 Solution

For Keplerian angular velocity, $R^2(d\Omega /dR{)}^2=9GM/(4R^3)$. The stated $F_{\mathrm{diss}}$ is the dissipation summed over both disk faces, so $F_{\mathrm{diss}}=2{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^4$. Inside $R_0$ this givesDifferentiating the factor $R^{-3}[1-(R_{\mathrm{ISCO}}/R{)}^{1/2}]$ shows that the maximum occurs atwhen this radius lies below $R_0$. In the ordinary inflowing region far from its inner edge, $T_{\mathrm{eff}}\propto R^{-3/4}$.

For the static angular-momentum sink outside $R_0$, part c instead givesso the genuinely large-radius behavior of the complete injected disk is $T_{\mathrm{eff}}\propto R^{-7/8}$.

Integrating $2\pi R{F}_{\mathrm{diss}}dR$ over both regions givesandHencewhich is exactly the loss of Keplerian orbital energy as matter moves from its injection orbit to the inner edge.

For a standard disk extending through a radius $R\gg R_{\mathrm{ISCO}}$,This is three times the binding-energy release $GM\dot{m}/(2R)$ available outside $R$. The excess is energy carried outward by the viscous torque in an accretion disk and dissipated at larger radii. In the injected model the nonaccreting outer disk is an especially direct example: it radiates despite having zero mean radial mass flux because it absorbs the angular momentum and mechanical work exported by the inner disk.