Virial Q convention (Q = T/|V|)

Virial Q convention¶

The virial Q parameter pins the dynamical state of an N-body initial condition. progenax adopts the convention

where $T$ is the total kinetic energy of the particles and $V$ is the total gravitational potential energy. This page documents the convention, justifies the choice, and lists the call sites.

Three conventions in the literature¶

Three definitions appear in the literature; readers should not assume any value without checking which is in use:

The Aarseth et al. (1974) $Q = T/|V|$ form is the dominant convention in star-cluster N-body work and is what every progenax convenience function expects. The factor-of-2 forms appear when the virial theorem is quoted directly from $2T + V = 0$.

Why $Q = 0.5$ is equilibrium¶

The scalar virial theorem for a gravitationally bound system in steady state reads

Substituting $V = -|V|$ (gravitational potential energy is negative) and solving for $T/|V|$:

A system at $Q_{\mathrm{vir}} = 0.5$ is in dynamical equilibrium: the kinetic energy is exactly the value that balances the gravitational self-binding. Departures from 0.5 are physically meaningful:

The subvirial regime is particularly important for fractal-substructure work: Allison et al. (2009) showed that cool ($Q_{\mathrm{vir}} \approx 0.3$) clumpy initial conditions produce dynamical mass segregation on $\sim 1$ Myr timescales — much faster than the classical relaxation time. Reproducing that result therefore requires the ability to seed $Q_{\mathrm{vir}}$ explicitly, which is what virial_scale provides.

Implementation¶

Every velocity DF in progenax produces velocities at exactly $Q_{\mathrm{vir}} = 0.5$ by construction (the equilibrium DFs of Plummer (1911), King (1966), and EFF). To reach a target $Q_{\mathrm{vir}}^\star$, progenax rescales the velocities by

The factor of 2 converts from the equilibrium $Q = 0.5$ baseline. The rescaling is implemented in progenax.builders.virial_scale:

@jax.jit
def virial_scale(positions, velocities, masses, Q_target, G, softening=0.0):
    T = compute_kinetic_energy(velocities, masses)
    V = compute_potential_energy(positions, masses, G=G, softening=softening)
    Q_current = T / jnp.abs(V)
    scale = jnp.sqrt(Q_target / Q_current)
    return velocities * scale   # only velocities are rescaled

Three properties hold by construction:

1. Energy ratio is enforced exactly (to floating-point precision). Tests in tests/integration/test_physics_validation.py::test_virial_scaling verify $|Q_{\mathrm{measured}} - Q_{\mathrm{target}}| < 10^{-12}$ for $N \le 10^4$.

2. Spatial structure is preserved. Only velocities scale; positions are untouched. The density profile, half-mass radius, and any substructure are unchanged.

3. Differentiable. virial_scale is JIT-compatible and survives jax.grad, so it can sit inside a posterior-evaluation chain.

Call sites and contract¶

All progenax IC builders accept Q_target and pipe it through virial_scale:

Validation¶

The progenax test suite includes physics-anchored regressions for the virial Q convention:

• tests/validation/test_plummer_physics.py::test_virial_ratio — sample $N = 10^4$ Plummer particles, measure Q post-rescaling, assert agreement to $\sim 0.5\%$ (statistical from finite-N kinetic-energy noise).

• tests/integration/test_physics_validation.py::test_virial_scaling — parametrise $Q_{\mathrm{target}} \in \{0.1, 0.3, 0.5, 0.7, 1.0\}$, assert Q recovered to 10^-12.

• See Plummer equilibrium for the rendered test results.

Why not $\alpha_{\mathrm{vir}} = 2T/|U|$?¶

The cloud-evolution literature uses $\alpha_{\mathrm{vir}} = 2T/|U|$ for GMC virial parameters, where $|U|$ is the gravitational binding energy Federrath & Klessen, 2012. progenax does not mirror that convention because the IC builders operate on point-mass particle distributions, not continuous gas. Using the cluster-N-body $Q = T/|V|$ convention keeps consistency with Aarseth et al. (1974), McLuster Küpper et al., 2011, NBODY6, and the rest of the star-cluster N-body ecosystem.

When converting between conventions: $\alpha_{\mathrm{vir}} = 2 \cdot Q_{\mathrm{vir}}$. Equilibrium is $\alpha_{\mathrm{vir}} = 1$ in cloud notation, $Q_{\mathrm{vir}} = 0.5$ in the progenax notation.

References¶

The convention follows Aarseth et al. (1974)’s star-cluster N-body tradition and matches McLuster Küpper et al., 2011. The out-of-equilibrium use cases are Allison et al. (2009) (subvirial cold collapse for primordial segregation) and Goodwin & Whitworth (2004) (supervirial post-gas-expulsion). For the substructure Q parameter that shares the letter, see JAX-native CW04 substructure Q parameter.