Cartwright & Whitworth (2004)

The big idea¶

How do you tell, objectively, whether a star cluster is centrally concentrated (a smooth radial density gradient) or substructured (multi-scale fractal clumping)? CW04 show that two normalised statistics together do it. The normalised correlation length $\bar{s}$ (mean separation / cluster radius) decreases both as a cluster becomes more centrally concentrated and as it becomes more fractal — so $\bar{s}$ alone is ambiguous. The normalised mean MST edge length $\bar{m}$ also decreases in both cases, but with a different sensitivity. Their ratio

breaks the degeneracy: $\mathcal{Q} > 0.8$ ⇒ centrally concentrated (radial gradient), $\mathcal{Q} < 0.8$ ⇒ substructured (fractal).

Core definitions¶

Project the cluster to 2-D. Let $N$ be the number of stars, $R_\mathrm{cluster}$ the distance from the mean position to the furthest star, and $A$ the cluster area.

Normalised correlation length. With $\langle s\rangle$ the mean of all pairwise separations,

Normalised mean MST edge length. The minimum spanning tree (MST) is the shortest set of edges connecting all $N$ stars with no loops. Its expected total length for $N$ points uniformly spread over area $A$ scales as $(N A)^{1/2}$ (Beardwood+ 1959), so the size-independent normalisation is

with $L_\mathrm{MST}$ the total MST length. The $\sqrt{N}$ is mandatory — omitting it collapses $\bar m$ (and was the root cause of a discredited $\mathcal{Q}\approx0.13$ headline elsewhere in this codebase).

Calibration (their Table 1; 3-D models projected to 2-D, $100\le N\le300$)¶

Increasing central concentration pushes $\mathcal{Q}$ above 0.8; increasing fractal sub-clustering (lower $D$) pushes it below 0.8. (The $\mathcal{Q}$ uncertainties are small — $\pm0.09$ at $D=1.5$; it is the correlation length $\bar s$ that carries the large $\pm0.18$ scatter at low $D$, Table 1 column 2.)

Radial models are sampled analytically via the inverse CDF $r = \{(3-\alpha)\mathcal{R}/3\}^{1/(3-\alpha)}$ (their Eq. 2) for $n\propto r^{-\alpha}$; fractal models use a discrete box-fractal tree with Bernoulli maturation probability $p = \mathcal{N}_\mathrm{div}^{D-3} = 2^{D-3}$ ($\mathcal{N}_\mathrm{div}=2$).

Real clusters (their Table 1 / abstract). Centrally concentrated: IC 348 $\mathcal{Q}=0.98$, $\rho$ Oph 0.85 (radial slopes $\alpha\simeq2.2,\,1.2$); mildly substructured: Chamaeleon 0.67, IC 2391 0.66 ($D'\simeq2.25$); strongly substructured: Taurus 0.45 ($D'\simeq1.55$; treating its binaries as single systems raises this to 0.58, $D'\simeq1.9$).

Use in progenax¶

• JAX-native CW04 substructure Q parameter — JAX-native kNN approximation + scipy reference.

• Fractal substructure — $\mathcal{Q}$ as the substructure truth metric.

• experimental/gravoturb/diagnostics/q.py — clean-room CW04 estimator (numpy/scipy, non-differentiable).

• tests/experimental/fixtures/cw04_models.py — analytic radial models for the AC5 validation.

Validation (AC5): the estimator reproduces the 3D0/3D1/3D2 anchors (0.79/0.84/0.93) within $\sim$Table-1 scatter and is monotone in central concentration.

Notes¶

• Area convention (important). CW04 normalise each cluster so that $R_\mathrm{cluster}=1$ (Fig. 3 caption: “scaled so that the distance from the centre to the most distant star is unity”) and normalise $\bar m$ by $(N A)^{1/2}/(N-1)$ with “the area $A$ of the cluster” (Section 4.1). Operationally this is the circle area $A=\pi R_\mathrm{cluster}^2$, not the convex-hull area: using the hull biases $\mathcal{Q}$ high by $\sim+0.08$–0.14, whereas $A=\pi R_\mathrm{cluster}^2$ reproduces Table 1 to $<0.01$ (our AC5: 0.782/0.832/0.924 vs 0.79/0.84/0.93). Other authors (Schmeja & Klessen 2006; Lomax+ 2018) instead set $R=\sqrt{A_\mathrm{hull}}$, giving a different absolute $\mathcal{Q}$ scale.

• The CW04 $\mathcal{Q}$ is distinct from the virial ratio $Q_\mathrm{vir}=T/|V|$; progenax separates these into different modules.

• $\mathcal{Q}$ is a poor estimator of fractional-Brownian-motion (FBM) parameters — see Lomax, Bates & Whitworth (2018) — a caution for any $\mathcal{Q}$-vs-structure calibration built on GRF/FBM density fields.