Stability of Encounterless Spherical Stellar Systems

Doremus, J. P.; Feix, Marc; Baumann, G.

doi:10.1103/physrevlett.26.725

Cited by 52 publications

(38 citation statements)

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“…He conjectured that this equivalence between thermodynamical and dynamical stability remains valid for all isotropic star clusters, not only for those described by the heavily truncated Maxwell-Boltzmann distribution. This is in sharp contrast with the Newtonian case where it has been shown [50][51][52][53][54][55] that all isotropic stellar systems are dynamically stable with respect to the Vlasov-Poisson equations, even those that are thermodynamically unstable. To solve this apparent paradox, one expects that the growth rate λ of the dynamical instability decreases as relativity effects decrease and that it tends to zero in the nonrelativistic limit c → +∞.…”

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confidence: 81%

“…It has to be noted that the gravothermal catastrophe is a thermodynamical instability, not a dynamical instabilty. Indeed, it has been shown that all the isotropic stellar systems with a distribution function of the form f = f ( ) with f ( ) < 0, like the truncated Maxwell-Boltzmann distribution, are dynamically stable with respect to a collisionless (Vlasov) evolution [50][51][52][53][54][55]. In particular, all the isothermal configurations on the series of equilibria are dynamically stable, including those deep into the spiral that are thermodynamically unstable.…”

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confidence: 99%

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Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

Chavanis

2020

Eur. Phys. J. Plus

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We study the statistical mechanics of classical self-gravitating systems confined within a box of radius R in general relativity. It has been found that the caloric curve T∞(E) has the form of a double spiral whose shape depends on the compactness parameter ν = GN m/Rc 2 . The double spiral shrinks as ν increases and finally disappears when νmax = 0.1764. Therefore, general relativistic effects render the system more unstable. On the other hand, the cold spiral and the hot spiral move away from each other as ν decreases. Using a normalization Λ = −ER/GN 2 m 2 and η = GN m 2 /RkBT∞ appropriate to the nonrelativistic limit, and considering ν → 0, the hot spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the Emden equation) exhibiting a single cold spiral, as found in former works. Using another normalization M = GM/Rc 2 and B = Rc 4 /GN kBT∞ appropriate to the ultrarelativistic limit, and considering ν → 0, the cold spiral goes to infinity and the caloric curve tends towards a limit curve (determined by the general relativistic Emden equation) exhibiting a single hot spiral. This result is new. We discuss the analogies and the differences between this asymptotic caloric curve and the caloric curve of the selfgravitating black-body radiation. Finally, we compare box-confined isothermal models with heavily truncated isothermal distributions in Newtonian gravity and general relativity.PACS numbers: 04.40. Dg, 05.70.Fh, 95.30.Sf, 95.35.+d I. INTRODUCTIONIn a preceding paper [1] (Paper I), relying on previous works on the subject [2-25], we have developed a general formalism to study the statistical mechanics of self-gravitating systems in general relativity. This formalism is valid for an arbitrary form of entropy. The statistical equilibrium state is obtained by maximizing the entropy S at fixed mass-energy M c 2 and particle number N . The extremization problem yields the Tolman-Oppenheimer-Volkoff (TOV) equations [2,26] expressing the condition of hydrostatic equilibrium together with the Tolman-Klein relations [2,27]. The precise form of entropy determines the distribution function and the equation of state of the system. For illustration, in Paper I, we considered a system of self-gravitating fermions. In that case, the statistical equilibrium state is obtained by maximizing the Fermi-Dirac entropy at fixed mass-energy and particle number. This yields the relativistic Fermi-Dirac distribution function. In the present paper, we consider the statistical mechanics of classical self-gravitating particles. 1 In that case, the statistical equilibrium state is obtained by maximizing the Boltzmann entropy at fixed mass-energy and particle number. This yields the relativistic Maxwell-Boltzmann (or Maxwell-Juttner) distribution function. We start by a brief history of the subject giving an exhaustive list of references. 2 The statistical mechanics of nonrelativistic classical self-gravitating systems was developed in connection with the dynamical evolution of globular clusters (see...

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Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

Chavanis

2020

Eur. Phys. J. Plus

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“…This is a sufficient condition of nonlinear stability. We are tempted to believe that it is also a necessary condition of nonlinear stability although this seems to go against the theorem of Doremus et al (1971) which states that all distribution functions with f ( ) < 0 are dynamically stable (even if they are minima or saddle points of S at fixed E and M). In fact, this criterion is only a condition of linear stability (Binney & Tremaine 1987) and it may not be valid for box confined models (its general applicability has also been criticized).…”

Section: The Vlasov-poisson Systemmentioning

confidence: 99%

Gravitational instability of isothermal and polytropic spheres

Chavanis

2003

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Abstract. We complete previous investigations on the thermodynamics of self-gravitating systems by studying the grand canonical, grand microcanonical and isobaric ensembles. We also discuss the stability of polytropic spheres in connexion with a generalized thermodynamical approach proposed by Tsallis. We determine in each case the onset of gravitational instability by analytical methods and graphical constructions in the Milne plane. We also discuss the relation between dynamical and thermodynamical stability of stellar systems and gaseous spheres. Our study provides an aesthetic and simple approach to this otherwise complicated subject.

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“…In a seminal work, Antonov (1960Antonov ( , 1962) used a variational principle to demonstrate that non-rotating spherical models with a phase-space distribution function f depending only on the energy E are stable to non-radial perturbations if d f /dE < 0. Subsequent works showed that this condition is also a sufficient condition for stability to radial perturbations (Dorémus et al 1971;Sygnet et al 1984;Kandrup & Sygnet 1985). In general, non-rotating spherical stellar systems are described by distribution functions f that depend on both the energy E and the magnitude of the angular momentum L. In such systems only the stability to radial modes can be tested by using the sufficient condition ∂ f /∂E < 0 (Dorémus & Feix 1973;Dejonghe & Merritt 1988).…”

Section: Introductionmentioning

confidence: 99%

Stability of rotating spherical stellar systems

Meza

2002

A&A

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Abstract. The stability of rotating isotropic spherical stellar systems is investigated by using N-body simulations. Four spherical models with realistic density profiles are studied: one of them fits the luminosity profile of globular clusters, while the remaining three models provide good approximations to the surface brightness of elliptical galaxies. The phase-space distribution function f (E) of each one of these non-rotating models satisfies the sufficient condition for stability d f /dE < 0. Different amounts of rotation are introduced in these models by changing the sign of the z-component of the angular momentum for a given fraction of the particles. Numerical simulations show that all these rotating models are stable to both radial and non-radial perturbations, irrespective of their degree of rotation. These results suggest that rotating isotropic spherical models with realistic density profiles might generally be stable. Furthermore, they show that spherical stellar systems can rotate very rapidly without becoming oblate.

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Stability of Encounterless Spherical Stellar Systems

Cited by 52 publications

References 2 publications

Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

Statistical mechanics of self-gravitating systems in general relativity: II. The classical Boltzmann gas

Gravitational instability of isothermal and polytropic spheres

Stability of rotating spherical stellar systems

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