Negative heat capacity of a virialized gravitational system ($dE/d\Theta =-3N{k}_B/2$)

An isolated Newtonian system in virial equilibrium has total energy $E=K+W=-K$. Along an equilibrium sequence at fixed particle number, define the kinetic temperature by $K=3N{k}_B\Theta /2$. Its heat capacity is therefore negative, as displayed: losing total energy increases the kinetic temperature. A collisionless stellar system need not have a thermodynamic temperature; this is a kinetic-temperature statement unless thermal equilibrium is additionally specified. Boundary pressure and external confinement can change the energy relation.