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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.

In a plane-parallel polytropic atmosphere of index $m$, let $z$ measure depth from a zero-pressure boundary and take constant gravitational acceleration $g$. The stellar hydrostatic equation and $p=K{\rho }^{1+1/m}$ integrate to $(m+1)p/\rho =gz$. Consequently the adiabatic sound speed satisfies $c_s^2=\gamma p/\rho =\gamma gz/(m+1)$, so $c_s\propto z^{1/2}$. This local relation does not justify extrapolating a surface model to the stellar core.

Consider steady isentropic flow toward a Newtonian gravitational potential $-GM/r$ through a tube of cross-sectional area $A(r)=C{r}^n$. Write $v>0$ for the inward speed and use a polytropic equation of state with specific-heat ratio $\gamma >1$. Mass conservation and the Euler equations for an inviscid fluid givewhere $c_s$ is the adiabatic sound speed. A regular sonic point therefore has $v_s=c_s$ and $r_s=GM/(n{c}_s^2)$. The Bernoulli equation, matched to a nearly stationary reservoir with sound speed $c_0$, givesFor $n>1/2$, a finite positive sonic point requires $1<\gamma <(2n+1)/(2n-1)$. Differentiating the flow equation at that point, with $x=r_s{v}_s^{\prime }/c_s$, givesIts discriminant is $4n[(2n+1)-(2n-1)\gamma ]$, so the same bound permits real regular slopes. The branch on which the Mach number rises inward selects the negative sign inFor a spherical tube $n=2$, this recovers the $5/3$ threshold of Bondi accretion; for a dipolar flux-tube area, $n=3$ gives $7/5$. The tube approximation must remain valid between the reservoir matching region and the accretor.

With reservoir density ${\rho }_{\infty }$ and adiabatic sound speed $c_{\infty }$, put $q=2/\beta -1/2$ and $h=1-q(\gamma -1)>0$. The selected mass accretion rate is $4\pi {\rho }_{\infty }(A\beta /2{)}^{2/\beta }{c}_{\infty }^{1-4/\beta }{h}^{q-1/(\gamma -1)}$. It follows by evaluating the conserved spherical flux at the sonic point.