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 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]

Sérsic models with different indices . The order of is reversed for large radii.

DefinitionEdit

The Sérsic profile has the form

 

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:

 

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]

ApplicationsEdit

 
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:

 

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]

Generalizations of the Sérsic profileEdit

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.

See alsoEdit

ReferencesEdit

  1. ^ J. L. Sérsic (1963), Influence of the atmospheric and instrumental dispersion on the brightness distribution in a galaxy
  2. ^ L. Ciotti and G. Bertin (1999) Analytical properties of the R1/mlaw
  3. ^ 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
  4. ^ G. Savorgnan et al. (2013),The supermassive black hole mass-Sérsic index relations for bulges and elliptical galaxies
  5. ^ N. Caon et al. (1993), On the Shape of the Light Profiles of Early Type Galaxies
  6. ^ C. Young & M. Currie (1994), A New Extragalactic Distance Indicator Based on the Surface Brightness Profiles of Dwarf Elliptical Galaxies
  7. ^ Trujillo, I., Graham, Alister W., Caon, N. (2001), On the estimation of galaxy structural parameters: the Sérsic model
  8. ^ A. van der Wel (2008), The morphology-density relation: a constant of nature
  9. ^ A. Graham & S. Driver (2007), A Log-Quadratic Relation for Predicting Supermassive Black Hole Masses from the Host Bulge Sérsic Index
  10. ^ D. Merritt et al. (2005), A Universal Density Profile for Dark and Luminous Matter?
  11. ^ D. Merritt et al. (2006), Empirical Models for Dark Matter Halos. III. Nonparametric Construction of Density Profiles and Comparison with Parametric Models
  12. ^ 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
  13. ^ I. Trujillo et al. (2004), Evidence for a New Elliptical-Galaxy Paradigm: Sérsic and Core Galaxies
  14. ^ B. Terzić & A. W. Graham (2005), Density-potential pairs for spherical stellar systems with Sérsic light profiles and (optional) power-law cores
  15. ^ A. Graham & R. Guzmán (2003), HST Photometry of Dwarf Elliptical Galaxies in Coma
  16. ^ P. Cote et al. (2006), The ACS Virgo Cluster Survey. VIII. The Nuclei of Early-Type Galaxies

External linksEdit