Sersic Profile

One curve, one knob

Photograph a galaxy and the first thing you measure is how its light fades as you move outward from the core. Plot that surface brightness against radius and you get a falling curve. Remarkably, almost every galaxy ever measured — a wispy dwarf, a grand-design spiral, a billion-star elliptical the size of a small cluster — has a curve that one simple formula can reproduce, and the entire difference between them collapses into a single dial.

That dial is the Sersic index, written n. Turn it down toward 1 and the curve becomes a straight line on a log plot: a flat-cored, fast-fading exponential disk, the signature of a spiral galaxy. Turn it up toward 4 and the centre spikes into a brilliant cusp while the outskirts stretch into a long, slowly dimming halo — the de Vaucouleurs profile of a giant elliptical. The galaxy's overall size and brightness are set by two other numbers, but its shape — the thing your eye reads as "fluffy disk" versus "concentrated blob" — is n alone. That compression of an entire two-dimensional light distribution into one interpretable parameter is why the Sersic profile is the workhorse of galaxy morphology.

The equation and what each symbol does

José Luis Sérsic wrote the profile down in 1963. In its standard form the projected surface brightness (light per unit area on the sky) at projected radius R is

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

Three parameters fully specify the curve, and each one moves it in an independent way:

• R_e — the effective (half-light) radius. The radius of the isophote that encloses exactly half the galaxy's total light. It sets the horizontal scale. By construction the profile passes through I_e at R = R_e.
• I_e — the surface brightness at R_e. A vertical scaling. It sets the absolute brightness without touching the shape.
• n — the Sersic index. The pure shape parameter. It controls how sharply the curve bends — the concentration of light toward the centre.

The fourth symbol, b_n, is not a free parameter: it is a number determined entirely by n, chosen so that the half-light definition of R_e holds. The defining condition is γ(2n, b_n) = ½ Γ(2n), where γ and Γ are the incomplete and complete gamma functions. There is no closed form, but the asymptotic expansion of Ciotti & Bertin (1999) is accurate to better than 0.1% for n ≳ 0.5:

b_n ≈ 2n − 1/3 + 4/(405 n) + 46/(25515 n²) + …

This yields b₁ ≈ 1.678 (exponential disk), b₄ ≈ 7.669 (de Vaucouleurs), b₀.₅ ≈ 0.693 (Gaussian). The exponential and de Vaucouleurs "laws" that predate Sérsic are simply this one formula evaluated at n = 1 and n = 4.

The special cases hiding inside

What made Sérsic's generalisation so powerful is that the empirical laws astronomers had already discovered turn out to be specific values of n. Setting 1/n to particular numbers recovers each one:

n	Name	I(R) reduces to	Typical hosts
0.5	Gaussian	I_e exp[−b(R/R_e)²]	Diffraction-limited point sources, some star clusters
1	Exponential	I₀ exp(−R/h), with h = R_e / b₁	Spiral & dwarf-irregular disks
2	—	I_e exp[−b₂((R/R_e)^½ − 1)]	Pseudobulges, low-mass spheroids
4	de Vaucouleurs (R^¼)	I_e exp[−7.669((R/R_e)^¼ − 1)]	Giant ellipticals, classical bulges
6–10	—	Extremely cuspy core, very extended wings	Brightest cluster galaxies, cD halos

The de Vaucouleurs case is worth dwelling on, because it gives the law its older name. In astronomical magnitudes (μ = −2.5 log₁₀ I + const), the n = 4 profile becomes

μ(R) = μ_e + 8.327 [ (R / R_e)^(1/4) − 1 ]   (mag/arcsec²)

which is a straight line when you plot μ against R^(1/4) — hence "the R^(1/4) law." For a general index the same magnitude form is μ(R) = μ_e + 1.0857 b_n [(R/R_e)^(1/n) − 1], straight against R^(1/n).

How n reshapes the light

The single most useful intuition is that n controls concentration. Two galaxies with identical R_e and identical total luminosity can look completely different depending on n. The high-n profile pulls light simultaneously inward (a brighter, sharper central cusp) and outward (a fainter but far more extended wing), at the expense of the light at intermediate radii. A clean way to quantify this is the fraction of light inside one R_e contributed by the very centre, or the ratio R₂₀/R₈₀ of the radii enclosing 20% and 80% of the light:

Sersic index n	I(0) / I_e (central cusp)	Light within 0.1 R_e	Concentration C = 5 log(R₈₀/R₂₀)
0.5 (Gaussian)	~2	~0.7%	~2.2
1 (exponential)	~5.4	~1.3%	~2.8
2	~40	~3%	~3.8
4 (de Vaucouleurs)	~2,140	~7%	~5.3
8	~6 × 10⁶	~14%	~7.4

The central-cusp column makes the point dramatically: going from n = 1 to n = 4 multiplies the predicted central surface brightness, relative to I_e, by roughly 400×. This is why high-n galaxies look like they have a blazing point at the core — and why fitting them demands careful deconvolution of the telescope's point-spread function, which smears that cusp.

Total light, and why it stays finite

Integrating the profile over the whole sky gives a clean closed form. The total luminosity is

L_total = 2π n I_e R_e² · e^(b_n) · Γ(2n) / b_n^(2n)

where Γ is the complete gamma function. The mean surface brightness inside R_e is ⟨I⟩_e = L_total / (2π R_e²), and the ratio ⟨I⟩_e / I_e = n · e^(b_n) Γ(2n) / b_n^(2n) grows from about 1.9 at n = 1 to about 3.6 at n = 4. Although the high-n wings fade slowly, the integral converges for every n > 0, so the model never predicts infinite light — a frequent misconception.

For a worked feel: an exponential disk (n = 1, b₁ = 1.678) has total light L = 2π I_e R_e² e^(1.678) Γ(2) / 1.678² ≈ 2π I_e R_e² × 1.901. Re-expressing R_e = 1.678 h in terms of the disk scale length h recovers the textbook result L = 2π I₀ h², a useful sanity check that the Sersic machinery collapses to the familiar disk formula at n = 1.

How a profile is measured from an image

Real galaxies are not measured by hand-fitting a one-dimensional curve. Modern practice fits the full two-dimensional image, because galaxies are elliptical on the sky and overlap with neighbours and foreground stars. The standard tools are GALFIT (Peng, Ho, Impey & Rix 2002), along with GIM2D, IMFIT, ProFit, and the python package statmorph. The model image is built from the Sersic formula with free parameters — centre (x, y), total magnitude, R_e, n, axis ratio b/a, and position angle — then convolved with the point-spread function of the telescope so that the blurred model can be compared like-for-like with the data. A Levenberg–Marquardt or MCMC optimiser minimises the pixel-by-pixel residual.

Crucially, a single Sersic component rarely fits a real spiral. The standard move is bulge–disk decomposition: fit two superposed components, an n ≈ 4 (or free-n) bulge plus an n = 1 disk, and let the code apportion the light. The bulge-to-total ratio B/T that falls out is one of the most physically meaningful numbers in extragalactic astronomy — it tracks where a galaxy sits on the Hubble sequence and how much of its stellar mass formed in a dissipative spheroid versus a quiescent disk.

Why n tracks formation history

The Sersic index is more than a curve-fitting convenience; it correlates with the physics of how galaxies were built. The empirical trend is unambiguous: more massive, more dispersion-supported, more dynamically relaxed systems have higher n.

• n ≈ 1 — disks. Late-type spirals (Sc, Sd) and dwarf irregulars are rotationally supported disks assembled by relatively smooth, ongoing gas accretion and in-situ star formation. They stay exponential.
• n ≈ 2–3 — composite systems. Earlier-type spirals (Sa, Sb) with substantial bulges, and lower-mass spheroidals, sit in between. Pseudobulges, grown secularly by bar-driven gas inflow rather than mergers, typically have n < 2 and are a key diagnostic distinguishing them from classical bulges.
• n ≈ 4 — classical ellipticals and bulges. Violent relaxation in dissipationless ("dry") mergers scatters stars into a centrally concentrated profile with extended wings — exactly the de Vaucouleurs shape. The R^(1/4) law is, in effect, the fingerprint of merger-driven assembly.
• n ≈ 6–10 — brightest cluster galaxies. The most massive cD galaxies at cluster centres, swollen by repeated minor mergers and by tidally stripped stars from cluster members, push n to the highest observed values and develop faint extended halos.

This is why a measured n is a first-cut classifier. A simple cut at n ≈ 2.5 separates disk-dominated (late-type) from spheroid-dominated (early-type) galaxies with surprising fidelity across the local Universe.

Famous examples and survey-scale figures

• M87 (NGC 4486). The giant elliptical at the heart of the Virgo Cluster, ~16.4 Mpc away. Its light is fit by n ≈ 4–6 over a half-light radius of roughly R_e ≈ 100–700 arcsec (≈ 8–40 kpc depending on the depth and whether the cD halo is included) — a textbook high-n profile with an enormous extended envelope.
• The Milky Way and M31 disks. Galactic disks follow n = 1 closely. The Milky Way's disk scale length is h ≈ 2.5–3 kpc; M31's is h ≈ 5–6 kpc. Their classical bulges, by contrast, are fit with n ≈ 2–4.
• Local dwarf spheroidals. Faint companions like Fornax and Sculptor are often fit with n < 1 — profiles even flatter-cored than an exponential, reflecting their pressure-supported but diffuse, weakly bound structure.
• SDSS and Galaxy Zoo at scale. Single-Sersic fits to roughly a million SDSS galaxies (e.g. Simard et al. 2011) show a clear bimodal distribution in n: a peak near n ≈ 1 (blue, star-forming disks) and a peak near n ≈ 4 (red, quiescent spheroids), one of the cleanest demonstrations that morphology and star-formation state are coupled.
• High-redshift compact galaxies. JWST and Hubble fits to z ≈ 2 quiescent "red nuggets" find de Vaucouleurs-like n ≈ 4 profiles but with R_e several times smaller than equally massive ellipticals today — evidence that early spheroids grew inside-out by accreting an extended high-n envelope over cosmic time.

Common misconceptions and edge cases

• "Higher n means a bigger galaxy." No. n is dimensionless and orthogonal to size. Two galaxies can share the same R_e and L while one (high n) has a sharp core and the other (low n) is flat-topped. Size is R_e; brightness is I_e; n is shape only.
• "The profile has infinite light." The integral converges for all n > 0 (see the Γ-function formula above). What can mislead is extrapolation: a high-n fit's slowly fading wing may over-predict faint outer light that, in a real galaxy, is truncated by tides or sky noise.
• "You can read n straight off the raw image." The central cusp of a high-n profile is severely blurred by the point-spread function. Without PSF convolution in the fit, the central concentration — and therefore n — is systematically underestimated. This is the single most common fitting error.
• "R_e is where the galaxy ends." R_e is the half-light radius, not an edge. Half the light still lies beyond it. For a de Vaucouleurs profile a substantial fraction of the total light sits outside several R_e in the extended wing.
• "A single Sersic always works." Composite galaxies (bulge + disk, plus bars, rings, and AGN point sources) need multi-component fits. Forcing a single Sersic onto a two-component galaxy biases n to an intermediate, physically meaningless value and inflates the residuals.
• "de Vaucouleurs and Sersic are different laws." They are the same equation. de Vaucouleurs (1948) is exactly the n = 4 case of Sérsic (1963); the exponential disk is the n = 1 case. Sérsic's contribution was to free the exponent.