The Sérsic profile (or Sérsic model or Sérsic's law) is a mathematical function that describes how the intensity of a galaxy varies with distance
from its center. It is a generalization of de Vaucouleurs' law. José Luis Sérsic first published his law in 1963.[1]
The Sérsic profile has the formor
where is the intensity at
. The parameter
, called the "Sérsic index," controls the degree of curvature of the profile (see figure). The smaller the value of
, the less centrally concentrated the profile is and the shallower (steeper) the logarithmic slope at small (large) radii is. The equation for describing this is:
Today, it is more common to write this function in terms of the half-light radius, Re, and the intensity at that radius, Ie, such that
where is approximately
for
.
can also be approximated to be
, for
.[2]It can be shown that
satisfies
, where
and
are respectively the Gamma function and lower incomplete Gamma function.Many related expressions, in terms of the surface brightness, also exist.[3]
Most galaxies are fit by Sérsic profiles with indices in the range 1/2 < n < 10.The best-fit value of n correlates with galaxy size and luminosity, such that bigger and brighter galaxies tend to be fit with larger n.[5][6]Setting n = 4 gives the de Vaucouleurs profile:which is a rough approximation of ordinary elliptical galaxies.Setting n = 1 gives the exponential profile:
which is a good approximation of spiral galaxy disks and a rough approximation of dwarf elliptical galaxies. The correlation of Sérsic index (i.e. galaxy concentration[7]) with galaxy morphology is sometimes used in automated schemes to determine the Hubble type of distant galaxies.[8] Sérsic indices have also been shown to correlate with the mass of the supermassive black hole at the centers of the galaxies.[9]
Sérsic profiles can also be used to describe dark matter halos, where the Sérsic index correlates with halo mass.[10][11]
The brightest elliptical galaxies often have low-density cores that are not well described by Sérsic's law. The core-Sérsic family of models was introduced[12][13][14] to describe such galaxies. Core-Sérsic models have an additional set of parameters that describe the core.
Dwarf elliptical galaxies and bulges often have point-like nuclei that are also not well described by Sérsic's law. These galaxies are often fit by a Sérsic model with an added central component representing the nucleus.[15][16]
The Einasto profile is mathematically identical to the Sérsic profile, except that is replaced by
, the volume density, and
is replaced by
, the internal (not projected on the sky) distance from the center.