 Solution

Write $\Omega =-aG{M}_{\odot }^2/R_{\odot }$ with a structure factor $a$ of order unity. Then the thermal radiation timescale isFor a uniform sphere $a=3/5$, this is about $9.4\times 10^6$ years. The conventional Kelvin-Helmholtz cooling time $G{M}_{\odot }^2/(R_{\odot }{L}_{\odot })$ is about $3.1\times 10^7$ years; a centrally concentrated solar model changes the coefficient. Both estimates are far shorter than the nuclear lifetime.

Ionization and dissociation can store compression energy without the full temperature rise of a monatomic gas, lowering the effective adiabatic response in partially ionized outer layers. Radiation pressure tends towards an adiabatic exponent $4/3$. These effects, changing composition and degeneracy make the effective exponent position dependent. The local dynamical exponent is ${\Gamma }_1=(\partial \log P/\partial \log \rho {)}_s$; it equals $C_P/C_V$ in the simple fixed-composition ideal-gas model, but need not do so for an ionizing mixture.

If an envelope has an effective exponent at or below $4/3$, its pressure response is soft and sufficiently extensive such regions can promote radial pulsation, envelope instability or mass loss. In the ideal homologous global model, $\gamma =4/3$ is marginal and $\gamma <4/3$ is unstable to contraction or expansion. A low value confined to a thin outer layer does not by itself prove collapse of the entire star: global stability depends on the pressure-weighted structure. Its location and mass extent matter.