Fractal substructure

Stars do not form in smooth, spherically symmetric distributions. Young clusters inherit clumpy, hierarchical spatial structure from the turbulent molecular clouds that birthed them, and that substructure controls early dynamical evolution Allison et al., 2009. The Goodwin & Whitworth (2004) recursive-tree fractal generator is the standard prescription for seeding this substructure into N-body initial conditions — and it is also a useful lens on why differentiable IC generation is hard, which is the theory this chapter keeps.

Why Goodwin–Whitworth doesn’t differentiate in JAX¶

The Goodwin & Whitworth (2004) algorithm is conceptually elegant — recursive subdivision with stochastic survival — but every step that gives it its distinctive clumpiness is incompatible with JAX’s differentiable model:

For pure forward Monte Carlo, GW04 is fine. For gradient-based inference,

every backpropagation step would zero out at the GW04 boundary — which is the motivation for building substructure from a smooth, differentiable field instead of a discrete tree.

The “statistics, not algorithm” insight¶

GW04 realisations are characterised observationally not by their recursion tree but by summary statistics:

• The Cartwright & Whitworth (2004) Q parameter (JAX-native CW04 substructure Q parameter).

• Azimuthal density variation $\sigma_\Sigma / \langle\Sigma\rangle$, with $\sigma_\Sigma/\langle\Sigma\rangle \approx 1.45 - 0.46\,D$ Küpper et al., 2011.

• Two-point correlation function $\xi(r)$ (rarely used directly).

Any generator that reproduces the same $(Q_{\mathrm{CW}},\,\sigma_\Sigma/\langle\Sigma\rangle)$ distributions as a function of one continuous parameter recovers everything observationally relevant about GW04 substructure. This is the key reason a differentiable generator can stand in for the discrete tree: match the statistics, not the algorithm.

Physical motivation: turbulent fragmentation¶

A differentiable substructure generator is arguably more physically motivated than the GW04 tree, not less. Stars form in supersonic-turbulent molecular clouds where the velocity field has a power-law spectrum

with $\beta = 11/3$ for incompressible Kolmogorov turbulence, $\beta = 4$ for highly compressible Burgers turbulence, and $\beta \approx 3.8$–4.0 for the observed ISM Kritsuk et al., 2011Federrath & Klessen, 2012. (Note this is the velocity-field spectrum; the gravoturbulent density spectrum behaves differently — it flattens with Mach, see Density PDFs and the freefall-density factor.) Stars inherit the spatial structure of the dense cores carved out of these turbulent flows: a smooth equilibrium profile perturbed by a power-law-spectrum field directly represents this picture, with steeper spectra producing clumpier (more small-scale) structure. The GW04 tree, by contrast, is a purely abstract construction with no direct connection to ISM physics.

Fractal dimension $D$ and its CW04 $Q$ signature¶

The observational handle on fractal substructure is the relationship between the Goodwin & Whitworth (2004) fractal dimension $D$ and the Cartwright & Whitworth (2004) $Q$ parameter: lower $D$ (more hierarchical clumping) gives lower $Q$; a uniform sphere sits at $Q \approx 0.79$. Representative anchors (Plummer base, $N_\star \sim 1000$):

These are schematic reference anchors illustrating the monotonic $Q$-vs-$D$ trend, not fresh output and not verbatim Cartwright & Whitworth (2004) Table 1 values — that table tabulates $Q = 0.45\pm0.09,\,0.61\pm0.08,\,0.73\pm0.06,\,0.80\pm0.02$ at $D = 1.5,\,2.0,\,2.5,\,3.0$ (3-D box-fractal models projected to 2-D, $100 \le N \le 300$). What progenax’s CW04 $Q$ estimator is quantitatively validated against is the Cartwright & Whitworth (2004) uniform-sphere anchor $Q = 0.79 \pm 0.02$ (reproduced to $<0.01$; see JAX-native CW04 substructure Q parameter). The experimental gravoturb package’s headline calibration reproduces the direction of this ladder — $Q$ decreasing as more stars are drawn from the dense turbulent tail — measured with realization bands (its VALIDATION_SUMMARY.md, AC7).

Non-equilibrium kinematics¶

Fractal/clumpy ICs are intentionally non-equilibrium: the standard equilibrium velocity DFs (Plummer, King) assume a smooth density field, so they are inconsistent with clumpy positions. The Allison et al. (2009) “cool clumpy” setup ($Q_{\mathrm{vir}} \approx 0.3$, where $Q_{\mathrm{vir}} \equiv T/|V|$ — see Virial Q convention (Q = T/|V|)) is a deliberate use of this non-equilibrium pathway to study rapid dynamical mass segregation — substructure that erases itself within $\sim 1$ Myr of evolution. See Mass segregation for the segregation side, which is released: the primordial energy_sorted_segregation generator and the equilibrium MultiComponentCluster.from_mass_segregation constructor.

Implementation, validation & references¶

• In code: the GW04/FDF generator was removed and has no released successor; what survives in released progenax is the diagnostic — src/progenax/diagnostics/substructure.py (compute_q_parameter) and src/progenax/diagnostics/q_approx.py (the differentiable kNN approximation). See the diagnostics API and the JAX-native substructure-$Q$ design. Turbulent-density ICs now live in the experimental, repo-only src/experimental/gravoturb/ package.

• Validated in: fractal substructure (the CW04 $Q$ uniform-sphere anchor $Q = 0.79 \pm 0.02$).

• Primary sources: the GW04 baseline is Goodwin & Whitworth (2004); the substructure diagnostic is Cartwright & Whitworth (2004) ($Q$ parameter) with the azimuthal-variation relation from Küpper et al. (2011); the turbulent-fragmentation picture follows Federrath & Klessen (2012) and Kritsuk et al. (2011); the cool-fractal dynamical-segregation pathway is Allison et al. (2009) and the primordially-segregated comparison case Baumgardt et al. (2008). Full notes in the bibliography.