Gravitational instability of finite isothermal spheres

Chavanis, Pierre‐Henri

doi:10.1051/0004-6361:20011438

Cited by 155 publications

(370 citation statements)

References 41 publications

(108 reference statements)

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“…These equations decrease the Boltzmann free energy at fixed mass and temperature. Self-gravitating Brownian particles can experience an "isothermal collapse" [25], which is the canonical version of the "gravothermal catastrophe" [26] experienced by globular clusters. A Brownian model has also been introduced in the case of a cosinusoidal potential of interaction in d = 1 [6].…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Chavanis¹

2006

Physica A: Statistical Mechanics and its Applications

204

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We discuss the kinetic theory of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system described by the Liouville equation from the canonical description of a stochastically forced Brownian system described by the FokkerPlanck equation. We show that the mean-field approximation is exact in a proper thermodynamic limit. For N → +∞, a Hamiltonian system is described by the Vlasov equation. In this collisionless regime, coherent structures can emerge from a process of violent relaxation. These metaequilibrium states, or quasi-stationary states (QSS), are stable stationary solutions of the Vlasov equation. To order 1/N , the collision term of a homogeneous system has the form of the Lenard-Balescu operator. It reduces to the Landau operator when collective effects are neglected. The statistical equilibrium state (Boltzmann) is obtained on a collisional timescale of order N or larger (when the Lenard-Balescu operator cancels out). We also consider the stochastic motion of a test particle in a bath of field particles and derive the general form of the Fokker-Planck equation describing the evolution of the velocity distribution of the test particle. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. For Brownian systems, in the N → +∞ limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ → +∞, or for large times t ≫ ξ −1 , it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, two-dimensional vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker-Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.

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Section: Introductionmentioning

confidence: 99%

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Chavanis¹

2006

Physica A: Statistical Mechanics and its Applications

204

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“…We assume spherical symmetry; that is, we only consider radial perturbations. The configurations are known to be stable against nonradial perturbations (Semelin et al 2001;Chavanis 2002;Binney & Tremaine 2008).…”

Section: Basic Equations Of the Dynamical Approachmentioning

confidence: 99%

“…Most of them are for modes of minimum L at given Ξ. Density profiles of higher modes exhibit oscillations not present in the lowest mode. For the case of marginal stability (L = 0), with the help of properties (12), the relevant eigenfunction can be expressed analytically (Chavanis 2002). The function…”

Section: Eulerian Representationmentioning

confidence: 99%

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Gravothermal catastrophe: The dynamical stability of a fluid model

Sormani

Bertin

2013

A&A

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A re-investigation of the gravothermal catastrophe is presented. By means of a linear perturbation analysis, we study the dynamical stability of a spherical self-gravitating isothermal fluid of finite volume and find that the conditions for the onset of the gravothermal catastrophe, under different external conditions, coincide with those obtained from thermodynamical arguments. This suggests that the gravothermal catastrophe may reduce to Jeans instability, rediscovered in an inhomogeneous framework. We find normal modes and frequencies for the fluid system and show that instability develops on the dynamical time scale. We then discuss several related issues. In particular, (1) for perturbations at constant total energy and constant volume, we introduce a simple heuristic term in the energy budget to mimic the role of binaries. (2) We outline the analysis of the two-component case and show how linear perturbation analysis can also be carried out in this more complex context in a relatively straightforward way. (3) We compare the behavior of the fluid model with that of the collisionless sphere. In the collisionless case the instability seems to disappear, which is at variance with the linear Jeans stability analysis in the homogeneous case. We argue that a key ingredient for understanding the difference lies in the role of the detailed angular momentum in a collisionless system. A spherical stellar system is expected to undergo the gravothermal catastrophe only in the presence of some collisionality, which suggests that the instability is dissipative and not dynamical. Finally, we briefly comment on the meaning of the Boltzmann entropy and its applicability to the study of the dynamics of selfgravitating inhomogeneous gaseous systems.

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“…In the matter dominated era, the velocity of gas' constituents is low and the Newtonian limit is appropriate. The thermodynamic stability of a self-gravitating gas in the Newtonian limit is a very old subject [17][18][19][20][21][22][23][24] (the relativistic case has recently earned a lot of attention [25][26][27][28][29][30][31][32][33][34]). Hence, this subject has not only a pure theoretical interest, but also an additional cosmological motivation.…”

Section: Introductionmentioning

confidence: 99%

Galaxy clusters and structure formation in quintessence versus phantom dark energy universe

et al. 2014

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The self-gravitating gas in the Newtonian limit is studied in the presence of dark energy with a linear and constant equation of state. Entropy extremization associates to the isothermal Boltzmann distribution an effective density that includes 'dark energy particles', which either strengthen or weaken mutual gravitational attraction, in case of quintessence or phantom dark energy, respectively, that satisfy a linear equation of state. Stability is studied for microcanonical (fixed energy) and canonical (fixed temperature) ensembles. Compared to the previously studied cosmological constant case, in the present work it is found that quintessence increases, while phantom dark energy decreases the instability domain under gravitational collapse. Thus, structures are more easily formed in a quintessence rather than in a phantom dominated Universe. Assuming that galaxy clusters are spherical, nearly isothermal and in hydrostatic equilibrium we find that dark energy with a linear and constant equation of state, for fixed radius, mass and temperature, steepens their total density profile! In case of a cosmological constant, this effect accounts for a 1.5% increase in the density contrast, that is the center to edge density ratio of the cluster. We also propose a method to constrain phantom dark energy.

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Gravitational instability of finite isothermal spheres

Cited by 155 publications

References 41 publications

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Hamiltonian and Brownian systems with long-range interactions: II. Kinetic equations and stability analysis

Gravothermal catastrophe: The dynamical stability of a fluid model

Galaxy clusters and structure formation in quintessence versus phantom dark energy universe

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