= Solution
Write $\Omega=-aGM_\odot^2/R_\odot$ with a structure factor $a$ of order unity. Then the thermal radiation timescale is
$$
\boxed{t_{\rm th}=\frac{U}{L_\odot}=\frac a2\frac{GM_\odot^2}{R_\odot L_\odot}\sim10^7\text{--}3\times10^7\,\mathrm{yr}.}
$$
For a uniform sphere $a=3/5$, this is about $9.4\times10^6$ years. The conventional <Kelvin-Helmholtz cooling time> $GM_\odot^2/(R_\odot L_\odot)$ is about $3.1\times10^7$ 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. \b[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.
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