Halo + core recovery (B3)

A real cluster is not one population. This demo builds a two-family cluster — a spatially extended, kinematically anisotropic halo plus a compact, isotropic core — in one shared self-consistent potential, and asks whether three structural parameters can be recovered jointly from its kinematics:

• $t$ — the halo mass fraction (the core carries $1-t$),

• $r_a$ — the halo’s Osipkov-Merritt anisotropy radius,

• $r_h$ — the halo’s Plummer half-mass radius.

The truth model (the science-headline mix of the Engine B validation suite):

sampled with $N = 3\times10^4$ stars. The EFF core shape $(a,\gamma,r_t)$, the stellar-mass labels $m_j$, and the core’s isotropy are held fixed; only $(t, r_a, r_h)$ are recovered.

The physics being recovered¶

Anisotropy (Osipkov-Merritt). The halo’s DF depends on energy and angular momentum only through $Q = E - L^2/(2 r_a^2)$, which makes the velocity ellipsoid isotropic in the centre and increasingly radial outside $r_a$. The Binney anisotropy has the closed form

so $\beta\to 0$ as $r\to 0$ (isotropic) and $\beta\to 1$ as $r\to\infty$ (radial). The core, with $r_a\to\infty$, has $\beta\equiv 0$. The OM curve Merritt, 1985 is the second observable, and it is what pins $r_a$.

Dispersion. The 1-D dispersion of each component is built from the speed moments of its Eddington DF $f_j$ in the shared potential, with the OM stretch folded in:

where $m_{p,j}(r) = \int_0^{\sqrt{2\Psi}} w^{p} f_j\!\big(\Psi - \tfrac12 w^2\big)\,dw$ are the stretched-frame speed moments. This is the verbatim oracle the Engine B validation suite checks against sampled velocities. Physical units come from the engine’s own velocity scale $v = \sqrt{G M_{\rm sampled}/(4\pi\mu)}\,s$, anchored to the measured total mass (never an input $M$).

Why $\sigma(r)$ and $\beta(r)$ together recover all three. The halo scale $r_h$ sets where $\sigma(r)$ falls off; the mass split $t$ sets the relative amplitude and shape of the two components’ dispersion profiles; $r_a$ is read almost entirely from the rising $\beta(r)$. The three are nearly orthogonal — borne out by the near-zero Fisher correlations below.

Inputs and assumptions¶

The fit recovers three halo parameters $(t, r_a, r_h)$; the core family, the domain radius, and the total mass are all assumed known. The split matters: the recovery looks clean partly because the hardest potential degeneracy ($r_h$ vs the cluster’s outer extent) is removed by fixing the domain.

Result — joint MLE + Fisher (freshly run, ALL PASS)¶

Measured 2026-06-11 ($N=3\times10^4$, three dispersed Adam starts; exit 0):

All three within $1\sigma$, comfortably inside the $3\sigma$ recovery gate. Supporting diagnostics, all measured in the same run:

• Robust optimum. All three dispersed initializations converged to the identical loss 19.1479 — the recovered minimum is not an initialization artifact.

• Realizability. Rebuilding Engine B at $\hat\theta$ gives DF positivity margins $f_{\min,j} = [0.085,\ 1.16\times10^{-4}]$, both $\ge -10^{-3}$ — the recovered model is a genuine equilibrium, not a negative-DF fiction.

• Self-consistency. The analytic prediction at truth matches the binned data to $\max|{\rm dev}/{\rm SE}| = 2.34$ over all 48 populated cells — no systematic oracle/binning bias.

• Near-orthogonality. Gauss-Newton Fisher correlations $\rho(r_a, r_h) = +0.03$, $\rho(t, r_a) = -0.04$ (condition number 13.4): the three parameters are independently constrained.

Figure¶

Caveats¶

How to run¶

env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_halo_core.py

References¶

The EFF profile is Elson et al. (1987); the Plummer model Plummer (1911); Osipkov-Merritt anisotropy Merritt (1985); Eddington inversion follows Binney & Tremaine (2008). The Engine B construction and its realizability/anisotropy validation are documented at Engine B (Eddington) and the multi-component theory pages.