Moe & Di Stefano (2017)

Abstract (paraphrased)¶

Compiles ~30 surveys of early-type (and re-analyses solar-type) main-sequence binaries across spectroscopy, eclipses, interferometry, AO, and common proper motion, correcting each for its selection effects. Measures the intrinsic joint distribution $f(M_1,q,P,e)$ and shows it is not separable: at short periods binaries have small $e$, modest $q$, and a small twin excess; at intermediate $\log P\approx3.5$ the companion frequency peaks with $q$ weighted to small values; at long periods companions approach random IMF pairings. The corrected statistics are fit with mathematical functions for use in binary population synthesis.

Multiplicity statistics — Table 13 (verified, p. 52)¶

Table 13 gives, per primary-mass bin (solar 0.8–1.2, A/late-B 2–5, mid-B 5–9, early-B 9–16, O-type >16 $M_\odot$):

The mass-ratio distribution is a three-parameter, period-dependent form (Table 1, Eq. 2): a small-$q$ slope $\gamma_{\rm smallq}$ ($0.1<q<0.3$), a large-$q$ slope $\gamma_{\rm largeq}$ ($0.3<q<1.0$), and a twin excess $\mathcal F_{\rm twin}$ ($q>0.95$) — all functions of $M_1$ and $\log P$ ($\gamma_{\rm largeq}$ steepens from -0.5 at $\log P=1$ to -2.0 at long $P$; $\mathcal F_{\rm twin}$ falls to $<0.03$ beyond $\log P\gtrsim3$). The multiplicity frequency $f_{\rm mult}>1$ for massive stars because O/B stars are commonly triples/quadruples.

Eccentricity distribution — §9.2 (verified, p. 38, Fig. 36)¶

The eccentricity follows a power law $p(e) \propto e^{\eta}$ on $0 \le e \le e_{\max}(P)$ (their Fig. 36), with the slope $\eta$ a function of orbital period and primary mass. The upper limit is the period-dependent Roche-lobe ceiling (their Eq. 3, p. 38),

which guarantees the components do not fill their Roche lobes at periapsis (e.g. $e_{\max}(10\,\mathrm{d})\approx0.66$, $e_{\max}(100\,\mathrm{d})\approx0.93$); $P \le 2$ d circularizes. $\eta = 0$ is uniform ($\langle e\rangle = 0.5$); $\eta = 1$ is thermal $f(e)=2e$ ($\langle e\rangle = 2/3$). The analytic $\eta$ fits are

with linear interpolation in $M_1$ for $3 \le M_1 \le 7\,M_\odot$ (Eqs. 17–18 valid $0.5 < \log P < 6$/5). Late-type binaries asymptote to $\eta \approx 0.5$ at long $P$; early-type intermediate-period binaries reach $\eta \approx 0.8$ (near-thermal); short periods circularize ($\eta$ “not well defined” for $\log P \lesssim 1$). Sana et al. (2012) likewise find short-period O-star binaries are eccentricity-poor ($\eta < 0$); their precise slope is tabulated in the supplementary Table S3 (paywalled, not in the held main report).

Use in progenax¶

• Mass-ratio distributions, Binary-aware IMF recovery.

• progenax.imf.MoeDiStefano2017 — a period-averaged single-slope reduction of the $\gamma_{\rm smallq}/\gamma_{\rm largeq}+\mathcal F_{\rm twin}$ model (captures the trend; appropriate for a mass-function analysis that marginalizes over period, e.g. the Confidently Wrong forward model).

• progenax.imf.MoeDiStefano2017Full / MoePeriod / MoeJointOrbit (Batch 4i) — the faithful two-slope, period-dependent $q\,|\,(M_1,P)$ from Table 13 (bilinear interpolation, grid inverse-CDF) and the joint $(P,q,e)$ interrelation sampler.

• progenax.binaries.MoeCompanions (Batch 4k) — wires MoeJointOrbit + Moe’s own $f_b(M_1)$ into build_binary_cluster as a CompanionModel; the same $q$ sets $m_2 = q\,m_1$, so the P–q correlation is self-consistent in the secondary masses.

• progenax.imf.MassDependentBinaryFraction — the multiplicity fraction $1-\mathcal F_{n=0}$ from Table 13 (e.g. solar $1-0.60=0.40$, O-type $1-0.06=0.94$), with $<0.8\,M_\odot$ bins from M-dwarf surveys.

• progenax.binaries.MoeEccentricity — faithful implementation of the $p(e)\propto e^{\eta}$ law with $\eta(\log P, M_1)$ from (2) (Eqs. 17–18) on $[0, e_{\max}(P)]$ with the period-dependent Roche ceiling (1) (Eq. 3); samples via inverse-CDF $e = e_{\max}(P)\,u^{1/(\eta+1)}$, with $\eta \le -1$ (very short $P$) → circular. A fixed numerical ceiling (e_max, default 0.99) caps the long-$P$ limit where Eq. 3 → 1.

• progenax.binaries.LogisticThermalEccentricity — a smooth circular→thermal heuristic (a logistic blend toward $f(e)=2e$); not Moe’s $\eta(P)$ law (see Duquennoy & Mayor (1991)).

Notes¶

The most-referenced paper across progenax binary modelling. The central result — that $P$, $q$, and $e$ are interrelated — is captured at two fidelities: the period-averaged MoeDiStefano2017 (for mass-function analyses that marginalize over period) and the faithful joint MoeJointOrbit / MoeCompanions (for dynamical ICs with realistic orbits; Batches 4i–4k). The composition into build_binary_cluster is documented in the API reference; the faithful two-slope q-axis follow-up is CLOSED.