Allison et al. (2009)

The big idea¶

Two questions, one MST toolkit. (MNRAS) How do you quantify whether the massive stars in a cluster are more spatially concentrated than average? (ApJ) Why are so many young clusters observed mass-segregated when two-body relaxation is far too slow? The answer to the second is that clusters born cool (subvirial) and clumpy (fractal) collapse into a short-lived dense core whose violent relaxation segregates the most massive stars in $\sim 1$ crossing time — orders of magnitude faster than the classical $t_{\mathrm{seg}}$.

The $\Lambda_{\mathrm{MSR}}$ metric (as described in the held ApJ L99 PDF, §3.2)¶

Compare the minimum spanning tree (MST) of the $N$ most massive stars to the MSTs of many random $N$-star subsets. The mass segregation ratio is the ratio of the average random-subset MST length to the massive-star MST length:

with the uncertainty the standard deviation (L99 calls it the “instantaneous standard deviation”) of the random-subset MST lengths. Interpretation:

• $\Lambda_{\mathrm{MSR}} \approx 1$ — no mass segregation (massive subset is a typical random subset);

• $\Lambda_{\mathrm{MSR}} > 1$ — massive stars more concentrated (segregated); $\gg 1$ ⇒ strong;

• $\Lambda_{\mathrm{MSR}} < 1$ — inverse segregation (rare).

The short-timescale result (held ApJ L99 PDF)¶

Initial conditions (§3.1, verified). $N=1000$ single stars; Kroupa (2002) three-part power-law MF, $m\in[0.08, 50]\,M_\odot$; fractal spatial distribution (dimension $D=1.6$; $D=3.0$ = uniform sphere) in a sphere of radius 1 pc; velocities coherent so nearby stars move together (Goodwin & Whitworth 2004); virial ratio $Q=0.3$ ($Q=T/|V|$, so 0.5 = virial). $D=1.6$ + $Q=0.3$ are chosen as the most extreme (fastest-segregating) case. Integrated with kira/starlab; stellar evolution neglected over the 4 Myr runs.

Result (Fig. 2, verified). $\Lambda_{\mathrm{MSR}}(t)$ for the $N=10, 20, 50, 100$ most massive stars. Initially $\Lambda=1$ (unsegregated); after $\sim 1$ Myr the 10 most massive reach $\Lambda\sim 3$. The 20 and 50 most massive segregate weakly; beyond the 50th, none. Because fractal ICs are seed-dependent, an ensemble is essential: of 50 clusters, 29 segregate within 1 Myr, 44 within 4 Myr, 6 never (segregation = an event lasting $>0.1$ Myr with significance $>1$).

Mechanism (§3.3, verified). Cool + clumpy ⇒ gravitational collapse + violent relaxation ⇒ a short-lived dense core (~half the mass within ~0.1 pc, lasting 0.1–0.2 Myr ≈ 10–20 core crossing times → dynamically old). Classical timescales: Spitzer (1969) $t_{\mathrm{seg}}(M)\approx (\langle m\rangle/M)\,t_{\mathrm{relax}}$ (L99 Eq. 1); $t_{\mathrm{relax}}\approx [N/(8\ln N)]\, t_{\mathrm{cross}}$ (Eq. 2); combined (Eq. 3). Core params $N\sim300$–500, $R\sim0.1$–0.2 pc, $\langle m\rangle=0.4\,M_\odot$, $\sigma\sim2$ km s$^{-1}$ give $t_{\mathrm{seg}}\sim0.1$ Myr ⇒ segregation only above $M\sim2$–$4\,M_\odot$ (the 50th most massive $\approx2\,M_\odot$).

Why smooth clusters don’t (verified). Collapse factor $R_0/R_f = (\alpha_0/\alpha_f)\,2(1-Q_0)$. A smooth Plummer ($\alpha_0\approx\alpha_f\approx0.75$) at $Q_0=0.3$ collapses only $\times1.4$ — too little. A fractal $D=1.6$ has $\alpha_0\approx1.5$ ⇒ collapses $\times2.5$ (1 pc → ~0.4 pc; core much smaller), reaching the dense state that segregates. This is the crux: substructure enables the deep collapse; subvirial supplies the cold start.

Use in progenax¶

• Mass segregation — Λ_MSR diagnostic theory.

• progenax.diagnostics.compute_lambda_msr — released-core estimator; its formula ($\langle L_{\mathrm{rand}}\rangle/L_{\mathrm{massive}}$, std error) matches the held L99 description. Validation still owed (see below) and the docstring citation needs the MNRAS fix.

• The cool-fractal pathway is the canonical alternative to Baumgardt et al. (2008) primordial segregation; the experimental gravoturb cluster-IC forward tool targets exactly this regime (turbulent substructure + chosen sub-virial $Q$).

Validation owed (planned)¶

• Tier A (analytic, released-core): $\Lambda_{\mathrm{MSR}}$ absolute correctness — unsegregated random masses $\to 1$; hand-constructed exact value; maximal $\to\gg1$; inverse $\to<1$; estimator convergence in $N_{\mathrm{random}}$; the binary-contamination caveat (tight massive binary → spurious spike).

• Tier B (literature): the end-to-end anchor is L99 Fig. 2 — a cool ($Q=0.3$) fractal ($D=1.6$) $N=1000$ cluster should evolve $\Lambda(N{=}10)$ from 1 to a few within ~1 Myr, segregating only down to $\sim2$–$4\,M_\odot$. (The exact ONC $\Lambda$ and the formal equation need the MNRAS 395,1449 PDF, not yet held.)

Notes¶

• Two distinct $Q$’s. Allison’s virial ratio $Q=T/|V|$ (0.5 = virial) is not the CW04 structure parameter $\mathcal{Q}=\bar m/\bar s$ — see Cartwright & Whitworth (2004). Both appear in this problem (cool and clumpy); keep them separate.

• Softening / collisionality. L99’s segregation is violent-relaxation (collapse-driven), not slow two-body relaxation — so a softened collisionless integrator captures the dominant effect over ~1 crossing time. State this when reproducing.

• Ensembles are mandatory for fractal ICs (seed-to-seed scatter is large) — L99 used 50.