OED for anisotropy — where to spend telescope time on r_a (B14)

This is the first worked example of optimal experimental design: given a fixed budget of stars, where on the sky and in which channel should you measure to pin one number — the cluster’s velocity anisotropy? The answer the optimiser returns, having been told none of the underlying physics, is the lesson: proper motions belong in the outskirts.

The physical question: how do you measure orbital anisotropy?¶

Our target is the velocity anisotropy of a star cluster — the degree to which stellar orbits are radially or tangentially biased, captured by the Binney parameter $\beta(r)$ Equation. We adopt the Osipkov–Merritt law Merritt, 1985,

isotropic in the core and increasingly radial outward, with a single knob: the anisotropy radius $r_a$, the radius where $\beta=\tfrac12$. Measuring the cluster’s orbital structure is measuring $r_a$.

Why does this matter physically? Anisotropy is a fossil of how a cluster formed and how it is being torn apart: violent relaxation, tidal stripping, and radial-orbit instability all leave their signature in $\beta(r)$. In dwarf-galaxy dynamics the same $\beta$ is the dominant nuisance in the mass–anisotropy degeneracy that limits dark-matter cusp-versus-core measurements. So $r_a$ is both a number worth measuring and a number that is notoriously hard to measure — which is what makes it a perfect OED target.

Why proper motions, and why the outskirts¶

The three sky channels — line-of-sight RV, radial PM, tangential PM — are the B&M82 projection Equation of one internal $\sigma_r(r)$ and $\beta(r)$. The key fact, derived there: in the isotropic limit the three channels are identical — the anisotropy lives only in their ratios. The tangential proper motion is the cleanest probe, $\sigma_{{\rm pm},T}^2\propto(1-\beta)$, so as orbits turn radial it collapses while the other channels stay up. And because $\beta(r)$ grows outward, the channels disagree most in the outskirts. A single $\sigma_{\rm los}(R)$ profile cannot break the tie — a cold outer LOS profile can mean “low $\sigma_r$” or “high $\beta$.” Proper motions break it. Keep that picture in mind: it is the physics the optimiser is about to rediscover, having been told none of it.

Inputs and assumptions¶

The design optimises over a fixed budget of $N_{\rm total}$ stars allocated across $K=12$ projected-radius bins $\times$ 3 kinematic channels (36 cells). The science target is $r_a$; total mass $M$ and half-mass radius $r_h$ are nuisances carried with fractional priors. Everything is computed pre-data — no mock catalogue enters the optimisation loop (a 64-draw calibration ensemble afterwards is a gate, not part of the design).

The Fisher is built in the dimensionless $\ln\theta$ metric Important, so $\sigma(r_a)/r_a$ is a fractional precision and the nuisance prior $\mathtt{PRIOR\_DIAG}=[0,\,1/0.3^2,\,1/0.3^2]$ is fractional too (none on the target).

The method, specialised: c-optimality on the additive Fisher¶

This example pours the additive backbone Equation straight into a c-optimal allocation. The per-star blocks $M_{b,c}$ come from one jacrev of project_dispersion at the truth; the design enters only through the softmax weights Equation, multiplied by the fixed completeness $c(R_b)$. The headline criterion minimises the $r_a$ variance with $M,r_h$ profiled out:

We also run D ($\max\log\det F$) and A ($\min\mathrm{tr}\,F^{-1}$) as the contrast — see why c and D should disagree. Under the $M\!\leftrightarrow\!r_a$ (through $GM$) and $r_h\!\leftrightarrow\!r_a$ degeneracies they put stars at different radii, which is the criterion-disagreement lesson made visible below.

Validating a pre-data calculation: the calibration ensemble¶

A Fisher forecast is a promise: “if you observe this way, your error bar will be this large.” A promise made before any data is only trustworthy if you check it, because the Cramér–Rao bound Equation is a local, Gaussian approximation — exact in the high-information limit, optimistic when the likelihood is curved or the estimator is biased.

So we close the loop once, as a gate (not inside the optimiser). We draw a 64-member ensemble of mock catalogues from the actual Osipkov–Merritt Plummer sampler, project each to the sky, bin it, and fit $\hat r_a$ by maximum a posteriori (with the same fractional prior the design Fisher used). The realised scatter should match the Fisher prediction,

both sides being fractional variances in the $\ln\theta$ metric. The tolerance is not a free knob: the Monte-Carlo error on a variance estimated from $n_{\rm draws}$ draws is $\approx\sqrt{2/n_{\rm draws}}$, so the gate is a principled $2\sigma$ band, $2\sqrt{2/64}\approx0.35$. The realised $\sigma(r_a)/r_a=0.109$ sits just below the Fisher’s 0.121 — the pre-data Fisher is mildly conservative (the binned-dispersion estimator loses a little information relative to the idealised per-star Fisher), and the two agree well inside the band. The promise holds.

Figures¶

The headline — PMs to the outskirts¶

c vs D vs A — why the criteria disagree¶

The precision frontier — fewer stars at equal precision¶

Calibration — the pre-data Fisher is trustworthy¶

Optimiser convergence¶

Quantitative results¶

All numbers are from the gated CLI’s run-record (--full, 64-draw calibration, $N_{\rm total}=4000$):

The equal-precision factor is exact at fixed $N$ (the fixed nuisance prior cancels in the $c_{\rm uniform}/c_{\rm designed}$ ratio, because $c\propto1/N$ when the prior is held fixed); the “$\approx3.9\times$ fewer stars” gloss on the frontier is the same physics read off the swept budget curve, where the prior no longer cancels and the slope departs mildly from $1/N$.

What the optimum means: science implications¶

The $3.66\times$ is not a free lunch — it is the optimiser discovering the physics of where information lives, and it generalises well beyond this mock.

1. Information is localised — in space and in channel. Anisotropy $\beta(r)$ grows outward, and the tangential-PM sensitivity to $r_a$ grows with it. So the value of a star for measuring $r_a$ depends sharply on where it sits and how you measure it. OED makes that quantitative instead of intuitive: the headline figure shows the optimiser routing PM stars to exactly the radii where $\beta(R)$ is turning over.

2. Proper motions break the mass–anisotropy degeneracy where it bites. A line-of-sight dispersion profile alone is degenerate — recall Equation: a cold outer $\sigma_{\rm los}$ can mean low $\sigma_r$ or high $\beta$. The two PM channels carry different $\beta$-weightings Equation, so they lift the degeneracy — and the OED tells you where the lift is largest: the outskirts. The result — RV in the core, PM in the outskirts — is the observing strategy this degeneracy demands, derived from first principles. That is directly actionable for real Gaia-PM + spectroscopic campaigns on globular clusters and dwarf spheroidals.

3. Design for your target, not for “everything.” The c-vs-D-vs-A divergence is the deepest lesson. If your science is one number — anisotropy as a probe of a dark-matter cusp/core, an IMBH’s kinematic signature, a cluster’s tidal state — a design that minimises the joint ellipsoid (D) wastes stars tightening nuisances. c-optimality can buy a multiplicative factor in the precision you actually care about.

4. The honest caveat is itself a research direction. The optimum is optimal for the assumed OM-Plummer model. That model-dependence is the central limitation of all OED — and it points straight at the most valuable extension: robust and model-discriminating design (see the section’s capability map), which progenax is unusually well-placed to do because it ships more than one differentiable forward model for the same observable.

Current scope and planned extensions¶

How to run¶

# quick (12-draw calibration, ~1 min)
env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_oed.py

# publication-grade (64-draw calibration, ~4 min) — regenerates the figures here
env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_oed.py --full

The CLI is gated (exit 0 only if the headline factor, PM-outskirts trend, and calibration all pass) and writes a JSON run-record alongside the five figures.

References¶

The shared Fisher / Cramér–Rao / projection theory and its references are on the OED formalism page. The Osipkov–Merritt anisotropy law is Merritt (1985); the line-of-sight projection into $\sigma_{\rm los}$, $\sigma_{{\rm pm},R}$, $\sigma_{{\rm pm},T}$ is Binney & Mamon (1982, MNRAS 200, 361). The differentiable project_dispersion forward model is documented on the velocity-DF kinematics pages; the anisotropy recovery counterpart is the anisotropy demo (B6), and B8 introduces the same sky-projection helper.