Gravitational instability of finite isothermal spheres

By Pierre-Henri Chavanis. 2001

139 citations recorded by the source corpus.

We investigate the stability of bounded self-gravitating systems in the canonical ensemble by using a thermodynamical approach. Our study extends the earlier work of Padmanabhan ([CITE]) in the microcanonical ensemble. By studying the second variations of the free energy, we find that instability sets in precisely at the point of minimum temperature in agreement with the theorem of Katz ([CITE]). The perturbation that induces instability at this point is calculated explicitly; it has not a “core-halo” structure contrary to what happens in the microcanonical ensemble. We also study Jeans type gravitational instability of isothermal gaseous spheres described by Navier-Stokes equations. The introduction of a container and the consideration of an inhomogeneous distribution of matter avoids the Jeans swindle. We show analytically the equivalence between dynamical stability and thermodynamical stability and the fact that the stability of isothermal gas spheres does not depend on the viscosity. This confirms the findings of Semelin et al. ([CITE]) who used numerical methods or approximations. We also give a simpler derivation of the geometric hierarchy of scales inducing instability discovered by these authors. The density profiles that trigger these instabilities are calculated explicitly; high order modes of instability present numerous oscillations whose nodes also follow a geometric progression. This suggests that the system will fragment in a series of “clumps” and that these “clumps” will themselves fragment in substructures. The fact that both the domain sizes leading to instability and the “clumps” sizes within a domain follow a geometric progression with the same ratio suggests a fractal-like behavior. This gives further support to the interpretation of de Vega et al. (1996) concerning the fractal structure of the interstellar medium.

Subfields: Astronomy and Astrophysics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b08b28c735aaaf1a15d9c106fa965cb7fbe279dc

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In Astronomy and Astrophysics

In Statistical and Nonlinear Physics

Citations in the corpus

Interstellar Turbulence I: Observations and Processes (2004, background): “Continued collapse of the gas may feed more turbulence on smaller scales (Semelin et al. 1999, Chavanis 2002, Huber & Pfenniger – 13 – 2002).”
Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: II. Numerical results (2011, background): “…non-zero temperature has been studied in the context of statistical mechanics by Hertel & Thirring (1971) [37], Messer (1981) [38] and more recently by Chavanis (2002) [39] (an exhaustive description of the phase diagrams in canonical and microcanonical ensembles is presented in the review [40]).”
Statistical Mechanics of Two-Dimensional Vortices and Stellar Systems (2002, background): “It also enables one to show a clear equivalence between thermodynamical stability in the canonical ensemble and dynamical stability with respect to the Navier-Stokes equations (Jeans problem) [37,44].”
Statistical Mechanics of Violent Relaxation in Stellar Systems (2002, methodology): “The computation of isothermal Fermi-Dirac spheres has been performed only recently by Chavanis & Sommeria [25].”
Star Formation from Galaxies to Globules (2001, background): “…Scalo 1990; Lazarian 1995; Goldman 2000) or turbulence combined with nonlinear self-gravitational instabilities (Fuchs & von Linden 1998; Semelin & Combes 2000; Crosthwaite, Turner, & Ho 2000; Bertin & Lodato 2001; Wada & Norman 2001; Huber & Pfenniger 2001; Vollmer & Beckert 2002; Chavanis 2002).”