Sérsic profile explained

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$$\backslash ln\; I(R)\; =\; \backslash ln\; I\_\; -\; k\; R^,$$or$$I(R)\; =\; I\_0\; \backslash exp,$$

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:$$\backslash frac\; =\; -(k/n)\backslash \; R^\; .$$

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( (R)/(R _{e}) \right) ^{1/n} -1 \right] \right\}, where b _{n} is approximately 2 n -1 / 3 for n> 8 . b _{n} can also be approximated to be 2 n - 1 / 3 + (4)/(405 n) + (46)/(25515 n ^{2}) + (131)/(1148175 n ^{3}) - (2194697)/(30690717750 n ^{4}) , for n > 0 . 36 .^[2] It can be shown that b _{n} satisfies $\backslash gamma(2n;\; b\_n)\; =\; \backslash frac\; \backslash Gamma(2n)$, 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

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 gives the de Vaucouleurs profile:$$I(R)\; \backslash propto\; e^$$which is a rough approximation of ordinary elliptical galaxies.Setting gives the exponential profile:$$I(R)\; \backslash propto\; e^$$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

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.

See also

• Elliptical galaxy
• Galactic bulge

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.

Notes and References

1. Sérsic . J. L. . 1963-02-01 . Influence of the atmospheric and instrumental dispersion on the brightness distribution in a galaxy . Boletin de la Asociacion Argentina de Astronomia . 6 . 41–43 . 1963BAAA....6...41S . 0571-3285.
2. Ciotti . L. . Bertin . G. . 1999 . Analytical properties of the R^(1/m) luminosity law . Astronomy and Astrophysics . 352 . 447–451 . astro-ph/9911078 . 1999A&A...352..447C .
3. Graham . Alister W. . Driver . Simon P. . 2005-01-01 . A Concise Reference to (Projected) Sérsic R1/n Quantities, Including Concentration, Profile Slopes, Petrosian Indices, and Kron Magnitudes . Publications of the Astronomical Society of Australia . 22 . 2 . 118–127 . 10.1071/AS05001 . astro-ph/0503176 . 2005PASA...22..118G . 1323-3580.
4. Caon . N. . Capaccioli . M. . D'Onofrio . M. . 1993-12-01 . On the Shape of the Light Profiles of Early Type Galaxies . Monthly Notices of the Royal Astronomical Society . 265 . 4 . 1013–1021 . 10.1093/mnras/265.4.1013 . free . astro-ph/9309013 . 0035-8711 . 1993MNRAS.265.1013C.
5. Young . C. K. . Currie . M. J. . 1994-05-01 . A New Extragalactic Distance Indicator Based on the Surface Brightness Profiles of Dwarf Elliptical Galaxies . Monthly Notices of the Royal Astronomical Society . 268 . L11–L15 . 10.1093/mnras/268.1.L11 . free . 0035-8711 . 1994MNRAS.268L..11Y .
6. Trujillo . I. . Graham . Alister W. . Caon . N. . 2001-09-01 . On the estimation of galaxy structural parameters: the Sérsic model . Monthly Notices of the Royal Astronomical Society . 326 . 3 . 869–876 . 10.1046/j.1365-8711.2001.04471.x . free . astro-ph/0102393 . 2001MNRAS.326..869T . 0035-8711.
7. van der Wel . Arjen . 2008-07-01 . The morphology-density relation: a constant of nature . Formation and Evolution of Galaxy Bulges . 245 . 59–62 . 10.1017/S1743921308017286. 2008IAUS..245...59V .
8. Graham . Alister W. . Driver . Simon P. . 2007-01-01 . A Log-Quadratic Relation for Predicting Supermassive Black Hole Masses from the Host Bulge Sérsic Index . The Astrophysical Journal . 655 . 1 . 77–87 . 10.1086/509758 . astro-ph/0607378 . 2007ApJ...655...77G . 0004-637X.
9. Merritt . David . David Merritt . Navarro . Julio F. . Ludlow . Aaron . Jenkins . Adrian . 2005-05-01 . A Universal Density Profile for Dark and Luminous Matter? . The Astrophysical Journal . 624 . 2 . L85–L88 . 10.1086/430636 . astro-ph/0502515 . 2005ApJ...624L..85M . 0004-637X.
10. Merritt . David . Graham . Alister W. . Moore . Ben . Diemand . Jürg . Terzić . Balša . 2006-12-01 . Empirical Models for Dark Matter Halos. I. Nonparametric Construction of Density Profiles and Comparison with Parametric Models . The Astronomical Journal . 132 . 6 . 2685–2700 . 10.1086/508988 . astro-ph/0509417 . 2006AJ....132.2685M . 0004-6256.
11. Graham . Alister W. . Erwin . Peter . Trujillo . I. . Asensio Ramos . A. . 2003-06-01 . A New Empirical Model for the Structural Analysis of Early-Type Galaxies, and A Critical Review of the Nuker Model . The Astronomical Journal . 125 . 6 . 2951–2963 . 10.1086/375320 . astro-ph/0306023 . 2003AJ....125.2951G . 0004-6256.
12. Trujillo . I. . Erwin . Peter . Asensio Ramos . A. . Graham . Alister W. . 2004-04-01 . Evidence for a New Elliptical-Galaxy Paradigm: Sérsic and Core Galaxies . The Astronomical Journal . 127 . 4 . 1917–1942 . 10.1086/382712 . astro-ph/0403659 . 2004AJ....127.1917T . 0004-6256.
13. Terzić . Balša . Graham . Alister W. . 2005-09-01 . Density-potential pairs for spherical stellar systems with Sérsic light profiles and (optional) power-law cores . Monthly Notices of the Royal Astronomical Society . 362 . 1 . 197–212 . 10.1111/j.1365-2966.2005.09269.x . free . astro-ph/0506192 . 2005MNRAS.362..197T . 0035-8711.
14. Graham . Alister W. . Guzmán . Rafael . 2003-06-01 . HST Photometry of Dwarf Elliptical Galaxies in Coma, and an Explanation for the Alleged Structural Dichotomy between Dwarf and Bright Elliptical Galaxies . The Astronomical Journal . 125 . 6 . 2936–2950 . 10.1086/374992 . astro-ph/0303391 . 2003AJ....125.2936G . 13284968 . 0004-6256.
15. Côté . Patrick . Piatek . Slawomir . Ferrarese . Laura . Jordán . Andrés . Merritt . David . Peng . Eric W. . Haşegan . Monica . Blakeslee . John P. . Mei . Simona . West . Michael J. . Milosavljević . Miloš . Tonry . John L. . 2006-07-01 . The ACS Virgo Cluster Survey. VIII. The Nuclei of Early-Type Galaxies . The Astrophysical Journal Supplement Series . 165 . 1 . 57–94 . 10.1086/504042 . astro-ph/0603252 . 2006ApJS..165...57C . 0067-0049.