 Homologous vertical structure of a constant-opacity alpha disk

Let $\mathcal{R}=k/({\mu }_m{m}_p)$ and choose $h^6=3\alpha \kappa {\Sigma }^2{\mathcal{R}}^4/(16\sigma {\Omega }^5)$. Set $z=h\zeta$, $\rho =(\Sigma /h)D$, $p=\Sigma {\Omega }^{2}hP$, $T=({\Omega }^2{h}^2/\mathcal{R})\Theta$ and $F=\alpha \Sigma {\Omega }^3{h}^{2}Q$. Vertical hydrostatic equilibrium, viscous dissipation, radiative diffusion and the ideal gas law reduce to $P^{\prime }=-D\zeta$, $Q^{\prime }=9P/4$, ${\Theta }^{\prime }=-DQ/{\Theta }^3$, $P=D\Theta$. Midplane symmetry gives $Q(0)=0$; a regular radiative-zero surface has $P=\Theta =0$ at a universal ${\zeta }_s$, with $2{\int }_0^{{\zeta }_s}Dd\zeta =1$. All physical parameters have disappeared. Uniqueness of this normalized boundary-value problem makes every physical profile a rescaling of one dimensionless solution.

// Target: astrophysics.bigb