Faber-Jackson Relation and the Fundamental Plane

The Fundamental Plane is now a precision instrument on two fronts. As a distance indicator it underpins large peculiar-velocity surveys: the 6dF Galaxy Survey measured the plane for roughly nine thousand early-type galaxies across the southern sky, and current campaigns extend this to hundreds of thousands, mapping the velocity field of the local universe to test structure growth and constrain the parameter combination $f\sigma_8$ against the predictions of general relativity. As a probe of galaxy evolution, the evolution of the plane's zero-point with redshift — traced with the Hubble and now the James Webb space telescopes out to $z\sim1$–$2$ — measures how the mass-to-light ratios of massive ellipticals brighten toward the past, dating the bulk of their star formation to the first few billion years. The tilt itself remains an active battleground: hydrodynamical cosmological simulations (IllustrisTNG, EAGLE) reproduce a Fundamental Plane, and disentangling how much of the tilt owes to stellar populations, a possibly non-universal initial mass function, dark-matter fraction, and structural non-homology is a central goal, sharpened by integral-field spectroscopy from surveys such as ATLAS3D, SAMI and MaNGA that resolve the internal dynamics the plane only summarizes.

Picture every giant elliptical in the local universe plotted as a point in a room whose three axes are size, brightness, and how fast the stars jitter. Your instinct says a shapeless cloud. Instead the points lie in a thin sheet — you could slide a flat table underneath and touch nearly all of them at once. That sheet is the Fundamental Plane, and its very existence is the physics: a galaxy has only two independent structural knobs, so fixing any two of $\{R_e,\sigma_0,I_e\}$ pins the third. Look at the sheet edge-on and it degrades to a fat line — that fat line is the Faber-Jackson relation, scattered precisely because the projection has flattened a real third dimension into what looks like measurement noise.

Why the plane is tilted is the deeper half of the story. The virial theorem alone, applied to a homologous family with a single mass-to-light ratio, predicts a sheet at a definite angle: $R_e\propto\sigma_0^{2}I_e^{-1}$, exponents $(2,-1)$. Nature hands us $(1.24,-0.82)$ instead. The plane is real and thin, but it is not sitting where the naive virial calculation says it should. The rotation of the sheet away from that prediction means that as ellipticals grow more massive, they become slightly "darker" per unit light — older stellar populations, a bottom-heavier initial mass function, more dark matter inside $R_e$, or a gentle departure from strict structural similarity. The second figure shows the projection that history discovered first: the Faber-Jackson relation itself, luminosity climbing as the fourth power of $\sigma_0$, with the factor-of-two scatter that is the price of ignoring the surface-brightness axis.

The whole subject is built from two physical statements — the virial theorem and the definition of an effective radius — and the algebra that combines them. We begin with the empirical relation that opened the field, then assemble the machinery beneath it. The Faber-Jackson relation is the original empirical scaling: luminosity as the fourth power of central velocity dispersion. It is a one-line summary of the pressure-supported galaxy sequence, and it is the projection the plane collapses to.

where: $L$ — total luminosity of the elliptical galaxy; $\sigma_0$ — central line-of-sight stellar velocity dispersion, in km/s. Reading it: a galaxy with twice the dispersion is roughly sixteen times as luminous, but with a factor-of-two vertical scatter that signals a hidden third variable. The Fundamental Plane, written in general form, is a flat relation among the three logarithmic observables. The coefficients $a$ and $b$ are what a survey measures; the zero-point $c$ carries the distance scale.

where: $R_e$ — effective (half-light) radius, in kpc; $\sigma_0$ — central velocity dispersion, in km/s; $I_e$ — mean surface brightness within $R_e$, in $L_\odot\,\mathrm{pc}^{-2}$; $a,b$ — empirical plane coefficients; $c$ — zero-point. Reading it: any two observables predict the third along a plane whose thinness is the physics. Fitted in the optical $B$-band, the plane has specific, non-virial exponents — the numbers that carry the tilt.

where: $1.24$ — the observed $\sigma_0$ exponent, whose pure-virial value would be $2$; $-0.82$ — the observed $I_e$ exponent, whose pure-virial value would be $-1$. Reading it: both exponents fall short of virial, and by nearly the same fractional amount — the signature of a smooth $M/L$–mass trend. The virial theorem supplies the dynamical mass from the two observables that dynamics cares about. The dimensionless factor $k$ absorbs the density and orbital structure.

where: $M$ — total (dynamical) mass within roughly $R_e$; $k$ — structure factor, $k\approx5$ for realistic elliptical profiles when $\sigma_0$ is the central dispersion and $R_e$ the half-light radius; $G$ — gravitational constant. Reading it: mass is set by how fast the stars move and how far apart they are, nothing else. The effective radius is defined so that half the light falls inside it; combined with the mean interior surface brightness this fixes the total luminosity. This is the bridge between photometry and the virial mass.

where: $I_e$ — mean surface brightness inside $R_e$; $R_e$ — radius enclosing half the light. Reading it: brightness times area returns luminosity, and the factor $2\pi$ is exactly what the half-light definition demands (derived below). The tilt is quantified by a single power law: the mass-to-light ratio rises weakly with mass. This one exponent is what a formation model must reproduce.

where: $M/L$ — dynamical mass-to-light ratio in the observed band; $0.2$ — the tilt exponent inferred from the plane's deviation from virial. Reading it: doubling the mass raises $M/L$ by only about 15%, so the trend is gentle but decisive across the two decades of mass ellipticals span. Because $\sigma_0$ and $I_e$ do not depend on distance, the plane predicts a physical $R_e$; matching it to the observed angular size returns a distance without a redshift.

where: $\theta_e$ — observed angular effective radius; $d$ — distance; $R_e(\sigma_0,I_e)$ — physical radius predicted by the plane from distance-blind observables. Reading it: spectroscopy plus photometry becomes a ruler. Finally, Bender, Burstein and Faber rotated the plane into orthogonal $\kappa$-coordinates aligned with physics: $\kappa_1$ measures mass, $\kappa_3$ measures the mass-to-light ratio.

where: $\kappa_1$ — proportional to $\log M$ (the mass axis); $\kappa_3$ — proportional to $\log(M/L)$ (the mass-to-light axis). Reading it: in this frame the plane's thinness in $\kappa_3$ is the statement that $M/L$ is a tight function of mass.

The relations change character depending on which regime you sit in and, crucially, on which projection of the plane you look at.

1 · The dynamical mass and mass-to-light ratio of a giant elliptical (fully worked)

A luminous elliptical has central velocity dispersion $\sigma_0 = 250\ \mathrm{km/s}$ and effective radius $R_e = 5\ \mathrm{kpc}$. Take the structure factor $k=5$. Find its virial mass within $R_e$, and if its luminosity is $L\approx5\times10^{10}\,L_\odot$, estimate the mass-to-light ratio.

Step 1. Put everything in SI. The dispersion is $\sigma_0 = 250\ \mathrm{km/s} = 2.5\times10^{5}\ \mathrm{m/s}$, and the radius is $R_e = 5\ \mathrm{kpc} = 5\times3.086\times10^{19}\ \mathrm{m} = 1.543\times10^{20}\ \mathrm{m}$, using $1\ \mathrm{kpc}=3.086\times10^{19}\ \mathrm{m}$.

Step 2. Insert into the virial estimator $M = k\,\sigma_0^2 R_e/G$ with $G = 6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$:

Step 3. Evaluate the numerator: $(2.5\times10^{5})^2 = 6.25\times10^{10}$; times $1.543\times10^{20}$ gives $9.64\times10^{30}$; times $5$ gives $4.82\times10^{31}$. Divide by $G$: $M = 4.82\times10^{31}/6.674\times10^{-11} \approx \mathbf{7.2\times10^{41}\ kg}$.

Step 4. Convert to solar masses with $M_\odot = 1.989\times10^{30}\ \mathrm{kg}$: $M = 7.2\times10^{41}/1.989\times10^{30} \approx \mathbf{3.6\times10^{11}\,M_\odot}$ — squarely in the giant-elliptical range.

Step 5. The mass-to-light ratio is $M/L = 3.6\times10^{11}/5\times10^{10} \approx \mathbf{7\ M_\odot/L_\odot}$ (in solar units, within $R_e$). This is a few times the value of a young stellar population, consistent with an old, metal-rich population plus a modest dark-matter contribution inside the half-light radius — exactly the ingredients the tilt encodes.

2 · The Fundamental Plane as a distance indicator

A cluster elliptical has a measured central dispersion $\sigma_0 = 220\ \mathrm{km/s}$, a mean surface brightness that (through the calibrated plane) implies a physical effective radius $R_e = 4.0\ \mathrm{kpc}$, and an observed angular effective radius $\theta_e = 5.0\ \mathrm{arcsec}$. Estimate its distance.

Step 1. The plane's job is done once it returns the physical radius: $\sigma_0$ and $I_e$ are distance-independent, so $R_e(\sigma_0,I_e)=4.0\ \mathrm{kpc}$ is a true length, not an angle. This is the whole trick — the distance-blind observables have predicted a distance-dependent one.

Step 2. Convert the angular size to radians: $\theta_e = 5.0\ \mathrm{arcsec} = 5.0/206265\ \mathrm{rad} = 2.42\times10^{-5}\ \mathrm{rad}$, using $1\ \mathrm{rad}=206265\ \mathrm{arcsec}$.

Step 3. Apply $d = R_e/\theta_e = 4.0\ \mathrm{kpc}/2.42\times10^{-5} \approx \mathbf{1.65\times10^{5}\ kpc} = \mathbf{165\ Mpc}$.

Step 4. Interpret. A single galaxy carries a plane-distance good to $\sim10$–$20\%$; averaging over the dozens of ellipticals in a cluster beats this down to a few percent. Comparing such distances to redshift-inferred distances is what exposes peculiar velocities — the streaming of clusters toward mass concentrations like the Great Attractor — because any mismatch between the plane distance and $cz/H_0$ is a line-of-sight peculiar velocity.

3 · Reading the tilt: from the $M/L$ slope to the plane angle

Suppose deep spectroscopy fixes the mass-to-light slope at $\gamma = 0.20$ (that is, $M/L\propto M^{0.20}$). Predict the Fundamental Plane exponents and compare with the observed $B$-band values.

Step 1. Use the tilted-plane result $R_e \propto \sigma_0^{(2-2\gamma)/(1+\gamma)}\,I_e^{-1/(1+\gamma)}$ from the first derivation.

Step 2. The $\sigma_0$ exponent: $(2-2\times0.20)/(1+0.20) = 1.60/1.20 = \mathbf{1.33}$.

Step 3. The $I_e$ exponent: $-1/(1+0.20) = -1/1.20 = \mathbf{-0.83}$.

Step 4. Compare with the observed $(1.24,-0.82)$. The predicted pair $(1.33,-0.83)$ reproduces the $I_e$ exponent almost exactly and the $\sigma_0$ exponent within the fitting scatter; nudging $\gamma$ up to $\approx0.23$ tightens the $\sigma_0$ match. The lesson: a single number, the $M/L$–mass slope $\gamma\approx0.2$, rotates the pure-virial plane $(2,-1)$ into the observed one. The residual imperfection is where structural non-homology and band-dependent stellar-population effects hide.

The Fundamental Plane is unusual among astronomical scaling laws in having a living, named discoverer whose career traces the whole arc of the subject — from a first empirical correlation to a precision cosmological tool.

Texts. Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §25.2–25.3 (Faber-Jackson and the Fundamental Plane) · Binney and Tremaine, Galactic Dynamics, 2nd ed., §4.3 (virial estimates) and §4.10 (scaling relations) · Mo, van den Bosch and White, Galaxy Formation and Evolution, §13.1 (elliptical scaling laws and the tilt) · Sparke and Gallagher, Galaxies in the Universe, §6.2 · Ryden, Introduction to Cosmology, 2nd ed. (distance indicators).

Papers. Faber and Jackson, ApJ 204, 668 (1976) · Kormendy, ApJ 218, 333 (1977) · Dressler et al., ApJ 313, 42 (1987) · Djorgovski and Davis, ApJ 313, 59 (1987) · Bender, Burstein and Faber, ApJ 399, 462 (1992). Key numerical checks: the virial-to-plane derivation gives exponents $(2,-1)$ for $\gamma=0$ and $(\approx1.28,-0.82)$ for $\gamma\approx0.22$; the worked virial mass from $\sigma_0=250\ \mathrm{km/s}$, $R_e=5\ \mathrm{kpc}$, $k=5$ is $\approx3.6\times10^{11}\,M_\odot$; and the Faber-Jackson index is $n\approx4$ with $B$-band plane coefficients $(1.24,-0.82)$. Constants used throughout: $G=6.674\times10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$, $1\ \mathrm{kpc}=3.086\times10^{19}\ \mathrm{m}$, $M_\odot=1.989\times10^{30}\ \mathrm{kg}$, $1\ \mathrm{rad}=206265\ \mathrm{arcsec}$.

Prerequisites. Elliptical Galaxies and the de Vaucouleurs Profile · Hubble Sequence and Galaxy Morphology · Evidence for Dark Matter in Galaxies · Stellar Populations. Leads to. Galactic Rotation Curves · Galactic Dark-Matter Halos and Density Profiles · Galactic Bulge, Bar, and Central Regions.

Ask the tutor: Ask me to walk you through why $\gamma\approx0.2$ rotates the pure-virial plane into the observed $(1.24,-0.82)$, or to work a Fundamental-Plane distance for a galaxy of your choosing.