Binary energy budget (B9)

A cluster with primordial binaries carries two energy scales, and conflating them is a classic mistake. build_binary_cluster virializes the system centres-of-mass to $Q=0.5$ treating each binary as a single point mass — the McLuster scale-separation convention Küpper et al., 2011 — and leaves the internal binary binding energy as a separate reservoir that the global virial scaling never touches. This demo makes that separation explicit and shows why the naive virial ratio of the resolved stars is not the cluster’s.

The two scales, from binary_energy_budget:

• $W_{\rm com}$ — the cluster’s bulk gravitational binding on the system COMs; the scale the cluster is virialized on, $Q_{\rm com} = T_{\rm com}/|W_{\rm com}| \approx 0.5$.

• $E_{\rm internal} = \sum_b \left(-G\,m_{1}m_{2}/2a_b\right)$ — the internal binary binding (vis-viva), set by the periods and masses and independent of where the COM sits in the cluster.

because the resolved ratio mixes the two scales. It is not the cluster’s virial ratio (audit S10): a hard binary’s internal virial is itself $\approx 0.5$ (time-averaged), so sampled at random orbital phases $Q_{\rm resolved}$ scatters around 0.5 rather than deflating monotonically. The robust, gated statement is the energy separation — $Q_{\rm com}$ cleanly recovers 0.5 while $|E_{\rm internal}|$ dwarfs $|W_{\rm com}|$.

Why a young cluster. Primordial binaries are the population present at birth, before dynamical processing ionizes the soft ones and hardens the rest. Their natural home is therefore a young cluster, for which the Elson–Fall–Freeman (1987) Elson et al., 1987 extended power-law profile is the standard parametrization — not the King model of an old, relaxed globular. So the primary cluster here is EFF ($a=1$ pc, $\gamma=2.5$, $r_t=15$ pc); a concentrated King ($W_0=7$) King, 1966 appears only in the environment figure (below).

What is built¶

A Plummer-masses-first EFF young cluster of $N_{\rm sys}=2000$ systems, virialized to $Q=0.5$, with a smooth (differentiable) Maschberger $\alpha=2.3$ primary IMF over $[0.08, 100]\,\Msun$ and companions from IndependentCompanions (or the full Moe & Di Stefano (2017) Moe & Di Stefano, 2017 model). Two controlled sweeps:

• Hardness — the companion period band LogUniformPeriod is slid from $\log_{10}(P/{\rm day})=1.5$ (hard, $\sim$30 d) to 5.5 (soft, $\sim$900 yr) at fixed $f_b=0.5$. Shorter periods $\Rightarrow$ smaller $a$ $\Rightarrow$ deeper binding (vis-viva).

• Binary fraction — $f_b$ swept $0.1\!\to\!1.0$ at a fixed broad period band.

Inputs and assumptions¶

This demo fits nothing — it is a controlled sweep over hardness and binary fraction, gating the energy separation and reporting diagnostics. Every quantity is a known/fixed physical input or a swept/numerical choice.

Result — freshly run, ALL PASS¶

Measured 2026-06-12 (CPU/float64, $N_{\rm sys}=2000$, seeds from PRNGKey(0); wall $\approx 32$ s; exit 0).

Hardness sweep (EFF, $f_b=0.5$). $Q_{\rm com}$ is pinned at 0.5 to four decimals at every point; $|E_{\rm internal}|$ spans three orders of magnitude; $Q_{\rm resolved}$ scatters (the contamination):

Across both sweeps the reservoir ratio $|E_{\rm internal}|/|W_{\rm com}|$ stays in $[11.6,\ 1.9\times10^{3}]$ — the internal binding always dwarfs the cluster potential — while $\max|Q_{\rm resolved}-Q_{\rm com}| = 0.257$ confirms the resolved ratio is a contaminated proxy. Gate summary:

The environment comparison¶

The same realistic Moe & Di Stefano population is laid into a young EFF and a concentrated King ($W_0=7$) potential using the same random key, so the masses and orbital elements — and therefore $E_{\rm internal}$ — are byte-identical between the two. Only the cluster structure differs:

The concentrated King is more tightly bound ($|W_{\rm com}|$ larger), so the same primordial binary population is a relatively larger energy store in the young, puffy cluster — the global importance of the binary reservoir depends on the birth environment. This is the Heggie (1975) hard/soft intuition at the level of the total energy budget; a per-binary hard/soft classification ($|E_{\rm bind}|$ vs the local $kT$) is a related but distinct quantity and is not claimed here.

Figures¶

Caveats¶

How to run¶

env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_binary_energy_budget.py

References¶

The EFF young-cluster profile is Elson et al. (1987); the King model King (1966); the binary statistics follow Moe & Di Stefano (2017); the COM/internal scale separation is the McLuster convention Küpper et al. (2011); the hard/soft intuition is Heggie (1975). The budget API and its vis-viva / scale-separation tests live in progenax.binaries.diagnostics.