Disk flaring ($d(H/r)/dr>0$)

A disk flares when its disk aspect ratio increases outward. This is stronger than the statement that its height increases: a disk with $H\propto r$ has constant aspect ratio and is not flaring by this definition. The radial temperature and surface density of a disk profiles determine flaring in a specified vertical equilibrium model.

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 $q^{\prime }=C{q}^8/R^3$ for $q=z_s/R$, where $C=4\pi \sigma (GM/a{)}^4/(L_{\ast }{f}^8)$. 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.