= Solution
An <accretion disk> forms when inward-moving gas retains enough <angular momentum> to orbit a compact object. In a geometrically thin <Keplerian disk>, radial <force> balance gives $\Omega\simeq\sqrt{GM/r^3}$ and vertical <hydrostatic equilibrium> gives a thickness $H\sim c_s/\Omega\ll r$. Matter cannot fall far inward while keeping its original <specific angular momentum> $r^2\Omega\propto r^{1/2}$: stresses must transport <angular momentum> outward.
The <viscous torque in an accretion disk> has outward magnitude
$$
\mathcal G=-2\pi r^3\nu\Sigma\frac{d\Omega}{dr}
=3\pi\nu\Sigma\sqrt{GMr},
$$
where $\nu$ is the effective <kinematic viscosity> and $\Sigma$ the <surface density>. With constant inward <accretion rate> $\dot M>0$ and zero inner torque at $r_{\rm in}$, <conservation of angular momentum> gives
$$
\nu\Sigma=\frac{\dot M}{3\pi}\left(1-\sqrt{\frac{r_{\rm in}}r}\right).
$$
The associated <standard thin-disk dissipation flux> from each face is
$$
F(r)=\frac{3GM\dot M}{8\pi r^3}\left(1-\sqrt{\frac{r_{\rm in}}r}\right).
$$
Orbital <gravitational potential energy> becomes <heat> and, in a radiatively efficient disk, escaping radiation. Integrating both faces gives the <standard thin-disk luminosity> $L=GM\dot M/(2r_{\rm in})$ under this boundary condition. A central stellar surface or an inner torque can release or redistribute additional <energy>.
The <alpha disk> prescription $\nu\sim\alpha c_sH$ summarizes an effective stress rather than a claim that molecular <viscosity> alone explains observed rates. In an ionized disk, <magnetorotational instability> can produce magnetic and fluid stresses. The inward evolution occurs on the <viscous timescale> $t_\nu\sim r^2/\nu$, much longer than an orbit for a thin disk with $\alpha\lesssim1$. Hot inner annuli and cooler outer annuli naturally give a range of emitted <temperatures>; the <blackbody> approximation additionally requires sufficient <optical depth>.
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