Kinetic Theory and the Mechanics of Isothermal Gas Spheres: Derivation of the Classical Emden--Chandrasekhar Equation via the Vlasov--Poisson Formalism
Steven D Miller
TL;DR
The paper derives the isothermal Emden--Chandrasekhar framework directly from the stationary Vlasov--Poisson equations by extremizing the Boltzmann entropy, yielding a Maxwell--Boltzmann distribution $\mathcal{F}_E \propto \exp[-\beta(\tfrac{|p|^2}{2m}+m\Phi)]$ and the Poisson--Boltzmann equation for self-gravitating Newtonian gases. Specializing to spherical symmetry recovers the Emden--Chandrasekhar equation, linking kinetic theory, thermodynamics, and stellar structure; the second variation of entropy reproduces the Antonov instability criteria with critical energy $E_c$, while the formalism naturally exhibits negative specific heat and shows uniform density is impossible for isothermal spheres. The work connects kinetic theory to classic stellar models, clarifying the self-consistency between microscopic phase-space dynamics and macroscopic hydrostatic equilibria, and extends the discussion to finite-size truncations (Bonnor--Ebert, King models) and relativistic generalizations via the Einstein--Vlasov system. Overall, the paper provides a unified kinetic-theoretic derivation of foundational isothermal-sphere results and situates them within the broader context of self-gravitating thermodynamics and stellar evolution.
Abstract
We present a derivation of the mechanics of isothermal gas spheres directly from the Vlasov--Poisson equation. By extremising the Boltzmann entropy, we obtain the Maxwell--Boltzmann distribution for a self-gravitating isothermal Newtonian gas, which is a stationary solution of the Vlasov--Poisson system. From this distribution, the corresponding Poisson--Boltzmann equation for the gravitational potential is deduced. The second variation of entropy reproduces the classical Antonov instability criterion: the critical energy is $E_c \simeq -0.335\,\frac{G M^2}{R}$, below which no local entropy maximum exists and the configuration becomes unstable (the so-called "gravothermal catastrophe"). In this work, we assume $E>E_c$, so all equilibria lie on the stable branch, and the Antonov instability does not affect the analysis. Specializing to spherical symmetry, we recover the classical equation for the isothermal gas sphere, as originally studied by Chandrasekhar, which has applications to the formation of red giant stars. We also derive the fundamental equation of hydrostatic equilibrium, the energy integral and the virial theorem directly from the stationary Vlasov--Poisson solution, demonstrating also that an isothermal gas exhibits negative specific heat. Furthermore, we show that an isothermal gas sphere of strictly constant density is an impossibility. This exposition emphasizes some of the deep connections and self-consistency between kinetic theory, statistical mechanics, and stellar structure, while highlighting some formal aspects of classical astrophysical models.