Surface brightness ($I$)

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.

For a diffraction-limited telescope, collecting area scales as $D^2$ while a native angular resolution element scales as $(\lambda /D{)}^2$. Counts from a smooth resolved source per native resolution element therefore do not increase with aperture, at fixed radiance, exposure and throughput. Counts in a fixed angular bin instead grow as $D^2$, and the photon-noise-limited signal-to-noise ratio grows as $D\sqrt{t}$. Conservation of etendue and cosmological surface-brightness dimming do not forbid this latter gain.

For freely propagating radiation with the usual distance duality relation, a fixed emitted bolometric surface brightness is observed smaller by $(1+z{)}^4$. One factor each comes from photon energy and arrival rate, and two from the relation of projected angular area to luminosity distance. In a fixed observed photometric passband, spectral shifts add a band-dependent correction; the bolometric power is not automatically the monochromatic one. A distortion created by scattering the local cosmic microwave background has a different redshift comparison, as in the thermal Sunyaev-Zeldovich effect.

For a circular projected profile with a positive power-law tail, an annulus contributes $dF=2\pi I(R)RdR$. Thus the outer integral converges only when $p>2$; $p=2$ diverges logarithmically and $p<2$ as $R^{2-p}$. A shallower tail measured over a finite range must steepen or terminate before an infinite-radius extrapolation is physical. This projected criterion already includes line-of-sight integration and must not be replaced by a three-dimensional $r^{2}dr$ volume factor.

A Sérsic profile models projected galactic light, with index $n$ controlling curvature and $b_n$ chosen so that $R_e$ encloses half the luminosity. Its logarithmic slope is $d\ln I/d\ln R=-(b_n/n)(R/R_e{)}^{1/n}$. The $n=1$ case is exponential; $n=4$ gives the traditional de Vaucouleurs form. Unlike a fixed power law, its slope steepens outward.

The $n=4$ case of the Sérsic surface-brightness profile, approximately useful for many bright elliptical galaxies: $I(R)=I_e\exp [-7.67((R/R_e{)}^{1/4}-1)]$. Real galaxies can have other Sérsic indices.

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.