Dyad: a binary-star dynamics and statistics library for Python

Based on the provided text, here is a detailed technical summary of the paper introducing Dyad:

Problem Statement

Current galactic dynamics models typically treat galaxies as populations of single stars moving within a gravitational potential generated by the stars and a dark matter halo. These models are fitted to observational data (e.g., on-sky positions and line-of-sight velocities) to infer physical properties like total mass. However, this approach is fundamentally misspecified because a significant fraction of stars exist in binary systems.

• The Consequence: The kinematics of stars in binary-rich galaxies differ from those in binary-free models. Ignoring binary dynamics leads to incorrect inferences about galactic properties.
• Severity: While the error is negligible for large disc galaxies, it becomes significant for small spheroidal galaxies, where total mass is directly proportional to velocity dispersion.
• The Gap: While the field of stellar population synthesis has robust software (e.g., COSMIC, COMPAS, binary_c) for simulating the evolution of binary populations, these tools are designed for long-term evolutionary tracking. They lack the specific functionality required for statistical dynamics analysis, such as evaluating probability density functions (PDFs) directly or handling correlations between variables without running full evolutionary simulations. No dedicated, publicly available library existed for the specific statistical analysis of binary-star populations in galactic dynamics.

Methodology

The author, Amery Gration, developed Dyad, a Python library designed to treat binary stars as gravitational two-body systems. The methodology relies on two core components:

1. Physical Modeling (dyad.TwoBody):

   • The library adopts the standard astrophysical convention where the brighter component is the primary (reference body) and the dimmer is the secondary (subject body).
   • In the primary's frame, the secondary moves on an elliptical orbit defined by eight parameters: primary mass, secondary mass, semi-major axis, eccentricity, true anomaly, longitude of the ascending node, inclination, and argument of pericentre.
   • The class computes dynamic quantities (angular momentum, total energy, eccentricity vectors, positions, velocities, kinetic/potential energies) in both the primary star's frame and the center-of-mass frame.

2. Statistical Modeling (dyad.stats):

   • This module treats the population parameters (masses, mass ratios, orbital elements) as random variables with specific probability distributions.
   • It implements these distributions following the SciPy (scipy.stats) architecture, providing methods to evaluate:
     • Probability Density Functions (PDFs)
     • Cumulative Distribution Functions (CDFs)
     • Inverse CDFs
     • Statistical moments (mean, variance, skewness, etc.)
   • It incorporates established astrophysical models for these distributions, including:
     • Stellar Mass: Kroupa (2001) and Salpeter (1955) Initial Mass Functions (IMF).
     • Binary Properties: Mass ratios and orbital elements based on Duquennoy & Mayor (1991) and Moe & Di Stefano (2017).

Key Contributions

• Specialized Library: Dyad fills a critical software gap by providing a tool dedicated specifically to the population dynamics of binary stars, distinct from evolutionary synthesis codes.
• Statistical Flexibility: Unlike existing packages that primarily offer sampling routines, Dyad allows users to directly evaluate the underlying probability distributions and compute statistical moments, enabling rigorous statistical testing and fitting.
• Correlation Handling: The library is designed to handle correlations between quantities (e.g., between mass ratio and orbital elements), a feature often missing or inconsistent across other synthesis packages.
• Integration: It seamlessly combines the physical two-body dynamics (TwoBody) with statistical sampling (stats), allowing researchers to generate custom representations of binary populations for specific research needs.

Results

• Application to Gration et al. (2025): The library was successfully utilized to quantify the error in inferred galactic masses for small spheroidal galaxies. By constructing velocity distributions using Dyad, the authors demonstrated that ignoring binary stars leads to significant over- or under-estimations of galactic mass in these systems.
• Validation: The implementation successfully reproduces the dynamics and statistical properties required to model binary-rich galaxies, proving the necessity of including binary kinematics in galactic dynamicist models.

Significance

• Correcting Astrophysical Inferences: Dyad provides the necessary tools to correct systematic errors in galactic mass measurements, particularly for dwarf and spheroidal galaxies, ensuring that dark matter and total mass estimates are accurate.
• Research Enabler: It serves as a "laboratory" for the growing field of binary-rich galaxy modeling, supporting active research areas cited in recent literature (Minor et al., 2020; Rastello et al., 2020; Arroyo-Polonio et al., 2023).
• Community Standard: By adopting the SciPy statistical interface, Dyad lowers the barrier to entry for astrophysicists, allowing them to easily integrate binary statistics into existing Python-based analysis pipelines without needing to learn proprietary or complex simulation frameworks.