Models of surface brightness profiles

No physical law exists which describes the surface brightness profile of a galaxy. Initially Patterson patterson1940 and later De Vaucouleurs vaucouleurs1959 showed that surface brightness profiles could be described by an exponential function. Even later Freeman freeman1970 surveyed the literature to show empirically this exponential behaviour on face-on disk galaxies and coupled the exponential function with a dynamical background, establishing its form as we know it up to today as

$$I (r)\ =\ I_0 e^{-r/h}\ ,$$ (2)

where $r$ is the radial position from the center of the disk and $h$ the radial scalelength, and $I_0$ the central surface brightness.
Van der Kruit vdkruit1979 showed that the surface brightness in $z$ is independent of $r$ and that the vertical profile can be described by an analogue exponential function using the vertical scaleheight as a parameter. In an attempt to fit the $z$ distribution of light at each $r$ by that of a locally isothermal sheet, Van der Kruit & Searle vdkruit1981a found that a sech$^2$ ($z/z_0$) function was a more appropriate description for the vertical surface brightness profile. Some years later Van der Kruit vdkruit1988 proposed to use a sech function as an intermediate solution to fit vertical profile as he showed that the exponential and the sech$^2$ function all belonged to the same family of density laws

$$I(z)\ =\ 2^{2/n}I_0\ \textrm{sech}^{2/n}(nz/2z_0)\ ,\ (n\ >\ 0)$$ (3)

where $I(z)$ is the observed vertical density profile and $z_0$ the scaleheight, while $z$ is the distance from the plane. The isothermal model is the extreme for $n$ = 1 and the exponential is the other extreme for $n\ =\ \infty$. With this family of density laws, there are several ways to determine structural parameters.

Figure: Radial profiles of NGC 5290 J at the midplane and at vertical positions above and below the plane. The magenta line shows $\mu_\textrm{\scriptsize crit}$.

Figure: Vertical profiles of NGC 5290 J at the center and at two radial positions on the left and right side from the center. The magenta line shows $\mu_\textrm{\scriptsize crit}$.