The Jeans’ criterion and the gravitational instability

  • Serge Bouquet
Plasma And Gravitation
Part of the Lecture Notes in Physics book series (LNP, volume 518)


In this paper, we derive a Jeans’ criterion for the dynamical spherical collapse of a gravitational polytropic medium with constant γ. In opposition to the classical (static configuration) Jeans’ criterion where a threshold for stability appears, it is shown that the stability of the dynamical solution depends on the value of γ. For γ=4/3, the existence of a threshold is also obtained. However, for γ≠4/3, the system is always unstable with an instability growing according to a time-power, instead of the usual exponential law. In addition, two types of instabilities are obtained: the large wavelengths are unstable from the very beginning of the evolution, whereas the small ones, produce oscillations on a finite range of time and, finally, the system becomes unstable (oscillations stop). In this stage the pressure can be neglected. Since the collapse is unstable, we examine the fragmentation problem. The dynamical Jeans’ criterion suggests a cascade in which the mass M and the radius R of the pieces are connected by a relation of the type M∞R 2.


Gravitational collapse Jeans’ instability Self-similar solutions 


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  1. Abramovitz, M., Stegun, I.A. (1964): Handbook of Mathematical Functions (Dover: New York)Google Scholar
  2. Blottiau, P., Bouquet, S., Chièze, J.P. (1988): An Asymptotic Self-Similar Solution for the Gravitational Collapse. Astron. Astrophys. 207, 24–36ADSGoogle Scholar
  3. Bonnor, W.B. (1957): Jeans’ Formula for Gravitational Instability, M.N.R.A.S. 116, 104–117ADSMathSciNetGoogle Scholar
  4. Bouquet, S., Feix, M.R., Fijalkow, E., Munier, A. (1985): Density Bifurcation in a Homogeneous Isotropic Collapsing Star. Ap.J., 293, 494–503CrossRefADSGoogle Scholar
  5. Chièze, J.P. (1987): The Fragmentation of Molecular clouds. Astron. Astrophys. 171, 225–232ADSGoogle Scholar
  6. Hunter, C. (1962): The Instability of the Collapse of a Self-Gravitating Gas Cloud. Ap. J., 136, 594–608CrossRefADSMathSciNetGoogle Scholar
  7. Hunter, C. (1967): Galactic Structure, in Relativity. Theory and Astrophysics Vol. 2 ed. J. Ehlers (American Mathematical Society: Providence)Google Scholar
  8. Jeans, J.H. (1929): Astronomy and Cosmology (Cambridge University Press: Cambridge)Google Scholar
  9. Larson, R.B. (1981): Turbulence and Star Formations in Molecular Clouds. M.N.R.A.S 194, 809–826ADSGoogle Scholar
  10. Mestel, M.L. (1965): Problem of Star formation — I. Quart. J. R. A. S. 6, 161–198ADSGoogle Scholar
  11. Penston, M.V. (1969): Dynamics of Self-Gravitating Gaseous Spheres. M.N.R.A.S 144, 425–448ADSGoogle Scholar
  12. Schwartz, L. (1979): Méthode Mathématique pour les Sciences Physiques (Hermann: Paris)Google Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • Serge Bouquet
    • 1
  1. 1.Commissariat à l’Energie Atomique, Centre de Bruyères-le-Châtel, DPTA/PPEBruyères-le-ChâtelFrance

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