Kormendy Relation

The Kormendy relation is an empirical photometric scaling relation introduced by American astronomer John Kormendy in 1977 that connects the effective half-light radius (R_e) of elliptical galaxies and classical stellar bulges to their mean surface brightness (⟨μ⟩_e) or surface brightness at the effective radius (μ_e). Specifically, the relation demonstrates that as the physical size and total luminosity of a giant elliptical galaxy increase, its average surface intensity I_e decreases—corresponding to a numerically larger (fainter) surface brightness in magnitudes per square arcsecond. In modern extragalactic astrophysics, the Kormendy relation is recognized as the two-dimensional photometric projection of the three-dimensional Fundamental Plane of early-type galaxies, complementing the kinematic Faber–Jackson relation and providing a diagnostic boundary that separates dissipationless dry-merger giant ellipticals from supernova-sculpted dwarf spheroidal galaxies and disk-grown pseudobulges.

For an early-type galaxy whose sky-projected starlight follows a de Vaucouleurs R^1/4 law or a generalized Sérsic profile, let R_e denote the effective radius (or half-light radius) enclosing exactly half of the total integrated flux L_tot:

L(<R_e) = 2π ∫₀^R_e I(R) R dR = (1/2) L_tot

The mean effective surface intensity within R_e is ⟨I⟩_e = (L_tot / 2) / (π R_e²) = L_tot / (2π R_e²). Converting linear intensity I into logarithmic surface brightness ⟨μ⟩_e (measured in mag arcsec⁻²) via ⟨μ⟩_e = −2.5 log₁₀ ⟨I⟩_e + const, Kormendy (1977) established that bright elliptical galaxies and the classical bulges of lenticular (S0) and spiral galaxies obey a tight linear relation between ⟨μ⟩_e and log₁₀ R_e:

⟨μ⟩_e = α + β log₁₀(R_e / 1 kpc)

Across optical bandpasses (B, V, g, and r bands) in the local universe (z < 0.1), empirical calibrations from the Sloan Digital Sky Survey (SDSS) and nearby cluster samples yield a slope β ≈ 2.9 to 3.3 (with Kormendy's original 1977 B-band fit giving β ≈ 3.02 for μ_e). In linear physical units, a slope β ≈ 3.0 corresponds to the power-law scaling:

⟨I⟩_e ∝ R_e^−1.2 and L_tot = 2π R_e² ⟨I⟩_e ∝ R_e^0.8, or R_e ∝ L_tot^1.25

Thus, more luminous giant elliptical galaxies above the knee L* of the Schechter luminosity function are structurally more diffuse in projected surface density than compact intermediate-luminosity ellipticals such as M32.

The physical origin of the Kormendy relation—and its intrinsic scatter of ~0.3–0.4 dex in log₁₀ R_e—is understood by viewing it as an edge-on projection of the three-dimensional Fundamental Plane discovered independently by Djorgovski & Davis (1987) and Dressler et al. (1987). For a self-gravitating spheroid in dynamical equilibrium governed by the Jeans equations, the scalar virial theorem relates the total gravitational potential energy W to the internal stellar rotational and random velocity dispersion kinetic energy K via 2K + W = 0, which implies:

M_dyn = c_n (σ₀² R_e / G)

where σ₀ is the central line-of-sight stellar velocity dispersion and c_n is a dimensionless structural virial coefficient that depends on the Sérsic index n and orbital anisotropy. Substituting L_tot = 2π R_e² ⟨I⟩_e and introducing the dynamical mass-to-light ratio (M_dyn / L), the virial theorem yields:

R_e = [c_n / (2π G)] σ₀² ⟨I⟩_e⁻¹ (M_dyn / L)⁻¹

In empirical observations across early-type galaxies, the dynamical mass-to-light ratio increases systematically with mass (M_dyn/L ∝ M_dyn^0.2) due to older stellar populations, higher metallicity, and rising dark-matter fractions within R_e from the host Navarro–Frenk–White profile or Einasto profile halo, while structural non-homology (c_n varying with Sérsic index n) tilts the virial scaling into the observed Fundamental Plane:

log₁₀ R_e = a log₁₀ σ₀ + b ⟨μ⟩_e + c, with a ≈ 1.24, b ≈ 0.328 (so ⟨I⟩_e ∝ σ₀^−1.52 R_e^−1.22)

Projecting the Fundamental Plane onto the (L, σ₀) axes eliminates R_e to yield the Faber–Jackson relation (L ∝ σ₀⁴), whereas projecting it onto the (R_e, ⟨μ⟩_e) photometric plane averages over velocity dispersion σ₀ to yield the Kormendy relation (⟨I⟩_e ∝ R_e^−1.2).

When plotted across the full dynamic range of spheroidal stellar systems—from ultra-faint dwarf spheroidals (M_V ≈ −9) up to cD cluster giants (M_V ≈ −24)—the Kormendy diagram splits into two nearly perpendicular branches intersecting near M_V ≈ −18 (M_* ≈ 10^9.5 M_☉):

1. The Giant Elliptical & Classical Bulge Branch (M_V < −18): Larger galaxies have fainter mean surface brightness (⟨μ⟩_e increases with log₁₀ R_e, β ≈ +3.0). In hierarchical assembly, giant ellipticals grow predominantly via dissipationless ("dry") mergers of stellar systems after central black hole feedback (linked via the M–sigma relation) quenches cold gas cooling. By energy conservation during dry minor mergers, orbital energy is deposited into the outer stellar halo (Hernquist profile envelope), causing the effective radius to expand rapidly as R_e ∝ M² and driving the mean surface density M / R_e² sharply downward.
2. The Dwarf Spheroidal Branch (M_V > −18): Diffuse dwarf elliptical (dE) and dwarf spheroidal (dSph) galaxies exhibit the opposite trend: lower-luminosity, smaller dwarfs have fainter surface brightness (⟨μ⟩_e decreases or stays flat with increasing R_e). In these shallow gravitational potentials, supernova winds and ultraviolet photoionization expel large fractions of gas, puffing up the stellar distribution to low surface densities. By contrast, rare tidally stripped compact ellipticals (such as M32) remain on the high-density upward extrapolation of the classical Kormendy sequence.

In disk galaxies, the Kormendy relation also serves as a primary structural diagnostic for separating merger-built classical bulges (which lie directly on the elliptical Kormendy sequence with Sérsic index n > 2) from secularly evolved pseudobulges formed through internal bar instabilities and spiral torque inflows, which exhibit lower surface brightness at fixed R_e and disk-like n ≈ 1–2 profiles.

Why is the Kormendy relation useful as a cosmological surface brightness test?

Because bolometric surface brightness dims with cosmological redshift strictly as I(z) ∝ (1 + z)⁻⁴ (Tolman surface brightness dimming), measuring the zero-point shift Δ⟨μ⟩_e(z) of the Kormendy relation at fixed physical radius R_e in high-redshift galaxy clusters separates geometric (1 + z)⁻⁴ cosmological expansion from passive stellar population luminosity evolution.

Why do giant elliptical galaxies have lower average density than small ellipticals?

Small and intermediate-mass ellipticals (M_* ≈ 10¹⁰ M_☉) formed through gas-rich ("wet") mergers where radiative cooling allowed gas to sink into dense central starbursts before turning into stars. Giant ellipticals (M_* > 10¹¹ M_☉) subsequently grew via gas-poor ("dry") stellar mergers and central supermassive black hole binary scouring, which puff up R_e without adding dense central star formation.

How does the Kormendy relation distinguish classical bulges from pseudobulges?

Classical bulges assemble through rapid early mergers and follow the same virial scaling as elliptical galaxies on the Kormendy diagram. Pseudobulges grow slowly out of the galactic disk via bar-driven gas inflows; they retain rotational support, exponential (n ≈ 1–2) surface brightness profiles, and lie systematically below (fainter in ⟨μ⟩_e than) the elliptical Kormendy relation for their effective radius.

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