Sersic 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]

Sérsic models with different indices ${\displaystyle n}$.

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, 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 2n-1/3. 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.[2]

Applications

Massive elliptical galaxies have high Sérsic indices and a high degree of central concentration. This galaxy, M87, has a Sérsic index n~ 4. [3]

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. [4] [5] 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[6]) with galaxy morphology is sometimes used in automated schemes to determine the Hubble type of distant galaxies.[7] Sérsic indices have also been shown to correlate with the mass of the supermassive black hole at the centers of the galaxies. [8]

Sérsic profiles can also be used to describe dark matter halos, where the Sérsic index correlates with halo mass.[9] [10]

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 [11][12][13] 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. [14] [15]

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.

See also

• Elliptical galaxies
• Bulges

References

1. J. L. Sérsic (1963), Influence of the atmospheric and instrumental dispersion on the brightness distribution in a galaxy
2. Graham, A.W. and Driver, S.P. (2005), A Concise Reference to (Projected) Sérsic R1/n Quantities, Including Concentration, Profile Slopes, Petrosian Indices, and Kron Magnitudes
3. G. Savorgnan et al. (2013),The supermassive black hole mass-Sérsic index relations for bulges and elliptical galaxies
4. N. Caon et al. (1993), On the Shape of the Light Profiles of Early Type Galaxies
5. C. Young & M. Currie (1994), A New Extragalactic Distance Indicator Based on the Surface Brightness Profiles of Dwarf Elliptical Galaxies
6. Trujillo, I., Graham, Alister W., Caon, N. (2001), On the estimation of galaxy structural parameters: the Sérsic model
7. A. van der Wel (2008), The morphology-density relation: a constant of nature
8. A. Graham & S. Driver (2007), A Log-Quadratic Relation for Predicting Supermassive Black Hole Masses from the Host Bulge Sérsic Index
9. D. Merritt et al. (2005), A Universal Density Profile for Dark and Luminous Matter?
10. D. Merritt et al. (2006), Empirical Models for Dark Matter Halos. III. Nonparametric Construction of Density Profiles and Comparison with Parametric Models
11. A. Graham et al. (2003), A New Empirical Model for the Structural Analysis of Early-Type Galaxies, and A Critical Review of the Nuker Model
12. I. Trujillo et al. (2004), Evidence for a New Elliptical-Galaxy Paradigm: Sérsic and Core Galaxies
13. B. Terzić & A. W. Graham (2005), Density-potential pairs for spherical stellar systems with Sérsic light profiles and (optional) power-law cores
14. A. Graham & R. Guzmán (2003), HST Photometry of Dwarf Elliptical Galaxies in Coma
15. P. Cote et al. (2006), The ACS Virgo Cluster Survey. VIII. The Nuclei of Early-Type Galaxies

External links

• 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.