EDDINGTON'S “Internal Constitution of the Stars” was published in 1926 and gives what now ranks as a classical account of his own researches and of the general state of the theory at that time. Since then, a tremendous amount of work has appeared. Much of it has to do with the construction of stellar models with different equations of state applying in different zones. Other parts deal with the effects of varying chemical composition, with pulsation and tidal and rotational distortion of stars, and with the precise relations between the interior and the atmosphere of a star. The striking feature of all this work is that so much can be done without assuming any particular mechanism of stellar energy-generation. Only such very comprehensive assumptions are made about the distribution and behaviour of the energy sources that we may expect future knowledge of their mechanism to lead mainly to more detailed results within the framework of the existing general theory. An Introduction to the Study of Stellar Structure By S. Chandrasekhar. (Astrophysical Monographs sponsored by The Astrophysical Journal.) Pp. ix+509. (Chicago: University of Chicago Press; London: Cambridge University Press, 1939.) 50s. net.

An Introduction to the Study of Stellar Structure

Astronomy and Cosmogony

The local thermodynamics of a system with long-range interactions in d dimensions is studied using the mean-field approximation. Long-range interactions are introduced through pair interaction potentials that decay as a power law in the interparticle distance. We compute the local entropy, Helmholtz free energy, and grand potential per particle in the microcanonical, canonical, and grand canonical ensembles, respectively. From the local entropy per particle we obtain the local equation of state of the system by using the condition of local thermodynamic equilibrium. This local equation of state has the form of the ideal gas equation of state, but with the density depending on the potential characterizing long-range interactions. By volume integration of the relation between the different thermodynamic potentials at the local level, we find the corresponding equation satisfied by the potentials at the global level. It is shown that the potential energy enters as a thermodynamic variable that modifies the global thermodynamic potentials. As a result, we find a generalized Gibbs-Duhem equation that relates the potential energy to the temperature, pressure, and chemical potential. For the marginal case where the power of the decaying interaction potential is equal to the dimension of the space, the usual Gibbs-Duhem equation is recovered. As examples of the application of this equation, we consider spatially uniform interaction potentials and the self-gravitating gas. We also point out a close relationship with the thermodynamics of small systems.

/pdf/local-thermodynamics-and-the-generalized-gibbs-duhem-3owbeup3dq.pdf

Local thermodynamics and the generalized Gibbs-Duhem equation in systems with long-range interactions.

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 di erent 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 di erence 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.

/pdf/gravothermal-catastrophe-the-dynamical-stability-of-a-fluid-18zyagu5s0.pdf

Gravothermal catastrophe: The dynamical stability of a fluid model ?

Aims. The dynamics of large-scale structure formation in the universe by gravitational instability still presents many open issues and is mostly studied through numerical simulations. This motivates the study of simpler models which can be investigated by analytical means in order to understand the main processes at work. Thus, we describe here a one-dimensional self-gravitating system derived from the cosmological context, which leads to an effective external potential in addition to the standard gravitational self-interaction. As a first step we consider small times so that the expansion can be neglected. Then we present a thermodynamical analysis of this system as well as the stability properties of the associated hydrodynamical and collisionless systems. Methods. We consider the mean field limit (i.e. continuum limit) to perform an analytical study. Results. We find a second-order phase transition at T c from an homogeneous equilibrium at high temperature to a clustered phase (with a density peak at one of the boundaries of the system) at low temperature. There also exists an infinite series of unstable equilibria which appear at lower temperatures T cn , reflecting the scale-free nature of the gravitational interaction and the usual Jeans instability. We find that, as for the similar HMF (Hamiltonian mean field) model, all three micro-canonical, canonical and grand-canonical ensembles agree with each other, as well as with the stability properties associated with a hydrodynamical approach. On the other hand, the collisionless dynamics governed by the Vlasov equation yields the same results except that at low T the equilibrium associated with two density peaks (one at each boundary) becomes stable.

/pdf/thermodynamics-and-dynamics-of-a-1-d-gravitational-system-2ngx0ifhdc.pdf

Thermodynamics and dynamics of a 1-D gravitational system

We study the statistical mechanics of the self-gravitating gas at thermal equilibrium with two kinds of particles. We start from the partition function in the canonical ensemble, which we express as a functional integral over the densities of the two kinds of particles for a large number of particles. The system is shown to possess an infinite volume limit when ${(N}_{1}{,N}_{2},V)\ensuremath{\rightarrow}\ensuremath{\infty},$ keeping ${N}_{1}{/V}^{1/3}$ and ${N}_{2}{/V}^{1/3}$ fixed. The saddle point approximation becomes here exact for ${(N}_{1}{,N}_{2},V)\ensuremath{\rightarrow}\ensuremath{\infty}.$ It provides a nonlinear differential equation for the densities of each kind of particle. For the spherically symmetric case, we compute the densities as functions of two dimensionless physical parameters: ${\ensuremath{\eta}}_{1}{=Gm}_{1}^{2}{N}_{1}{/V}^{1/3}T$ and ${\ensuremath{\eta}}_{2}{=Gm}_{2}^{2}{N}_{2}{/V}^{1/3}T$ (where G is Newton's constant, ${m}_{1}$ and ${m}_{2}$ the masses of the two kinds of particles, and T the temperature). According to the values of ${\ensuremath{\eta}}_{1}$ and ${\ensuremath{\eta}}_{2}$ the system can be either in a gaseous phase or in a highly condensed phase. The gaseous phase is stable for ${\ensuremath{\eta}}_{1}$ and ${\ensuremath{\eta}}_{2}$ between the origin and their collapse values. We have thus generalized the well-known isothermal sphere for two kinds of particles. The gas is inhomogeneous and the mass $M(R)$ inside a sphere of radius R scales with R as $M(R)\ensuremath{\propto}{R}^{d}$ suggesting a fractal structure. The value of d depends in general on ${\ensuremath{\eta}}_{1}$ and ${\ensuremath{\eta}}_{2}$ except on the critical line for the canonical ensemble in the $({\ensuremath{\eta}}_{1},{\ensuremath{\eta}}_{2})$ plane where it takes the universal value $d\ensuremath{\simeq}1.6$ for all values of ${N}_{1}{/N}_{2}.$ The equation of state is computed. It is found to be locally a perfect gas equation of state. The thermodynamic functions (free energy, energy, entropy) are expressed and plotted as functions of ${\ensuremath{\eta}}_{1}$ and ${\ensuremath{\eta}}_{2}.$ They exhibit a square root Riemann sheet with the branch points on the critical canonical line. The behavior of the energy and the specific heat at the critical line is computed. This treatment is further generalized to the self-gravitating gas with n types of particles.

Statistical mechanics of the self-gravitating gas with two or more kinds of particles.

The self-gravitating gas in the presence of the cosmological constant

The self-gravitating gas in the presence of dark energy: Monte Carlo simulations and stability analysis

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Statistical mechanics of the self-gravitating gas with two or more kinds of particles.

The self-gravitating gas in the presence of the cosmological constant

The self-gravitating gas in the presence of dark energy: Monte Carlo simulations and stability analysis