Faber-Jackson Relation

The idea: bigger galaxies hold faster stars

An elliptical galaxy has no spiral arms and almost no ordered rotation. Its stars are not riding around a disk; they swarm on randomly oriented orbits, like bees in a hive, held together only by their mutual gravity. The single number that captures how fast that swarm moves is the velocity dispersion σ — the spread, in kilometres per second, of the stars' line-of-sight speeds about the galaxy's mean. A small dwarf elliptical might have σ ≈ 50 km/s; a giant elliptical at the heart of a cluster can reach σ ≈ 350–400 km/s.

In 1976, Sandra Faber and Robert Jackson took spectra of 25 elliptical galaxies and noticed something clean: the brighter the galaxy, the faster its stars moved — and not linearly, but steeply. Doubling the velocity dispersion went hand in hand with roughly a sixteen-fold jump in luminosity. Written as a power law,

L ∝ σ⁴

This is the Faber-Jackson relation. It says you can stand outside a galaxy, measure how much its starlight is Doppler-smeared, and infer how intrinsically luminous the galaxy must be — without ever knowing how far away it is. That single fact is what makes it one of the workhorse scaling laws of extragalactic astronomy.

Why the exponent is (almost) four

The relation is not a coincidence; it falls out of basic gravitational physics. A self-gravitating, relaxed stellar system obeys the virial theorem, which for a galaxy of mass M and characteristic radius R relates the kinetic and potential energy as 2K + U = 0. In terms of the velocity dispersion that reads

M ≈ k σ² R / G

where k is a structure constant of order unity. Now write the luminosity as the surface brightness I₀ integrated over the projected area, L ≈ 2π I₀ R², so that R ∝ (L / I₀)^(1/2). Substitute that into the virial mass and replace M with (M/L)·L:

(M/L) L ∝ σ² (L / I₀)^(1/2)
L^(1/2) ∝ σ² / [ (M/L) I₀^(1/2) ]
L ∝ σ⁴ / [ (M/L)² I₀ ]

If the mass-to-light ratio M/L and the surface brightness I₀ were exactly the same for every elliptical, the bracket would be a constant and you would get the clean L ∝ σ⁴. They are nearly constant — elliptical galaxies are an old, dynamically homogeneous population — which is why the relation works at all. The small systematic drift of M/L and I₀ with galaxy size is precisely what tilts the real exponent and what the three-dimensional fundamental plane captures completely.

How σ is measured: line broadening

Velocity dispersion is read directly off a spectrum, and this is the elegant part. A galaxy's integrated light is the sum of millions of stellar spectra, each carrying the same absorption features — the Ca II H&K doublet near 3950 Å, the magnesium Mg b triplet at 5175 Å, the Ca II infrared triplet near 8550 Å. Each star's lines are Doppler-shifted by its own line-of-sight velocity. Because the stars move with a spread σ, the lines of all of them, added together, are broadened into a near-Gaussian profile whose width is σ.

The conversion is the ordinary Doppler formula. A dispersion of σ = 200 km/s broadens a line at rest wavelength λ by

Δλ / λ = σ / c = 200 / 299,792 ≈ 6.7 × 10⁻⁴
→ at λ = 5175 Å, Δλ ≈ 3.5 Å (1σ Gaussian width)

Modern pipelines fit the observed galaxy spectrum against a library of stellar templates convolved with a line-of-sight velocity distribution — the penalized pixel-fitting code (pPXF) by Cappellari & Emsellem is the standard tool — and recover σ to a few percent. Crucially, this broadening does not depend on distance: a galaxy twice as far has dimmer lines, but the lines are broadened by exactly the same fraction. That distance-independence is the whole point.

Real galaxies on the relation

The relation spans nearly the entire elliptical population, from compact dwarfs to brightest-cluster giants. Approximate values (B-band, for orientation):

Galaxy / type	σ (km/s)	M_B	L_B (L☉)	Notes
Dwarf elliptical (e.g. NGC 205)	~50	~−16	~3 × 10⁸	Low-luminosity end
M32 (compact dE)	~75	~−16.5	~5 × 10⁸	Satellite of Andromeda
NGC 3379 (M105)	~210	~−20.9	~3 × 10¹⁰	Textbook normal elliptical
M87 (Virgo cD)	~320	~−22.4	~1.2 × 10¹¹	Hosts a 6.5 × 10⁹ M☉ black hole
IC 1101 (cD giant)	~360	~−24	~5 × 10¹¹	Among the most luminous known

Run the scaling as a sanity check. M32 (σ ≈ 75) to M87 (σ ≈ 320) is a factor of 4.3 in σ. Raised to the fourth power that predicts a luminosity ratio of 4.3⁴ ≈ 340, i.e. about 6.3 magnitudes — and M_B goes from −16.5 to −22.4, a difference of 5.9 magnitudes. The relation gets you within a magnitude over a factor of 300 in luminosity. That is remarkable for a single observable, and also a reminder that the intrinsic scatter (~0.5 mag) is real.

Reading off distance and mass

Two practical uses follow immediately.

As a distance indicator. Measure σ from the line widths; predict the absolute luminosity L (hence absolute magnitude M) from L ∝ σ⁴; measure the apparent magnitude m; then the distance modulus m − M gives the distance via

m − M = 5 log₁₀(d / 10 pc)
→ d = 10^[(m − M + 5)/5]  parsecs

Because the raw Faber-Jackson scatter of ~0.5 mag translates to ~25% distance error per galaxy, astronomers in practice use its tighter relatives — the Dⁿ–σ relation (Dressler et al. 1987), which replaces luminosity with a brightness-defined diameter, and the full fundamental plane — to reach ~15–20% per galaxy and a few percent for a cluster average. These methods were central to mapping the peculiar-velocity flows of the 1980s, including the discovery of the "Great Attractor."

As a dynamical mass estimator. The virial mass within the effective radius R_e is

M ≈ 5 σ² R_e / G    (the common k ≈ 5 calibration)

For M87, σ ≈ 320 km/s and R_e ≈ 7 kpc give M ≈ 5 × (3.2 × 10⁵ m/s)² × (2.2 × 10²⁰ m) / (6.67 × 10⁻¹¹) ≈ 1.7 × 10⁴² kg ≈ 8 × 10¹¹ M☉ within one effective radius — consistent with detailed dynamical models. So a single line width, plus a size, weighs a galaxy.

Faber-Jackson is a shadow of the fundamental plane

By the early 1980s it was clear that L ∝ σ⁴ had more scatter than measurement error alone could explain. The resolution, found independently by Djorgovski & Davis (1987) and Dressler et al. (1987), is that elliptical galaxies live on a tight two-dimensional surface inside the three-dimensional space of (size R_e, surface brightness I_e, velocity dispersion σ). That surface is the fundamental plane:

R_e ∝ σ^1.24 I_e^(−0.82)        (Jorgensen et al. 1996, optical)

Faber-Jackson is simply the projection of this plane onto the luminosity–σ axis. When you ignore surface brightness and look only at L versus σ, you are squashing a thin plane edge-on, and the plane's finite thickness in the discarded direction becomes the scatter you see. View it edge-on along the right direction and the scatter drops to ~10%. This is why the fundamental plane, not Faber-Jackson itself, is the precision tool — but Faber-Jackson is the conceptual ancestor, and σ remains the dominant variable in both.

Tully-Fisher, M–σ, and the family of scaling laws

Faber-Jackson belongs to a family of luminosity–velocity laws, each tied to the virial theorem but applied to a different kind of system:

Relation	System	Velocity measured	Scaling	What it predicts
Faber-Jackson	Elliptical galaxies, bulges	Velocity dispersion σ (random)	L ∝ σ⁴	Spheroid luminosity / mass
Tully-Fisher	Spiral galaxies	Flat rotation speed V_max (ordered)	L ∝ V⁴	Disk luminosity / mass
M–σ relation	Galaxy with central SMBH	Velocity dispersion σ	M_BH ∝ σ^4–5	Black-hole mass
Fundamental plane	Elliptical galaxies	σ + size + brightness	R_e ∝ σ^1.24 I_e^−0.82	Distance (~10% scatter)
Dⁿ–σ	Elliptical galaxies	σ + isophotal diameter	Dⁿ ∝ σ^1.2–1.3	Distance indicator
Baryonic Tully-Fisher	Spirals (gas + stars)	V_flat	M_baryon ∝ V⁴	Total baryonic mass

The conceptual unity is striking: ellipticals store their kinetic energy in disordered motion (pressure support), spirals in ordered rotation, and both obey a luminosity ∝ velocity⁴ law because both are virialised systems with roughly constant mass-to-light ratios. The M–σ relation extends the same variable, σ, all the way down to the central black hole — evidence that the growth of the spheroid and the growth of its black hole were physically coupled.

Where the relation shows up in practice

• Galaxy surveys. The Sloan Digital Sky Survey measured σ for hundreds of thousands of ellipticals, letting astronomers populate Faber-Jackson and the fundamental plane statistically and study how the relations evolve with cosmic time.
• Peculiar-velocity cosmology. The "7 Samurai" collaboration (Dressler, Faber, and colleagues) used the Dⁿ–σ form to map the local velocity field in the 1980s and identified the bulk flow toward the Great Attractor — a ~10¹⁶ M☉ mass concentration behind the Milky Way's disk.
• Black-hole demographics. Because σ tracks both spheroid luminosity (Faber-Jackson) and black-hole mass (M–σ), a single dispersion measurement places a galaxy in both relations and lets surveys estimate black-hole masses for objects too distant to resolve dynamically.
• Strong-lensing mass calibration. The Einstein radius of a galaxy-scale gravitational lens depends on σ²; Faber-Jackson lets lens modellers cross-check the lensing mass against the stellar luminosity, e.g. in the SLACS survey.
• High-redshift evolution. Comparing the L–σ zero point at z ≈ 1 with today tests how the stellar populations of ellipticals have aged and brightened, a sensitive probe of when massive galaxies assembled.

Common misconceptions and edge cases

• "σ is a rotation speed." No — for ellipticals σ measures random stellar velocities (pressure support), not bulk rotation. Massive ellipticals rotate slowly; their support against gravity is the velocity dispersion, not centrifugal force. This is the structural difference from Tully-Fisher's rotation velocity.
• "The exponent is exactly 4." Four is the clean virial-theorem result under idealised constant M/L and surface brightness. Real fits give γ ≈ 3.7 in blue bands, climbing toward ~5 in the near-infrared, because M/L and I₀ vary systematically with galaxy size — the same tilt the fundamental plane encodes.
• "Faber-Jackson is a precise distance ruler." On its own it has ~0.5 mag scatter, too loose for cosmology. Precision comes only from the fundamental plane or Dⁿ–σ, which add the surface-brightness dimension Faber-Jackson throws away.
• "It applies to all galaxies." It applies to spheroids — ellipticals and the bulges of spirals — that are dynamically relaxed and pressure-supported. Gas-rich, rotation-dominated disks follow Tully-Fisher instead. Recent mergers and disturbed systems scatter off both relations until they relax.
• "Aperture doesn't matter." σ depends on the aperture you measure it through, because the dispersion profile changes with radius. Catalog values are corrected to a standard aperture (often R_e/8) before being placed on the relation; mixing apertures introduces a spurious systematic.
• "Faber-Jackson gives the black-hole mass." That is the M–σ relation, a different law sharing the same σ. Faber-Jackson gives the luminosity (and stellar mass) of the whole spheroid, which is typically about a thousand times more massive than the central black hole.