False. Stellar evaporation removes stars above the escape energy after two-body relaxation has redistributed their energies. The remaining self-gravitating cluster need not become less dense; losing mass is not the same as holding its radius fixed.

For a slowly evolving isolated cluster, apply the virial theorem at each approximate equilibrium:

E = T + W = W /2 = - T

. Stars evaporating with nearly zero energy at infinity remove little total energy, so the bound cluster has nearly constant negative

E

R_{g}

defined by

W = - G M^{2} / R_{g}

R_{g} = \frac{G M ^{2}}{2∣ E ∣} \propto M^{2} .

For homologous structure, its characteristic mean density therefore scales as

\overset{ρ}{ˉ} \propto \frac{M}{R _{g}^{3}} \propto M^{- 5} .

It increases as mass is lost. Escaping stars carrying positive energy make the bound energy more negative and strengthen the contraction tendency. This is virial contraction under stellar evaporation, related to the negative heat capacity of bound gravitational systems. A tidally limited globular cluster has additional radius constraints, so this scaling is not universal, but the claimed inevitable density decrease is false even in the simplest isolated evaporation model.

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