King profile

The King profile¶

The King model King, 1966 is the canonical tidally truncated spatial density profile for old globular clusters. Unlike Plummer, it has no closed-form $\rho(r)$ — the profile is defined implicitly via an ODE in the dimensionless gravitational potential. In exchange, King provides what Plummer cannot: a finite outer radius matching the observed tidal cutoff, a single dimensionless concentration parameter $W_0$ that captures the full cluster geometry, and the family of King (1966) model density profiles fit to most Galactic globular clusters in the second half of the 20th century.

This chapter derives the King profile from its defining lowered-isothermal DF, sets up the ODE that progenax integrates with diffrax, lists the $W_0 \to (r_c, r_t, c, r_h)$ mapping, and notes the Gieles & Zocchi (2015) lowered-model-family generalisation — progenax’s planned own differentiable extension when multi-mass anisotropy matters (not currently implemented).

The lowered-isothermal distribution function¶

The King profile starts from a distribution function, not a density profile. Define a dimensionless potential

where $\sigma_0$ is the central one-dimensional velocity dispersion and $r_t$ is the tidal (truncation) radius — the location where the gravitational potential of the cluster matches the host galaxy’s tidal potential. By construction $W(r_t) = 0$ and $W$ increases monotonically inward, with $W_0 \equiv W(0)$ the central value.

The lowered-isothermal DF is

↗ model card

The “-1” in the bracket is the lowering: it removes the high-energy tail of the Maxwell-Boltzmann distribution that would otherwise extend to $E \to \infty$, producing a finite cluster with a sharp tidal cutoff. King (1966) derived this form to model the observation that globular clusters appear truncated rather than extending as infinite-mass isothermal spheres.

Integrating $f(E)$ over velocity space gives the density profile:

↗ model card

where $\mathrm{erf}$ is the error function. $\rho(W = 0) = 0$ (cluster edge) and $\rho(W = W_0) = \rho_0$ at the centre. Note that $\rho$ is not yet a function of radius — that requires solving the Poisson equation for $W(r)$.

The defining ODE¶

Combining (3) with the spherical Poisson equation,

and non-dimensionalising via $\xi = r / r_c$ where $r_c$ is the King core radius

↗ model card

yields the dimensionless King ODE:

with boundary conditions $W(0) = W_0$ and $W'(0) = 0$. The ODE integrates outward from $\xi = 0$ until $W$ reaches zero — that defines the tidal radius $r_t = \xi_t \cdot r_c$. The $W_0$ parameter fully determines the cluster’s dimensionless structure.

The concentration parameter $c$¶

The shape of a King cluster is captured by the dimensionless concentration parameter

↗ model card

For Galactic globular clusters, observational fits give $c$ in the range $\sim 0.7$–2.5, with $W_0 \sim 5$–9 for most clusters (the high-concentration / near-core-collapse tail reaching higher):

The $r_h$ column shows that — unlike Plummer’s $a/r_h \approx 0.766$ constant — the King profile’s scale-to-half-mass mapping depends on $W_0$. progenax stores this mapping as a precomputed lookup table and exposes it through progenax.profiles.solve_king_profile(W0), which returns the 3-tuple (xi_grid, psi_clamped, psi_raw): the dimensionless radius grid, the potential clamped to $\psi \ge 0$ (used for the density and CDF), and the unclamped potential (negative just past $r_t$; used only for the differentiable tidal-radius crossing). The profile implementation consumes the first two.

Inverse-CDF sampling¶

Once $W(\xi)$ is integrated, the cumulative mass

is computed by trapezoidal integration on the same grid the ODE solver produced. progenax then inverts this CDF to sample radii:

u = jax.random.uniform(key, (N,))                        # u ~ U(0,1)
xi_samples = jnp.interp(u, cumulative_mass_normalized, xi_grid)
r_samples = xi_samples * r_c                              # r in physical units

The whole chain is differentiable in $r_c$ (and therefore in $r_h$ via the $r_h(W_0, r_c)$ mapping). It is also differentiable in $W_0$: diffrax propagates $\partial\psi/\partial W_0$ through the ODE solve, so the density profile and any shape-based observable carry correct $W_0$ gradients (validated against finite differences in King profile validation). The one exception is the scalar tidal radius $r_t$: its $W_0$-derivative is zeroed by the argmax zero-crossing in _find_tidal_radius, so $W_0$ inference should target the profile shape (differentiable) rather than the scalar $r_t$ readout. This makes joint gradient-based / HMC inference of $(W_0, r_c)$ — and, with the velocity DF, $M_{\rm tot}$ — feasible.

The lowered-model family¶

Gieles & Zocchi (2015) showed that King is one member of a one-parameter family of lowered-isothermal models: a continuous truncation parameter $g$ interpolates between Woolley ($g = 0$), King ($g = 1$), and Wilson ($g = 2$) models, and the framework extends to multi-mass and anisotropic clusters. progenax will implement this family natively as its own differentiable, JAX-native generalization — so that $g$ (and the multi-mass / anisotropy parameters) can be inferred — rather than depending on the external (non-differentiable) limepy package. See the roadmap.

For most production work — single-mass populations or coarse mass binning — the released King profile suffices. The unified family becomes the right tool when the science target is the radial velocity-anisotropy profile or the multi-mass equipartition state.

Tidal physics¶

The King profile’s “tidal radius” $r_t$ is a defining property of the model — the place where $W = 0$ and the cluster terminates. In real space, the physical tidal radius (Jacobi radius) of a cluster on a circular orbit at galactocentric radius $R$ is

progenax.tidal.jacobi_radius computes $r_J$ given a Galactic mass model; equating $r_t = r_J$ then fixes the King concentration self-consistently. Tidal physics is documented in detail at Tidal physics.

Implementation in progenax¶

from progenax.profiles import KingProfile, solve_king_profile
from progenax.kinematics import KingVelocityDF
from jaxstro.units import STELLAR

# Solve the King ODE once for W_0 = 7 (returns a 3-tuple; psi_raw unused here)
xi_grid, psi_grid, _ = solve_king_profile(W0=7.0)   # diffrax Tsit5
profile = KingProfile.from_W0_rc(W0=7.0, r_c=1.0)
df = KingVelocityDF(W0=7.0, r_c=1.0)                 # r_t derived from W0

masses = jnp.ones(1000)
positions = profile.sample_positions(masses, key_pos)
velocities = df.sample_velocities(positions, masses, key_vel, G=STELLAR.G)

The solve_king_profile call is the ODE helper for the King profile; it returns the 3-tuple (xi_grid, psi_clamped, psi_raw) — here we keep the clamped potential and discard psi_raw. KingVelocityDF takes only (W0, r_c) and re-solves the ODE internally (deriving $r_t$ from $W_0$), so no r_t argument is passed.

Domain of validity and limitations¶

1. Single-mass only. The lowered-isothermal DF assumes one mass species. Multi-mass clusters in equilibrium have radially-varying velocity dispersions per mass group; the planned lowered-model family will handle this natively, while progenax’s standard King does not.

2. Spherical and isotropic. No rotation, no anisotropy. For anisotropic generalisations see Anisotropy and rotation and the anisotropic Michie model; rotating King-like profiles are part of the planned lowered-model family.

3. No primordial substructure. King is a smooth equilibrium profile. Turbulent/fractal substructure — relevant to cool-clumpy initial conditions Allison et al., 2009 — is provided separately by the experimental gravoturb subsystem (Fractal substructure), not by the released King profile, and deliberately breaks the equilibrium assumption.

4. Tidal radius is fixed at IC time. A cluster’s tidal radius evolves with its galactocentric orbit; progenax’s King IC fixes $r_t$ at $t = 0$ and the user must apply post-evolution truncation manually if the orbit drives the cluster across a tidal-radius threshold.

Implementation, validation & references¶

• In code: src/progenax/profiles/king.py — the King ODE solve (solve_king_profile), $W_0 \to (r_c, r_t, c)$ mapping, and inverse-CDF sampling; the paired DF is in src/progenax/kinematics/king_df.py. See the KingProfile API.

• Validated in: King profile — validates $c(W_0)$ against King (1966) Table II and the volume density against an independent direct-velocity-integral oracle.

• Primary sources: King (1966) (the lowered-isothermal model), with the Gieles & Zocchi (2015) LIMEPY generalisation (see the lowered-model family); the integration scheme follows the standard Aarseth et al. (1974) approach (progenax uses diffrax as the JAX-native ODE backend). Full notes in the bibliography.