Küpper, Maschberger, Kroupa & Baumgardt (2011)

Abstract (paraphrased)¶

Introduces and publicly releases McLuster, a code for generating star-cluster initial conditions, and uses models of the young massive cluster R136 to calibrate the methods used to detect and quantify mass segregation and substructure in (non-seeing-limited) $N$-body data. The paper compares mass-segregation diagnostics — the mass-function-slope-vs-radius method, Allison’s $\Lambda$ minimum-spanning-tree parameter, colour gradients, and local stellar surface density — and substructure diagnostics — the projected radial density profile, the azimuthal density profile, and the Cartwright & Whitworth $\mathcal{Q}$ parameter. It finds the mass-function-slope method and the azimuthal-profile method most practical for large data sets, and discusses how binaries bias each measure (notably that $\mathcal{Q}$ is binary-sensitive and depends on the radial density gradient). McLuster is progenax’s primary cross-validation reference for cluster ICs.

What progenax actually uses¶

Partial mass segregation — the energy-ordered shuffle (§A6, Eq. A1, verified)¶

McLuster applies any degree of primordial mass segregation to any density profile using the method of Baumgardt et al. (2008a): it builds $N$ energy-ordered orbits and assigns masses to them. Perfect ordering (most-massive star on the lowest-energy orbit) gives full segregation; no ordering gives none. Intermediate (partial) segregation comes from a controlled shuffle (Eq. A1):

where $X \in [0,1)$ is random, $i$ indexes the mass-ranked stars (most massive first), and $S \in [0,1]$ is the mass-segregation parameter: $S = 1$ reproduces the perfectly ordered (fully segregated) array, $S = 0$ gives a random (unsegregated) assignment. Crucially, this shuffle does not change the chosen density profile as the segregation degree increases.

Fractal substructure (§A7, verified)¶

Two constructions: (i) a homogeneous box-fractal (after Goodwin & Whitworth 2004) where each parent cell spawns children with probability $2^{\,(D-3)}$, set by the fractal dimension $D$ (option -D); $D = 3.0$ gives no fractality, smaller $D$ gives more substructure; and (ii) fractal substructure folded into any smooth density profile (Plummer, King, EFF) by rescaling radii. The paper is explicit that this is an ad hoc (non-physical-origin) substructure generator giving a smooth spherical→substructured transition.

Binaries: COM particle then resolve (§A8, verified)¶

After masses are drawn from the IMF, chosen binaries are replaced by a single centre-of-mass (CoM) particle for the rest of the IC build; only at the very end, after the density profile is established and the velocities virialised, are the CoM particles replaced by their two component stars with sampled orbital elements. This COM-then-resolve / scale-separation convention is the one progenax follows: a binary is virialised as one CoM body and its internal binding energy is kept as a separate reservoir. Semi-major axes can be drawn flat (adis=0), from the Kroupa (1995a) period law (adis=1, default), or from the Duquennoy & Mayor (1991) Galactic-field period law (adis=2); eccentricities are thermal $f(e) = 2e$ (Duquennoy & Mayor 1991), with pre-main-sequence eigenevolution (Kroupa 1995b) circularising short-period orbits.

Fill factor and diagnostics¶

McLuster’s R136-calibrated comparison sample motivates the tidally-filling fill-factor range $r_h/r_t \approx 0.05$–0.3 progenax quotes, and the $\mathcal{Q}$-parameter range used in the fractal-substructure validation. The $\mathcal{Q}$ diagnostic itself is from Cartwright & Whitworth (2004).

Use in progenax¶

• Mass segregation — the partial-segregation $S$-shuffle (1).

• Fractal substructure — fractal-substructure conventions and the $\mathcal{Q}$ range.

• Tidal physics — the tidally-filling fill-factor range.

• Binary populations — the §A8 COM-then-resolve binary virialisation convention.

• Multi-component populations — layered / multi-component cluster set-up.

• Substructure (CW04 Q) validation — McLuster $\mathcal{Q}$-range anchor for the substructure validation.

Notes¶

The McLuster code paper and the dynamical-diagnostics calibration paper are the same publication (MNRAS 417, 2300). progenax’s spatial IC machinery is an independent JAX-native re-implementation, cross-validated against McLuster’s conventions and calibrated ranges rather than ported from it. The fractal/substructure ICs themselves now live in the experimental gravoturb subsystem; the Cartwright & Whitworth $\mathcal{Q}$ diagnostic survives in progenax.diagnostics.