= Flaring passive disk
{title2=$z_s/R\propto R^{2/7},\quad T\propto R^{-3/7}$}
Assume a <passive disk> has local <isothermal sound speed> squared $c_s^2=aT$, <disk scale height> $H=c_s/\Omega_K$, and irradiated height $z_s=fH$ with constant $f$. Its <irradiation grazing angle of a disk> and <Stefan–Boltzmann law> give $q'=Cq^8/R^3$ for $q=z_s/R$, where $C=4\pi\sigma(GM/a)^4/(L_*f^8)$. Integrating gives $q^{-7}=R_0^2/R^2+B$, with $R_0^2=7C/2$. Where the first term dominates and the <thin disk> assumptions hold, $q\simeq(R/R_0)^{2/7}$, $H\propto R^{9/7}$ and $T\propto R^{-3/7}$. The law cannot be extrapolated into a geometrically thick region using the same grazing-angle approximation.
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