King (1966)

Abstract (paraphrased)¶

Presents spatially-limited dynamical models of star clusters from steady-state solutions of the Fokker–Planck equation, corresponding to the tidal cutoff imposed by the Galaxy. The velocity distribution is a lowered (truncated) Maxwellian; solving Poisson’s equation self-consistently yields a family of single-mass, isotropic density profiles parameterised by the dimensionless central potential $W_0$. The projected densities resemble those of open clusters, globular clusters, and elliptical galaxies.

Model in brief (verified against the paper)¶

• Distribution function (Eq. 3): at the centre $f(0,v) = k[\exp(-j^2 v^2) - \exp(-j^2 v_e^2)]$ — a Maxwellian lowered by a constant so $f$ and the density vanish at the escape velocity $v_e^2 = -2V$ (Eq. 5). Energy $E = \tfrac12 v^2 + V(r)$ (Eq. 4).

• Dimensionless potential $W \equiv -2j^2 V$ (Eq. 9), with central value $W_0$; $W \to 0$ at the tidal boundary. The density follows $\Psi(W) = e^{W}\!\int_0^{W} e^{-\eta}\eta^{1/2}\,d\eta$ (Eq. 17), proportional to $\rho$.

• Poisson equation in the dimensionless radius $R = r/r_c$ (Eq. 13): $\dfrac{d^2 W}{dR^2} + \dfrac{2}{R}\dfrac{dW}{dR} = -9\,\dfrac{\rho}{\rho_0}$ (Eq. 16). The factor of 9 comes from fixing the central $\nabla^2 W = -9$, i.e. $8\pi G j^2 r_c^2 \rho_0 = 9$ (Eq. 15); King notes the scale factor $r_c$ is then very close to the observational “core radius”. Central boundary conditions: $dW/dR = 0$ and $\lim_{R\to 0}(2/R)\,dW/dR = -6$ (Eq. 18).

• Concentration. The model truncates at the radius $R = r_t/r_c$ where $W$ (and $\rho$) reach zero. The concentration parameter is $c \equiv r_t/r_c$, and the quantity usually quoted is $\log_{10} c = \log_{10}(r_t/r_c)$ (King’s Table II tabulates both). Higher $W_0$ ⇒ higher concentration; as $W_0 \to \infty$ the model approaches the isothermal sphere.

Table II concentrations (King 1966, p. 73)¶

Verbatim from Table II — the canonical $\log c$ values progenax validates against:

progenax.profiles.KingProfile.from_W0_rc reproduces this $\log c(W_0)$ to $\le 0.03$ — see King profile validation. (Note Table II begins at $W_0 = 2.5$; values below that are not tabulated in the paper.)

Use in progenax¶

• King profile — full ODE derivation and progenax implementation

• King velocity distribution functions — lowered-Maxwellian velocity DF and sampling

• progenax.profiles.KingProfile — tidally-truncated spatial profile (from_W0_rc(W0, r_c)), solving Eq. 16 with the factor-of-9 nondimensionalisation

• The dimensionless potential $W(R)$ on the solution grid is the model’s true relative potential, reused for King energetics elsewhere in the package.

Notes¶

The right choice for old globular clusters where the tidal-radius constraint is observationally well-defined. Gieles & Zocchi (2015) (LIMEPY) generalises the lowered-isothermal family to multi-mass, anisotropic, and rotating populations.