Issue |
A&A
Volume 451, Number 1, May III 2006
|
|
---|---|---|
Page(s) | 109 - 123 | |
Section | Galactic structure, stellar clusters, and populations | |
DOI | https://doi.org/10.1051/0004-6361:20054008 | |
Published online | 25 April 2006 |
Dynamical stability of collisionless stellar systems and barotropic stars: the nonlinear Antonov first law
Laboratoire de Physique Théorique (UMR 5152 du CNRS), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France e-mail: chavanis@irsamc.ups-tlse.fr
Received:
8
August
2005
Accepted:
22
December
2005
We complete previous investigations on the dynamical
stability of barotropic stars and collisionless stellar systems. A
barotropic star that minimizes the energy functional at a fixed mass is
a nonlinearly dynamically stable stationary solution of the
Euler-Poisson system. Formally, this minimization problem is similar
to a condition of “canonical stability” in thermodynamics. A
stellar system that maximizes an H-function at fixed mass and
energy is a nonlinearly dynamically stable stationary solution of the
Vlasov-Poisson system. Formally, this maximization problem is similar
to a condition of “microcanonical stability” in
thermodynamics. Using a thermodynamical analogy, we provide a
derivation and an interpretation of the nonlinear Antonov first law in
terms of “ensembles inequivalence”: a spherical stellar system with
and
is nonlinearly dynamically stable
with respect to the Vlasov-Poisson system if the corresponding
barotropic star with the same equilibrium density distribution is
nonlinearly dynamically stable with respect to the Euler-Poisson
system. This is similar to the fact that “canonical stability
implies microcanonical stability” in thermodynamics. The converse is
wrong in case of “ensembles inequivalence” which is generic for
systems with long-range interactions like gravity. We show that
criteria of nonlinear dynamical stability can be obtained very simply
from purely graphical constructions by using the method of series of
equilibria and the turning point argument of Poincaré, as in
thermodynamics.
Key words: stellar dynamics / hydrodynamics / instabilities / gravitation / galaxies: structure / methods: analytical
© ESO, 2006
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