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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 I {\displaystyle I} of a galaxy varies with distance R {\displaystyle R} from its center. It is a generalization of de Vaucouleurs' law. José Luis Sérsic first published his law in 1963.

Sérsic models with different indices n {\displaystyle n} . The order of n {\displaystyle n} is reversed for large radii.

Definition

The Sérsic profile has the form ln ⁡ I ( R ) = ln ⁡ I 0 − k R 1 / n , {\displaystyle \ln I(R)=\ln I_{0}-kR^{1/n},} or I ( R ) = I 0 exp ⁡ ( − k R 1 / n ) , {\displaystyle I(R)=I_{0}\exp {\!\left(-kR^{1/n}\right)},}

where I 0 {\displaystyle I_{0}} is the intensity at R = 0 {\displaystyle R=0} . The parameter n {\displaystyle n} , called the "Sérsic index," controls the degree of curvature of the profile (see figure). The smaller the value of n {\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: d ln ⁡ I d ln ⁡ R = − ( k / n )   R 1 / n . {\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

I ( R ) = I e exp ⁡ { − b n [ ( R R e ) 1 / n − 1 ] } , {\displaystyle I(R)=I_{e}\exp \left\{-b_{n}\left\right\},}

where b n {\displaystyle b_{n}} is approximately 2 n − 1 / 3 {\displaystyle 2n-1/3} for n > 8 {\displaystyle n>8} . b n {\displaystyle 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 {\displaystyle 2n-1/3+{\frac {4}{405n}}+{\frac {46}{25515n^{2}}}+{\frac {131}{1148175n^{3}}}-{\frac {2194697}{30690717750n^{4}}}} , for n > 0.36 {\displaystyle n>0.36} . It can be shown that b n {\displaystyle b_{n}} satisfies γ ( 2 n ; b n ) = 1 2 Γ ( 2 n ) {\textstyle \gamma (2n;b_{n})={\frac {1}{2}}\Gamma (2n)} , where Γ {\displaystyle \Gamma } and γ {\displaystyle \gamma } are respectively the Gamma function and lower incomplete Gamma function. Many related expressions, in terms of the surface brightness, also exist.

Applications

Massive elliptical galaxies have high Sérsic indices and a high degree of central concentration. This galaxy, M87, has a Sérsic index n~ 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. Setting n = 4 gives the de Vaucouleurs profile: I ( R ) ∝ e − b R 1 / 4 {\displaystyle I(R)\propto e^{-bR^{1/4}}} which is a rough approximation of ordinary elliptical galaxies. Setting n = 1 gives the exponential profile: I ( R ) ∝ e − b R {\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) with galaxy morphology is sometimes used in automated schemes to determine the Hubble type of distant galaxies. Sérsic indices have also been shown to correlate with the mass of the supermassive black hole at the centers of the galaxies.

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

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 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.

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

See also

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.

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