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

Definition

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

where $I_{0}$ is the intensity at $R=0$ . The parameter $n$ , called the "Sérsic index," controls the degree of curvature of the profile (see figure). The smaller the value of $n$ , 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: ${\frac {\mathrm {d} \ln I}{\mathrm {d} \ln R}}=-(k/n)\ R^{1/n}.$

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

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

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

Applications

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: $I(R)\propto e^{-bR^{1/4}}$ which is a rough approximation of ordinary elliptical galaxies. Setting $n = 1$ gives the exponential profile: $I(R)\propto e^{-bR}$ 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 $I$ is replaced by $\rho$ , the volume density, and $R$ is replaced by $r$ , the internal (not projected on the sky) distance from the center.