= Stellar adiabatic exponent
{title2=$\Gamma_i$}
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> $s$,
$$
\Gamma_1=\left(\frac{\partial\log P}{\partial\log\rho}\right)_s,
\qquad
\Gamma_3-1=\left(\frac{\partial\log T}{\partial\log\rho}\right)_s,
\qquad
\frac{\Gamma_2}{\Gamma_2-1}=\left(\frac{\partial\log P}{\partial\log T}\right)_s.
$$
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 $\nabla_{\rm ad}=(\Gamma_2-1)/\Gamma_2$, while $\Gamma_1$ determines the squared <adiabatic sound speed> $c_s^2=\Gamma_1P/\rho$.<Stellar Structure and Evolution by Onno Pols>[Section 3.4, printed p. 34, footnote on the three exponents]{ref}
For a fixed-composition monatomic <ideal gas>, all three exponents equal $5/3$ and $\nabla_{\rm ad}=2/5$. For equilibrium <blackbody radiation>, all three equal $4/3$ and $\nabla_{\rm ad}=1/4$. 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>.
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