Astronomy: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 [math]\displaystyle{ I }[/math] of a galaxy varies with distance [math]\displaystyle{ R }[/math] 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 [math]\displaystyle{ n }[/math]. The order of [math]\displaystyle{ n }[/math] is reversed for large radii.

Definition

The Sérsic profile has the form [math]\displaystyle{ \ln I(R) = \ln I_{0} - k R^{1/n} , }[/math] or [math]\displaystyle{ I(R) = I_0 \exp{\!\left(-k R^{1/n}\right)}, }[/math]

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

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

[math]\displaystyle{ I(R)=I_e \exp\left\{ -b_n\left[ \left( \frac{R}{R_e}\right) ^{1/n} -1\right] \right\}, }[/math]

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

Applications

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: [math]\displaystyle{ I(R) \propto e^{-bR^{1/4}} }[/math] which is a rough approximation of ordinary elliptical galaxies. Setting n = 1 gives the exponential profile: [math]\displaystyle{ I(R) \propto e^{-bR} }[/math] 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 [math]\displaystyle{ I }[/math] is replaced by [math]\displaystyle{ \rho }[/math], the volume density, and [math]\displaystyle{ R }[/math] is replaced by [math]\displaystyle{ r }[/math], 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 La Plata Argentina 6: 41–43. ISSN 0571-3285. https://ui.adsabs.harvard.edu/abs/1963BAAA....6...41S. 
  2. ↑ Ciotti, L.; Bertin, G. (1999). "Analytical properties of the R^(1/m) luminosity law". Astronomy and Astrophysics 352: 447–451. doi:10.48550/ARXIV.ASTRO-PH/9911078. Bibcode: 1999A&A...352..447C. https://arxiv.org/abs/astro-ph/9911078. 
  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: 118–127. doi:10.1071/AS05001. ISSN 1323-3580. https://ui.adsabs.harvard.edu/abs/2005PASA...22..118G. 
  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: 387–397. doi:10.1093/mnras/stt1027. ISSN 0035-8711. https://ui.adsabs.harvard.edu/abs/2013MNRAS.434..387S. 
  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: 1013–1021. doi:10.1093/mnras/265.4.1013. ISSN 0035-8711. Bibcode: 1993MNRAS.265.1013C. https://ui.adsabs.harvard.edu/abs/1993MNRAS.265.1013C. 
  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. doi:10.1093/mnras/268.1.L11. ISSN 0035-8711. Bibcode: 1994MNRAS.268L..11Y. https://ui.adsabs.harvard.edu/abs/1994MNRAS.268L..11Y. 
  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: 869–876. doi:10.1046/j.1365-8711.2001.04471.x. ISSN 0035-8711. https://ui.adsabs.harvard.edu/abs/2001MNRAS.326..869T. 
  8. ↑ van der Wel, Arjen (2008-07-01). "The morphology-density relation: a constant of nature". 245. pp. 59–62. doi:10.1017/S1743921308017286. https://ui.adsabs.harvard.edu/abs/2008IAUS..245...59V. 
  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: 77–87. doi:10.1086/509758. ISSN 0004-637X. https://ui.adsabs.harvard.edu/abs/2007ApJ...655...77G. 
  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: L85–L88. doi:10.1086/430636. ISSN 0004-637X. https://ui.adsabs.harvard.edu/abs/2005ApJ...624L..85M. 
  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: 2685–2700. doi:10.1086/508988. ISSN 0004-6256. https://ui.adsabs.harvard.edu/abs/2006AJ....132.2685M. 
  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: 2951–2963. doi:10.1086/375320. ISSN 0004-6256. https://ui.adsabs.harvard.edu/abs/2003AJ....125.2951G. 
  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: 1917–1942. doi:10.1086/382712. ISSN 0004-6256. https://ui.adsabs.harvard.edu/abs/2004AJ....127.1917T. 
  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: 197–212. doi:10.1111/j.1365-2966.2005.09269.x. ISSN 0035-8711. https://ui.adsabs.harvard.edu/abs/2005MNRAS.362..197T. 
  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: 2936–2950. doi:10.1086/374992. ISSN 0004-6256. https://ui.adsabs.harvard.edu/abs/2003AJ....125.2936G. 
  16. ↑ Côté, Patrick; Piatek, Slawomir; Ferrarese, Laura; Jordán, Andrés; Merritt, David; Peng, Eric W.; Haşegan, Monica; Blakeslee, John P. et al. (2006-07-01). "The ACS Virgo Cluster Survey. VIII. The Nuclei of Early-Type Galaxies". The Astrophysical Journal Supplement Series 165: 57–94. doi:10.1086/504042. ISSN 0067-0049. https://ui.adsabs.harvard.edu/abs/2006ApJS..165...57C. 

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