Supernovae: An example of complexity in the physics of compressible fluids

  • Yves Pomeau
  • Martine Le Berre
  • Pierre-Henri Chavanis
  • Bruno Denet
Regular Article
Part of the following topical collections:
  1. Irreversible Dynamics: A topical issue dedicated to Paul Manneville


Because the collapse of massive stars occurs in a few seconds, while the stars evolve on billions of years, the supernovae are typical complex phenomena in fluid mechanics with multiple time scales. We describe them in the light of catastrophe theory, assuming that successive equilibria between pressure and gravity present a saddle-center bifurcation. In the early stage we show that the loss of equilibrium may be described by a generic equation of the Painlevé I form. This is confirmed by two approaches, first by the full numerical solutions of the Euler-Poisson equations for a particular pressure-density relation, secondly by a derivation of the normal form of the solutions close to the saddle-center. In the final stage of the collapse, just before the divergence of the central density, we show that the existence of a self-similar collapsing solution compatible with the numerical observations imposes that the gravity forces are stronger than the pressure ones. This situation differs drastically in its principle from the one generally admitted where pressure and gravity forces are assumed to be of the same order. Moreover it leads to different scaling laws for the density and the velocity of the collapsing material. The new self-similar solution (based on the hypothesis of dominant gravity forces) which matches the smooth solution of the outer core solution, agrees globally well with our numerical results, except a delay in the very central part of the star, as discussed. Whereas some differences with the earlier self-similar solutions are minor, others are very important. For example, we find that the velocity field becomes singular at the collapse time, diverging at the center, and decreasing slowly outside the core, whereas previous works described a finite velocity field in the core which tends to a supersonic constant value at large distances. This discrepancy should be important for explaining the emission of remnants in the post-collapse regime. Finally we describe the post-collapse dynamics, when mass begins to accumulate in the center, also within the hypothesis that gravity forces are dominant.

Graphical abstract


Topical issue: Irreversible Dynamics: A topical issue dedicated to Paul Manneville 


  1. 1.
    A. Burrows, Rev. Mod. Phys. 85, 245 (2013).ADSCrossRefGoogle Scholar
  2. 2.
    H.A. Bethe, Rev. Mod. Phys. 62, 801 (1990).ADSCrossRefGoogle Scholar
  3. 3.
    R. Thom, Stabilité Structurelle et Morphogénese (Benjamin, New York, 1972).Google Scholar
  4. 4.
    M.V. Penston, Mon. Not. R. Astron. Soc. 144, 425 (1969).ADSCrossRefGoogle Scholar
  5. 5.
    R.B. Larson, Mon. Not. R. Astron. Soc. 145, 271 (1969).ADSCrossRefGoogle Scholar
  6. 6.
    M.P. Brenner, T.P. Witelski, J. Stat. Phys. 93, 863 (1998).ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    C. Josserand, Y. Pomeau, S. Rica, J. Low Temp. Phys. 145, 231 (2006).ADSCrossRefGoogle Scholar
  8. 8.
    J. Sopik, C. Sire, P.H. Chavanis, Phys. Rev. E 74, 011112 (2006).ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    R.D. Peters, M. Le Berre, Y. Pomeau, Phys. Rev. E 86, 026207 (2012).ADSCrossRefGoogle Scholar
  10. 10.
    A.A. Dorodnicyn, Am. Math. Soc. Transl. Series One 4, 1 (1953) (translated from Priklad Mat. i Mek. 11.Google Scholar
  11. 11.
    P. Coullet, C.R. Mécanique 340, 777 (2012).ADSCrossRefGoogle Scholar
  12. 12.
    L.D. Landau, E.M. Lifshitz, Statistical Physics, second edition (Pergamon, Oxford, 1987) chapt. IX, p. 317.Google Scholar
  13. 13.
    R. Emden, Gaskugeln Anwendungen der Mechanischen Wärmetheorie auf Kosmologische und Meteorologie Probleme (Teubner, Leipzig, 1907).Google Scholar
  14. 14.
    R. Ebert, Z. Astrophys. 37, 217 (1955).ADSMATHGoogle Scholar
  15. 15.
    W.B. Bonnor, Mon. Not. R. Astron. Soc. 116, 351 (1956).ADSCrossRefMathSciNetGoogle Scholar
  16. 16.
    W.H. McCrea, Mon. Not. R. Astron. Soc. 117, 562 (1957).ADSCrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    V.A. Antonov, Vest. Leningr. Gos. Univ. 7, 135 (1962).Google Scholar
  18. 18.
    D. Lynden-Bell, R. Wood, Mon. Not. R. Astron. Soc. 138, 495 (1968).ADSCrossRefGoogle Scholar
  19. 19.
    P.H. Chavanis, Astron. Astrophys. 381, 340 (2002).ADSCrossRefMATHGoogle Scholar
  20. 20.
    P.H. Chavanis, Astron. Astrophys. 401, 15 (2003).ADSCrossRefMATHGoogle Scholar
  21. 21.
    P.H. Chavanis, Phys. Rev. E 65, 056123 (2002).ADSCrossRefMathSciNetGoogle Scholar
  22. 22.
    P.H. Chavanis, Astron. Astrophys. 432, 117 (2005).ADSCrossRefMATHGoogle Scholar
  23. 23.
    P.H. Chavanis, Int. J. Mod. Phys. B 20, 3113 (2006).ADSCrossRefMATHGoogle Scholar
  24. 24.
    J.R. Oppenheimer, G.M. Volkoff, Phys. Rev. 55, 374 (1939).ADSCrossRefMATHGoogle Scholar
  25. 25.
    M. Colpi, S.L. Shapiro, I. Wasserman, Phys. Rev. Lett. 57, 2485 (1986).ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    P.H. Chavanis, Phys. Rev. D 84, 043531 (2011).ADSCrossRefMathSciNetGoogle Scholar
  27. 27.
    P.H. Chavanis, T. Harko, Phys. Rev. D 86, 064011 (2012).ADSCrossRefGoogle Scholar
  28. 28.
    P. Painlevé, Bull. Soc. Math. Phys. France 28, 201 (1900) E.L. Ince, Ordinary Differential Equations.MATHGoogle Scholar
  29. 29.
    P. Hoflich, P. Kumar, J.C. Wheeler, Cosmic Explosions in Three Dimensions: Asymmetries in Supernovae and Gamma Ray Bursts (Cambridge University Press, Cambridge, 2004) p. 276.Google Scholar
  30. 30.
    P.H. Chavanis, Astron. Astrophys. 451, 109 (2006).ADSCrossRefMATHGoogle Scholar
  31. 31.
    H. Nessyahu, E. Tadmor, J. Comput. Phys. 87, 408 (1990) J. Balbas, E. Tadmor, CentPack, Scholar
  32. 32.
    P.H. Chavanis, C. Sire, Phys. Rev. E 70, 026115 (2004).ADSCrossRefGoogle Scholar
  33. 33.
    A. Yahil, Astrophys. J. 265, 1047 (1983).ADSCrossRefGoogle Scholar
  34. 34.
    G.I. Barenblatt, Y.B. Zel’dovich, Annu. Rev. Fluid Mech. 4, 285 (1972).ADSCrossRefGoogle Scholar
  35. 35.
    L. Mestel, Q. J. R. Astron. Soc. 6, 161 (1965).ADSGoogle Scholar
  36. 36.
    F.H. Shu, Astrophys. J. 214, 488497 (1977).Google Scholar
  37. 37.
    P. Goldreich, S.V. Weber, Astrophys. J. 238, 991 (1980).ADSCrossRefGoogle Scholar
  38. 38.
    J. Bricmont, A. Kupiainen, G. Lin, Comm. Pure Appl. Math. 47, 285 (1994).CrossRefMathSciNetGoogle Scholar
  39. 39.
    G.L. Eyink, J. Xin, J. Stat. Phys. 100, 679 (2000).CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    C. Sire, P.H. Chavanis, Phys. Rev. E 69, 066109 (2004).ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Yves Pomeau
    • 1
  • Martine Le Berre
    • 2
  • Pierre-Henri Chavanis
    • 3
  • Bruno Denet
    • 4
  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Institut des Sciences Moléculaires d’Orsay ISMO - CNRSUniversité Paris-SudOrsay CedexFrance
  3. 3.Laboratoire de Physique Théorique (UMR 5152 du CNRS)Université Paul SabatierToulouse Cedex 4France
  4. 4.UMR 7342 CNRS et Centrale Marseille, Technopole de Château-GombertUniversité Aix-Marseille, IRPHEMarseille Cedex 13France

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