Sérsic profile

The Sérsic profile has the form

$$\backslash ln\; I(R)\; =\; \backslash ln\; I\_\{0\}\; -\; k\; R^\{1/n\}\; ,$$

or

$$I(R)\; =\; I\_0\; \backslash exp\{\backslash !\backslash left(-k\; R^\{1/n\}\backslash right)\},$$

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 $2n\; -\; 1/3\; +\; \backslash frac\{4\}\{405\; n\}\; +\; \backslash frac\{46\}\{25515\; n^2\}\; +\; \backslash frac\{131\}\{1148175\; n^3\}\; -\; \backslash frac\{2194697\}\{30690717750\; n^4\}$, for $n\; >\; 0.36$.{{Cite journal |last1=Ciotti |first1=L. |last2=Bertin |first2=G. |date=1999 |title=Analytical properties of the R^(1/m) luminosity law | journal = Astronomy and Astrophysics | volume = 352 | pages = 447–451 |arxiv=astro-ph/9911078 | bibcode = 1999A&A...352..447C }}

It can be shown that $b\_n$ satisfies $\backslash gamma(2n;\; b\_n)\; =\; \backslash frac\{1\}\{2\}\; \backslash Gamma(2n)$, where $\backslash Gamma$ and $\backslash gamma$ are respectively the Gamma function and lower incomplete Gamma function.

Many related expressions, in terms of the surface brightness, also exist.{{Cite journal |last1=Graham |first1=Alister W. |last2=Driver |first2=Simon P. |date=2005-01-01 |title=A Concise Reference to (Projected) Sérsic R1/n Quantities, Including Concentration, Profile Slopes, Petrosian Indices, and Kron Magnitudes |url=https://ui.adsabs.harvard.edu/abs/2005PASA...22..118G |journal=Publications of the Astronomical Society of Australia |volume=22 |issue=2 |pages=118–127 |doi=10.1071/AS05001 |arxiv=astro-ph/0503176 |bibcode=2005PASA...22..118G |issn=1323-3580}}

File:Messier 87 Hubble WikiSky.jpg

File:M33HunterWilson09.jpg, 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.{{Cite journal |last1=Caon |first1=N. |last2=Capaccioli |first2=M. |last3=D'Onofrio |first3=M. |date=1993-12-01 |title= On the Shape of the Light Profiles of Early Type Galaxies |journal=Monthly Notices of the Royal Astronomical Society |volume=265 |issue=4 |pages=1013–1021 |doi=10.1093/mnras/265.4.1013 |doi-access=free |arxiv=astro-ph/9309013 |issn=0035-8711 | bibcode = 1993MNRAS.265.1013C}}{{Cite journal |last1=Young |first1=C. K. |last2=Currie |first2=M. J. |date=1994-05-01 |title= A New Extragalactic Distance Indicator Based on the Surface Brightness Profiles of Dwarf Elliptical Galaxies |journal=Monthly Notices of the Royal Astronomical Society |volume=268 |pages=L11–L15 |doi=10.1093/mnras/268.1.L11 |doi-access=free |issn=0035-8711 | bibcode = 1994MNRAS.268L..11Y }}

Setting {{math|1=n = 4}} gives the de Vaucouleurs profile:

$$I(R)\; \backslash propto\; e^\{-bR^\{1/4\}\}$$

which is a rough approximation of ordinary elliptical galaxies.

Setting {{math|1=n = 1}} gives the exponential profile:

$$I(R)\; \backslash propto\; e^\{-bR\}$$

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{{Cite journal |last1=Trujillo |first1=I. |last2=Graham |first2=Alister W. |last3=Caon |first3=N. |date=2001-09-01 |title=On the estimation of galaxy structural parameters: the Sérsic model |journal=Monthly Notices of the Royal Astronomical Society |volume=326 |issue=3 |pages=869–876 |doi=10.1046/j.1365-8711.2001.04471.x |doi-access=free |arxiv=astro-ph/0102393 |bibcode=2001MNRAS.326..869T |issn=0035-8711}}) with galaxy morphology is sometimes used in automated schemes to determine the Hubble type of distant galaxies.{{Cite journal |last=van der Wel |first=Arjen |date=2008-07-01 |title=The morphology-density relation: a constant of nature |journal=Formation and Evolution of Galaxy Bulges |url=https://ui.adsabs.harvard.edu/abs/2008IAUS..245...59V |volume=245 |pages=59–62 |doi=10.1017/S1743921308017286|bibcode=2008IAUS..245...59V }} Sérsic indices have also been shown to correlate with the mass of the supermassive black hole at the centers of the galaxies.{{Cite journal |last1=Graham |first1=Alister W. |last2=Driver |first2=Simon P. |date=2007-01-01 |title=A Log-Quadratic Relation for Predicting Supermassive Black Hole Masses from the Host Bulge Sérsic Index |url=https://ui.adsabs.harvard.edu/abs/2007ApJ...655...77G |journal=The Astrophysical Journal |volume=655 |issue=1 |pages=77–87 |doi=10.1086/509758 |arxiv=astro-ph/0607378 |bibcode=2007ApJ...655...77G |issn=0004-637X}}

Sérsic profiles can also be used to describe dark matter halos, where the Sérsic index correlates with halo mass.{{Cite journal |last1=Merritt |first1=David |author1-link = David Merritt |last2=Navarro |first2=Julio F. |last3=Ludlow |first3=Aaron |last4=Jenkins |first4=Adrian |date=2005-05-01 |title=A Universal Density Profile for Dark and Luminous Matter? |url=https://ui.adsabs.harvard.edu/abs/2005ApJ...624L..85M |journal=The Astrophysical Journal |volume=624 |issue=2 |pages=L85–L88 |doi=10.1086/430636 |arxiv=astro-ph/0502515 |bibcode=2005ApJ...624L..85M |issn=0004-637X}}{{Cite journal |last1=Merritt |first1=David |last2=Graham |first2=Alister W. |last3=Moore |first3=Ben |last4=Diemand |first4=Jürg |last5=Terzić |first5=Balša |date=2006-12-01 |title=Empirical Models for Dark Matter Halos. I. Nonparametric Construction of Density Profiles and Comparison with Parametric Models |url=https://ui.adsabs.harvard.edu/abs/2006AJ....132.2685M |journal=The Astronomical Journal |volume=132 |issue=6 |pages=2685–2700 |doi=10.1086/508988 |arxiv=astro-ph/0509417 |bibcode=2006AJ....132.2685M |issn=0004-6256}}

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{{Cite journal |last1=Graham |first1=Alister W. |last2=Erwin |first2=Peter |last3=Trujillo |first3=I. |last4=Asensio Ramos |first4=A. |date=2003-06-01 |title=A New Empirical Model for the Structural Analysis of Early-Type Galaxies, and A Critical Review of the Nuker Model |url=https://ui.adsabs.harvard.edu/abs/2003AJ....125.2951G |journal=The Astronomical Journal |volume=125 |issue=6 |pages=2951–2963 |doi=10.1086/375320 |arxiv=astro-ph/0306023 |bibcode=2003AJ....125.2951G |issn=0004-6256}}{{Cite journal |last1=Trujillo |first1=I. |last2=Erwin |first2=Peter |last3=Asensio Ramos |first3=A. |last4=Graham |first4=Alister W. |date=2004-04-01 |title=Evidence for a New Elliptical-Galaxy Paradigm: Sérsic and Core Galaxies |url=https://ui.adsabs.harvard.edu/abs/2004AJ....127.1917T |journal=The Astronomical Journal |volume=127 |issue=4 |pages=1917–1942 |doi=10.1086/382712 |arxiv=astro-ph/0403659 |bibcode=2004AJ....127.1917T |issn=0004-6256}}{{Cite journal |last1=Terzić |first1=Balša |last2=Graham |first2=Alister W. |date=2005-09-01 |title=Density-potential pairs for spherical stellar systems with Sérsic light profiles and (optional) power-law cores |journal=Monthly Notices of the Royal Astronomical Society |volume=362 |issue=1 |pages=197–212 |doi=10.1111/j.1365-2966.2005.09269.x |doi-access=free |arxiv=astro-ph/0506192 |bibcode=2005MNRAS.362..197T |issn=0035-8711}} 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.{{Cite journal |last1=Graham |first1=Alister W. |last2=Guzmán |first2=Rafael |date=2003-06-01 |title=HST Photometry of Dwarf Elliptical Galaxies in Coma, and an Explanation for the Alleged Structural Dichotomy between Dwarf and Bright Elliptical Galaxies |url=https://ui.adsabs.harvard.edu/abs/2003AJ....125.2936G |journal=The Astronomical Journal |volume=125 |issue=6 |pages=2936–2950 |doi=10.1086/374992 |arxiv=astro-ph/0303391 |bibcode=2003AJ....125.2936G |s2cid=13284968 |issn=0004-6256}}{{Cite journal |last1=Côté |first1=Patrick |last2=Piatek |first2=Slawomir |last3=Ferrarese |first3=Laura |last4=Jordán |first4=Andrés |last5=Merritt |first5=David |last6=Peng |first6=Eric W. |last7=Haşegan |first7=Monica |last8=Blakeslee |first8=John P. |last9=Mei |first9=Simona |last10=West |first10=Michael J. |last11=Milosavljević |first11=Miloš |last12=Tonry |first12=John L. |date=2006-07-01 |title=The ACS Virgo Cluster Survey. VIII. The Nuclei of Early-Type Galaxies |url=https://ui.adsabs.harvard.edu/abs/2006ApJS..165...57C |journal=The Astrophysical Journal Supplement Series |volume=165 |issue=1 |pages=57–94 |doi=10.1086/504042 |arxiv=astro-ph/0603252 |bibcode=2006ApJS..165...57C |issn=0067-0049}}

The Einasto profile is mathematically identical to the Sérsic profile, except that $I$ is replaced by $\backslash rho$, the volume density, and $R$ is replaced by $r$, the internal (not projected on the sky) distance from the center.

• Elliptical galaxy
• Galactic bulge

{{reflist}}

• [http://adsabs.harvard.edu/abs/1991A%26A...249...99C Stellar systems following the R exp 1/m luminosity law] A comprehensive paper that derives many properties of Sérsic models.
• [http://adsabs.harvard.edu/abs/2005PASA...22..118G A Concise Reference to (Projected) Sérsic R^1/n Quantities, Including Concentration, Profile Slopes, Petrosian Indices, and Kron Magnitudes].

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