A. Shuvalova, “Three-body problem for the Earth–Moon system under photo-gravitational influence of the Sun,” Adv. Astron. Sci. 161, 201–208 (2018).

    Google Scholar 

  18. 18

    T. Salnikova and S. Stepanov, “Effect of electromagnetic field on Kordylewski clouds formation,” AIP Conf. Proc. 1959 (1), 020004–1 (2018).

    Article  Google Scholar 

  19. 19

    T. V. Salnikova and S. Ya. Stepanov, “Study of periodic trajectories of a particle near triangular libration points of the Earth–Moon system with allowance for solar disturbances,” in Proceedings of the 11th All-Russia Conference on Fundamental Problems in Theoretical and Applied Mechanics (2015), pp. 3330–3331.

  20. 20

    T. Salnikova and S. Stepanov, “Polish contribution to the discovery of cosmical dust clouds,” Proceedings of the International Conference on Geometry, Integrability, Mechanics, and Quantization, Varna, Bulgaria, June 6–11,2014 (Sofia, Bulgaria, 2015), pp. 350–355.

  21. 21

    A. Chatterjee and A. Ruina, “A new algebraic rigid-body collision law based on impulse space consideration,” J. Appl. Mech. 65, 939–951 (1998).

    Article  Google Scholar 

  22. 22

    A. P. Ivanov, Dynamics of Systems with Mechanical Collisions (Mezhdunarodnaya Programma Obrazovaniya, Moscow, 1997) [in Russian].

    Google Scholar 

  23. 23

    D. B. Marghitu and Y. Hurmuzlu, “Three-dimensional rigid body collisions with multiple contact points,” J. Appl. Mech. 62, 725–732 (1995).

    Article  MATH  Google Scholar 

  24. 24

    J. Barnes and P. Hut, “A hierarchical O(NlogN) force-calculation algorithm,” Nature 324, 446–449 (1986).

    Article  Google Scholar 

  25. 25

    C. O. Ahn and S. H. Lee, “A new treecode for long-range force calculation,” Comput. Phys. Commun. 178 (2), 121–127 (2008).

    Article  Google Scholar 

  26. 26

    R. Belleman, J. Bédorf, and S. P. Zwart, “High performance direct gravitational N-body simulations on graphics processing units II: An implementation in CUDA,” New Astron. 13, 103–112 (2008).

    Article  Google Scholar 

  27. 27

    J. Makino, “A fast parallel treecode with GRAPE,” Publ. Astron. Soc. Jpn. 56, 521–531 (2004).

    Article  Google Scholar 

  28. 28

    T. V. Salnikova, S. Ya. Stepanov, and E. I. Kugushev, “Possible models of the planetary systems formations,” Int. J. Modern Phys. A 35 (02n03) (2020).

Download references

Author information



Corresponding authors

Correspondence to S. Ya. Stepanov or T. V. Salnikova.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Stepanov, S.Y., Salnikova, T.V. Simulation of the Interaction of Oppositely Directed Particle Flows. Comput. Math. and Math. Phys. 60, 1730–1736 (2020).

Download citation


  • cosmogony
  • planetary system
  • evolution
  • impact