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Take the positive constant ratio $\lambda =p_r/p_g$. The equation of state impliesThis is a polytropic equation of state of index three. Differentiating the pressure and using vertical hydrostatic equilibrium gives $4(1+\lambda )\mathcal{R}\rho T_z=-\rho {\Omega }^{2}z$. HenceDefine the semi-thickness $H$ by $T(H)=0$ and put $s=1-z^2/H^2$. Then the constant gas-radiation ratio vertical disc structure iswithThe surface density is the integral across both faces. The supplied polynomial integral gives ${\int }_0^1(1-x^2{)}^{3}dx=16/35$, so $\Sigma =(32/35){\rho }_{0}H$. Substitution yieldsAll coefficients here are local in radial position; the derivation fixes the vertical profile rather than a global accretion rate. The zero-temperature surface is a formal boundary of this bulk model; optically thick diffusion ultimately fails in a thin photospheric layer.

For the constant gas-radiation ratio vertical disc structure, the integrated alpha prescription $\bar{\nu }\Sigma =\int \mu dz=(\alpha /\Omega )\int pdz$ gives the displayed mean kinematic viscosity and $\lambda /(1+\lambda )=\alpha H\Omega \kappa \Sigma /(8c)$. Uniform dynamic viscosity and nonuniform pressure are compatible because this prescription is an integrated equality. Imposing the corresponding pointwise equality would not preserve the exact vertical solution.