= 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 $z=h\zeta$, $\rho=(\Sigma/h)D$, $p=\Sigma\Omega^2hP$, $T=(\Omega^2h^2/\mathcal R)\Theta$ and $F=\alpha\Sigma\Omega^3h^2Q$. Vertical <hydrostatic equilibrium>, <viscous dissipation>, <radiative diffusion> and the <ideal gas> law reduce to $P'=-D\zeta$, $Q'=9P/4$, $\Theta'=-DQ/\Theta^3$, $P=D\Theta$. Midplane <symmetry> gives $Q(0)=0$; a regular radiative-zero surface has $P=\Theta=0$ at a universal $\zeta_s$, with $2\int_0^{\zeta_s}D\,d\zeta=1$. 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
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