Faber-Jackson Relation

Faber-Jackson relation (FJR) is an empirical scaling law in extragalactic astronomy that connects the intrinsic total luminosity L of a pressure-supported elliptical galaxy (or the classical spheroidal bulge of a lenticular/spiral galaxy) to approximately the fourth power of its central stellar line-of-sight velocity dispersion σ₀ (L ∝ σ₀^γ, with γ ≈ 4). Discovered in 1976 by the American astronomers Sandra M. Faber and Robert E. Jackson at Lick Observatory—one year prior to the discovery of the analogous Tully-Fisher relation (L ∝ V_rot⁴) for rotationally supported spiral disks—the Faber–Jackson relation provided the first quantitative kinematical tool to measure the distances, dynamical masses, and mass-to-light ratios (M / L) of early-type galaxies across galaxy clusters. In modern galaxy evolution theory, the Faber–Jackson relation is understood as a two-dimensional edge-on projection of the three-dimensional Fundamental Plane of elliptical galaxies (R_e, σ₀, I_e) and serves as the galactic host predecessor to the M_BH–σ₀ relation linking spheroid kinematics to central supermassive black hole masses.

Unlike spiral galaxies—where cold gas and stars orbit in a coherent flat plane at circular velocity V_rot—elliptical galaxies (such as Messier 87 and Messier 60 in the Virgo Cluster) contain very little cold neutral hydrogen and minimal net rotation. Instead, an elliptical galaxy is supported against gravitational collapse by the random, three-dimensional thermal-like orbital motions of hundreds of billions of old K and M giant stars orbiting within the collective gravitational potential Φ(r), governed by the collisionless Boltzmann and Jeans equations.

To measure the central stellar velocity dispersion σ₀ (typically integrated within an aperture of 1/8th of the galaxy's effective half-light radius, R_e / 8):

1. Absorption-Line Doppler Broadening: Astronomers obtain a high-resolution optical or near-infrared spectrum of the galaxy's bright core, focusing on prominent stellar atmospheric metal absorption features such as the Fraunhofer Mg b triplet (5167, 5173, 5184 Å), the Ca II H and K lines (3934, 3968 Å), the G-band (4300 Å), or the near-IR Ca II triplet (8498, 8542, 8662 Å).
2. Template Fourier Cross-Correlation / Penalized Pixel-Fitting (pPXF): Because the observed galaxy spectrum G(ln λ) is the sum of billions of individual red-giant spectra Doppler-shifted by each star's line-of-sight radial velocity v_z along the line of sight, the absorption lines are broadened compared to the narrow spectrum S(ln λ) of a single resting K-giant template star. Deconvolving G(ln λ) = S(ln λ) ∗ B(v_z) via the Fourier quotient method (Sargent et al., 1977; Tonry & Davis, 1979) or penalized pixel-fitting (Michele Cappellari) extracts the Gaussian line-of-sight velocity dispersion σ₀, which ranges from σ₀ ≈ 40–60 km/s in dwarf ellipticals up to σ₀ ≈ 300–400 km/s in giant cD cluster BCGs (Brightest Cluster Galaxies).

Why does the luminosity L of an elliptical galaxy scale roughly with the fourth power of its velocity dispersion σ₀?

Applying the scalar Virial Theorem (2K + W = 0) to a self-gravitating spheroidal stellar system of total dynamical mass M_dyn, effective half-light radius R_e, and three-dimensional velocity dispersion ⟨v²⟩ = 3σ₀² yields:

• M_dyn = k_v · (σ₀² R_e) / G (where k_v ≈ 4 to 5 is a dimensionless virial structure constant that depends mildly on the galaxy's Sérsic index n).

Squaring the velocity dispersion expression to introduce σ₀⁴, substituting the mean effective surface brightness I_e = L / (2π R_e²) to eliminate R_e, and replacing M_dyn with L · (M_dyn / L) yields the exact virial scaling relation:

• L = [ k_v² / (2π G²) ] · [ I_e · (M_dyn / L)² ]⁻¹ · σ₀⁴.

If all elliptical galaxies were strict homologous systems sharing a constant mean surface brightness I_e and a universal stellar + dark-matter mass-to-light ratio (M_dyn / L = constant), then L ∝ σ₀⁴ would hold with zero intrinsic scatter! In practice, however, observations across 5 orders of magnitude in early-type galaxy luminosity reveal systematic curvature and scatter (~0.8–1.0 magnitudes in M_V, corresponding to ~35–45% distance error per galaxy using the raw Faber–Jackson relation alone):

Why does the two-parameter Faber–Jackson relation (L, σ₀) have twice the intrinsic scatter (~0.8 mag) of the spiral Tully-Fisher relation (~0.35 mag)? In 1987, two independent research teams—the "Seven Samurai" (Alan Dressler, Donald Lynden-Bell, David Burstein, Roger Davies, Sandra Faber, Roberto Terlevich, and Gary Wegner) and George Djorgovski & Marc Davis—discovered that elliptical galaxies do not populate a one-dimensional line in parameter space, but rather lie on a tight two-dimensional plane embedded in the three-dimensional logarithmic space of (effective radius log R_e, central velocity dispersion log σ₀, and mean effective surface brightness log I_e), known as the Fundamental Plane (FP):

• R_e ∝ σ₀^{1.24 ± 0.07} · I_e^{−0.82 ± 0.02} (or in logarithmic form: log R_e = 1.24 log σ₀ − 0.82 log I_e + constant).

Because an elliptical galaxy's total luminosity is L = 2π I_e R_e², the Faber–Jackson relation (L vs. σ₀) is simply an oblique projection of the Fundamental Plane that ignores variations in surface brightness I_e (or radius R_e) at fixed σ₀! Including the third parameter (I_e or R_e via the D_n–σ relation of Dressler et al.) collapses the scatter in galaxy distance measurements from ~40% down to ~15–18% per galaxy—the exact breakthrough that enabled the Seven Samurai to discover the Great Attractor, a large supercluster gravitational anomaly pulling the Local Group and Hydra–Centaurus superclusters at ~600 km/s relative to the cosmic microwave background rest frame.

Furthermore, notice that the empirical Fundamental Plane exponents (1.24 and −0.82) deviate from the pure virial theorem prediction (R_e ∝ σ₀^2.0 · I_e^−1.0). This discrepancy—known as the Tilt of the Fundamental Plane—proves that the dynamical mass-to-light ratio (M_dyn / L) of elliptical galaxies is not constant, but increases with galaxy mass as (M_dyn / L) ∝ M_dyn^0.20 ∝ L^0.25, driven by three combined effects: (1) a higher fraction of central dark matter within R_e in giant ellipticals, (2) higher stellar metallicity [Z/H] and older stellar ages in high-mass spheroids, and (3) a more bottom-heavy stellar Initial Mass Function (IMF, enriched in dim low-mass M dwarfs) in high-σ₀ cores (van Dokkum & Conroy, 2010).

In 2000, Laura Ferrarese & David Merritt and Karl Gebhardt, Sandra Faber, and collaborators discovered that the mass M_BH of the central supermassive black hole residing at the heart of an elliptical galaxy or spiral bulge correlates more closely with the spheroid's central stellar velocity dispersion σ₀ than with its total luminosity L:

• M_BH ∝ σ₀^4.4–5.1 (the M_BH–σ₀ relation).

Combining the Faber–Jackson relation (L_bulge ∝ σ₀⁴) with the M_BH–σ₀ relation (M_BH ∝ σ₀^4–5) directly yields the Magorrian relation (M_BH ≈ 0.0015 – 0.005 M_bulge). Theoretical models of active galactic nucleus (AGN) quasar feedback (Andrew King, 2003; Silk & Rees, 1998) explain both scaling laws simultaneously: when a growing supermassive black hole accretes gas near the Eddington limit, its momentum- or energy-driven radiation winds sweep the surrounding interstellar gas out of the galaxy's gravitational potential well once the black hole reaches a critical mass M_BH,crit ∝ f_gas σ₀⁴ / (G² κ), simultaneously shutting down further star formation across the spheroid and locking in both the M_BH–σ₀ relation and the Faber–Jackson relation L ∝ σ₀⁴.

1. Faber, S. M., & Jackson, R. E. (1976). Velocity dispersions and mass-to-light ratios for elliptical galaxies. The Astrophysical Journal, 204, 668–683. https://doi.org/10.1086/154215
2. Dressler, A., Lynden-Bell, D., Burstein, D., Davies, R. L., Faber, S. M., Terlevich, R., & Wegner, G. (1987). Spectroscopy and photometry of elliptical galaxies. I. A new distance estimator. The Astrophysical Journal, 313, 42–58. https://doi.org/10.1086/164947
3. Djorgovski, S., & Davis, M. (1987). Fundamental properties of elliptical galaxies. The Astrophysical Journal, 313, 59–68. https://doi.org/10.1086/164948
4. Gebhardt, K., Bender, R., Bower, G., Dressler, A., Faber, S. M., Filippenko, A. V., ... & Richstone, D. (2000). A relationship between nuclear black hole mass and galaxy velocity dispersion. The Astrophysical Journal, 539(1), L13–L16. https://doi.org/10.1086/312840