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An optically thin spherical source with volume emissivity $j(r)=j_0{r}^{-p}$ has surface brightnessSubstitute $z=Ru$ 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.

Surface brightness is received flux per unit solid angle on the sky. It describes the brightness of an extended source independently of a particular aperture's area. An isophote is a contour of constant surface brightness; central surface brightness sets the available photons for resolved galaxy spectroscopy.