Maschberger & Clarke (2011)

The problem¶

Allison’s $\bar\Lambda$ (mean-MST; see Allison et al. (2009)) is fragile to outliers in substructured regions: a single isolated massive star adds one long MST edge to the massive-star tree, which can make $\bar\Lambda < 1$ (“inversely segregated”) even when the massive stars are overwhelmingly in subcluster centres (their Fig. 2). M&C give two more robust diagnostics.

Diagnostic 1 — median-MST $\tilde\Lambda$ (Eqs. 1–3)¶

Over a moving window of 40 mass-ranked stars, $\bar\Lambda_{(i)}(\bar m_{(i)}) = \bar\ell_{40}/\bar\ell_{(i),(i+40)}$ (mean edge length, Eq. 1, following Parker et al. 2011). The robust variant replaces the mean edge length with the median (Eq. 3),

which is far less sensitive to a few isolated massive outliers. Using mean and median together flags both genuine segregation of the majority and the presence of outliers.

Diagnostic 2 — local surface density / the m–Σ plane (§4, Eq. 4) ← used in progenax¶

For each star, the projected local stellar surface number density (von Hoerner 1963; Casertano & Hut 1985) is

with $r_k$ the distance to the $k$-th nearest neighbour ($k=6$ is “a good compromise between locality and low-number fluctuations” for $N\in[30,1000]$). Mass segregation is read off the m–Σ plane: massive stars sit at systematically higher Σ. M&C quantify the difference between the high-mass and low-mass Σ distributions with a two-sample KS test (e.g. their $1000\,M_\odot$ run: high- vs low-mass Σ differ at $p=0.025$). This estimator is robust to substructure and to outliers — a single isolated massive star is simply one low-Σ point and does not dominate, unlike the MST mean.

Physical results (for context)¶

SPH star-formation sims (Bonnell et al. 2003, 2008) show global mass segregation from very early times, only mildly affected by subcluster merging. Up to $\approx2$–3 per cent of “massive” sinks ($m>2.5\,M_\odot$) are in relative isolation (formed in situ), but the majority are in dense centres. (Softened gravity suppresses ejections, so isolated-massive counts are a lower bound.)

Use in progenax¶

• gravoturb.diagnostics.mass_density (experimental) — implements Eq. 4 (local_surface_density) and the m–Σ diagnostic (mass_density_segregation: Spearman ρ(m,Σ), high/low median-Σ ratio, KS p). This is the substructure-robust mass-weighted metric for the FDF cluster IC — it detects primordial mass–density correlation (massive stars placed in dense BM19 clumps by gravoturb.realization.mass_assignment.correlated_mass_assignment) where CW04 $\mathcal{Q}$ on small massive subsets is too noisy. Validated in tests/experimental/unit/test_mass_density.py (exact-formula + uniform-density recovery + primordial-correlation detection).

• Complements Allison et al. (2009) ($\Lambda_{\mathrm{MSR}}$ concentration) and Cartwright & Whitworth (2004) ($\mathcal{Q}$ geometry): Σ is the local-density axis of the mass-segregation picture.

Notes¶

• The metric is computed in projection (2-D), matching M&C; for 3-D one would adapt the neighbour count / normalisation.

• $k=6$ and the $(k-1)$ numerator are the Casertano & Hut (1985) convention; do not “simplify” the -1 away (it is the unbiased surface-density estimator, not $k/\pi r_k^2$).