OED for concentration — where to spend telescope time on W₀ (Stage 3)

OED for concentration — where to spend telescope time on $W_0$ (Stage 3)¶

You have a cluster and a star budget, and you want to know how centrally concentrated it is. The anisotropy design asked where on the sky and in which channel to measure to pin the orbital anisotropy $r_a$; the dynamical-mass design asked how deep to survey to weigh the cluster. This page asks the same kind of question for a third target — the cluster’s concentration $W_0$ — and the optimiser returns an answer that is the mirror image of the anisotropy result. Anisotropy wanted the outskirts; concentration wants the core. Same machinery, opposite strategy, and the contrast is the lesson.

Why this arc exists: closing the W₀-differentiability loop¶

A short piece of history makes the rest of the page land. The forward models that turn a cluster’s structure into observable dispersions — the Osipkov–Merritt Jeans projection project_dispersion and the exact Michie second moment df_moment_dispersion — were made differentiable in the concentration $W_0$ by two pieces of internal work: a $C^1$ PCHIP interpolation of the King/Michie equilibrium-solver tables and a $W_0$ lock on the df_moment path. Those changes made $\partial\sigma/\partial W_0$ exist and be smooth.

But they were validated only at the gradient-audit level — an automatic-versus-finite-difference check that the derivative is numerically correct (98 entry points, 0 hazards). They were never exercised through an actual Fisher / OED inference that treats $W_0$ as a parameter. The stated goal of that whole line of work — “$W_0$-differentiable $\sigma$ for OED/Fisher” — had an open loop. This demo closes it: it builds the design Fisher with $W_0$ in the parameter vector, optimises an observing strategy for it, and confirms the forecast against real mock catalogues. It is the end-to-end use of the capability those ADRs promised.

The physical question: how do you measure concentration?¶

Our target is the concentration of a star cluster — how steeply its density climbs from the tidal boundary to the centre. For the King (1966) lowered-isothermal family the concentration is governed by a single dimensionless number, the central potential depth $W_0$ (documented on the King profile theory page): a small $W_0$ is a diffuse, weakly-bound cluster; a large $W_0$ is a deep, centrally-peaked one with an extended outer envelope. $W_0$ sets the King concentration $c=\log_{10}(r_t/r_c)$ — the ratio of tidal to core radius — so measuring how concentrated a cluster is means measuring $W_0$.

Why does this matter physically? Concentration is a dynamical clock and a structural fingerprint: it tracks core collapse, two-body relaxation, and the cluster’s tidal history, and it is the first parameter any structural model of a globular cluster or a dwarf must pin. It is usually read off the surface-density / star-count profile — that is the count-channel recovery demo (B11). Here we ask the complementary question: what does the kinematic dataset say, and where in it does the concentration information live?

Why concentration is a core probe — the physical intuition¶

The key fact, which the optimiser is about to rediscover unaided: concentration is an isotropic, central property. Changing $W_0$ rescales the depth of the potential well, and its strongest observable fingerprint is the velocity dispersion contrast between the deep core and the cold outskirts. A more concentrated cluster has a hotter, more sharply peaked $\sigma(r)$ in the core, falling to a colder envelope; a diffuse one is flatter. The core is where the dispersion is highest and the projected density is greatest, so it is where a fixed number of stars pins that contrast most tightly. This is the opposite of anisotropy, which is zero in the core by construction ($\beta_{\rm OM}(r)=r^2/(r^2+r_a^2)\to0$) and only grows outward — so the anisotropy signal lives in the outskirts. Hold that contrast in mind: concentration → core, anisotropy → outskirts. It is the physics behind the headline figure.

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 $W_0$; the anisotropy radius $r_a$ is a co-determined kinematic parameter and the total mass $M$ is a nuisance carried with a fractional prior. Everything is computed pre-data — no mock catalogue enters the optimisation loop (a real-star calibration ensemble afterwards is a gate, not part of the design).

The Fisher is built in the dimensionless $\ln\theta$ metric Important, so $\sigma(\ln W_0)$ is a fractional precision and the nuisance prior $\mathtt{PRIOR\_DIAG}=[0,\,0,\,1/0.3^2]$ is fractional too — none on the target $W_0$, none on $r_a$, only on $M$. That the data alone (three velocity channels $\times$ 12 radial bins) keep the $(W_0, r_a)$ 2-block positive-definite — so no prior on $W_0$ or $r_a$ is needed for a well-posed inverse — is a measured property of this design, not an assumption (it is checked across random designs in the test suite).

The modelling choice: OM-King and OM-on-Michie-density¶

Two facts about the forward model set up the whole demo, and stating them plainly is the honest way to read the result:

1. project_dispersion is Osipkov–Merritt only. It imposes $\beta_{\rm OM}(r)=r^2/(r^2+r_a^2)$ Equation on whatever density it is handed — it does not read a profile’s own intrinsic anisotropy. So both models here are projected under the same OM anisotropy law; they differ only in the density profile, and hence in the concentration-dependent map $r_t(W_0)$. King is isotropic in density (OM is layered on top); Michie carries its own native anisotropy in its density shape, but it is still projected under OM. The headline is OM-King; Michie is added to exercise the native $r_t(W_0)$ path explicitly and to test that the qualitative answer is robust to the density model.

2. The calibration’s sampler equals the Fisher model. The real-star gate (below) must draw stars from the exact distribution project_dispersion projects — “that density under OM” — or it would validate a mismatched model. So the OM particle sampler is assembled to be self-consistent with the Fisher forward model: for King via the Engine B MultiComponentCluster.from_density_profiles path (a single-component King density in its own shared-$\Psi$ potential with an Eddington/OM DF, documented on the Engine B validation page); for Michie via the generic Eddington inversion ($\rho_Q=(1+r^2/r_a^2)\rho$, Merritt 1985 augmented density) on the Michie density, since Engine B does not ingest MichieProfile. In both cases sampler $\equiv$ Fisher-model, which is what makes the calibration a real test rather than a tautology. (Deliberately we do not use MichieVelocityDF’s native $\beta$ — that would mismatch the OM projection and bias the gate.)

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

This example pours the additive backbone Equation straight into a c-optimal allocation, exactly as the anisotropy design does — only the target index changes. The per-star blocks $M_{b,c}$ come from one jacrev of project_dispersion at the truth (reverse-mode by policy — the King/Michie equilibrium solvers hit a custom_vjp ODE with no forward-mode rule); the design enters only through the softmax weights Equation, multiplied by the fixed completeness $c(R_b)$. The headline criterion minimises the $W_0$ variance with $r_a$ and $M$ 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. Because the whole Fisher is additive and design-linear, the expensive projection is never re-differentiated inside the optimiser loop: one Jacobian, then $F=\sum n\,M\to$ invert a $3\times3\to$ read one element.

Validating a pre-data calculation: two gates¶

A Fisher forecast is a promise — “if you observe this way, your error bar will be this large” — and a promise made before any data is only trustworthy if you check it, because the Cramér–Rao bound Equation is a local, Gaussian approximation. This demo checks it twice.

Gate 1 — the gradient is correct (AD-vs-FD on $\partial\sigma/\partial\ln W_0$)¶

The load-bearing differentiability claim is the $W_0$ column of the Jacobian. We verify it against finite differences at every radial bin. For OM-King the automatic and finite-difference gradients agree to better than 10^-3 across all bins. For OM-Michie the inner bins ($R\lesssim r_a$) agree to the same 10^-3 with a fixed step, but at mid-to-outer radii the proximity of Michie’s $r_t(W_0)$ truncation makes a fixed-step finite difference an unreliable proxy — the function has high curvature there (the truncation-curvature signature). So those bins are gated by Richardson finite differences instead: we confirm that as the step $h\downarrow$, the finite difference converges toward the automatic gradient. It does, to $\sim10^{-6}$, at every bin — the automatic gradient is correct everywhere; the mid-radius $>10^{-3}$ values are pure $O(h^2\sigma''')$ truncation, not a code defect.

Gate 2 — the forecast is calibrated (real-star King calibration)¶

A Fisher forecast must predict the realized scatter of the estimator. We close the loop with a real-star ensemble: for the King model we draw independent mock catalogues from the OM sampler (which equals the Fisher forward model), project each to the sky, bin it by projected radius, subsample the design counts, add per-star measurement error, and fit $\hat W_0$ by maximum a posteriori in the $\ln\theta$ Gauss–Newton metric (a Levenberg–Marquardt MAP fit, started at the truth). The realised fractional 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. The realised / Fisher variance ratio for King is 0.976 — comfortably inside the band, with no significant bias. The promise holds.

The headline result: concentration wants the core¶

The pre-registered hypothesis (H1) was: $W_0$’s optimal allocation differs from $r_a$’s — more core/intermediate weight, more channel-balanced, with only a modest outward pull. The result splits cleanly into a confirmed radial prediction and a refuted channel sub-prediction. Reporting both honestly — the hit and the miss — is the point; a wrong pre-registered sub-prediction is a genuine finding, not a failure.

RADIAL: confirmed — the mirror image of the anisotropy design¶

The $W_0$ c-optimal design puts $\sim71\%$ of its stars in the core half of the radial range (King; $\sim71\%$ Michie). The Stage-1 anisotropy design put $\sim99\%$ in the outskirts. They are mirror images, and the physical reason is exactly the intuition above: concentration is an isotropic, central property imprinted where the dispersion is highest and the density greatest (the core), while anisotropy lives in $\beta(r)$'s outward growth (the outskirts). The optimiser was told none of this — it discovered where the concentration information lives.

CHANNEL: refuted — concentration is strongly PM-dominated¶

H1 also predicted that $W_0$ would be more channel-balanced than $r_a$ (RV pulling more of its weight). It is not. The $W_0$ c-optimal design is strongly proper-motion dominated — RV carries only $\sim1\%$ of the King budget. The reason is structural and the same one the anisotropy page documents: at the assumed RV/PM error parity, a proper-motion star delivers two information-bearing components (PM$_R$ and PM$_T$) for one measurement, a 2-for-1 efficiency that the optimiser exploits almost everywhere. The error parity is per component, so PM wins on count. This sub-prediction was wrong, and we say so: it is a null result reported with integrity, and it sharpens the real lesson — the radial contrast (core vs outskirts) is the robust discriminator between the two targets, not the channel mix.

The precision gain and the criterion contrast¶

How much does designing for concentration buy you? At fixed budget the King c-optimal design reaches a fractional precision $\sigma(\ln W_0)=0.056$, versus 0.104 for a uniform design — a $1.85\times$ gain in precision, equivalent to reaching the same error bar with $\approx3.4\times$ fewer stars (the precision gain squared, since variance scales as $1/N$ when the prior is held fixed). The Michie design improves $0.095\to0.057$, a $1.67\times$ gain. As in the anisotropy example, the c-, D-, and A-optimal designs genuinely differ — the c-design, which only cares about $W_0$, beats the others on its objective and would waste stars if it chased the whole ellipsoid.

The broken degeneracy: concentration is separable from anisotropy¶

The deepest science payoff is that this design breaks the $W_0\leftrightarrow r_a$ degeneracy — it measures concentration and anisotropy independently. The Fisher correlation at the c-optimal design,

is the covariance off-diagonal normalised to $[-1,1]$: $|\rho|\to1$ would mean the two are nearly degenerate (you cannot tell them apart), $|\rho|\to0$ that the design measures them cleanly. We find $\rho(W_0, r_a)\approx-0.02$ for King and -0.12 for Michie — both far from $\pm1$. Concentration is cleanly separable from anisotropy in this kinematic design space, which is exactly what makes a $W_0$-targeted observing strategy worth computing: the core-PM stars that pin $W_0$ are not the same stars that pin $r_a$, and the design knows it.

King vs Michie: is the answer robust to the density model?¶

Yes, qualitatively. The headline — core plus proper motions — is the same for both density models, and the precision gains ($1.85\times$ vs $1.67\times$) and core fractions ($\sim71\%$ vs $\sim71\%$) are close. The differences are second-order and model-dependent: Michie wants modestly more of the RV channel ($\sim16\%$ vs $\sim1\%$) and a slightly more outward balance, because its native-anisotropy density shape moves the $\sigma(r)$ contrast a little. The robust, model-independent statement is the one to take away: measure concentration in the core, with proper motions. The RV fraction and the fine radial balance are where the model choice shows up.

Quantitative results¶

All numbers are from the gated CLI’s run-record ($N_{\rm total}=400$, $K=12$ bins, the King calibration operating point):

The precision gain 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 “$\approx3.4\times$ fewer stars” gloss is the same physics read as a star factor ($\text{gain}^2$).

What the optimum means: science implications¶

The result is the optimiser discovering where concentration information lives, and it generalises well beyond this mock.

1. The observing strategy is the opposite of an anisotropy campaign. To measure concentration, put the proper-motion budget in the core; to measure anisotropy, put it in the outskirts. Same instrument, same channels, opposite radial allocation — because concentration is a central, isotropic property and anisotropy is an outer, radial one. That is directly actionable: a Gaia-PM + spectroscopic campaign optimised for $W_0$ should not reuse an anisotropy campaign’s footprint, and OED tells you so quantitatively.

2. Concentration and anisotropy are independently measurable. The near-zero $\rho(W_0, r_a)$ Equation means the kinematic dataset can pin both without one contaminating the other — the core-PM stars that constrain $W_0$ are not the outskirts-PM stars that constrain $r_a$. A joint structural+orbital fit is well-posed in this design space, and the design is what makes it so.

3. Differentiable OED is a general tool. This is the third distinct science target ($r_a$, $M$, now $W_0$) run through the same additive-Fisher machinery, each with a different physical answer the optimiser found unaided. Because progenax’s forward models are differentiable, $\partial(\text{information})/\partial(\text{observing strategy})$ is computable for essentially any cluster-science parameter — which is the throughline of the whole OED section.

4. The honest caveat is itself a research direction. The optimum is optimal for the assumed OM-King/Michie model, and it rides the Jeans path, not the exact-moment one. That model-dependence is the central limitation of all OED, and it points at the same valuable extensions the sibling pages flag: robust, Bayesian, and model-discriminating (T-optimal) design — for which progenax is well-placed because it ships more than one differentiable forward model for the same observable.

Current scope and planned extensions¶

How to run¶

# the cheap design + figures (one jacrev per model + 3x3 linalg; ~1 min)
env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_oed_concentration.py

# regenerate into the docs figure directory
env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_oed_concentration.py \
    --outdir docs/website/60-science-demos/optimal-design/figures

The CLI computes only the cheap parts — one jacrev per model at the truth, the c/D/A $3\times3$-linear-algebra optimisation, the H1 radial/channel split contrasted with Stage-1, and the five figures — and writes a JSON run-record alongside them. The expensive real-star King calibration is the env-gated @slow test test_W0_fisher_calibration_matches_realized_scatter; the King ratio of 0.976 quoted here is cited from that gate, not re-run in the CLI (the Michie MAP-MC is intentionally never run — see the caveat).

References¶

The shared Fisher / Cramér–Rao / projection theory and its references are on the OED formalism page. The King (1966) lowered-isothermal model and its $W_0$ parameter are documented on the King profile theory page; the Michie–King anisotropic model on the Michie–King theory page; the Engine B Eddington/OM sampler on the Engine B validation 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). The count-channel concentration recovery — the complementary way to measure $W_0$, from star counts rather than kinematics — is the King-concentration demo (B11); the companion OED designs are anisotropy (where) and dynamical mass (how deep).