Rotation & anisotropy validation - progenax — differentiable initial conditions for N-body simulations

Two kinematic structures layered on the isotropic ICs: streaming rotation (an additive velocity transform) and Osipkov-Merritt radial anisotropy (a velocity-DF shape). Test file: tests/validation/test_rotation_anisotropy_physics.py (see the test dashboard for the live per-suite count); figures: scripts/validate_rotation_anisotropy.py. Theory at Anisotropy and rotation.

What is verified¶

Each row maps to assertions in test_rotation_anisotropy_physics.py. Measured values are regenerated by scripts/validate_rotation_anisotropy.py. Rotation is an additive transform, so the added field $v_{\rm rot}=v_{\rm after}-v_{\rm before}$ is validated exactly (no isotropic-scatter noise).

Property	Tolerance (as tested)	Measured	Anchor
Solid-body $v_\phi(R)=\Omega R$	slope $=\Omega$ to 10^-6	0.300000 (resid $3\times10^{-14}$ )	$v_{\rm rot}=\Omega\times r$
Angular-momentum budget $L_z=\Omega\sum m R^2$	10^-6 rel	exact	rigid rotation
Rotation is purely azimuthal ( $v_R$ unchanged)	10^-9 abs	unchanged	$\Omega\times r \perp \hat R$
Differential $v_\phi(R)=v_{\rm peak}\frac{R}{R_p}e^{1-R/R_p}$	10^-6 abs	max dev $<10^{-6}$	phenomenological curve
OM anisotropy $\beta(r)=r^2/(r^2+r_a^2)$ — Plummer	$<0.04$ per bin	max dev 0.024	Merritt (1985) stretch
OM anisotropy $\beta(r)$ — EFF	$<0.05$ per bin	max dev 0.016	same stretch
$r_a=\mathrm{None}$ is isotropic	$\|\beta\|<0.04$	$\beta\approx0$	no stretch
Differentiability ( $\Omega, v_{\rm peak}, r_a$ )	AD $=$ central-FD	rel err $3\times10^{-9}$ , $7\times10^{-9}$ , $8\times10^{-6}$	pure transforms + fixed-key stretch

Figures¶

Generated by scripts/validate_rotation_anisotropy.py (PASS/FAIL per panel).

Osipkov-Merritt anisotropy. Both Plummer (circles) and EFF (squares) sampled
\beta(r)=1-\sigma_t^2/2\sigma_r^2 land exactly on the OM target
r^2/(r^2+r_a^2) (max dev 0.024, 0.016), reaching \beta=0.5 at r=r_a. The
direction-stretch realizes OM exactly — unlike the self-consistent
Michie-King \beta, which is suppressed below this curve. — Figure 1:**Osipkov-Merritt anisotropy.** Both Plummer (circles) and EFF (squares) sampled $\beta(r)=1-\sigma_t^2/2\sigma_r^2$ land **exactly** on the OM target $r^2/(r^2+r_a^2)$ (max dev 0.024, 0.016), reaching $\beta=0.5$ at $r=r_a$ . The direction-stretch realizes OM exactly — unlike the self-consistent Michie-King $\beta$ , which is suppressed below this curve.

Face-on velocity field. (a) The isotropic IC has random in-plane velocity
directions (no net rotation). (b) After solid-body rotation \Omega=0.3 the
streaming field v_{\rm rot}=\Omega\times r is coherent circulation (arrows
colored by v_\phi, all co-rotating). — Figure 2:**Face-on velocity field.** (a) The isotropic IC has random in-plane velocity directions (no net rotation). (b) After solid-body rotation $\Omega=0.3$ the streaming field $v_{\rm rot}=\Omega\times r$ is coherent circulation (arrows colored by $v_\phi$ , all co-rotating).

Solid-body rotation. The added streaming v_\phi(R) is linear in cylindrical
R with fitted slope =\Omega=0.300000 (residual 3\times10^{-14}); the injected
angular momentum equals \Omega\sum m R^2 exactly. — Figure 3:**Solid-body rotation.** The added streaming $v_\phi(R)$ is linear in cylindrical $R$ with fitted slope $=\Omega=0.300000$ (residual $3\times10^{-14}$ ); the injected angular momentum equals $\Omega\sum m R^2$ exactly.

Differential rotation. The added v_\phi(R)=v_{\rm peak}(R/R_p)e^{1-R/R_p}
peaks at R_{\rm peak} and decays inward and outward (a phenomenological curve,
not from a specific paper). Per-particle match to <10^{-6}. — Figure 4:**Differential rotation.** The added $v_\phi(R)=v_{\rm peak}(R/R_p)e^{1-R/R_p}$ peaks at $R_{\rm peak}$ and decays inward and outward (a phenomenological curve, not from a specific paper). Per-particle match to $<10^{-6}$ .

Differentiability¶

Gradient validation (autodiff vs central finite difference). (a)
\partial T_{\rm rot}/\partial\Omega, (b) \partial T_{\rm rot}/\partial v_{\rm peak},
(c) \partial\langle v_r^2\rangle/\partial r_a: autodiff and finite difference agree
to max rel err 3\times10^{-9}, 7\times10^{-9}, 8\times10^{-6}. The rotation
amplitudes and the anisotropy radius are all differentiable for inference. — Figure 5:**Gradient validation (autodiff vs central finite difference).** (a) $\partial T_{\rm rot}/\partial\Omega$ , (b) $\partial T_{\rm rot}/\partial v_{\rm peak}$ , (c) $\partial\langle v_r^2\rangle/\partial r_a$ : autodiff and finite difference agree to max rel err $3\times10^{-9}$ , $7\times10^{-9}$ , $8\times10^{-6}$ . The rotation amplitudes and the anisotropy radius are all differentiable for inference.

How to run¶

pytest tests/validation/test_rotation_anisotropy_physics.py -q   # ~7 s
python scripts/validate_rotation_anisotropy.py                   # 5 figures + PASS/FAIL

What this suite does not test¶

Self-consistent rotating equilibria — the rotation here is a streaming transform added to a non-rotating equilibrium, not a self-consistent rotating DF (that is part of the lowered-model-family roadmap).

References¶

Rotation: Binney & Tremaine (2008), Galactic Dynamics (2nd ed.) Sec 4.8; Lynden-Bell (1960), MNRAS 120, 204. Anisotropy: Merritt (1985) (the stretched-isotropic Osipkov-Merritt split, Eq. 15). Theory at Anisotropy and rotation.

References¶

Merritt, D. (1985). Spherical stellar systems with spheroidal velocity distributions. The Astronomical Journal, 90, 1027–1037. 10.1086/113810