Glossary

Definitions of common terms used in the progenax docs.

Cluster dynamics¶

half-mass radius

The radius $r_h$ enclosing half the total cluster mass: $M(<r_h) = M_{\mathrm{total}}/2$. progenax parameterises every spatial profile by $r_h$ for cross-profile comparability. See What is an initial condition?.

scale radius

The internal length scale of a spatial profile (Plummer’s $a$, King’s $r_c$, EFF’s $a$). Profile-specific; converted to/from $r_h$ via closed-form or numerical mappings.

tidal radius

The Jacobi radius $r_J$ at which the cluster’s gravity balances the host galaxy’s tidal field. Stars beyond $r_J$ are stripped. Computed by progenax.tidal.jacobi_radius. See Tidal physics.

virial Q

$Q_{\mathrm{vir}} \equiv T/|V|$, the ratio of kinetic to absolute potential energy. Equilibrium value: 0.5 (virial theorem). progenax convention. See Virial Q convention (Q = T/|V|).

CW04 Q

Cartwright & Whitworth (2004) substructure parameter: $Q = \bar m / \bar s$ from the minimum spanning tree. Distinct from the virial Q. See JAX-native CW04 substructure Q parameter.

virial equilibrium

Dynamical state in which $2T + V = 0$, equivalently $Q_{\mathrm{vir}} = 0.5$. progenax’s default IC state.

subvirial

$Q_{\mathrm{vir}} < 0.5$; cluster is collapsing. The Allison et al. (2009) cool-fractal setup uses $Q \approx 0.3$.

supervirial

$Q_{\mathrm{vir}} > 0.5$; cluster is expanding. Models post-gas-expulsion states.

centre-of-mass (COM) frame

Frame in which $\sum_i m_i\,\mathbf{r}_i = 0$ and $\sum_i m_i\,\mathbf{v}_i = 0$. progenax always returns ICs in this frame.

Distribution functions & anisotropy¶

distribution function (DF)

The phase-space density $f(\mathbf{r}, \mathbf{v})$: mass (or number) per unit volume of position–velocity space. For spherical isotropic equilibria the DF depends on energy alone, $f(E)$ — the object every progenax velocity sampler draws from. See Velocity distribution functions and the model cards.

Eddington inversion

The Abel-integral inversion that recovers the unique ergodic DF $f(\mathcal{E})$ from a density profile $\rho(\Psi)$ in a known potential — including the truncation boundary term. Walked through on Plummer velocity distribution functions; the engine behind EFFVelocityDF and Engine B.

relative potential ($\Psi$) and binding energy ($\mathcal{E}$)

$\Psi \equiv -\Phi$ (positive inside a bound system, $\to 0$ at the boundary/infinity) and $\mathcal{E} \equiv -E = \Psi - v^2/2$ (positive for bound orbits). Theory pages write the King/Michie dimensionless potential as $\psi$ or $W = \Psi/\sigma_0^2$, with central value $W_0$.

$W_0$ (King concentration)

The dimensionless central potential depth $W_0 = \Psi(0)/\sigma_0^2$ of the King/LIMEPY/Michie families; sets the concentration $c = \log_{10}(r_t/r_c)$ (typical globulars: $W_0 \approx 3$–12).

anisotropy radius ($r_a$)

The single knob of the Osipkov–Merritt and Michie constructions: velocities are isotropic well inside $r_a$ and increasingly RADIAL outside it. OM overlays keep the density fixed; Michie re-solves Poisson. See Anisotropy and rotation.

$\beta$ — four different symbols on this site

Context decides: (1) velocity anisotropy $\beta(r) = 1 - \sigma_t^2/\sigma_r^2$ (the usual dynamics meaning); (2) the turbulence spectral slope $P(k) \propto k^{-\beta}$ (Fractal substructure); (3) the Maschberger IMF low-mass exponent ($\beta = 1.4$, Classical IMFs (Salpeter, Kroupa, Chabrier, Maschberger)); (4) the experimental projected-inference summary $\beta$ (gravoturbulence pages). Theory pages define which one at first use.

Mass functions¶

IMF

Initial mass function $\xi(m) = \mathrm{d}N/\mathrm{d}m$. The birth-mass distribution of stars. See Initial mass functions.

Salpeter slope

$\alpha = 2.35$. The high-mass slope of the IMF, established by Salpeter (1955).

mass ratio

$q = m_2/m_1 \in [0, 1]$ for a binary. Distribution $g(q | M_1)$ follows Moe & Di Stefano (2017).

binary fraction

Probability that a primary has at least one companion. Mass-dependent; $\sim 0.5$ for solar-type, $\sim 0.9$ for O-type.

twin excess

Narrow Gaussian peak at $q \approx 1$ in the Moe & Di Stefano (2017) mass-ratio distribution. Solar-type stars show the strongest excess, $f_{\mathrm{twin}} \approx 0.10$.

confidently wrong

The regime where a misspecified likelihood produces a posterior whose 95% CI shrinks below the bias and excludes the true parameter value. Demonstrated for binary IMF inference at $N \gtrsim 10^4$ in Binary-aware IMF recovery.

Substructure¶

fractal dimension

$D \in [1.6, 3.0]$. Parameter of the Goodwin & Whitworth (2004) fractal IC. $D = 3$ uniform; $D = 1.6$ highly clumpy.

Fractal Displacement Field

A differentiable fractal-IC generator that once lived in progenax, removed in the 2026-06 clean-room rewrite with no released successor. (Not to be confused with the freefall-density factor below, also abbreviated FDF.) Turbulent-density ICs are now the experimental gravoturb package. See Fractal substructure.

mass segregation

Spatial arrangement where massive stars preferentially occupy central / low-energy orbits. Primordial (set at IC time, see Baumgardt et al. (2008)) vs dynamical (emerges via two-body relaxation, Allison et al. (2009)).

Λ_MSR

Allison et al. (2009) MST ratio for quantifying mass segregation. $\Lambda \sim 1$ for unsegregated; $\Lambda > 1$ for segregated.

Gravoturbulence¶

density PDF

Volume-density distribution $p_V(\rho)$ in a turbulent self-gravitating cloud. Lognormal core + power-law tail per Federrath & Klessen (2012)Burkhart (2018).

Mach number

Sonic Mach $\mathcal{M} = v_{\mathrm{turb}}/c_s$. Sets the lognormal variance via $\sigma_s^2 = \ln(1 + b^2\,\mathcal{M}^2)$.

forcing parameter

$b \in [1/3, 1]$. Turbulence-driving geometry: $b = 1/3$ solenoidal, $b = 1$ compressive, $b \approx 0.4$ natural mix.

freefall-density factor (FDF)

The kernel $\rho/t_{\mathrm{ff}}(\rho) \propto \rho^{3/2}$ that weights local density by its star-forming efficiency. See Density PDFs and the freefall-density factor.

magnification factor

$\zeta$ = SFR boost a centrally-concentrated cloud gets over a uniform top-hat. Parmentier & Pasquali (2020) Eq. 6 gives the closed form for power-law profiles. See The magnification factor ζ — three ways to compute it.

BM19 framework

Burkhart (2018)Burkhart & Mocz (2019) forward model: turbulence parameters → density PDF → SFR. See BM19 dense-gas SFR framework.

JAX programming¶

PyTree

A nested Python structure (dict, list, tuple, custom class) that JAX can trace through. progenax classes are PyTrees via equinox.Module.

JIT

Just-in-time compilation via @jax.jit. Compiles a Python function to XLA, eliminating Python overhead for hot paths.

vmap

jax.vmap. Vectorises a function over an axis without writing a loop. progenax uses this extensively for parallelisation over particles.

grad

jax.grad. Automatic differentiation. The foundation of progenax’s HMC inference capability.

scan

jax.lax.scan. Fixed-iteration sequential loop primitive. Used instead of while_loop for differentiability. See Differentiability rules.

while-loop antipattern

Using a data-dependent jax.lax.while_loop in code that needs gradients. Fixed-shape JAX loops are acceptable when gradients and static-shape compilation remain well-defined. See Differentiability rules.

Architecture¶

SpatialProfile protocol

Runtime-checkable protocol every spatial profile satisfies: sample_positions and characteristic_radius. See Protocol-based composition.

VelocityDF protocol

Runtime-checkable protocol every velocity DF satisfies: sample_velocities. See Protocol-based composition.

IMFProtocol

Runtime-checkable protocol every IMF satisfies: logpdf, cdf, ppf, sample, and mean_mass. See Protocol-based composition.

three-brick state

A planned architecture pattern described in the design docs. The current public code does not expose SystemParams or ParticleSystem. See Three-brick state pattern.

DEFAULT_UNITS

Per-package default unit system (STELLAR for progenax) used only by convenience wrappers. Core APIs require explicit units. See Units policy.