# 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 ${\displaystyle I}$ 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]

## Definition

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, Re, and the intensity at that radius, Ie, 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]

## 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.[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]

## 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 ${\displaystyle I}$ 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.

## 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. Bibcode:1963BAAA....6...41S. ISSN 0571-3285.
2. ^ Ciotti, L.; Bertin, G. (1999). "Analytical properties of the R^(1/m) luminosity law". Astronomy and Astrophysics. 352: 447–451. arXiv:astro-ph/9911078. Bibcode: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. arXiv:astro-ph/0503176. Bibcode:2005PASA...22..118G. doi:10.1071/AS05001. ISSN 1323-3580.
4. ^ Savorgnan, G.; Graham, A. W.; Marconi, A.; Sani, E.; Hunt, L. K.; Vika, M.; Driver, S. P. (2013-09-01). "The supermassive black hole mass-Sérsic index relations for bulges and elliptical galaxies". Monthly Notices of the Royal Astronomical Society. 434 (1): 387–397. arXiv:1306.2679. Bibcode:2013MNRAS.434..387S. doi:10.1093/mnras/stt1027. ISSN 0035-8711.
5. ^ 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. arXiv:astro-ph/9309013. Bibcode:1993MNRAS.265.1013C. doi:10.1093/mnras/265.4.1013. ISSN 0035-8711.
6. ^ 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. Bibcode:1994MNRAS.268L..11Y. doi:10.1093/mnras/268.1.L11. ISSN 0035-8711.
7. ^ 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. arXiv:astro-ph/0102393. Bibcode:2001MNRAS.326..869T. doi:10.1046/j.1365-8711.2001.04471.x. ISSN 0035-8711.
8. ^ van der Wel, Arjen (2008-07-01). "The morphology-density relation: a constant of nature". Formation and Evolution of Galaxy Bulges. 245: 59–62. Bibcode:2008IAUS..245...59V. doi:10.1017/S1743921308017286.
9. ^ 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. arXiv:astro-ph/0607378. Bibcode:2007ApJ...655...77G. doi:10.1086/509758. ISSN 0004-637X.
10. ^ Merritt, David; 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. arXiv:astro-ph/0502515. Bibcode:2005ApJ...624L..85M. doi:10.1086/430636. ISSN 0004-637X.
11. ^ 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. arXiv:astro-ph/0509417. Bibcode:2006AJ....132.2685M. doi:10.1086/508988. ISSN 0004-6256.
12. ^ 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. arXiv:astro-ph/0306023. Bibcode:2003AJ....125.2951G. doi:10.1086/375320. ISSN 0004-6256.
13. ^ 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. arXiv:astro-ph/0403659. Bibcode:2004AJ....127.1917T. doi:10.1086/382712. ISSN 0004-6256.
14. ^ 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. arXiv:astro-ph/0506192. Bibcode:2005MNRAS.362..197T. doi:10.1111/j.1365-2966.2005.09269.x. ISSN 0035-8711.
15. ^ 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. arXiv:astro-ph/0303391. Bibcode:2003AJ....125.2936G. doi:10.1086/374992. ISSN 0004-6256. S2CID 13284968.
16. ^ 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. arXiv:astro-ph/0603252. Bibcode:2006ApJS..165...57C. doi:10.1086/504042. ISSN 0067-0049.