Elliptical Galaxies and the de Vaucouleurs Profile

An elliptical galaxy is what nature gives when a stellar system settles into equilibrium with almost no net spin. Where a spiral disk is thin, cold, and rotationally supported — its stars sweeping around on nearly circular orbits — an elliptical is a hot, three-dimensional swarm whose stars plunge through the center on randomly oriented orbits. Support against self-gravity comes not from rotation but from the pressure of that random motion: the velocity dispersion σ plays the role a sound speed plays in a gas. For most bright ellipticals the ratio of ordered to random speed, v/σ, sits well below unity — they are "slow rotators," and their flattening (the E0–E7 sequence) is a signature of anisotropic velocity structure and triaxial shape, not of centrifugal bulging. The observational miracle is how regular these objects look. In 1948 Gérard de Vaucouleurs found that the projected surface brightness of giant ellipticals falls off as log I ∝ R^(1/4), straight as a ruler over many magnitudes. Sérsic later embedded this in a one-parameter family, I ∝ exp[−(R/R_e)^(1/n)], with the de Vaucouleurs law recovered at n = 4 and an exponential disk at n = 1. A single scale — the effective radius R_e enclosing half the light — plus the surface brightness I_e there fixes the whole profile, once the constant b_4 ≈ 7.669 is pinned by the half-light condition. These smooth, "red and dead" systems then obey astonishingly tight scaling laws — Faber–Jackson (L ∝ σ⁴) and its unification into the Fundamental Plane — that turn ellipticals into standard-ish distance indicators and fossil records of galaxy assembly.

Key takeaways

Elliptical galaxies are smooth, ellipsoidal stellar systems held up against gravity by random stellar motion (velocity dispersion σ), not by ordered rotation — dynamically the opposite of a spiral disk.
Their projected surface brightness follows the de Vaucouleurs r^(1/4) law, I(R) = I_e·exp{−7.669[(R/R_e)^(1/4) − 1]}, the n = 4 member of the Sérsic r^(1/n) family (exponential disk is n = 1).
The effective radius R_e encloses exactly half the total light; the constant b_4 ≈ 7.669 is fixed by that half-light condition through an incomplete-gamma equation.
Ellipticals obey tight scaling relations — Faber–Jackson L ∝ σ⁴ and the Fundamental Plane R_e ∝ σ^1.24·⟨I⟩_e^(−0.82) — that tie together size, luminosity, dispersion, and surface brightness.
The "tilt" of the Fundamental Plane away from the naive virial prediction (2, −1) toward (1.24, −0.82) means the mass-to-light ratio Υ rises with mass, tracking dark matter and stellar-population trends.
The Sérsic index tracks luminosity and formation history: giant slow-rotator ellipticals have n ≳ 4 with scoured cores, while faint dwarf ellipticals have n ≈ 1–2.
Key idea: An elliptical galaxy is a collisionless stellar system held up by velocity dispersion, whose projected light follows the de Vaucouleurs r^(1/4) law — the n = 4 Sérsic profile — with R_e and σ obeying the Faber–Jackson and Fundamental Plane scaling relations.

The core insight

An elliptical galaxy is a collisionless stellar system held up by velocity dispersion, whose projected surface brightness follows the de Vaucouleurs r^(1/4) law — the n = 4 member of the Sérsic family — with the half-light radius R_e and dispersion σ obeying the Faber–Jackson and Fundamental Plane scaling relations.

Where this lives today

Ellipticals remain central to how astronomers weigh galaxies, measure distances, and read the fossil record of galaxy assembly. The Fundamental Plane and its cousin the D_n–σ relation are still working distance indicators: because σ₀ and ⟨μ⟩_e are redshift-independent, they predict the physical R_e, and comparing to the angular size yields a distance — used to map the peculiar-velocity field of galaxies streaming toward the Great Attractor. Integral-field spectrographs such as ATLAS³D, MaNGA, and SAMI resolved the two-dimensional velocity fields of hundreds of ellipticals and sharpened the slow-rotator / fast-rotator dichotomy the old E0–E7 sequence had blurred, confirming that most flattening is anisotropy, not spin. Hubble and JWST photometry, fit with core-Sérsic profiles, reveal the scoured cores of the brightest cluster galaxies — kpc-scale evidence of binary supermassive black holes. Strong lensing by massive ellipticals (the SLACS survey) confirms the near-isothermal total mass profiles and the rising dark-matter fraction the Fundamental Plane tilt first hinted at. And how compact "red nuggets" at redshift 2 grow into today's extended ellipticals through dry minor mergers — building the r^(1/4) envelope while barely changing the stellar mass — is an active frontier of galaxy-formation modeling.

The physical picture

Picture a few hundred billion stars orbiting a common center of mass on randomly oriented, mostly plunging orbits. No preferred rotation axis emerges, so the system relaxes into a roughly ellipsoidal blob held up by the "pressure" of stellar motion — like a self-gravitating gas cloud in hydrostatic balance, except that stars essentially never collide. The velocity dispersion σ is the effective sound speed, and the shape of the ellipsoid is set by how that dispersion is distributed among the three axes (the anisotropy), not by spin. The first figure contrasts the two ways a stellar system resists gravity.

Figure

Two ways to hold up a galaxy: an elliptical is pressure-supported by random stellar motion (v/σ ≈ 0.1, a slow rotator whose flattening comes from anisotropy), while a spiral disk is rotation-supported (v/σ greater than 1, flattened by spin)

The second thing an elliptical does — the reason it has its own article — is obey the de Vaucouleurs profile. The light is fiercely concentrated at the center yet trails into a stubbornly extended faint envelope, and de Vaucouleurs' trick was to find the axes on which that curve becomes a straight line. On a plot of log I against R^(1/4), a giant elliptical falls on a ruler-straight line of slope b_4/ln10 ≈ 3.33; a spiral disk instead straightens on log I against R (an exponential, n = 1). That "straightening" is the fingerprint of n = 4, and the second figure shows why the exponential refuses to lie flat on these axes.

Figure

The de Vaucouleurs straightening: on axes of log₁₀(I/Iₑ) versus (R/Rₑ)^¼ the n = 4 profile is a perfectly straight line of slope b₄/ln10 = 3.33 (b₄ = 7.669), while an n = 1 exponential disk curves away — both pass through the effective radius Rₑ where I = Iₑ

The single number R_e carries a clean meaning: the radius of the circle that captures half the total light. Fix R_e, I_e, and n and everything else follows. And because ellipticals live on a thin Fundamental Plane, knowing any two of {R_e, σ, I_e} predicts the third — they are very nearly a one-parameter family wearing a three-parameter costume.

The core equations

The whole photometric description rests on one profile with one shape parameter — the Sérsic law, generalizing both the de Vaucouleurs r^(1/4) form and the exponential disk.

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

where: $I(R)$ — projected surface brightness at projected radius $R$ ; $I_e$ — surface brightness at the effective radius; $R_e$ — effective (half-light) radius; $n$ — Sérsic index (shape), $n=4$ de Vaucouleurs and $n=1$ exponential; $b_n$ — constant fixed by the half-light condition, $b_4\approx7.669$ . Reading: $I=I_e$ exactly at $R=R_e$ , and larger $n$ means a more concentrated core with a more extended faint wing. Setting $n=4$ gives the historical de Vaucouleurs law, the profile that fits giant ellipticals and classical bulges.

I(R)=I_e\,\exp\!\left\{-7.669\left[\left(\frac{R}{R_e}\right)^{1/4}-1\right]\right\} \qquad (2)

where: $7.669$ — the value of $b_4$ (sometimes rounded to $7.67$ ), fixed by requiring $R_e$ to enclose half the light. Reading: take the logarithm and this is a straight line in $R^{1/4}$ with slope $b_4/\ln10\approx3.33$ per unit — the "straightening" of the second figure. Integrating the Sérsic profile over the sky gives a closed form for the total luminosity in terms of the gamma function.

L_{\rm tot}=2\pi\,I_e\,R_e^{2}\,e^{b_n}\,\frac{n\,\Gamma(2n)}{b_n^{\,2n}} \qquad (3)

where: $L_{\rm tot}$ — luminosity integrated to infinity; $\Gamma(2n)$ — complete gamma function. Reading: the luminosity scales as $I_e R_e^2$ times a pure number depending only on $n$ . A quantity observers actually report is the mean surface brightness inside $R_e$ , because it appears directly in the Fundamental Plane.

\langle I\rangle_e = \frac{L_{\rm tot}/2}{\pi R_e^2}=f(n)\,I_e,\qquad f(4)=e^{b_4}\frac{4\,\Gamma(8)}{b_4^{8}}\approx3.61 \qquad (4)

where: $\langle I\rangle_e$ — mean surface brightness interior to $R_e$ ; $f(n)$ — the ratio $\langle I\rangle_e/I_e$ . Reading: for a de Vaucouleurs galaxy the average brightness inside $R_e$ is about $3.6$ times the brightness at $R_e$ , because so much light is piled into the core. The classical Hubble label En quantifies only the projected flattening, not any dynamical fact.

\mathrm{E}n:\quad n = 10\left(1-\frac{b}{a}\right) \qquad (5)

where: $a,b$ — projected semi-major and semi-minor axes; $n$ — integer $0$ – $7$ (E0 round, E7 the most flattened observed). Reading: a shape-on-the-sky classifier that says nothing about whether the flattening is caused by rotation. The tightest one-parameter scaling relation is Faber–Jackson, linking luminosity to central dispersion.

L \propto \sigma_0^{4} \qquad (6)

where: $L$ — total luminosity; $\sigma_0$ — central line-of-sight velocity dispersion. Reading: dispersion is a distance-independent spectroscopic observable, so measuring $\sigma_0$ estimates $L$ — the basis of using ellipticals as standard candles. Faber–Jackson is the edge-on view of a two-dimensional relation, the Fundamental Plane, in the space of size, dispersion, and surface brightness.

\log R_e = a\,\log\sigma_0 + b\,\langle\mu\rangle_e + c,\qquad R_e \propto \sigma_0^{1.24}\,\langle I\rangle_e^{-0.82} \qquad (7)

where: $\langle\mu\rangle_e$ — mean surface brightness within $R_e$ in magnitudes; $\langle I\rangle_e$ — the same in linear units; $a\approx1.24,\ b\approx0.33$ — empirical, band-dependent coefficients. Reading: the scatter about this plane is only $\sim20\%$ in $R_e$ , tight enough to serve as a distance indicator. Dispersion and size together give a dynamical mass through the virial theorem, up to a structure constant.

M \simeq \frac{\alpha\,\sigma_0^2\,R_e}{G},\qquad \alpha \sim 5 \qquad (8)

where: $M$ — dynamical mass within a few $R_e$ ; $\alpha$ — structure factor ( $\approx5$ for de Vaucouleurs); $G$ — Newton's constant. Reading: the estimator that, compared to stellar mass, reveals dark matter. Finally, converting the observed 2-D brightness into a 3-D density requires the Abel inversion, valid for a spherical system.

\rho(r)=-\frac{1}{\pi}\int_r^\infty \frac{\mathrm{d}I}{\mathrm{d}R}\,\frac{\mathrm{d}R}{\sqrt{R^2-r^2}} \qquad (9)

where: $\rho(r)$ — 3-D luminosity density (or, times $\Upsilon$ , mass density) at radius $r$ ; $I(R)$ — projected surface brightness. Reading: this deprojection feeds the Jeans equation that turns dispersion profiles into mass profiles.

Derivation AFrom r to the fourth-root light to total luminosity and the constant b_n

The goal is to show precisely where the number 7.669 comes from, and in the same stroke to obtain the total-luminosity formula (3) and the mean-brightness ratio (4). We integrate the Sérsic profile over the projected plane, convert it into a gamma function by one substitution, impose the definition of the effective radius as the half-light radius, and solve the resulting incomplete-gamma equation for n = 4. Every constant in the photometric toolkit falls out of this one calculation.

Step 1. Write the light enclosed within projected radius R as the surface-brightness integral over a face-on disk, $L(<R)=\int_0^{R} I_e\exp\{-b_n[(R'/R_e)^{1/n}-1]\}\,2\pi R'\,\mathrm{d}R'$ . Why: integrating surface brightness over the sky-plane annuli recovers the enclosed luminosity, whose half-value defines R_e.

Step 2. Substitute $x\equiv b_n(R'/R_e)^{1/n}$ , so that $R'=R_e(x/b_n)^n$ and $\mathrm{d}R'=(nR_e/b_n)(x/b_n)^{n-1}\mathrm{d}x$ . The integrand collapses to $x^{2n-1}e^{-x}$ up to constants. Why: this change of variables turns the stretched-exponential profile into the integrand of the gamma function — why the Sérsic law has closed-form integrals.

Step 3. Collecting constants gives $L(<R)=2\pi n\,I_e\,e^{b_n}R_e^2 b_n^{-2n}\,\gamma\!\left(2n,\,b_n(R/R_e)^{1/n}\right)$ , where $\gamma(s,x)=\int_0^x t^{s-1}e^{-t}\mathrm{d}t$ is the lower incomplete gamma function. Why: the enclosed light at any radius is now one special function evaluated at the scaled radius.

Step 4. Let $R\to\infty$ : the incomplete gamma tends to the complete gamma, giving $L_{\rm tot}=2\pi n\,I_e\,e^{b_n}R_e^2 b_n^{-2n}\,\Gamma(2n)$ , which is equation (3). Why: the total luminosity is finite even though the profile extends to infinity, because the stretched exponential kills the outer wing fast enough.

Step 5. Impose the definition of R_e as the half-light radius: $L(<R_e)=\tfrac12 L_{\rm tot}$ . Since $R=R_e$ maps to $x=b_n$ , this reads $\gamma(2n,b_n)=\tfrac12\Gamma(2n)$ — an implicit equation for $b_n$ alone. Why: R_e is a definition, not a fitted constant; requiring it to bisect the light is what fixes b_n once n is chosen.

Step 6. For n = 4, $\Gamma(8)=7!=5040$ , so the condition is $\gamma(8,b_4)=2520$ ; solving numerically gives $b_4=7.6692$ , matching the Ciotti–Bertin asymptotic $b_n\approx 2n-\tfrac13+\tfrac{4}{405n}=7.6692$ . Feeding $b_4$ back into (4) gives $\langle I\rangle_e/I_e=3.607$ . Why: this pins the two anchor numbers of the subject — 7.669 and 3.61 — from first principles rather than a table. *The anchor numbers fall out cleanly: $b_4 = 7.6692$ from $\gamma(8,b_4)=\tfrac{1}{2}\Gamma(8)=2520$ ; $f(4)=3.607$ ; and $b_1=1.6783$ for the exponential disk ( $n=1$ ).*

Go deeper · Derivation B · 2 minFaber–Jackson from the virial theorem and the tilt of the Fundamental Plane

Derivation BFaber–Jackson from the virial theorem and the tilt of the Fundamental Plane

The goal is to show that a single physical statement — a self-gravitating system in virial equilibrium with roughly constant mass-to-light ratio and surface brightness — predicts the Faber–Jackson relation L ∝ σ⁴, and then that the observed Fundamental Plane is exactly this virial relation "tilted" by a mass dependence of Υ. This connects a measured empirical law to the deepest structural fact about ellipticals: they are near-homologous equilibrium systems whose only systematic variation is in how much dark matter they carry.

Step 1. Write the scalar virial theorem $2K+U=0$ with $K=\tfrac12 M\langle v^2\rangle$ and $U=-\alpha GM^2/R_e$ , and use $\langle v^2\rangle\propto\sigma_0^2$ for a homologous velocity field. This gives $\sigma_0^2=\alpha GM/R_e$ , the mass estimator (8). Why: for a system that has phase-mixed into equilibrium, the virial theorem is the exact global constraint linking dispersion to mass and size.

Step 2. Introduce the two photometric bridges $M=\Upsilon L$ and $L=k\,\langle I\rangle_e R_e^2$ , where $\Upsilon\equiv M/L$ and $k$ absorbs the profile shape. Why: dynamics knows only mass, observations deliver light; these relations translate between them and introduce Υ, which carries the physics of the tilt.

Step 3. Eliminate M and R_e. From $L=k\langle I\rangle_e R_e^2$ we get $R_e=(L/k\langle I\rangle_e)^{1/2}$ ; substituting into $\sigma_0^2=\alpha G\Upsilon L/R_e$ yields $\sigma_0^2=\alpha G\Upsilon\,k^{1/2}\langle I\rangle_e^{1/2}L^{1/2}$ . Why: this collapses three structural quantities into one relation between the two most easily measured — dispersion and luminosity.

Step 4. Hold Υ and ⟨I⟩_e constant across the population. Then $\sigma_0^2\propto L^{1/2}$ , i.e. $L\propto\sigma_0^4$ — Faber–Jackson, with no free parameters. Why: the observed slope near 4 is therefore direct evidence that ellipticals are a near-homologous virial family with only mild variation in Υ.

Step 5. Now allow Υ to vary. Solving the same relations for R_e gives $R_e\propto\sigma_0^2\,\langle I\rangle_e^{-1}\,\Upsilon^{-1}$ , so the virial Fundamental Plane has exponents $(2,-1)$ . The observed plane has $(1.24,-0.82)$ . Why: the mismatch is not error — it is a systematic "tilt," and matching the two requires Υ to grow with mass roughly as $\Upsilon\propto M^{0.2}$ .

Step 6. Interpret the tilt: $\Upsilon\propto M^{0.2}$ means more massive ellipticals have progressively more mass per unit light — a rising dark-matter fraction within R_e and/or a more bottom-heavy stellar initial mass function. Why: the Fundamental Plane thus becomes a probe of the dark and stellar mass budget versus galaxy mass. *The virial exponents $(2,-1)$ versus the observed $(1.24,-0.82)$ quantify the Fundamental Plane tilt, and the Faber–Jackson slope near 4 follows directly from holding $\Upsilon$ and $\langle I\rangle_e$ constant.*

Go deeper · Derivation C · 1 minThe virial mass estimator and why the structure constant is of order five

Derivation CThe virial mass estimator and why the structure constant is of order five

The goal is to justify the workhorse dynamical mass formula M ≈ α σ₀² R_e / G and to show why the dimensionless factor α comes out near 5 for a realistic elliptical rather than being an adjustable fudge. This matters because the estimator is the most-used tool for weighing ellipticals, and its reliability rests entirely on how α depends on the density profile and the velocity anisotropy.

Step 1. Start from the spherical, isotropic Jeans equation $\frac{\mathrm{d}(\nu\sigma_r^2)}{\mathrm{d}r}=-\nu\frac{GM(<r)}{r^2}$ , relating the tracer density ν and radial dispersion σ_r to the enclosed mass. Why: the Jeans equation is the first velocity moment of the collisionless Boltzmann equation — the analogue of hydrostatic equilibrium for a pressure-supported system.

Step 2. Integrate over the whole galaxy to recover the global virial statement $M\langle\sigma^2\rangle=\alpha' GM^2/R_e$ , so $M=\langle\sigma^2\rangle R_e/(\alpha' G)$ . Why: the global moment removes any need to know the density at every radius and packages the structure into one number.

Step 3. The observed quantity is not the 3-D $\langle\sigma^2\rangle$ but the central line-of-sight σ₀; converting between them and folding in the de Vaucouleurs shape and mild anisotropy absorbs everything into one constant, $M\simeq\alpha\,\sigma_0^2 R_e/G$ . Why: observers measure a projected, aperture-averaged dispersion, so the estimator lumps projection and profile corrections into α.

Step 4. Calibrating α for a de Vaucouleurs profile in a realistic potential against detailed dynamical models yields $\alpha\approx5$ — close to the uniform-sphere value and stable at the tens-of-percent level. Why: this near-constancy across profiles is what makes the estimator trustworthy, whether the galaxy is cored or cuspy. *As a check, the estimator $M\approx 5\sigma_0^2 R_e/G$ with $\sigma_0=250\,\text{km/s}$ and $R_e=5\,\text{kpc}$ yields $3.6\times10^{11}\,M_\odot$ — the correct scale for a giant elliptical — and detailed dynamical models calibrate $\alpha\approx5.0$ .*

Limiting cases

The Sérsic index n is a dial that interpolates between the two great families of galaxy light profiles and pins down the extremes:

Case	Result	Meaning
de Vaucouleurs (n = 4)	I(R) = I_e·exp[−7.669((R/R_e)^(1/4) − 1)]	Giant ellipticals and classical bulges: steep core, extended faint envelope.
Exponential disk (n = 1)	I(R) = I_0·e^(−R/h), with b_1 ≈ 1.678 and R_e ≈ 1.678 h	Spiral disks and many dwarf ellipticals; straightens on log I versus R, not R^(1/4).
Gaussian-like (n = 1/2)	I(R) ∝ exp[−b_(1/2)(R/R_e)²]	Very flat-cored, sharply truncated; rare, but bounds the family from below.
Giant / cD envelope (n ≳ 6)	I(R) ∝ exp[−b_n(R/R_e)^(1/n)]	Brightest cluster galaxies: enormous low-surface-brightness halos, very high n.
High mass, cored center	core-Sérsic: outer Sérsic with a central deficit	Slow rotators scoured by a binary supermassive black hole; a flat core replaces the cusp.

Worked examples

1 · Dynamical mass and dark matter of a giant elliptical (fully worked)

A giant elliptical has an effective radius R_e = 5 kpc and a central velocity dispersion σ₀ = 250 km/s. Estimate its dynamical mass within a few R_e, and — given a stellar luminosity L ≈ 5×10¹⁰ L_⊙ and a stellar population with M/L ≈ 5 (solar units) — its dark-matter fraction.

Step 1. Convert to SI. σ₀ = 250 km/s = 2.50×10⁵ m/s, so σ₀² = 6.25×10¹⁰ m²/s². R_e = 5 kpc = 5 × 3.086×10¹⁹ m = 1.543×10²⁰ m. G = 6.674×10⁻¹¹ m³ kg⁻¹ s⁻².

Step 2. Apply the virial estimator (8) with α = 5: M ≈ 5 σ₀² R_e / G = 5 × (6.25×10¹⁰) × (1.543×10²⁰) / (6.674×10⁻¹¹) = 7.22×10⁴¹ kg. Dividing by M_⊙ = 1.989×10³⁰ kg gives M ≈ 3.6×10¹¹ M_⊙ — a textbook giant-elliptical mass.

Step 3. Stellar mass: M_⋆ ≈ (M/L) × L = 5 × 5×10¹⁰ = 2.5×10¹¹ M_⊙.

Step 4. The dark (plus any IMF-excess) fraction is f_dark = 1 − M_⋆/M ≈ 1 − 2.5/3.6 = 0.31. Bottom line: M_dyn ≈ 3.6×10¹¹ M_⊙, ~30% dark within R_e — consistent with the rising Υ that tilts the Fundamental Plane.

2 · Reading the de Vaucouleurs profile (numeric)

For a de Vaucouleurs galaxy, compare the surface brightness at the center, at R_e/16, and at 16 R_e to the value I_e, and give the drop in magnitudes per unit (R/R_e)^(1/4).

Step 1. Central brightness: at R = 0, (R/R_e)^(1/4) = 0, so I(0)/I_e = e^7.669 = 2141 — the pure law makes the center over 2000× brighter than the effective isophote (real cores flatten).

Step 2. At R = R_e/16, (R/R_e)^(1/4) = 1/2, so I/I_e = e^3.835 = 46.3. At R = 16 R_e, (R/R_e)^(1/4) = 2, so I/I_e = e^(−7.669) = 4.7×10⁻⁴ — the faint envelope.

Step 3. In logs, log₁₀ I = log₁₀ I_e + (b_4/ln10)(1 − (R/R_e)^(1/4)), so the slope is −b_4/ln10 = −3.33 per unit. In magnitudes the surface brightness fades by 2.5 × 3.33 = 8.33 mag per unit (R/R_e)^(1/4) — the constant slope that makes the r^(1/4) plot a straight line.

3 · Faber–Jackson as a distance-free luminosity ratio (numeric)

Two ellipticals have central dispersions σ_A = 300 km/s and σ_B = 150 km/s. Without knowing either distance, what is the ratio of their luminosities?

Step 1. Faber–Jackson gives L ∝ σ⁴, so L_A/L_B = (σ_A/σ_B)⁴ = (300/150)⁴ = 2⁴ = 16: a factor-2 spread in σ implies a factor-16 spread in L. The steep fourth power, plus the distance-independence of σ, is what lets the Fundamental Plane serve as a rung on the extragalactic distance ladder.

Physics of the portrait

The de Vaucouleurs profile is one of those rare empirical laws that arrived decades before anyone could derive it, and it carries the name of the astronomer who first straightened the light of an elliptical.

Origins · 1 minGérard de Vaucouleurs and how this was discovered

Gérard de Vaucouleurs

1918–1995 · extragalactic astronomy and the r to the fourth-root law

Gérard de Vaucouleurs was a French-American astronomer whose life's work was the quantitative cartography of galaxies. In 1948, in Paris, he published the empirical law that bears his name: the surface brightness of an elliptical falls off as log I ∝ R^(1/4), a fit so good over so many magnitudes that it turned "elliptical galaxy" from a morphological impression into a measurable profile with a single scale radius. He never claimed a physical derivation — the r^(1/4) law was, and largely remains, a phenomenological triumph — but it became the backbone of galaxy photometry, later generalized by Sérsic to the r^(1/n) family. At the University of Texas at Austin, de Vaucouleurs and his wife and collaborator Antoinette compiled the Reference Catalogues of Bright Galaxies (RC1, RC2, RC3) that underpin extragalactic statistics to this day, and he refined the Hubble classification into the finer "revised Hubble–de Vaucouleurs" system. He was an early champion of the Local Supercluster, and stood on the high side (near 100 km/s/Mpc) of the twentieth-century controversy over the Hubble constant against Allan Sandage. His enduring lesson is methodological: measure galaxies exhaustively, and let the regularities reveal the physics.

History

1948

— de Vaucouleurs finds the r to the fourth-root surface-brightness law for elliptical galaxies

1963

— Sérsic generalizes the profile to the r to the one-over-n family with a free shape index

1976

— Faber and Jackson discover L ∝ σ⁴ for elliptical galaxies

1987

— Djorgovski and Davis, and Dressler and collaborators, establish the Fundamental Plane

1999

— Ciotti and Bertin derive the asymptotic expansion b_n ≈ 2n − 1/3 + 4/(405n)

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 25 · Binney & Tremaine, Galactic Dynamics, 2nd ed., Ch. 1 & 4 (photometry and equilibria of collisionless systems) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 11–13 · Ryden, Introduction to Cosmology, 2nd ed. · Dodelson, Modern Cosmology.

Primary sources. de Vaucouleurs, G. (1948), Ann. d'Astrophysique 11, 247 · Sérsic, J. L. (1963), Bol. Asoc. Argentina Astron. 6, 41 · Faber & Jackson (1976), ApJ 204, 668 · Djorgovski & Davis (1987) and Dressler et al. (1987), ApJ 313 (Fundamental Plane) · Ciotti & Bertin (1999), A&A 352, 447 (b_n asymptotics) · Cappellari et al. (2006), MNRAS 366, 1126 (α ≈ 5). Key numerical anchors: $b_4 = 7.6692$ from $\gamma(8,b_4)=\tfrac{1}{2}\Gamma(8)=2520$ , consistent with the Ciotti–Bertin asymptotic $b_n\approx 2n-\tfrac{1}{3}+\tfrac{4}{405n}$ ; $f(4)=\langle I\rangle_e/I_e=3.607$ ; the Faber–Jackson slope $L\propto\sigma^4$ follows from the virial theorem with constant $\Upsilon$ and $\langle I\rangle_e$ ; the observed FP exponents $(1.24,-0.82)$ versus the virial prediction $(2,-1)$ ; and the sample virial mass $\approx3.6\times10^{11}\,M_\odot$ for $\sigma_0=250\,\text{km/s}$ , $R_e=5\,\text{kpc}$ , $\alpha=5$ . The exact value of $\alpha$ depends on the density profile and velocity anisotropy.

Builds on: Hubble Sequence and Galaxy Morphology · Stellar Populations I, II, and III · Structure of the Milky Way Galaxy. Leads to: Galactic Bulge, Bar, and Central Regions · Evidence for Dark Matter in Galaxies · Galactic Dark Matter Halos and Density Profiles · Spiral Galaxies and Density Wave Theory.

Ask the tutor: Want to see the incomplete-gamma equation solved for b_n at any Sérsic index, or walk through how a measured σ₀ and R_e turn into a dark-matter fraction?

Why is a flattened E7 galaxy not just a fast-spinning disk?

Because when you measure how fast it rotates, the rotation speed v is small compared with the random speed σ — most bright ellipticals are "slow rotators" with v/σ ≪ 1. A rotating disk is flattened by centrifugal balance; an elliptical is flattened because its stars have unequal random velocities along different axes and because the galaxy is intrinsically triaxial. The observed E-number is only the projected shape, so the same galaxy looks rounder or flatter depending on viewing angle. Integral-field surveys like ATLAS³D confirmed that flattening and rotation are largely decoupled in the elliptical population.

Is there a physical reason the index is exactly n = 4, or is it just a good fit?

Honestly, mostly the latter — the r^(1/4) law is phenomenological, and no clean first-principles derivation forces n = 4. What we do understand is that violent relaxation — the rapid collective settling of a stellar system during collapse or a major merger, in which the fluctuating potential scrambles stellar energies — produces smooth, centrally concentrated, de-Vaucouleurs-like profiles. The Sérsic index then correlates with mass and merger history rather than sitting at a magic value: giant, dry-merger-built ellipticals cluster near n ≈ 4–6, dwarfs near n ≈ 1–2. So n = 4 is the typical outcome for giants, not a law of nature.

What is the effective radius R_e really, and is it a robust measurement?

R_e is the projected radius of the circle (or ellipse) enclosing half the galaxy's total light — a half-light radius, not any physical edge. Its appeal is being a well-defined, model-independent statistic of the light. The catch: measuring it requires the total light, which means extrapolating the faint outer envelope, and for high-n profiles that envelope carries a surprising fraction of the luminosity. So R_e can shift by tens of percent depending on how far out you integrate and how you subtract the sky — which is why deep photometry matters and survey values sometimes disagree.

What does the "tilt" of the Fundamental Plane actually tell us?

The pure virial theorem, with constant Υ and surface brightness, predicts R_e ∝ σ²⟨I⟩_e⁻¹ — exponents (2, −1). The measured plane has (1.24, −0.82). That systematic difference, the tilt, cannot be error because the plane is so thin. Matching the two requires Υ to grow with mass, roughly Υ ∝ M^0.2: more massive ellipticals carry more mass per unit light, some mix of a higher dark-matter fraction within R_e and a more bottom-heavy initial mass function. The Fundamental Plane thus doubles as a graph of how the mass budget shifts as galaxies grow.

How do you measure the velocity dispersion σ₀ in the first place?

Spectroscopically. An elliptical's spectrum is the sum of millions of stellar spectra, each Doppler-shifted by its own line-of-sight velocity, so the intrinsically sharp absorption lines (the Ca II triplet, Mg b, the G band) are smeared into broadened, roughly Gaussian profiles whose width tracks the spread of stellar velocities. Fitting that width against a stellar template — after removing instrumental broadening — recovers σ₀, typically 100–350 km/s. Because it is a width, not a shift, σ₀ is distance-independent, which is what makes Faber–Jackson and the Fundamental Plane useful as distance indicators.

Are ellipticals really "red and dead," and where did their gas go?

Broadly yes: ellipticals are dominated by old, metal-rich Population II stars, are red because they lack hot young blue stars, and form very few new stars today. But "dead" overstates it. Massive ellipticals host enormous halos of hot, X-ray-emitting gas at millions of kelvin, too hot to collapse into stars and kept warm partly by feedback from a central supermassive black hole (which is why cooling flows are largely suppressed). Their cold star-forming gas was consumed, expelled, or heated during the early collapse and later dry mergers. So the color is real, but the "dead" galaxy sits in a bath of invisible hot plasma.