Gravothermal catastrophe

A collisional self-gravitating stellar core can lose heat to its halo, contract and become hotter because it has negative heat capacity of a virialized gravitational system. The increased temperature contrast drives further outward heat flow and further contraction. The classical gravothermal catastrophe is the loss of a finite-density thermodynamic equilibrium at a finite total energy along the confined three-dimensional Newtonian isothermal sequence. It concerns slow two-body relaxation, not simply a collisionless instability. The dimensional form of the gravitational interaction and the thermodynamic ensemble matter: a two-dimensional logarithmic gravitational gas need not have the same finite-energy instability.

For a mean-field gas of parallel infinite rods with total mass per unit length $M=Nm$ confined to radius $R_e$, put $q=GM\beta /2$, $\beta =m/(kT)$ and $F=1-q+q{R}^2/R_e^2$. The isothermal solution is $\psi =\psi (R_e)-2\log F/\beta$ with mass density $\rho =M(1-q)/(\pi R_e^2{F}^2)$ and enclosed line mass $M(<R)=M(R^2/R_e^2)/F$. Direct differentiation verifies the cylindrical Poisson equation for Newtonian gravity. Regularity requires $q<1$, or $T>T_{\min }$; equality is a singular core-concentration limit, not a regular equilibrium.

With $\Theta =T/T_{\min }>1$, the two-dimensional self-gravitating rod equilibrium has the displayed energy per unit length for the potential reference $\psi (R_e)=-GM\log (V/V_0)$. Integration of $-\rho \psi /2$ gives its gravitational energy; its two-dimensional Maxwell-Boltzmann velocity distribution has kinetic energy $NkT$. At fixed area, differentiation gives $C=Nk[2+2\Theta \log (1-{\Theta }^{-1})+\Theta /(\Theta -1)]$. Expanding the logarithm shows $C/(Nk)=1+{\sum }_{j\ge 3}(j-2)/(j{\Theta }^{j-1})>1$. The energy increases continuously from $-\infty$ to $+\infty$, so there is no finite-energy endpoint or turning point of the regular sequence. This excludes the classical gravothermal catastrophe in this model, while leaving a distinct canonical collapse at or below its minimum temperature.