Fundamental Plane

Three numbers, one surface

Pick an elliptical galaxy out of any spectroscopic survey and measure three things from its image and its spectrum. From the image you get the half-light or "effective" radius R_eff — the radius within which half of the total light is enclosed — and the mean surface brightness inside that radius, I_e (flux per square arcsecond, conventionally expressed as a magnitude per arcsec², but on the plane we work with the linear quantity). From the spectrum you get σ, the central stellar velocity dispersion: the width of the absorption lines, broadened by the random orbital motions of stars in the galaxy's potential. Three numbers per galaxy.

If those three numbers were uncorrelated, ellipticals would fill a 3D cloud in (log R_eff, log σ, log I_e) space. They do not. Instead they trace a thin sheet — a plane in the literal geometric sense — whose perpendicular thickness is only about 15 percent in log R_eff, smaller than the measurement scatter on any single axis. That is the fundamental plane, and its compact equation is

log R_eff = a · log σ + b · log I_e + c

with the best B-band slopes near a ≈ 1.24, b ≈ -0.82. Other bands give slightly different a and b (the slopes shift redward because the M/L gradient runs with band), but the existence of a tight plane survives.

Why Faber-Jackson alone was not enough

Before the fundamental plane there was Faber-Jackson (1976), the elliptical analogue of Tully-Fisher: a one-parameter scaling between luminosity and velocity dispersion,

L ∝ σ⁴

If you plot log L versus log σ for several hundred ellipticals, you get a clear correlation — but with a scatter of roughly 0.3 dex (factor of two) in L at fixed σ. That scatter is too big to be useful as a distance indicator. Worse, it is not random noise: it correlates with surface brightness. A high-surface-brightness elliptical at given σ is more luminous than average; a low-surface-brightness one is less. So one variable was clearly missing.

The fundamental plane is what you get when you add that missing variable. Move from the (L, σ) plane into the (R_eff, σ, I_e) cube — equivalent to splitting L = 2π R_eff² I_e into its two physical components — and the scatter collapses. Geometrically, Faber-Jackson is the projection of the plane onto the L-σ plane; the residuals you see in Faber-Jackson are the depth of the plane along the I_e direction.

The virial expectation, and why reality differs

The simplest theoretical motivation for a plane is the virial theorem. For a self-gravitating, dynamically relaxed stellar system,

2 T + W = 0     →     M = k₁ · σ² · R / G

where k₁ is a structure-dependent dimensionless constant of order unity, and R is some characteristic radius. Combine this with the definition of the mean surface brightness, I ∝ L / R², and the definition of mass-to-light ratio M/L, and you can eliminate variables to give

R ∝ σ²  ·  I⁻¹  ·  (M/L)⁻¹  ·  k₁⁻¹

If all ellipticals are homologous (same k₁, same M/L), then R ∝ σ² I⁻¹: a plane with a = 2 and b = -1. That is the "virial plane".

The observed plane has a ≈ 1.24 and b ≈ -0.82. The difference is the famous "tilt". Both slopes are systematically rotated from the virial prediction by an amount that says the implicit factors of k₁ and M/L are not constant across the galaxy population — they scale with mass. Tracking down which is the dominant culprit is one of the open questions in elliptical-galaxy structure.

What the tilt is telling us

Three physical effects can rotate the virial plane into the observed one. They are not mutually exclusive, and in practice all three contribute at some level.

• Non-homology. More massive ellipticals are systematically more centrally concentrated. The Sérsic index n rises from about 2 in low-mass ellipticals to about 6–8 in bright cluster cores. That changes k₁ in the virial relation. Models with mass-dependent structure reproduce roughly half of the tilt without invoking any change in stellar populations or dark matter.
• Stellar M/L variation. More massive ellipticals are older, more metal-rich, and on some evidence (gravity-sensitive spectral features) have bottom-heavy initial mass functions. All three push their stellar M/L up with mass. The empirical scaling M/L ∝ M^{0.2} is typical for the B band.
• Dark-matter fraction. The dynamical mass inferred from σ and R_eff includes whatever dark matter lies within R_eff. Strong-lensing and dynamical studies (SLACS, ATLAS3D) find that the dark-matter fraction inside R_eff rises from about 10 percent in low-σ ellipticals to 30–40 percent in the brightest cluster galaxies. That trend tilts the plane in the right direction.

The current best estimates apportion the tilt roughly half to non-homology and half to combined M/L variation. The plane is therefore not a pure dynamical relation; it is a dynamical relation with a mass-correlated structural and stellar-population correction baked in.

The plane as a distance indicator

Two of the three observables — σ and I_e — are independent of distance. σ is measured from line widths in the spectrum, which do not change with distance. I_e is a surface brightness, flux per solid angle, which is also distance-independent for nearby galaxies (and only weakly dependent at higher redshifts, where (1+z)⁴ cosmological dimming kicks in). R_eff in physical units — kiloparsecs — does depend on distance through the angular-diameter relation R_eff = θ_eff × D_A.

So the recipe is:

1. Measure σ from the spectrum and I_e from the image.
2. Plug into log R_eff (kpc) = a log σ + b log I_e + c to predict R_eff.
3. Measure the apparent angular size θ_eff from the image.
4. Distance D_A = R_eff / θ_eff.

The per-galaxy precision is about 0.08 dex in log R_eff — roughly 20 percent in distance. For a sample of N cluster ellipticals, the precision improves as 1/√N, giving roughly 1–2 percent on a cluster distance from a few hundred members. That makes the fundamental plane competitive with type Ia supernovae and surface-brightness fluctuations in the local universe (z < 0.05) where ellipticals are plentiful.

The D_n-σ short-cut

Dressler and the rest of the Seven Samurai realised in 1987 that you can collapse the plane into a single one-parameter relation by clever choice of size definition. Instead of R_eff (the half-light radius), use D_n, the angular diameter inside which the mean surface brightness exceeds a fixed threshold (typically the diameter at which the B-band surface brightness reaches 20.75 mag arcsec⁻²).

D_n combines size and surface brightness in just the right way to lie almost exactly along the plane: galaxies with high I_e have larger D_n at the same R_eff, exactly compensating the surface-brightness term in the fundamental plane. The result is a one-axis relation,

D_n ∝ σ^{1.33}

tight enough that no I_e measurement is needed. D_n-σ was the workhorse of the late-1980s peculiar-velocity surveys, including the survey that revealed the Great Attractor.

The Great Attractor

If the universe were perfectly homogeneous, every galaxy's redshift would be exactly cz = H₀ × d, and fundamental-plane distances would match Hubble-law distances. They almost do — but not quite. Subtract the Hubble flow from the measured FP distances of several hundred nearby elliptical galaxies, and what is left over is the peculiar velocity field — the bulk motions induced by gravitational attraction toward overdensities.

In 1987–1988 the Seven Samurai team (Lynden-Bell, Faber, Dressler, Davies, Burstein, Terlevich and Wegner) used D_n-σ on about 400 ellipticals out to ~ 60 Mpc and discovered something remarkable: every galaxy in a region 100 Mpc across, including the Milky Way, was being pulled at ~ 600 km/s toward a point at galactic coordinates roughly (l, b) = (309°, +18°) — behind the obscuring plane of our own galaxy in Centaurus. They named the implied mass concentration the Great Attractor. Later X-ray work identified the Norma Cluster (Abell 3627) at the heart of it, and CMB-dipole and 2MASS surveys showed that the deeper flow extends to the Shapley supercluster behind. The map of the Laniakea Supercluster published by Tully et al. in 2014 traces the same flow lines on a larger canvas.

This was the moment when the fundamental plane stopped being a curiosity of elliptical-galaxy phenomenology and became a tool for mapping mass on cosmological scales.

Worked example: predicting R_eff for a giant elliptical

Take a typical giant elliptical with σ = 250 km/s and a B-band mean surface brightness inside R_eff of µ_e = 21.0 mag/arcsec² (equivalently I_e ≈ 800 L☉/pc² in linear units). Use the B-band fundamental plane in the calibration of Jorgensen, Franx & Kjaergaard (1996):

log R_eff (kpc) = 1.24 · log σ (km/s) − 0.82 · log I_e (L☉/pc²) − 0.182

Plugging in σ = 250 and I_e = 800:

log R_eff = 1.24 · 2.398  −  0.82 · 2.903  −  0.182
          = 2.974 − 2.380 − 0.182
          = 0.412
R_eff ≈ 10^0.412 ≈ 2.58 kpc

A 2.6 kpc half-light radius — consistent with a moderately concentrated giant E galaxy. If we now measure its apparent angular half-light radius and find θ_eff = 5 arcsec, the implied distance is

D_A = R_eff / θ_eff
    = 2.58 kpc / (5 / 206265 rad)
    = 2.58 × 206265 / 5 kpc
    ≈ 106,400 kpc  ≈ 106 Mpc

That is roughly the distance to the Coma Cluster. Repeat this exercise for 30 ellipticals in Coma, take the geometric mean, and you have the cluster distance to about 4 percent — the kind of measurement the FP routinely delivers.

Forms in use

Form	Free variables	Scatter	Notes
Faber-Jackson	L, σ	~ 0.3 dex in L	Original 1976 elliptical relation; projection of FP onto L-σ
Fundamental Plane (linear)	R_eff, σ, I_e	~ 0.08 dex in log R_eff	Djorgovski-Davis & Seven Samurai 1987
Fundamental Plane (κ-space)	κ₁, κ₂, κ₃	~ same	Bender, Burstein & Faber 1992 — axes are linear combinations of (M, M/L, fine structure)
D_n-σ relation	D_n, σ	~ 0.10 dex in D_n	Dressler et al. 1987; effectively 2D projection onto the FP edge
Mass plane	R_eff, σ, M_⋆/L	tighter, ~ virial	Cappellari et al. 2013 — replace I_e with M⋆/L; tilt nearly vanishes
Fundamental manifold	R, V, M/L_dyn	tightest	Zaritsky, Gonzalez & Zabludoff 2006 — unifies E, S, dSph onto one surface

The progression visible in this table is from observable to physical: each successive form replaces a directly measurable quantity with one closer to the underlying dynamics. The "mass plane" of Cappellari and ATLAS3D, in which I_e is replaced by the dynamically-measured M_⋆/L, is essentially flat — the tilt is absorbed into the M/L term, confirming that M/L variation is the dominant cause of the tilt.

The plane at higher redshift

Push the fundamental plane out in redshift and the zero point shifts in a controlled way. At fixed σ and R_eff the surface brightness I_e is higher in the past, because the stars were younger and more luminous. The offset Δ log I_e (z) measured against the local plane gives a direct constraint on stellar-population age, since

Δ log I_e  ≈  -0.4  ·  d log (M/L) / dz · z

Observations out to z ≈ 1.3 (van Dokkum & Stanford 2003; Holden et al. 2010) give Δ log (M/L)_B ≈ -0.5 z, consistent with a passively evolving population that formed at z_form ≈ 2–3 with a Salpeter or Chabrier IMF. This is one of the cleanest constraints on the bulk-of-mass formation epoch for elliptical galaxies.

Where the plane shows up

• Cluster distances and H₀. Coma, Virgo, Fornax, Centaurus and Hydra cluster distances from the FP are part of the secondary distance-ladder calibration that feeds H₀ estimates from the local universe.
• Bulk-flow surveys. The original Seven Samurai survey, the ENEAR survey (da Costa et al. 2000), SFI++ and 6dFGSv (Springob et al. 2014) — all built their peculiar-velocity catalogues on FP or Tully-Fisher distances and used them to map bulk flows including the Great Attractor, the Shapley flow, and constraints on the dipole velocity of the Local Group.
• Cosmography. The Cosmicflows-3 and -4 catalogues (Tully, Courtois, Hoffman, Pomarède) reconstruct the velocity field of the local 200 Mpc and produce 3D density maps in which the Great Attractor, Perseus-Pisces, and Shapley appear as identifiable mass concentrations. FP distances are a key input.
• Galaxy-evolution diagnostics. Comparing the offset of cluster-elliptical FPs from the field FP gives a handle on environmental quenching and age. Comparing low-mass and high-mass ellipticals tracks the build-up of dark-matter fraction in their cores.
• Tests of modified gravity. MOND-flavoured theories predict different M/L scalings at low accelerations. The FP and especially the mass plane are stringent tests of those predictions.

Common pitfalls

• Confusing the plane with virial homology. The plane is the data; the virial expectation is a model that almost — but not quite — matches it. The 'tilt' is precisely the discrepancy and contains physical content.
• Forgetting that I_e and σ are aperture-dependent. Velocity dispersions measured in apertures of different physical size differ. A correction to a standard aperture (e.g. σ₀ at R = R_eff / 8) is needed before plugging into the plane. The same applies to I_e, which is by definition the mean within R_eff.
• Treating the slopes as universal. The slopes vary by ~ 10 percent between B, V, R, I, K bands because of M/L gradients in stellar populations. Use the calibration that matches the band you measured in.
• Mixing classical and pseudo-bulges. Pseudo-bulges (disk-like central regions in spirals) do not lie on the elliptical FP. Including them inflates the scatter. A clean sample requires morphological vetting.
• Ignoring rotation. Faint, low-mass ellipticals can be substantially rotation-supported. Replacing σ² with σ² + (V_rot)²/2 (the kinetic energy proxy) puts them back onto the plane.
• Selection biases at the high-luminosity end. Brightest cluster galaxies (BCGs) lie systematically above the FP for normal ellipticals because of cluster-scale dark matter and a different formation history. Excluding or separately fitting BCGs is standard practice in modern surveys.