Eccentricity distributions

The orbital eccentricity $e$ is the third free parameter (alongside $P$ and $q$) in the Moe & Di Stefano (2017) joint binary-population calibration. Unlike $P$ and $q$, the eccentricity distribution is period-dependent: short-period binaries are tidally circularised (small $e$); long-period binaries retain whatever eccentricity they formed with (typically thermal). progenax implements three families:

This chapter derives each form, sets out the period-dependent transitions, and documents progenax’s sampling implementation.

Thermal: $f(e) = 2e$¶

The thermal eccentricity distribution dates to Heggie (1975) and represents the steady-state distribution of a population of binaries with isotropic angular-momentum vectors:

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Physically, the thermal $f(e)$ arises because a binary’s specific angular momentum scales as $L \propto \sqrt{1 - e^2}$, and an isotropic distribution in $L$ produces $f(e) \propto e$. The factor of 2 normalises so that $\int_0^1 f(e)\,\mathrm{d}e = 1$.

The mean and median:

Most thermal binaries are highly eccentric. progenax’s ThermalEccentricity() samples via inverse-CDF:

u = jax.random.uniform(key, (N,))
e = jnp.sqrt(u)            # Inverse of F(e) = e^2

Differentiable analytically.

Uniform: $f(e) = 1$¶

The uniform distribution $f(e) = 1$ for $e \in [0, 1]$ is the maximum-entropy distribution in the absence of any prior information about the angular-momentum distribution. It is the correct choice for binaries whose formation channel does not preserve a thermal distribution — e.g. binaries formed via core fragmentation in turbulent clouds, where the angular momentum is set by the local flow and is not isotropic.

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Sampling: $e = u$ for $u \sim \mathcal{U}(0, 1)$. Mean and median both 0.5.

UniformEccentricity() is the default progenax option for short-period binaries that have not yet been processed by tidal evolution.

Moe & Di Stefano (2017): the period- and mass-dependent power law¶

Moe & Di Stefano (2017) find (their §9.2, Fig. 36) that the eccentricity distribution is a power law $p(e) \propto e^{\eta}$ on $0 \le e \le e_{\max}(P)$, with the slope $\eta$ depending on both orbital period and primary mass — not the $\delta$-plus-thermal blend that older qualitative pictures suggest. The qualitative Duquennoy & Mayor (1991)-style picture (short-period orbits tidally circularised, long-period orbits thermal) is a useful intuition for the trend, but progenax implements Moe’s quantitative law:

The slope follows Moe & Di Stefano (2017) Eqs. 17–18 (verified against their Fig. 36):

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with linear interpolation in $M_1$ across 3–$7\,\Msun$. Here $\eta = 0$ is uniform ($\langle e\rangle = 0.5$) and $\eta = 1$ is thermal ($\langle e\rangle = 2/3$); short-period massive binaries are driven to $\eta < 0$ (circularising), and $\eta \le -1$ (very short $P$, $e^{\eta}$ non-normalisable) returns $e \approx 0$ by construction. The upper limit is the period-dependent Roche-lobe ceiling (their Eq. 3),

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so the components do not overflow their Roche lobes at periapsis (e.g. $e_{\max}(10\,\mathrm{d}) \approx 0.66$, $e_{\max}(100\,\mathrm{d}) \approx 0.93$); $P \le 2$ d circularises. Sampling uses the inverse CDF $e = e_{\max}(P)\,u^{1/(\eta+1)}$, which is continuous in $P$ and $M_1$ — important for HMC inference where they may be free parameters.

Sampling implementation¶

from progenax.binaries import (
    ThermalEccentricity,
    UniformEccentricity,
    MoeEccentricity,
)

# Thermal (long-period or "no information" choice)
thermal = ThermalEccentricity()
e_thermal = thermal.sample(key, 10000)

# Uniform (short-period unprocessed)
uniform = UniformEccentricity()
e_uniform = uniform.sample(key, 10000)

# Moe17 period- and mass-dependent
moe = MoeEccentricity()
e_moe = moe.sample(key, periods_days, primary_masses)  # masses REQUIRED

The MoeEccentricity sampler signature is sample(key, periods, masses) — both the per-binary periods (in days) and the primary masses are required, because the slope $\eta(\log P, M_1)$ ((4)) depends on both. The module’s only field is e_max (the long-$P$ numerical ceiling, default 0.99); the physical cap is the period-dependent Roche relation (5).

Tidal circularisation in detail¶

Tidal circularisation operates on a timescale Hurley et al., 2002

with $R_\star$ the primary’s radius. The strong $a^8$ dependence makes circularisation effective only for very short orbits — typically $P \le 4$ d for solar-type primaries on the main sequence. For massive stars (larger radii) the circularisation cutoff extends to slightly longer periods.

progenax’s MoeEccentricity handles short-period circularisation intrinsically: where the Roche ceiling (5) and the $\eta \le -1$ branch drive $e \to 0$ (at $P \le 2$ d the ceiling is exactly 0). There is no separate circular-cutoff knob — the module’s only field is e_max, the long-$P$ numerical ceiling. Surveys targeting evolved stars (red giants with $R_\star \gg R_\odot$) need a longer effective cutoff than the main-sequence relation provides; that would require a stellar-evolution-aware Roche radius (a future startrax coupling), not a constructor argument here.

Joint $f(P, e, q, M_1)$ when consistency matters¶

For population studies that need mass-dependent period/eccentricity prescriptions, progenax provides MassDependentBinaryConfig and sample_mass_dependent_orbits:

import jax.numpy as jnp
from progenax.binaries import (
    LogNormalPeriod,
    MassDependentBinaryConfig,
    MoeEccentricity,
    SanaOBPeriod,
    ThermalEccentricity,
    sample_mass_dependent_orbits,
)

config = MassDependentBinaryConfig(
    m_break=8.0,
    low_mass_period=LogNormalPeriod(),
    high_mass_period=SanaOBPeriod(),
    low_mass_eccentricity=ThermalEccentricity(),
    high_mass_eccentricity=MoeEccentricity(),
)

primary_masses = jnp.array([1.0, 5.0, 10.0, 20.0])
periods, eccentricities = sample_mass_dependent_orbits(
    primary_masses, config, key
)

This samples periods and eccentricities. The fully joint Moe & Di Stefano mass-ratio and multiplicity model lives in the IMF binary module, not in this eccentricity sampler.

Domain of validity¶

1. Bound orbits only ($e < 1$). Marginally-bound or hyperbolic encounters are not represented — they are not “binaries” in the sense progenax models.

2. Single-event tidal cutoff — the circular cutoff is set by the primary’s current radius, ignoring stellar evolution. Stars that evolve through giant phases briefly have $R_\star \gg R_\odot$, which extends the cutoff temporarily. progenax’s IC-time treatment uses ZAMS radii; subsequent evolution should be handled by the integrator (gravax) and stellar-evolution code (startrax, planned).

3. Metallicity assumption — the Moe & Di Stefano (2017) calibration is solar-metallicity. Tidal-circularisation efficiency varies with internal stellar structure, which is metallicity-dependent. For metal-poor populations expect the cutoff period to shift slightly.

4. No magnetic-braking effects — magnetic braking circularises short-period binaries on timescales comparable to $\tau_{\mathrm{circ}}$. The progenax default lumps both effects into a single empirical cutoff, but for science targeting magnetic activity (CV-progenitor binaries, RS CVn analogues) the effects should be modelled separately.

Implementation, validation & references¶

• In code: src/progenax/binaries/eccentricity.py (ThermalEccentricity, UniformEccentricity, MoeEccentricity), with mixed-population routing in src/progenax/binaries/mass_dependent.py. See the binaries API.

• Validated in: binary-aware recovery.

• Primary sources: the thermal distribution is Heggie (1975); the period-dependent power law is Moe & Di Stefano (2017) (Eqs. 17–18, verified against their Fig. 36); the tidal-circularisation timescale follows Hurley et al. (2002). Full notes in the bibliography; for population-synthesis context see Sana et al. (2012).