Kinetic Theory and the Mechanics of Isothermal Gas Spheres: Derivation of the Classical Emden--Chandrasekhar Equation via the Vlasov--Poisson Formalism

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.