Stellar gas-pressure fraction ($\beta =P_{\mathrm{gas}}/(P_{\mathrm{gas}}+P_{\mathrm{rad}})$)

The gas-pressure fraction describes the relative support by an ideal gas and radiation pressure. For monatomic gas plus equilibrium radiation, it determines the adiabatic exponents of a monatomic gas-radiation mixture. Its value at the unperturbed state may be specified, but it generally changes during an isentropic parcel displacement.

For a radiative envelope, $d{P}_{\mathrm{rad}}/dP=\kappa L_r/(4\pi cGm)$. Exactly uniform stellar gas-pressure fraction instead requires this derivative to be $1-\beta$. Constant opacity therefore requires $L_r$ to increase in proportion to enclosed mass. A positive-density finite-mass envelope with no local energy generation has constant $L_r$ and cannot satisfy all these conditions pointwise. The Eddington-model convective-core mass fraction uses constant $\beta$ as a global approximation, rather than an exact detailed envelope constraint.