# Sérsic profile

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The Sérsic profile (or Sérsic model or Sérsic's law) is a mathematical function that describes how the intensity ${\displaystyle I}$ of a galaxy varies with distance ${\displaystyle R}$ from its center. It is a generalization of de Vaucouleurs' law. José Luis Sérsic first published his law in 1963.[1]

Sérsic models with different indices ${\displaystyle n}$. The order of ${\displaystyle n}$ is reversed for large radii.

## Definition

The Sérsic profile has the form

${\displaystyle \ln \ I(R)=\ln \ I_{0}-kR^{1/n},}$

where ${\displaystyle I_{0}}$  is the intensity at ${\displaystyle R=0}$ . The parameter ${\displaystyle n}$ , called the "Sérsic index," controls the degree of curvature of the profile (see figure). The smaller the value of ${\displaystyle n}$ , the less centrally concentrated the profile is and the shallower (steeper) the logarithmic slope at small (large) radii is:

${\displaystyle {\frac {\mathrm {d} \ln I}{\mathrm {d} \ln R}}=-(k/n)\ R^{1/n}.}$

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

${\displaystyle I(R)=I_{e}\exp \left\{-b_{n}\left[\left({\frac {R}{R_{e}}}\right)^{1/n}-1\right]\right\},}$

where ${\displaystyle b_{n}}$  is approximately ${\displaystyle 2n-1/3}$  for ${\displaystyle n>8}$ . ${\displaystyle b_{n}}$  can also be approximated to be ${\displaystyle 2n-1/3+{\frac {4}{405n}}+{\frac {46}{25515n^{2}}}+{\frac {131}{1148175n^{3}}}-{\frac {2194697}{30690717750n^{4}}}}$ , for ${\displaystyle n>0.36}$ .[2] It can be shown that ${\displaystyle b_{n}}$  satisfies ${\displaystyle \gamma (2n;b_{n})={1 \over 2}\Gamma (2n)}$ , where ${\displaystyle \Gamma }$  and ${\displaystyle \gamma }$  are respectively the Gamma function and lower incomplete Gamma function. Many related expressions, in terms of the surface brightness, also exist.[3]

## Applications

Massive elliptical galaxies have high Sérsic indices and a high degree of central concentration. This galaxy, M87, has a Sérsic index n~ 4. [4]

Discs of spiral galaxies, such as the Triangulum Galaxy, have low Sérsic indices and a low degree of central concentration.

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:

${\displaystyle I(R)\propto e^{-bR^{1/4}}}$

which is a rough approximation of ordinary elliptical galaxies. Setting n = 1 gives the exponential profile:

${\displaystyle I(R)\propto e^{-bR}}$

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]

## Generalizations of the Sérsic profile

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 ${\displaystyle I}$  is replaced by ${\displaystyle \rho }$ , the volume density, and ${\displaystyle R}$  is replaced by ${\displaystyle r}$ , the internal (not projected on the sky) distance from the center.