= Observed flux of an unresolved spherical emitter
For a uniformly emitting spherical body of <radius> $R$ at <Euclidean distance> $d\gg R$, integration of its <specific intensity> over the apparent <solid angle> gives $f_\lambda=\pi I_\lambda(R/d)^2$. For a <blackbody>, $I_\lambda=B_\lambda(T)$ and the bolometric result is $f=\sigma_{\rm SB}T^4(R/d)^2=L/(4\pi d^2)$. A grey <emissivity> factor multiplies this expression; isothermality alone does not imply blackbody emission.
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