King velocity distribution functions

The King velocity DF is the lowered-Maxwellian that defines the King spatial profile in the first place. Unlike Plummer — where the DF follows from the density profile via Eddington inversion — King’s DF is the input, and the density profile follows from integrating the DF over velocity space (King profile). Sampling King velocities is therefore conceptually simpler than sampling Plummer velocities: at each particle’s position, draw from the truncated Maxwellian directly.

The lowered-Maxwellian DF¶

For a particle of specific energy $E$ at position $\mathbf{r}$, the King DF is

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where $\sigma_0$ is the central one-dimensional velocity dispersion, $\Phi_t \equiv \Phi(r_t)$ is the potential at the tidal radius, and $\rho_1$ is a normalisation constant. The “-1” subtraction is the lowering: it ensures the DF vanishes at $E = \Phi_t$ rather than extending exponentially to $E \to \infty$ as a pure Maxwellian would.

In terms of the dimensionless central potential $W_0 \equiv (\Phi_t - \Phi(0))/\sigma_0^2$ (King profile, (1)), the DF becomes

at any radius where the local dimensionless potential is $W$. The escape speed at the tidal radius is zero ($W_t = 0$, $v_{\mathrm{esc}} = 0$), so no particles populate that boundary — the cluster is sharply truncated.

Marginal speed distribution¶

Integrating the DF over velocity directions gives the speed distribution at fixed position:

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The $v^2$ prefactor is the spherical-shell volume element in velocity space. The bracketed factor is positive throughout $0 \le v \le \sigma_0\sqrt{2W}$ and zero at the upper bound — which means inverse-CDF sampling on $f(v \mid W)$ produces velocities in $[0, v_{\mathrm{esc}}]$ with no rejection.

Sampling¶

progenax samples King velocities via a per-particle inverse-CDF on (3):

# At each particle's radius r, look up W(r) from the King ODE solution
W_per_particle = jnp.interp(radii / r_c, xi_grid, psi_grid, left=W0, right=0.0)

# Per particle: build a 1-D speed CDF on [0, sqrt(2W)] and invert it (vmap'd)
def sample_unit_speed(key, W):
    u_grid = jnp.linspace(0.0, jnp.sqrt(2.0 * W), N_SPEED_GRID)   # 256 points
    g = u_grid**2 * (jnp.exp(W - u_grid**2 / 2.0) - 1.0)         # eq:king-fv
    cdf = jnp.cumsum(0.5 * (g[1:] + g[:-1])) * du                # trapezoid
    return jnp.interp(jax.random.uniform(key), cdf / cdf[-1], u_grid)

u = jax.vmap(sample_unit_speed)(keys, W_per_particle)            # speed / sigma_0
v_vec = (sigma_0 * u)[:, None] * isotropic_unit_vector(key2, N)

Each particle builds its own fixed-size (256-point) speed grid scaled to its local escape speed $\sigma_0\sqrt{2W}$ and inverts the trapezoidal CDF by interpolation — there is no precomputed cross-$W$ table. The grid size is fixed and the operations are jnp.interp/vmap only (no while_loop), so the whole pipeline is JIT-compatible and differentiable in $r_h$ via the chain $r_h \to r_c \to \sigma_0 \to v$.

Velocity dispersion profile¶

The King DF’s velocity dispersion at radius $r$ is

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(in units where $\sigma_0 = 1$). The integrals do not have closed form in general, but progenax precomputes $\sigma(W)$ on the same $W$-grid used for the speed-CDF table inside the sampler. The current public KingVelocityDF API exposes sample_velocities; it does not export a separate velocity-dispersion method. The dispersion is monotonically increasing in $W$: $\sigma^2 \to 0$ as $W \to 0$ and $\sigma^2 \to 3\sigma_0^2$ as $W \to \infty$ (the untruncated-Maxwellian limit). Since $W$ decreases outward — from $W_0$ at the centre to 0 at the tidal radius — $\sigma$ falls with radius and vanishes at the tidal boundary, where the escape speed $\to 0$. King clusters therefore have cold outskirts, in contrast to a Plummer sphere whose dispersion also declines outward but stays finite at all radii.

Check yourself¶

Pairing with the King profile¶

The King DF is in equilibrium with the King density profile by construction: both are derived from the same lowered-isothermal Maxwellian. Pair them by using the same King concentration and tidal radius:

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

profile = KingProfile.from_W0_rc(W0=7.0, r_c=1.0)
df = KingVelocityDF(W0=7.0, r_c=1.0)   # r_t is derived from W0 internally

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

KingVelocityDF takes only (W0, r_c) — it re-solves the King ODE internally and derives the tidal radius from $W_0$, so there is no r_t argument. Using the same W0 and r_c in the profile and DF keeps them consistent. Pairing a King profile at $W_0 = 7$ with a King DF at $W_0 = 5$ would produce a mismatched non-equilibrium IC.

Differentiability¶

The King velocity DF is differentiable in $r_c$ (via $\sigma_0$) and in the total mass $M_{\rm tot}$ ($\sigma_0\propto\sqrt{GM}$). It is also differentiable in $W_0$: diffrax propagates $\partial\psi/\partial W_0$ through the ODE solve, so the dispersion profile and other shape observables carry correct $W_0$ gradients (gradient-validated in King profile validation). The lone non-differentiable quantity is the scalar tidal radius $r_t$, whose $W_0$-derivative is zeroed by the argmax zero-crossing in _find_tidal_radius; inference should therefore use the profile shape rather than the scalar $r_t$.

Consequently progenax supports joint gradient-based / HMC inference of the King structural parameters $(W_0, r_c, M_{\rm tot})$ directly — e.g. fitting a cluster’s number-density and velocity-dispersion profiles. The lowered-model family formalized by Gieles & Zocchi (2015) (where the truncation parameter $g$ enters analytically) remains the natural extension for multi-mass / anisotropic generalisations; progenax plans to implement this family natively as its own differentiable generalization (see The differentiable lowered-model family (Engine A)), but it is not yet available.

Domain of validity¶

1. Single-mass equilibrium. Multi-mass equilibrium is described by the lowered-model family formalized by Gieles & Zocchi (2015), not by the standard King DF here.

2. Spherical and isotropic by default. Anisotropic and rotating variants live in Anisotropy and rotation.

3. Tidal cutoff is sharp. Real clusters have a smooth transition from the bound population to the tidal tail; King’s mathematical cutoff at $E = \Phi_t$ is an idealisation. For studies where the tidal-tail kinematics matter, post-evolution analysis is the right approach.

Implementation, validation & references¶

• In code: src/progenax/kinematics/king_df.py (the per-particle lowered-Maxwellian speed sampler; the King ODE solve it shares with the profile is in src/progenax/profiles/king.py) — see the KingVelocityDF API.

• Validated in: King profile — the regression suite that locks the true-DF equilibrium ($Q_{\mathrm{vir}} \approx 0.5$ unscaled) and the $W_0$ gradients.

• Primary sources: the lowered-isothermal model is King (1966); the multi-mass LIMEPY generalisation is Gieles & Zocchi (2015), and the diffrax integration follows the standard Aarseth et al. (1974) numerical prescription — full notes in the bibliography.