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The chosen opacity law makes $\kappa L_r/m_r=\alpha L/M$ spatially constant. This closes the Eddington standard model relation between its radiation-pressure and total-pressure gradients. The four roman-numbered proofs use the whole model, not only this opacity assumption.

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.

The radiative-pressure equation isDividing by hydrostatic equilibrium and substituting
$\kappa ={\kappa }_0(M_r/L_r)(L/M)$ givesa constant. Since both pressures vanish at the surface,
$P_{\mathrm{rad}}=(1-\beta )P$ and $\beta$ is constant throughout the star. Part i therefore haswith spatially constant $K$.

Writing $\rho ={\rho }_c{\theta }^3$ andgives the $n=3$ Lane-Emden equationThe surface is the first zero $\theta ({\xi }_1)=0$. The $n=3$ equation has no elementary closed-form solution; among nonnegative indices, the standard analytic cases are $n=0,1,5$. This Eddington standard model approximates chemically homogeneous massive main-sequence stars with substantial radiation pressure.