= Solution
For Keplerian angular velocity, $R^2(d\Omega/dR)^2=9GM/(4R^3)$. The stated $F_{\rm diss}$ is the dissipation summed over both disk faces, so $F_{\rm diss}=2\sigma_{\rm SB}T_{\rm eff}^4$. Inside $R_0$ this gives
$$
\boxed{T_{\rm eff}^4(R)=\frac{3GM\dot m_0}{8\pi\sigma_{\rm SB}R^3}
\left[1-\left(\frac{R_{\rm ISCO}}R\right)^{1/2}\right].}
$$
Differentiating the factor $R^{-3}[1-(R_{\rm ISCO}/R)^{1/2}]$ shows that the maximum occurs at
$$
\boxed{R_{T,\max}=\frac{49}{36}R_{\rm ISCO}}
$$
when this radius lies below $R_0$. In the ordinary inflowing region far from its inner edge, $T_{\rm eff}\propto R^{-3/4}$.
For the static angular-momentum sink outside $R_0$, part c instead gives
$$
\boxed{T_{\rm eff}^4(R)=\frac{3GM\dot m_0}{8\pi\sigma_{\rm SB}R^3}
\left[\left(\frac{R_0}R\right)^{1/2}
-\left(\frac{R_{\rm ISCO}}R\right)^{1/2}\right],}
$$
so the genuinely large-radius behavior of the complete injected disk is $T_{\rm eff}\propto R^{-7/8}$.
Integrating $2\pi R F_{\rm diss}\,dR$ over both regions gives
$$
L_{<R_0}=GM\dot m_0\left[
\frac1{2R_{\rm ISCO}}-\frac3{2R_0}
+\frac{\sqrt{R_{\rm ISCO}}}{R_0^{3/2}}\right]
$$
and
$$
L_{>R_0}=\frac{GM\dot m_0}{R_0}
\left[1-\left(\frac{R_{\rm ISCO}}{R_0}\right)^{1/2}\right].
$$
Hence
$$
\boxed{L_{\rm disk}=\frac{GM\dot m_0}{2}
\left(\frac1{R_{\rm ISCO}}-\frac1{R_0}\right),}
$$
which 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 $R\gg R_{\rm ISCO}$,
$$
L(>R)=\frac{3GM\dot m}{2R}
\left[1-\frac23\left(\frac{R_{\rm ISCO}}R\right)^{1/2}\right]
\simeq\frac{3GM\dot m}{2R}.
$$
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.
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