Sérsic Profile

The Sérsic profile (also known as the Sérsic law or the R^1/n law) is a three-parameter mathematical function introduced by Argentine astronomer José Luis Sérsic in his 1963 atlas Atlas de Galaxias Australes (and formalized in 1968) to describe the radial variation of sky-projected surface intensity I(R) across galaxies of all morphological types. By replacing the fixed exponent 1/4 of Gérard de Vaucouleurs's 1948 elliptical galaxy profile with a continuous structural curvature parameter—the Sérsic index n—the Sérsic law unifies Gaussian cores (n = 0.5), exponential spiral disks and dwarf spheroidals (n = 1), and classical bulges and giant elliptical galaxies (n = 4 to 10) within a single analytical framework. Whereas the Einasto profile, Hernquist profile, and Navarro–Frenk–White profile model three-dimensional spatial mass density ρ(r), the Sérsic profile serves as the standard two-dimensional photometric parameterization for galaxy morphology decomposition, structural scaling laws such as the Kormendy relation, and weak gravitational lensing shape measurements.

Let R denote the projected radial distance from the photometric center of a galaxy on the plane of the sky (or the semi-major axis length along elliptical isophotes of ellipticity ε = 1 − b/a). The Sérsic intensity profile expresses the surface intensity I(R) (in units of flux per unit solid angle, such as L_☉ pc⁻²) in terms of three independent parameters: the effective radius R_e, the effective intensity I_e ≡ I(R_e), and the Sérsic index n > 0:

I(R) = I_e exp{ −b_n [(R / R_e)^1/n − 1] }

Evaluating I(R) at the center (R = 0) yields the central surface intensity I₀ = I(0) = I_e exp(b_n), allowing the distribution to be written compactly in terms of its scale length h = R_e / b_nⁿ as I(R) = I₀ exp[−(R/h)^1/n]. Taking the logarithmic derivative with respect to R shows that the local logarithmic slope of the projected intensity steepens as a pure power law of radius:

d ln I(R) / d ln R = −(b_n / n) (R / R_e)^1/n

Converting linear intensity I(R) into astronomical surface brightness μ(R) (measured in magnitudes per square arcsecond) via μ(R) = μ_e − 2.5 log₁₀[I(R) / I_e] yields the linear-in-R^1/n magnitude equation:

μ(R) = μ_e + 1.085736 b_n [(R / R_e)^1/n − 1]

where 2.5 / ln(10) ≈ 1.085736 and the central surface brightness is μ₀ = μ_e − 1.085736 b_n. Higher values of the Sérsic index n produce both a steeper inner cusp (a much brighter central peak I₀ / I_e = exp(b_n)) and a more extended outer low-surface-brightness halo wing at R ≫ R_e.

The dimensionless coupling constant b_n is not a free parameter; rather, it is defined implicitly so that the effective radius R_e encloses half of the total integrated luminosity L_tot of the galaxy. Integrating I(R) over circular isophotes up to projected radius R via the substitution x = b_n (R / R_e)^1/n gives the cumulative luminosity L(<R) in terms of the lower incomplete Euler gamma function γ(2n, x) = ∫₀^x t²ⁿ⁻¹ exp(−t) dt:

L(<R) = 2π ∫₀^R I(R′) R′ dR′ = 2π n I_e R_e² [exp(b_n) / b_n²ⁿ] γ(2n, x)

Letting R → ∞ (x → ∞) replaces γ(2n, x) with the complete gamma function Γ(2n) to yield the total luminosity L_tot. Enforcing the half-light condition L(<R_e) = (1/2) L_tot at R = R_e (x = b_n) produces the transcendental defining equation for b_n:

Γ(2n) = 2 γ(2n, b_n)

In 1999, Luca Ciotti and Giuseppe Bertin derived a high-precision asymptotic series expansion for the root b_n valid across all astrophysically relevant indices n > 0.36 (accurate to better than 10⁻⁶ for n ≥ 1):

b_n ≈ 2n − 1/3 + 4 / (405*n*) + 46 / (25515*n*²) + 131 / (1148175*n*³)

For quick analytical estimates when 0.5 ≤ n ≤ 10, Capaccioli (1989) showed that the linear approximation b_n ≈ 1.9992*n* − 0.3271 is accurate to within ~0.1%.

Two integer values of the Sérsic index correspond to historic profiles established before 1963:

• The Freeman Exponential Disk (n = 1): Setting n = 1 yields b₁ = 1.67835 and the pure exponential intensity law I(R) = I₀ exp(−R/R_d), where the disk scale length R_d is related to the half-light radius by R_e = 1.67835 R_d. Established by Kenneth Freeman (1970) for spiral disks formed through angular-momentum-conserving gas collapse and star formation obeying the Kennicutt–Schmidt law, the n = 1 profile also describes low-mass dwarf elliptical galaxies.
• The de Vaucouleurs Profile (n = 4): Setting n = 4 yields b₄ = 7.66925 (so μ(R) = μ_e + 8.3268 [(R/R_e)^1/4 − 1]), reproducing Gérard de Vaucouleurs's 1948 empirical law for bright elliptical galaxies formed via violent relaxation and hierarchical mergers.

In modern two-dimensional surface photometry codes (such as GALFIT and Profiler), galaxies are routinely decomposed into a superposition of an outer n = 1 exponential disk and an inner spheroidal bulge with free Sérsic index n_bulge. Furthermore, Graham & Driver (2005) demonstrated that for early-type galaxies and bulges, the Sérsic index n correlates tightly with both absolute luminosity (n increases from ~1 in dwarf ellipticals to >6 in cD cluster giants) and central supermassive black hole mass M_BH, providing a purely photometric counterpart to the kinematic M–sigma relation.

Because the Sérsic law is defined on the two-dimensional sky plane, recovering the intrinsic three-dimensional stellar luminosity density ν(r) required for the Jeans equations involves solving the Abel integral inversion:

ν(r) = −(1/π) ∫_r^∞ (dI / dR) [dR / √(R² − r²)]

Although this integral lacks an elementary closed form for arbitrary n, Prugniel & Simien (1997) and Lima Neto et al. (1999) proved that the three-dimensional deprojection of a Sérsic profile is accurately represented across all radii by a modified Einasto profile multiplied by a weak inner power-law core factor, ν(r) = ν₀ (r/R_e)^−p_n exp[−b_n (r/R_e)^1/n], where p_n ≈ 1 − 0.6097/n + 0.05463/n² (giving p₄ ≈ 0.851 for a de Vaucouleurs sphere, closely matching the three-dimensional Hernquist profile).

What does a higher Sérsic index n mean physically for a galaxy?

A higher Sérsic index n indicates that the galaxy's light is both more centrally concentrated at small radii (R ≪ R_e) and possesses a more extended, shallow outer stellar envelope at large radii (R ≫ R_e). Disk-dominated galaxies built by quiescent star formation have n ≈ 1, whereas giant elliptical galaxies assembled through repeated dry mergers have n ≈ 4 to 10.

How is the effective radius Re related to the exponential disk scale length Rd?

For a pure exponential disk (n = 1) with intensity I(R) = I₀ exp(−R/R_d), the half-light condition γ(2, b₁) = (1/2) Γ(2) = 1/2 reduces to the equation 1 − (1 + b₁) exp(−b₁) = 1/2, whose numerical root is b₁ = 1.678347. Therefore, the half-light effective radius is always R_e = 1.67835 R_d.

What is the difference between μe and ⟨μ⟩e in Sérsic photometry?

The quantity μ_e is the local surface brightness measured along the isophote at the exact radius R = R_e, whereas ⟨μ⟩_e (used in the Kormendy relation) is the mean surface brightness averaged over the entire circular aperture inside R_e. Because intensity increases steeply inward, ⟨μ⟩_e is always brighter (numerically smaller) than μ_e, by 0.70 mag for n = 1 and by 1.39 mag for n = 4.

1. Sérsic, J. L. (1963). "Influence of the atmospheric and instrumental dispersion on the brightness distribution in a galaxy." Boletín de la Asociación Argentina de Astronomía, 6, 41–43. https://ui.adsabs.harvard.edu/abs/1963BAAA....6...41S
2. de Vaucouleurs, G. (1948). "Recherches sur les nébuleuses extragalactiques." Annales d'Astrophysique, 11, 247–287. https://ui.adsabs.harvard.edu/abs/1948AnAp...11..247D
3. Ciotti, L., & Bertin, G. (1999). "Analytical properties of the R^1/m law." Astronomy & Astrophysics, 352, 447–451. https://ui.adsabs.harvard.edu/abs/1999A%26A...352..447C
4. Graham, A. W., & Driver, S. P. (2005). "A concise reference to (projected) Sérsic R^1/n quantities, including concentration, profile slopes, Petrosian indices, and Kron magnitudes." Publications of the Astronomical Society of Australia, 22(2), 118–127. https://doi.org/10.1071/AS05001
5. Prugniel, P., & Simien, F. (1997). "The fundamental plane of early-type galaxies: non-homology of the spatial structure." Astronomy & Astrophysics, 321, 111–122. https://ui.adsabs.harvard.edu/abs/1997A%26A...321..111P