Multiplicity statistics (Moe & Di Stefano 2017)

Multiplicity statistics: the Moe & Di Stefano (2017) joint distribution¶

A practical binary-aware IMF requires knowing the joint distribution of binary parameters: the probability that a primary of mass $M_1$ has a companion at all, and conditional on having one, the joint distribution of mass ratio $q$, orbital period $P$, and eccentricity $e$. Moe & Di Stefano (2017) compiled the most comprehensive such census to date, combining spectroscopic, eclipsing, long-baseline interferometric, adaptive-optics, and common-proper-motion samples, each carefully corrected for its own selection function. The result is the calibration progenax depends on.

This chapter unpacks two of the most consequential findings from that work — that binary properties are not universal, and that the joint distribution is not separable — then catalogues the per-mass and per-period numbers progenax stores as backing data.

Two properties that complicate the naive picture¶

Property 1: binary properties vary systematically with primary mass. Lower-mass stars and higher-mass stars have qualitatively different binary populations. Treating “the binary fraction” as a constant $f_b \approx 0.5$ (a common shortcut in earlier work) is correct on average for solar-type primaries but wrong by factors of $\sim 2$ at the mass-spectrum extremes.

Property 2: the joint distribution is not separable.

The mass-ratio distribution depends on both $M_1$ and $P$; eccentricity depends on $P$; companion frequency depends on both $M_1$ and $P$. Treating these as independent factors introduces correlated errors that are subtle but systematic — a common pitfall in older population-synthesis work.

progenax respects this non-separability by sampling the joint $f(M_1, q, P, e)$ from the Moe & Di Stefano (2017) piecewise fits: the master parameter set (their Table 1, Eq. 2) plus the per-mass multiplicity statistics (Table 13), the period-conditional mass-ratio slopes (Table 11), and the eccentricity slopes (Tables 10 and 12).

Terminology: companion frequency vs binary fraction¶

For O-type primaries, $f_{\mathrm{mult}} \sim 2.1$ — the typical O-star has 2+ companions. The “binary fraction” in the sense of “probability of at least one companion” is then close to 1, while the companion frequency exceeds 2.

In simplified models (like progenax’s BinaryIMF) that allow at most one companion per primary, the “binary fraction” $f_b(m_1)$ is treated as a probability — i.e. it is bounded above by 1. This is a deliberate simplification; the implications for inferred IMF slopes are documented at Limitations.

Mass-dependent binary fraction¶

```{list-table} Multiplicity fraction $f_b = 1 - \mathcal{F}_{n=0}$ ($\le 1$; not the companion frequency $f_{\mathrm{mult}}$): Moe & Di Stefano (2017) Table 13 single-star row for $M_1 \ge 0.8\,\Msun$, M-dwarf surveys below. :header-rows: 1

• • Primary mass

  • $f_b$

  • Stellar type

• • $M_1 < 0.1\,\Msun$

  • 0.22

  • Very low mass / brown dwarfs

• • 0.1–$0.5\,\Msun$

  • 0.26

  • M-dwarfs

• • 0.5–$1.0\,\Msun$

  • 0.44

  • K/G-dwarfs

• • 1.0–$2.0\,\Msun$

  • 0.50

  • F/A-stars

• • 2.0–$5.0\,\Msun$

  • 0.60

  • B-stars

• • 5–$10\,\Msun$

  • 0.80

  • Early B

• • $M_1 > 10\,\Msun$

  • 0.90

  • O-stars

The trend is steep: M-dwarfs are mostly single, O-stars are almost
always in multiples. The factor-of-4 dynamic range in $f_b$ across
the mass spectrum is the leading-order reason that single-IMF
inferences from binary-rich populations are biased — see
[](binary.md) for the inference-side consequences.

```{note}
**These bins are not identical to Moe & Di Stefano's Table 13 bins.**
Table 13 quotes $f_b = 1 - \mathcal{F}_{n=0} = 0.40$ for solar-type
primaries ($0.8$–$1.2\,\Msun$) and $0.94$ for O-stars ($9$–$16\,\Msun$);
the table above lists $0.44$ at $0.5$–$1.0\,\Msun$ and $0.90$ at
$M_1 > 10\,\Msun$. The differences are a **binning** artefact, not a
disagreement: the $0.5$–$1.0\,\Msun$ bin extends *below* Table 13's
$0.8\,\Msun$ floor, where lower-fraction M-dwarf surveys fill in, and
the open-ended $M_1 > 10\,\Msun$ bin averages across O and early-B
rather than picking out Table 13's $9$–$16\,\Msun$ O-star value.
progenax samples the **Table 13 grid itself** (see
[](../../99-bibliography/per-paper/moe-distefano-2017.md)); the rows
here are a coarser pedagogical summary.

Three orbital-period regimes¶

Moe & Di Stefano (2017) identify three qualitatively distinct period regimes:

The three regimes have distinct mass-ratio distributions, which is the source of the period-conditional structure in $g(q \mid M_1, P)$. progenax’s default BinaryIMF uses period-averaged mass-ratio parameters reduced from Moe & Di Stefano (2017) Table 13; surveys sensitive to a specific period range (spectroscopic = short-period, visual = wide) need a full period-conditional likelihood layer. That layer is described conceptually in Mass-ratio distributions, but the current BinaryIMF does not export a with_period_conditional() constructor.

How the calibration was derived¶

Moe & Di Stefano (2017) analysed dozens of binary samples, each spanning a narrow interval of $M_1$ and $P$. For each sample they:

1. Identified the relevant selection function of the survey technique (spectroscopic surveys are biased toward large $q$ and short $P$; visual surveys are biased toward wide separations).

2. Corrected for incompleteness using the known selection function of each technique.

3. Fit the intrinsic mass-ratio distribution as a power-law plus twin excess (see Mass-ratio distributions).

4. Identified a $\sim 30\%$ contamination rate from white-dwarf companions masquerading as main-sequence binaries in spectroscopic samples, and corrected for it.

The resulting joint $f(M_1, q, P, e)$ is the closest thing the field has to a “ground truth” binary-population calibration. progenax encodes the Table 13 grids in progenax.imf.MoeDiStefano2017 (and the faithful period-dependent MoeDiStefano2017Full / MoeJointOrbit in progenax.imf.binary), and exposes the mass-dependent multiplicity fraction through MassDependentBinaryFraction and the higher-level BinaryIMF API.

Solar-mass calibration and the twin excess¶

For solar-type primaries (0.8–$1.2\,\Msun$), the twin fraction $f_{\mathrm{twin}} = 0.10$ is the highest in the mass spectrum. The twin excess is a narrow peak in $g(q)$ near $q = 1$ with width $\sigma_{\mathrm{twin}} \approx 0.03$, sitting on top of the power-law $q^\gamma$ background. Solar-type primaries therefore have two distinct populations of binaries: the bulk power-law and a $\sim 10\%$ excess of near-equal-mass twins. The relative weights shift across the mass spectrum:

The full $g(q \mid M_1)$ form combining the power law and twin excess is documented in Mass-ratio distributions.

Higher-order multiples¶

progenax’s default BinaryIMF model caps at one companion per primary. Real populations include triples, quadruples, and higher: $\sim 10\%$ of solar-type systems are triples, rising to $> 50\%$ for O-type Moe & Di Stefano, 2017Sana et al., 2012. The single-companion approximation underestimates the high-mass distortion (a triple with masses $m_1 + m_2 + m_3$ inflates the system mass more than the binary $m_1 + m_2$). For O-star-dominated populations this is a substantial effect; for solar-type-dominated populations it is a small correction.

The architectural extension to triples is straightforward — replace the single $f_b(m_1)$ with a Poisson-companion model — but is deferred to a future progenax version.

Check yourself¶

Implementation, validation & references¶

• In code: the joint statistics live in src/progenax/imf/binary/moe_di_stefano.py (MoeDiStefano2017, MoeDiStefano2017Full) and the mass-dependent multiplicity in src/progenax/imf/binary/binary_fraction.py (MassDependentBinaryFraction) — see the IMF API.

• Validated in: IMF statistics and binary-aware recovery, which exercise the Table 13 grids and the binary-fraction trend.

• Primary sources: Moe & Di Stefano (2017) (the comprehensive joint-distribution census), Sana et al. (2012) (foundational massive-star multiplicity), and Moe et al. (2019) (metallicity dependence) — full notes in the bibliography. The period-distribution side is documented at Binary period distributions; the mass-ratio side at Mass-ratio distributions.