Effective-temperature profile of a zero-torque disk ($T_{\mathrm{eff}}(r)$)

A Keplerian accretion disk with a zero-torque inner boundary condition and local blackbody cooling has $T_{\mathrm{eff}}^4=3GM\dot{M}(1-\sqrt{r_{\mathrm{in}}/r})/(8\pi \sigma r^3)$. The Stefan–Boltzmann law uses the flux through one face; the standard thin-disk dissipation flux includes both faces. The scale $T_{\mathrm{in}}=[3GM\dot{M}/(8\pi \sigma r_{\mathrm{in}}^3){]}^{1/4}$ is a normalization, not the actual temperature at the stress-free inner edge.

For the effective-temperature profile of a zero-torque disk, differentiation of $y^{-3}(1-y^{-1/2})$ with $y=r/r_{\mathrm{in}}$ gives $y_{\max }=49/36$. Thus $T_{\max }=(6/7{)}^{3/2}{7}^{-1/4}{T}_{\mathrm{in}}\simeq 0.488T_{\mathrm{in}}$. This distinguishes a genuine maximum effective temperature from the normalization used in a power-law approximation.