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For fixed-composition monatomic ideal gas plus equilibrium radiation pressure, specific internal energy is $u=3\mathcal{R}T/2+a_{\mathrm{r}}{T}^4/\rho$. An isentropic process satisfies $du=Pd\rho /{\rho }^2$, giving ${\Gamma }_3-1=(8-6\beta )/(24-21\beta )$ andThe stellar gas-pressure fraction must be allowed to change along the adiabat. The pure-radiation and pure-gas limits give ${\Gamma }_1={\Gamma }_2=4/3$ and $5/3$, respectively. Partial ionization or different gas heat capacities change these formulas.

The global Eddington closure $L=(1-\beta )4\pi cGM/\kappa$, combined with a representative stellar gas-pressure fraction, makes the exterior stellar radiative temperature gradient approximately $M/(4m)$ when the luminosity is constant outside the core. Matching it to the adiabatic temperature gradient estimates the displayed boundary mass. This is a closure estimate: imposing exactly uniform $\beta$ throughout a finite-mass radiative envelope also conflicts with constant luminosity, as described by the constant-beta radiative-envelope obstruction.

Put $\mathcal{R}=k_B/m_u$, with $m_u$ the atomic mass constant and $a_r$ the radiation energy-density constant. The ideal gas and radiation pressure contributions obey $P_g=\rho \mathcal{R}T/\mu$ and $P_r=a_r{T}^4/3$. With the constant stellar gas-pressure fraction, eliminate $T=\mu \beta P/(\mathcal{R}\rho )$ to getHenceSince $\mu$ and $\beta$ are spatially constant, so is $K$. Comparing with $P=K{\rho }^{1+1/n}$ gives polytropic index $n=3$. This structural stellar polytrope relation does not assert that every perturbed fluid parcel has stellar adiabatic exponent $4/3$.

Divide the radiation-pressure gradient by the total-pressure gradient. Radiative diffusion in a star and hydrostatic equilibrium giveThis last quantity is independent of radius under the specified opacity assumption. Integration gives $P_r=fP+C$. With the standard idealized zero-pressure outer boundary, where both components vanish, $C=0$. The stellar gas-pressure fraction is thereforeconstant throughout the model. This is the Eddington standard model idealization. A finite photospheric pressure offset would need its boundary treatment; the differential relation alone does not set the integration constant to zero.

Use the printed stellar gas-pressure fraction, $\beta =P_g/P$, with $P=P_g+P_{\mathrm{rad}}$, and keep composition fixed. For a monatomic perfect gas plus equilibrium blackbody radiation, the specific heats of a monatomic gas-radiation mixture follow from its specific internal energy and specific enthalpy:At fixed total pressure, differentiating the equation of state givesIn particular, $\beta$ is not held constant during that derivative. The specific heat capacity at constant pressure is $c_P=(\partial h/\partial T{)}_P$, soSince $a{T}^3/(3\rho )=\mathcal{R}(1-\beta )/\beta$, this simplifies toThe pure-gas limit is $5\mathcal{R}/2$. For a gas with unspecified molecular degrees of freedom, replace $3\mathcal{R}/2$ in $u$ by its gas heat capacity $c_{V,g}$: then $c_P=c_{V,g}+\mathcal{R}+4\mathcal{R}(1-\beta )(4+\beta )/{\beta }^2$. The boxed formula uses the conventional monatomic stellar-gas interpretation.

Let $S$ denote specific entropy, reserving $s$ from the first question for envelope thickness. The stellar adiabatic exponents are defined at fixed composition byIt is useful to introduce ${\Gamma }_3-1=(\partial \log T/\partial \log \rho {)}_S$, so $({\Gamma }_2-1)/{\Gamma }_2=({\Gamma }_3-1)/{\Gamma }_1$.

Take the stellar gas to be a fully ionized, nonrelativistic monatomic ideal gas, with fixed specific gas constant $\mathcal{R}$. Its mixture with thermal radiation hasHere $u$ is specific internal energy, and the stellar gas-pressure fraction is $\beta =\mathcal{R}\rho T/P$. The assumption of monatomic gas fixes its heat capacity; the perfect-gas pressure law alone would not determine it.

At fixed $T$ and fixed $\rho$, respectively, the pressure derivatives areFor an isentropic process, the first law of thermodynamics gives $du=Pd\rho /{\rho }^2$. Differentiate the internal energy rather than artificially holding $\beta$ fixed during the perturbation:The parenthesis is $(P/\rho T)[(3/2)\beta +12(1-\beta )]$, givingSince $d\log P={\chi }_{\rho }d\log \rho +{\chi }_{T}d\log T$,Using the relation between the exponents gives the adiabatic exponents of a monatomic gas-radiation mixtureThe requested values, including the adiabatic temperature gradient, areThe pure-radiation and pure-gas rows are understood as limits of the mixture.

For uniform composition, the Schwarzschild criterion for stellar convective instability isTo test whether a radiative configuration becomes unstable, use its required stellar radiative temperature gradient ${\nabla }_{\mathrm{rad}}$ for $\nabla$. Since $dP/dr<0$, the equivalent radial condition isAn outward-displaced parcel then cools less than its new surroundings, remains less dense at the same pressure, and is further accelerated outward.

The requested Eddington-model convective-core mass fraction follows from the usual global constant-$\beta$ closure. At the outer radiative surface, put $m=M$, $L_r=L$, and $P_{\mathrm{rad}}=(1-\beta )P$. Dividing the radiation-pressure gradient by the stellar hydrostatic equation givesThere is no energy generation outside the convective core, so use $L_r=L$ there. The local radiative-gradient formula isThe same representative $\beta$ has been used in the global luminosity closure and at the boundary. Setting this gradient equal to ${\nabla }_{\mathrm{ad}}$ at $m=M_c$ producesThis is $1$, $77/80$, and $5/8$ for $\beta =0,1/2,1$, respectively, with $\beta =0$ a marginal radiation-dominated limit.

There is a consistency qualification to this last model estimate. It cannot be an exact stellar solution if all the printed assumptions are enforced pointwise. An exactly uniform nonzero radiation fraction would give $d{P}_{\mathrm{rad}}/dP=1-\beta$ at every radiative point. The exact equations would then requireIf $L_r$ is constant throughout an envelope with positive mass density, $m$ increases with radius, so this equality cannot hold there. Equivalently uniform $\beta$ fixes the actual gradient to $1/4$, whereas ${\nabla }_{\mathrm{ad}}>1/4$ for $\beta >0$. The boxed core fraction is the intended Eddington closure and boundary estimate, which relaxes exact constancy of $\beta$ in the detailed envelope. The constant-beta radiative-envelope obstruction identifies the missing approximation; it is not legitimate to silently assert exact compatibility.

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.