Michie (1963) - progenax — differentiable initial conditions for N-body simulations

Abstract (paraphrased)¶

Derives a distribution function $f(r,v,\mu;t)$ for an isolated spherical stellar system from the Boltzmann equation with Fokker–Planck encounters, self-consistently with Poisson’s equation. The orbits are not assumed isotropic: the model becomes increasingly radially anisotropic at large radius. The analysis focuses on the high-energy tail (stars near the escape energy), where the velocity-space flux is shown to be nearly constant, giving the energy cutoff. This is the origin of the anisotropic lowered-Maxwellian now universally combined with King’s (1966) cutoff as the “Michie–King” model.

The distribution function (verified against the paper, §4–5)¶

Michie writes (Eqs. 4.0–4.3, 5.0)

f(r,v,\mu;t) = A\,\exp\!\left[-\tfrac{m}{m_0}\left(\alpha E + \beta J^2\right)\right] Q, \qquad E = \tfrac12 v^2 + \Phi(r),\quad J^2 = r^2 v^2 (1-\mu^2),

(1)

with $\mu = \cos\theta$ ( $\theta$ the angle between $\mathbf r$ and $\mathbf v$ ), $\Phi(0)=0$ , and $\alpha,\beta$ model constants. The two ingredients:

Anisotropy — the Gaussian factor $\exp(-\tfrac{m}{m_0}\beta J^2)$ depopulates high-angular-momentum (circular) orbits, the more so at large $r$ (since $J^2\propto r^2$ ). This makes the velocity ellipsoid increasingly radial outward — Michie’s central result. The strength is set by $\beta/\alpha$ .
Cutoff — $Q(E)$ (Eq. 4.8) is a Fokker–Planck-derived high-energy depopulation function ( $Q\to1$ at $E=0$ , $Q\to0$ at the escape energy $E_e$ ), not a lowered Maxwellian. It carries a slight $J^2$ -dependent correction (Eq. 5.3).

What “Michie-King” means — and what progenax implements

Michie’s literal cutoff $Q$ (Eq. 4.8) is more elaborate than King’s. Modern IC codes (Gunn & Griffin 1979; LIMEPY, Gieles & Zocchi 2015) use the Michie anisotropy term $\exp(-J^2/2r_a^2\sigma^2)$ with King’s (1966) lowered-Maxwellian cutoff $[\exp(-E/\sigma^2)-1]$ :

f(E,J) \propto \exp\!\left(-\frac{J^2}{2 r_a^2 \sigma^2}\right) \left[\exp\!\left(-\frac{E}{\sigma^2}\right) - 1\right],\qquad E \le 0,

(2)

This is the standard Michie–King model. progenax adopts (2) (Michie 1963 anisotropy + King 1966 cutoff), not Michie’s full $Q$ — the $Q$ refinement is a high-energy-tail correction irrelevant to equilibrium ICs.

Self-consistency: the anisotropic King ODE (verified, §5)¶

In dimensionless variables $\phi=\alpha\Phi$ , $z^2 = r^2 A\alpha G(4\pi)^2(2/\alpha)^{3/2}m_0$ , $\eta = v\sqrt{\alpha/2}$ (Eqs. 5.4–5.6), Poisson’s equation becomes (Eq. 5.8)

\frac{1}{z^2}\frac{d}{dz}\!\left(z^2\frac{d\phi}{dz}\right) = e^{-\phi}\int_0^{\eta_e}\!\!\int_{-1}^{1} e^{-\eta^2}\,e^{-C z^2\eta^2(1-\mu^2)}\,Q\,\eta^2\,d\mu\,d\eta.

(3)

The right-hand side is the density, which now depends on $z$ explicitly through the anisotropy term $e^{-Cz^2\eta^2(1-\mu^2)}$ ( $J^2\propto z^2\eta^2(1-\mu^2)$ ) — so the King ODE becomes radius-dependent and the resulting density profile differs from the isotropic King model. The single model parameter is (Eq. 5.9)

C = \frac{\beta}{A(2\alpha)^{1/2}G(4\pi)^2}.

(4)

$C\to 0$ recovers the isotropic King model; anisotropy becomes important where $Cz^2\sim\tfrac12$ . In modern units $C \propto 1/r_a^2$ : the anisotropy radius $r_a$ (the radius where $\beta\to 0.5$ ) is the same parameter as Michie’s $C$ , rescaled.

Use in progenax¶

progenax.kinematics / progenax.profiles — the implemented Michie–King anisotropic model (MichieProfile + MichieVelocityDF): solve (3) for $\rho(r),\Phi(r),r_t$ given $(W_0, r_a)$ , then sample $(v_r,v_t)$ from (2). Distinct from the isotropic King (1966) (different density) and from the Merritt (1985) Osipkov–Merritt construction (which holds the density fixed; Michie does not).
The realised anisotropy increases outward, $\beta(r)\to 1$ at large $r$ inside $r_t$ .

Notes¶

Michie’s $\exp(-\beta J^2)$ Gaussian-in- $J^2$ anisotropy is the same functional form Merritt (1985) notes for Eddington’s (1914) generalised isothermal sphere; the difference is that Osipkov–Merritt holds a given density fixed and inverts for $f(Q)$ , whereas Michie specifies $f(E,J)$ and solves Poisson for a new, self-consistent (more radial) density. The Michie–King model is the anisotropic, tidally-truncated workhorse for globular-cluster ICs.