= Torque transport of energy in a steady accretion disk
{title2=$D_{a,b}=B_{a,b}+\Omega(a)\mathcal G(a)-\Omega(b)\mathcal G(b)$}
For a steady <Keplerian accretion disk> with inward <accretion rate> $\dot M$ and vanishing inner <torque>, the outward angular-momentum torque is $\mathcal G=\dot M(\sqrt{GMr}-\sqrt{GMr_{\rm in}})$. It transports <energy> outward at rate $\Omega\mathcal G$. The binding-energy acquisition rate between radii $b$ and $a$ is $B_{a,b}=GM\dot M(1/a-1/b)/2$. Integrating the vertical <viscous dissipation> and then the annulus gives
$$
D_{a,b}=GM\dot M\left[\frac32\left(\frac1a-\frac1b\right)-\sqrt{r_{\rm in}}(a^{-3/2}-b^{-3/2})\right].
$$
Since $\Omega\mathcal G=GM\dot M[r^{-1}-\sqrt{r_{\rm in}}r^{-3/2}]$, subtraction proves the displayed energy balance. Far from the inner edge $D_{a,b}\simeq3B_{a,b}$, whereas the boundary torque powers vanish for the complete disk and $D=B=GM\dot M/(2r_{\rm in})$. Local dissipation is therefore not supplied solely by local changes in orbital binding energy.
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