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 $f={\sigma }_{\mathrm{SB}}{T}^4(R/d{)}^2=L/(4\pi d^2)$. A grey emissivity factor multiplies this expression; isothermality alone does not imply blackbody emission.