= Solution
An <accretion disk> forms when infalling material has enough <angular momentum> to orbit before it can reach the central object. In a thin disc dominated by a central <mass> $M$, the orbital motion is approximately <Keplerian rotation>:
$$
\Omega_K=\sqrt{\frac{GM}{R^3}},\qquad
\ell=R^2\Omega_K=\sqrt{GMR}.
$$
Vertical <hydrostatic equilibrium> gives the <disk scale height> $H\sim c_s/\Omega_K$; a geometrically thin disc has $H/R\ll1$. The <surface density> is $\Sigma=\int\rho\,dz$, and a steady inward <mass flow rate> is $\dot M=-2\pi R\Sigma v_R>0$.
Moving inward requires losing specific <angular momentum>. Internal stresses transport <angular momentum> outward; much of the mass can then accrete while an outer fraction spreads to larger radius. Molecular <viscosity> alone is generally inadequate for rapid astrophysical accretion. Turbulent and magnetic stresses, including those generated by the <magnetorotational instability>, provide effective transport. A common parametrization is $\nu=\alpha c_sH$ for an effective <kinematic viscosity>, with $\alpha$ a dimensionless stress parameter.
For a thin <Keplerian accretion disk>, the evolution equation is
$$
\frac{\partial\Sigma}{\partial t}
=\frac3R\frac\partial{\partial R}\left[R^{1/2}\frac\partial{\partial R}
\left(\nu\Sigma R^{1/2}\right)\right].
$$
This <viscous evolution of an accretion disk> occurs on the <viscous timescale> $t_\nu\sim R^2/\nu$, much longer than an orbital period in a thin disc with moderate effective stress.
The orbital specific <energy> is $-GM/(2R)$, so inward motion releases binding energy. In a steady, optically thick, radiatively efficient disc with zero torque at its inner edge, the emergent flux from \b[each face] is
$$
\boxed{F(R)=\sigma_{\rm SB}T_{\rm eff}^4
=\frac{3GM\dot M}{8\pi R^3}\left[1-\left(\frac{R_{\rm in}}R\right)^{1/2}\right].}
$$
The <Stefan–Boltzmann law> identifies the local effective <temperature>. Far from the inner boundary, $T_{\rm eff}\propto R^{-3/4}$, so the emitted spectrum combines hotter inner and cooler outer annuli. Integrating both faces gives $L_{\rm disc}=GM\dot M/(2R_{\rm in})$ under these assumptions. This is not a universal spectrum or efficiency: optical depth, inner torque, energy advection and outflows can change the radiation. The essential connection is \b[outward angular-momentum transport, inward accretion, and release of binding energy].
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