Unresolved-binary mass function (B4) - progenax — differentiable initial conditions for N-body simulations

A binary that a survey cannot resolve appears as a single, brighter source, and its inferred mass is biased high — so unresolved binaries distort the observed stellar mass function. This demo recovers the binary fraction $f_b$ from that distortion, and makes the central point: getting $f_b$ right requires the correct Moe & Di Stefano (2017) period–mass-ratio coupling Moe & Di Stefano, 2017, because the blended (close) binaries are a biased, high- $q$ subset that an independent- $(P,q)$ model gets wrong.

The forward model¶

Each binary has a primary $m_1$ (from a Maschberger IMF), a mass ratio $q$ ( $m_2 = q\,m_1$ ) and a period $P$ ; the semimajor axis $a$ follows Kepler’s third law. A survey resolves wide pairs and blends close ones, $a < a_{\rm crit}$ (here $50\,\mathrm{AU}$ ):

\text{single} \to m_1, \qquad \text{resolved} \to \{m_1, m_2\}, \qquad \text{blended} \to m_{\rm obs} = L^{-1}\!\big(L(m_1) + L(m_2)\big) > m_1,

(1)

where $L(m)$ is the Tout et al. (1996) ZAMS mass–luminosity relation Tout et al., 1996 (provided in-package by progenax.stellar). The observed mass function is a linear mixture in $f_b$ ,

\mu_k(f_b) = N_{\rm sys}\big[(1-f_b)\,S_k + f_b\,B_k\big],

(2)

with the single template $S_k$ (the IMF) and the binary template $B_k$ (per-binary expected catalogue stars after resolution + blending) precomputed from a large pool. $f_b$ is recovered by a per-bin Poisson MLE, with the Fisher variance from the (linear, ODE-free) information.

The controlled coupling test¶

The “independent” comparison uses the same pool with $q$ shuffled against $P$ — this preserves the $q$ and $P$ marginals exactly and changes only the $P$ – $q$ correlation. So any bias is attributable to the coupling alone. The truth is Moe-coupled; we fit $f_b$ with the (correct) Moe template and the (wrong) independent template.

Inputs and assumptions¶

The fit recovers one parameter, the binary fraction $f_b$ ; everything else is an assumed-known input. The recovered-vs-known split is the key onboarding lesson — the demo’s punchline (that the $P$ – $q$ coupling must be right) is a statement about one of those assumed inputs.

Table 1:Model inputs

Input	Meaning and role	Status (fiducial)
$f_b$	Binary fraction — the science target, recovered from the Poisson mixture $\mu_k=N_{\rm sys}[(1-f_b)S_k+f_b B_k]$ (`FB_BOX=(0.05,0.95)`).	recovered (truth 0.5)
Moe $P$ – $q$ – $e$ coupling	The joint orbital model (`MoeJointOrbit`) that builds the correct binary template $B_k$ ; its $P$ – $q$ correlation is the whole point.	known / fixed (assumed correct)
$q_{\min}$	Minimum mass ratio in `MoeDiStefano2017Full`; truncates the secondary masses feeding the blend.	known / fixed (0.1)
$a_{\rm crit}$	Resolution limit: pairs with semimajor axis $a<a_{\rm crit}$ blend, wider ones resolve (selects which binaries distort the MF).	known / fixed (`A_CRIT_AU=50` AU)
$Z$	Metallicity for the Tout ZAMS $L(m)$ used to compute the blend mass $m_{\rm obs}=L^{-1}(L_1+L_2)$ .	known / fixed (`Z_MET=0.02`, solar)
IMF	Maschberger ( $\alpha=2.3$ , 0.08– $100\,M_\odot$ ): draws primaries $m_1$ , hence the single template $S_k$ and the blend masses.	known / fixed
$N_{\rm sys}$	Systems in the mock cluster; sets the Poisson counts and the significance of the wrong-model systematic ( $\propto\sqrt{N_{\rm sys}}$ ).	known / fixed ( $3\times10^5$ )
$N_{\rm pool}$ , `M_BINS`, `SEED`	Template pool size ( $6\times10^5$ ), mass-function bins (30 log bins), and PRNG seeds (templates / data / $q$ -shuffle).	numerical choices

Result — freshly run, ALL PASS¶

Measured 2026-06-12 ( $N_{\rm sys}=3\times10^5$ , $N_{\rm pool}=6\times10^5$ , $a_{\rm crit}=50$ AU, solar $Z$ ; wall $\approx 10$ s; exit 0).

The mechanism. The blended (close-pair) subset is more equal-mass under Moe than under the decoupled model — median $q = 0.540$ vs 0.479 — because Moe couples short periods to high $q$ . That is the whole effect.

model fitted	recovered $f_b$	vs truth 0.500
Moe template (correct)	$0.5016 \pm 0.0074$	$+0.22\sigma$ ✓
independent template (wrong)	$0.4749 \pm 0.0070$	$-3.6\sigma$ (≈5% low)

Ignoring the $P$ – $q$ coupling biases $f_b$ low by ~5%. This is a systematic (a wrong-model error), so its fractional size is fixed while its significance grows as $\sqrt{N_{\rm sys}}$ : only $-1.5\sigma$ at $N_{\rm sys}=5\times10^4$ , but $-3.6\sigma$ at the $3\times10^5$ survey scale here. At a real cluster survey it is unambiguous. Gate summary:

Check	Gate	Status
mechanism: blended median $q$ (Moe > independent)	coupling present	PASS
self-consistency (Moe template @ truth)	$<4\sigma$ Poisson (2.76)	PASS
$f_b$ recovery (Moe template)	$<3\sigma$ (+0.22)	PASS
$f_b$ bias (independent template)	$>3\sigma$ (-3.6)	PASS

Figure¶

Unresolved-binary mass function (scripts/demo_binary_mass_function.py, ALL
PASS). (a) The observed mass function (black) decomposed into singles (blue) and
the binary contribution (vermilion) — the blended pairs push a bump to higher
inferred mass. (b) The mechanism: the blended subset’s median q is higher under
the Moe coupling (vermilion) than the P-shuffled independent model (sky) at the
same marginals. (c) Recovered f_b: the Moe template recovers the truth (dashed)
while the independent template is biased low. — Figure 1:**Unresolved-binary mass function** (`scripts/demo_binary_mass_function.py`, ALL PASS). **(a)** The observed mass function (black) decomposed into singles (blue) and the binary contribution (vermilion) — the blended pairs push a bump to higher inferred mass. **(b)** The mechanism: the blended subset’s median $q$ is higher under the Moe coupling (vermilion) than the $P$ -shuffled independent model (sky) at the same marginals. **(c)** Recovered $f_b$ : the Moe template recovers the truth (dashed) while the independent template is biased low.

Caveats¶

Warning

Photometry is the Tout (1996) ZAMS relation ( $L$ – $M$ at fixed metallicity), from progenax.stellar (a startrax placeholder) — a clean main-sequence stand-in. Real surveys add band-dependent blending, extinction, and unequal evolutionary states; full SED/bandpass photometry is the fluxax package’s job, not this demo.
No external dependency. The ZAMS relations are in-package, so B4 runs in a clean environment / CI — nothing to install beyond progenax.
f_b only. The demo recovers the binary fraction and shows (not fits) that the $q$ -coupling drives the bias; jointly recovering the $q$ -distribution shape is a natural extension. The recovery assumes a known IMF and resolution limit.
Clean census + flat $f_b$ . A constant binary fraction and complete catalogue; Moe’s own mass-dependent $f_b(m_1)$ and selection are not modelled here.
$a_{\rm crit}$ is a single sharp, period-only cut. Blending is a hard step at $a<a_{\rm crit}$ on the physical semimajor axis — no distance, projection, or flux-contrast dependence, and eccentricity is drawn but unused (resolution depends only on $a$ , so eccentric pairs that would be resolvable near apocenter are not modelled).
Self-consistent by construction. Templates and data share the same IMF, $q_{\min}$ , $Z$ (=solar), $a_{\rm crit}$ and pool family; only the $P$ – $q$ coupling is varied between the right/wrong models. The mechanism is shown, not stress-tested against IMF / $a_{\rm crit}$ / metallicity misspecification.
No photometric noise. The blend mass is the exact $L^{-1}(L_1+L_2)$ with no measurement scatter; binning is the only smoothing. The $f_b$ uncertainty is the inverse-Hessian (fisher_cov) Gaussian approximation, and the MLE is seeded at the truth (a clean-start convergence check, not a multimodality test).

How to run¶

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

References¶

The binary $P$ – $q$ – $e$ coupling is Moe & Di Stefano (2017); the ZAMS mass–luminosity relation is Tout et al. (1996). The Moe and companion models are documented on the binary statistics theory pages.

References¶

Moe, M., & Di Stefano, R. (2017). Mind your Ps and Qs: The interrelation between period (P) and mass-ratio (Q) distributions of binary stars. The Astrophysical Journal Supplement Series, 230, 15. 10.3847/1538-4365/aa6fb6
Tout, C. A., Pols, O. R., Eggleton, P. P., & Han, Z. (1996). Zero-age main-sequence radii and luminosities as analytic functions of mass and metallicity. Monthly Notices of the Royal Astronomical Society, 281, 257. 10.1093/mnras/281.1.257