Binary populations

A binary population is the kinematic complement to the binary IMF discussed at Binary-aware IMF recovery. The IMF chapter covers the statistical framework — what fraction of stars are in binaries, what mass ratios they have, and how that biases inferred IMF slopes. This chapter family covers the orbital mechanics: given the binary fraction and mass ratios, how to assign Keplerian orbits to each binary, sample the orbital phase, and produce the resolved component positions and velocities that an N-body integrator consumes.

Map of the section¶

What a “binary” is in progenax¶

A binary in progenax is two particles whose relative orbit is a set of KeplerElements. Both components live in the same (positions, velocities, masses) arrays as single stars — secondaries are not “second-class” entries. The primordial pairing is recorded as provenance at $t=0$ on the ICResult returned by build_binary_cluster:

positions                 # (N, 3) component positions in the cluster frame
velocities                # (N, 3) component velocities
masses                    # (N,)   every star's mass, including secondaries
primordial_system_id      # (N,)   members of one system share an id
is_primordial_secondary   # (N,)   True on the secondary of each primordial binary

This labelling is the birth pairing. It goes stale under dynamical evolution (encounters ionise soft binaries, three-body captures form new ones, exchanges swap partners), so the current binary population of an evolved snapshot must be measured from the phase-space state, not read off the labels: progenax.binaries.find_bound_pairs(positions, velocities, masses, G=...) returns the energy-bound mutual-nearest-neighbour pairs, and primordial_survival compares them to the $t=0$ ids (survived / disrupted / newly-formed). A single binary’s elements are recovered with KeplerElements.from_state(r_rel, v_rel, M_total, G=...).

Composability¶

build_binary_cluster composes five independent axes:

The companion_model has two implementations: IndependentCompanions (versatile independent $f_b \times q \times P \times e$ marginals — the period-averaged default) and MoeCompanions (the faithful Moe & Di Stefano (2017) joint $P$–$q$–$e$ interrelation, where the same $q$ sets $m_2$, so the coupling is self-consistent). f_b lives inside the companion model — in Moe it is part of the model, set by the IMF masses.

Resolved vs unresolved¶

resolve_binary_components places each binary’s two components around its COM using the barycentric split $m_1\,\delta r_1 + m_2\,\delta r_2 = 0$, so the COM (and hence the cluster phase space) is preserved exactly. build_binary_cluster returns one of two forms:

from progenax.builders import build_binary_cluster, Systems
from progenax.binaries import MoeCompanions

# compact=True (default): eagerly compacted ICResult of real particles
ic = build_binary_cluster(profile, velocity_df, primary_imf,
                          MoeCompanions(), Systems(1000), key, units=STELLAR)

# compact=False: the masked, fixed-shape ResolvedBinaries (2N slots + is_real
# mask) — jit/grad-safe; required for differentiable IC generation
rb = build_binary_cluster(profile, velocity_df, primary_imf,
                         MoeCompanions(), Systems(1000), key, units=STELLAR,
                         compact=False)

For N-body integration the resolved components are what you want — the integrator evolves the orbital motion (binaries are collisional: integrate with a collisional scheme, $\varepsilon = 0$). The COM virialisation treats each binary as a single CoM particle (the McLuster convention, Küpper et al. (2011) §A8) and leaves the internal binary binding energy as a separate reservoir, which binary_energy_budget reports explicitly.

Connection to the binary IMF¶

The binary fraction $f_b(m_1)$ and the mass ratios from Binary-aware IMF recovery decide which stars are paired and the secondary masses; the chapters in this section decide the orbital properties. When consistency matters they come from the same Moe & Di Stefano (2017) calibration — periods from Binary period distributions, mass ratios from Mass-ratio distributions, eccentricities from Eccentricity distributions — and MoeCompanions samples them jointly to preserve the period-conditional non-separability noted at Multiplicity statistics (Moe & Di Stefano 2017).

Implementation, validation & references¶

• In code: the binary subsystem lives under src/progenax/binaries/ (kepler.py, period.py, eccentricity.py, companions.py, assembly.py, diagnostics.py); composition with the IMF + spatial profile + velocity DF is build_binary_cluster in src/progenax/builders.py. See the binaries API and the builders API; each chapter below carries its exact module path.

• Validated in: binary-aware recovery (the binary-IMF + composition regression suite).

• Primary sources: Kepler-element machinery is standard textbook material; period distributions Sana et al. (2012) (OB-type), Duquennoy & Mayor (1991) (solar), Moe & Di Stefano (2017) (joint $f(P)$); eccentricity distributions Moe & Di Stefano (2017) and earlier compilations. Full notes in the bibliography; each chapter below points at the specific result(s) used.