= Projection of a spherical power-law emissivity
{title2=$j\propto r^{-p}\Longrightarrow I\propto R^{1-p}$}
An optically thin spherical source with volume emissivity $j(r)=j_0r^{-p}$ has <surface brightness>
$$
I(R)=\frac{j_0}{4\pi}R^{1-p}\sqrt\pi\,\frac{\Gamma((p-1)/2)}{\Gamma(p/2)},\qquad p>1.
$$
Substitute $z=Ru$ in the line-of-sight <integral> and evaluate $\int_{-\infty}^\infty(1+u^2)^{-p/2}du$ using the <beta function>. The condition $p>1$ is convergence at large distance. A core or finite outer boundary changes the projection near that boundary; the stated scaling applies between those scales.
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