King Profile (Astrophysics)

A King profile (or King model) refers to two closely related analytical frameworks introduced by American astronomer Ivan R. King in 1962 and 1966 that characterize the radial surface-brightness distribution, three-dimensional space density, and phase-space structure of self-gravitating stellar systems—most notably globular clusters, old open clusters, and dwarf spheroidal galaxies. In an isolated, unbounded space, a gas or stellar system in thermal equilibrium follows the classical Maxwell–Boltzmann singular isothermal sphere, whose density falls asymptotically as ρ(r) ∝ r⁻² at large radii and therefore possesses infinite total mass, whereas the n = 5 polytropic Plummer model possesses finite mass but lacks a finite tidal boundary. Real star clusters orbit inside the gravitational potential of a host galaxy such as the Milky Way (often modeled with an axisymmetric Miyamoto–Nagai potential or integrable Stäckel potential), which strips away any star wandering beyond a finite Jacobi tidal boundary while two-body relaxation drives steady stellar evaporation. King resolved the infinite-mass divergence of the isothermal sphere by introducing a "lowered isothermal" phase-space distribution function solving the collisionless Boltzmann equation and Jeans equations (related to the Eddington inversion formula) whose density and velocity dispersion drop smoothly to zero at a finite tidal truncation radius (r_t), contrasting with the cuspy Dehnen profile, Hernquist profile, and Jaffe profile of elliptical galaxies or the Einasto profile and cored Burkert profile of dark-matter halos.

Before constructing his dynamical phase-space models, Ivan King (1962) analyzed photographic star counts and surface photometry across Milky Way globular clusters, open clusters, and dwarf elliptical galaxies. He demonstrated that their sky-projected surface number density or surface brightness I(R) as a function of projected radius R from the cluster center is accurately represented by a three-parameter empirical law characterized by a flat central core and a sharp outer cutoff:

I(R) = k { [ 1 + (R / r_c)² ]^−1/2 − [ 1 + (r_t / r_c)² ]^−1/2 }² (for R ≤ r_t)

Here:

• Core Radius (r_c): The inner scale radius at which the projected surface brightness in a high-concentration cluster (r_t ≫ r_c) drops to approximately half (specifically 2⁻¹ = 0.50 in the Hubble-modified limit) of its central value I(0). Typical Milky Way globular clusters exhibit r_c ≈ 0.5 to 3.0 parsecs.
• Tidal Radius (r_t): The outer truncation radius where the projected surface density I(r_t) and three-dimensional space density ρ(r_t) vanish completely, set physically by the tidal field of the host galaxy (r_t ≈ 30 to 100 pc for globular clusters).
• Concentration Parameter (c): Defined logarithmically as c = log₁₀(r_t / r_c), which measures the structural compactness of the cluster across a single dimensionless sequence.

Deprojecting the empirical 1962 surface-brightness law into three-dimensional spherical coordinates via an Abel integral inversion yields the analytical 3D space density ρ(r):

ρ(r) = [ k / (π r_c z²) ] [ z⁻¹ arccos(z) − (1 − z²)^1/2 ] where z(r) = [ (1 + r² / r_c²) / (1 + r_t² / r_c²) ]^1/2

In the limit of infinite tidal radius (r_t → ∞), the empirical King law reduces to the modified Hubble law I(R) = I₀ [ 1 + (R / r_c)² ]⁻¹, which serves as the standard baseline for the non-singular isothermal core of galaxies and galaxy clusters.

In 1966, Ivan King derived the physical foundation for his empirical profile directly from the Fokker–Planck equation governing two-body gravitational encounters in a star cluster (originally investigated by Michie in 1963). Inside a relaxed cluster, small-angle Coulomb scattering continually pushes the stellar velocity distribution toward a Maxwellian Gaussian exp(−E / σ²), while any star accelerated above the tidal escape energy E_esc = Φ(r_t) escapes across the Jacobi surface within a single crossing time. To satisfy the boundary condition that the phase-space distribution function f(E) must equal zero at and above the escape energy (E ≥ Φ(r_t)), King defined the relative gravitational potential Ψ(r) = Φ(r_t) − Φ(r) and relative binding energy ε = E_esc − E = Ψ(r) − ½ v², and subtracted a constant offset from the Maxwellian exponential to obtain the King lowered isothermal distribution:

f_King(ε) = ρ₁ (2π σ²)^−3/2 [ exp(ε / σ²) − 1 ] (for ε > 0; and f_King = 0 for ε ≤ 0)

Integrating f_King(ε) over all bound velocities from v = 0 to the local escape speed v_esc(r) = [2 Ψ(r)]^1/2 yields the three-dimensional mass density ρ(Ψ) as an analytical function of the dimensionless relative potential W(r) = Ψ(r) / σ²:

ρ(W) = ρ₁ [ exp(W) erf(W^1/2) − (4 W / π)^1/2 (1 + 2W / 3) ]

Substituting this density ρ(W) into the spherically symmetric Poisson equation ∇²Ψ = −4πG ρ(Ψ) and introducing the King radius r₀ ≡ (9 σ² / 4πG ρ₀)^1/2 (which closely approximates the empirical core radius r_c for W₀ ≥ 5) produces a single second-order nonlinear ordinary differential equation parameterized solely by the dimensionless central potential depth W₀ = Ψ(0) / σ²:

d²W / d(r/r₀)² + (2 r₀ / r) dW / d(r/r₀) = −9 [ ρ(W) / ρ(W₀) ]

Integrating outward from the boundary conditions W(0) = W₀ and (dW/dr)_r=0 = 0 until W(r_t) = 0 uniquely fixes the tidal radius r_t/r₀ and concentration c = log₁₀(r_t/r₀) as a one-to-one monotonic function of W₀.

In the deep central core where W(r) ≈ W₀ ≫ 1, the escape velocity v_esc = (2W)^1/2 σ greatly exceeds the thermal velocity dispersion σ, so exp(W) ≫ 1 and the King model behaves like a classical isothermal sphere satisfying the virial theorem. Moving outward toward the tidal boundary (r → r_t, W → 0), the truncation at v_esc(r) removes the high-velocity tail of the Maxwellian distribution, causing the local one-dimensional velocity dispersion σ_loc²(r) = (1/5) W(r) σ² to fall smoothly to zero at r = r_t.

Over subsequent decades, several dynamical generalizations of the single-mass isotropic King (1966) model were developed to capture additional cluster physics:

• Michie–King Anisotropic Models (Michie, 1963; Gunn & Griffin, 1979): Multiply f_King(ε) by an angular momentum factor exp(−J² / 2r_a²σ²) so that orbits beyond the anisotropy radius r_a become radially biased, matching stars scattered from the core into the halo before phase mixing and tidal stripping remove them.
• Multi-Mass King Models (Da Costa & Freeman, 1976): Assign distinct mass bins m_i with velocity dispersions scaled toward kinetic energy equipartition (σ_i² ∝ m_i⁻¹), accurately modeling mass segregation and dynamical friction of heavy white dwarfs, neutron stars, and black holes into the cluster core.
• Wilson Models (1975): Subtract a linear energy term [exp(ε/σ²) − 1 − ε/σ²] so the derivative df/dε also vanishes at ε = 0, producing more extended outer halos that match clusters surrounded by nascent tidal tails.

When the central potential of a single-mass King sequence exceeds W₀ ≈ 7.4 to 8.5 (c ≈ 2.12, corresponding to Antonov's density contrast threshold ρ₀ / ρ(r_t) > 708.81), the negative heat capacity of the self-gravitating core triggers the gravothermal catastrophe (Lynden-Bell & Wood, 1968). At this point, no stable equilibrium connects the core and halo, and the cluster undergoes runaway core collapse into a post-core-collapse power-law cusp (ρ ∝ r^−2.23).

1. King, I. (1962). "The structure of star clusters. I. An empirical density law." The Astronomical Journal, 67, 471–485. https://doi.org/10.1086/108756
2. King, I. R. (1966). "The structure of star clusters. III. Some simple dynamical models." The Astronomical Journal, 71, 64–75. https://doi.org/10.1086/109857
3. Lynden-Bell, D., & Wood, R. (1968). "The gravo-thermal catastrophe in isothermal spheres and the onset of red-giant structure for stellar systems." Monthly Notices of the Royal Astronomical Society, 138(4), 495–525. https://doi.org/10.1093/mnras/138.4.495
4. Gunn, J. E., & Griffin, R. F. (1979). "Dynamical studies of globular clusters based on photoelectric radial velocities of individual stars. I. M3." The Astronomical Journal, 84, 752–773. https://doi.org/10.1086/112477