Sérsic profile

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]

The Sérsic profile has the form $${\displaystyle \ln I(R)=\ln I_{0}-kR^{1/n},}$$ or $${\displaystyle I(R)=I_{0}\exp {\!\left(-kR^{1/n}\right)},}$$

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. The equation for describing this 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, R_e, and the intensity at that radius, I_e, 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 ${\textstyle \gamma (2n;b_{n})={\frac {1}{2}}\Gamma (2n)}$, 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]

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]

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.

• Elliptical galaxy
• Galactic bulge

• Stellar systems following the R exp 1/m luminosity law A comprehensive paper that derives many properties of Sérsic models.
• A Concise Reference to (Projected) Sérsic R^1/n Quantities, Including Concentration, Profile Slopes, Petrosian Indices, and Kron Magnitudes.