= Solution
For a plane-parallel atmosphere with optical depth increasing downward, the <radiative transfer equation> is
$$
\mu\frac{dI_\nu}{d\tau_\nu}=I_\nu-S_\nu.
$$
Integrating over solid angle gives the first <radiation-field moment>
$$
\frac{dF_\nu}{d\tau_\nu}=4\pi(J_\nu-S_\nu).
$$
In <radiative equilibrium>, matter has no net local radiative heating, so the opacity-weighted frequency integral of $J_\nu-S_\nu$ vanishes. After converting each optical-depth derivative to physical depth and integrating over frequency,
$$
\boxed{\frac{dF}{dz}=0}.
$$
Thus the bolometric internal flux is constant with depth, as stated by <constant flux in a plane-parallel radiative-equilibrium atmosphere>.
In local thermal equilibrium $S_\nu=B_\nu(T)$. Define the planet's <internal effective temperature of a planet> by the constant outward flux. The standard <Planck law> integral gives
$$
\boxed{F_{\rm int}=\pi\int_0^\infty B_\nu(T_{\rm eff})\,d\nu
=\sigma T_{\rm eff}^4}.
$$
The corresponding intrinsic luminosity is $L_{\rm int}=4\pi R_p^2\sigma T_{\rm eff}^4$.
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