Michie-King anisotropic model

The isotropic King model is the workhorse truncated globular-cluster model, but real clusters show radial velocity anisotropy in their outskirts. The Michie-King model adds Michie’s (1963) angular-momentum term to King’s (1966) lowered-Maxwellian cutoff, giving a one-parameter family (the anisotropy radius $r_a$) of self-consistent, tidally truncated, radially anisotropic models.

The distribution function¶

↗ model card

with $E=\tfrac12 v^2 + \Phi$, $J = r v_t$. The Gaussian $\exp(-J^2/2r_a^2\sigma^2)$ depopulates high-angular-momentum (tangential) orbits — increasingly so at large $r$ (since $J^2\propto r^2$) — so the velocity ellipsoid is isotropic at the centre and becomes radial outward:

Here $\sigma_t^2$ is a single tangential component (the convention used throughout progenax). Binney & Tremaine write $\beta = 1 - \sigma_t^2/(2\sigma_r^2)$ with $\sigma_t^2 \equiv \sigma_\theta^2 + \sigma_\phi^2$ the two-component tangential dispersion; the factor of 2 cancels, so both give the same $\beta(r)$ (see the Merritt 1985 note).

Unlike the Osipkov-Merritt construction (which holds a given density fixed and inverts for $f$), Michie specifies $f$ and re-solves Poisson, so the density itself changes — hence $\beta(r)$ is the DF-implied profile, not the OM form $r^2/(r^2+r_a^2)$.

Self-consistency¶

The density is the velocity integral of (1), which depends on radius explicitly through the anisotropy term, giving a radius-dependent King ODE — the King-1966 nondimensionalization of Michie (1963) Eq. 5.8, whose anisotropy weight $e^{-Cz^2\eta^2(1-\mu^2)}$ is our $e^{-J^2/2r_a^2\sigma^2}$ ($\xi=r/r_c$, $\hat r_a = r_a/r_c$):

↗ model card

progenax solves this with diffrax (differentiable), the velocity scale fixed self-consistently as $\sigma^2 = GM/(9 r_c \mu)$ — so the ICs are virial ($Q\approx0.5$) without any external rescale.

Usage¶

from jaxstro.units import STELLAR
from progenax import MichieProfile, MichieVelocityDF

profile = MichieProfile.from_W0_rc(W0=7.0, r_c=1.0, r_a=8.0)   # positions
df = MichieVelocityDF(W0=7.0, r_c=1.0, r_a=8.0)               # velocities (2-D sampler)

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

Velocities are drawn by a 2-D marginal-then-conditional inverse-CDF over $(v_r, v_t)$ (no Osipkov-Merritt stretch trick — (1) is not a function of one integral). The whole pipeline (ODE solve + sampler) is JIT-compatible and differentiable in $W_0$, $r_c$, $r_a$ (FD-verified).

Validation¶

• Isotropic limit — $r_a\to\infty$ reproduces the validated King profile and DF (matching $\xi_t$ and density).

• Anisotropy — realised $\beta(r)$ isotropic at the centre, increasing outward.

• Virial — $Q=T/|V|\approx0.5$ unscaled.

• Differentiability — finite, FD-matching gradients w.r.t. the model parameters.

Implementation, validation & references¶

• In code: src/progenax/kinematics/michie_df.py (the 2-D marginal-then-conditional sampler) paired with src/progenax/profiles/michie.py (solve_michie_profile, MichieProfile) — see the kinematics API and the profiles API.

• Validated in: Michie anisotropy — the isotropic-limit (→ King), the realised $\beta(r)$ profile, the unscaled virial $Q \approx 0.5$, and FD-matching gradients.

• Primary sources: Michie (1963) (the angular-momentum anisotropy term) and King (1966) (the lowered-Maxwellian cutoff) — full notes in the Michie (1963) and King (1966) per-paper bibliography pages.