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Onno Pols developed a systematic teaching account of stellar structure and stellar evolution. His 2011 notes connect hydrostatic equilibrium, energy transport and stellar thermodynamics, including the distinct adiabatic exponents needed when gas and radiation pressure both contribute.^[ref]

For example, a derivative describing how pressure responds to compression is not interchangeable with one describing the temperature response. Distinguishing them makes stability and energy-transport calculations usable beyond a single fixed-index ideal gas.

The stellar adiabatic exponents describe how pressure $P$, temperature $T$ and mass density $\rho$ respond along an isentropic process at fixed chemical composition. With specific entropy $s$,The chain rule gives ${\Gamma }_3-1={\Gamma }_1({\Gamma }_2-1)/{\Gamma }_2$. Thus the three names do not denote three independent thermodynamic responses. The adiabatic temperature gradient is ${\nabla }_{\mathrm{ad}}=({\Gamma }_2-1)/{\Gamma }_2$, while ${\Gamma }_1$ determines the squared adiabatic sound speed $c_s^2={\Gamma }_{1}P/\rho$.^[ref]

For a fixed-composition monatomic ideal gas, all three exponents equal $5/3$ and ${\nabla }_{\mathrm{ad}}=2/5$. For equilibrium blackbody radiation, all three equal $4/3$ and ${\nabla }_{\mathrm{ad}}=1/4$. In a gas-radiation mixture they generally differ; the adiabatic exponents of a monatomic gas-radiation mixture keep that distinction explicit. Partial ionization can soften the compressional response because some work changes ionization rather than immediately raising the temperature. The relevant exponent in stellar convective stability therefore need not equal a constant ideal-gas specific-heat ratio.