Advertisement

Earth, Moon, and Planets

, Volume 121, Issue 1–2, pp 1–12 | Cite as

Patterns Formation in a Self-Gravitating Isentropic Gas

  • Mayer Humi
Article
  • 99 Downloads

Abstract

In this paper we consider a hydrodynamic model for the matter density distribution in a self gravitating, isentropic 2-d disk of gas where the isentropy coefficient is allowed to be a function of position. For this model we prove analytically the existence of steady state and time dependent solutions in which the matter density in the disk is oscillatory and pattern forming. This research is motivated in part by recent astronomical observations and Laplace conjecture (made in 1796) that planetary systems evolve from a family of isolated rings that are formed within a primitive interstellar gas cloud.

Keywords

Interstellar cloud Pattern formation Isentropic gas 

Notes

Acknowledgements

The author is indebted to Prof. A. Prentice whose input and comments improved the quality of this paper.

References

  1. H.P. Belrage, The Origin of the Solar System (Pergamon Press, Oxford, 1968)Google Scholar
  2. S. Chandrasekhar, On a new theory of Weizsacker on the origin of the solar system. Rev. Mod. Phys. 18, 94–102 (1946)ADSCrossRefGoogle Scholar
  3. M. Humi, Steady states of self gravitating incompressible fluid. J. Math. Phys. 47, 093101 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. M. Humi, Steady states of self gravitating incompressible fluid with axial symmetry. Int. J. Mod. Phys. A 24(23), 4287–4303 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. M. Humi, A model for pattern formation under gravity. Appl. Math. Model. 40(2016), 41–49 (2016)MathSciNetCrossRefGoogle Scholar
  6. M. Kiguchi, S. Narita, S.M. Miyama, C. Hayashi, The equilibria of rotating isothermal clouds. Astrophys. J. 317, 830–845 (1987)ADSCrossRefGoogle Scholar
  7. P.S. Laplace, Exposition du Systeme du Monde (Reprinted by Cambridge University Press, Paris, 2010). ISBN 0511693338zbMATHGoogle Scholar
  8. J. Li, T. Zhang, Y. Zheng, Simple waves and a characteristic decomposition of the two dimensional compressible Euler equations. Commun. Math. Phys. 267, 1–12 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. J.J. Lissauer, Planet formation. Annu. Rev. Astron. Astrophys. 31, 120–174 (1993)ADSCrossRefGoogle Scholar
  10. T. Matsumoto, T. Hanawa, Bar and disk formation in gravitationally collapsing clouds. Astrophys. J. 521(2), 659–670 (1999)ADSCrossRefGoogle Scholar
  11. A.L.M.A. Partnership, Astrophys. J. Lett. 808, L3 (2015).  https://doi.org/10.1088/2041-8205/808/1/L3 ADSCrossRefGoogle Scholar
  12. E.A. Petigura, A.W. Howard, G.W. Marcy, Prevalence of Earth-size planets orbiting Sun-like stars. Proc. Nat. Acad. Sci. 110(48), 19273 (2013)ADSCrossRefGoogle Scholar
  13. A.J.R. Prentice, Origin of the solar system. Earth Moon Planet 19, 341–398 (1978)ADSCrossRefGoogle Scholar
  14. L. Spitzer Jr., Diffuse Matter in Space (Interscience Publishers, New-York, 1968)Google Scholar
  15. C.F. Von Weizsacker, Z. Astrophys. 22, 319–355 (1944)ADSGoogle Scholar
  16. C.E. Wayne, Periodic solutions of nonlinear partial differential equations. Not. AMS 44, 895–902 (1997)MathSciNetzbMATHGoogle Scholar
  17. M.M. Woolfson, Astron. Geophys. 41(1), 1.12–1.19 (2000).  https://doi.org/10.1046/j.1468-4004.2000.00012.x ADSCrossRefGoogle Scholar
  18. M. Ya Marov, A.V. Kolesnichenko, Turbulence and Self-Organization (Modeling Astrophysical Objects) (Springer, New York, 2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA

Personalised recommendations