[Paper Review] Eccentricity dynamics of wide binaries -- II. The effect of stellar encounters and constraints on formation channels

Chris Hamilton, Shaunak Modak|arXiv (Cornell University)|Nov 7, 2023

Stellar, planetary, and galactic studies4 citations

TL;DR

This paper demonstrates that stellar encounters in the Milky Way thermally redistribute the eccentricity distribution of wide binaries (a > 10³ AU) on a timescale of ~4 Gyr per 10⁴ AU, ruling out dynamical evolution as the origin of the observed superthermal eccentricity distribution (α ≈ 1.2). Instead, the data imply that wide binaries must have formed with a significantly more superthermal initial distribution (α > 1.2), favoring formation via turbulent fragmentation over cluster dissolution.

ABSTRACT

GAIA wide stellar binaries (separations $\sim 10^3-10^{4.5}$ AU) are observed to have a superthermal eccentricity distribution function (DF), well-fit by $P(e) \propto e^α$ with $α\sim 1.2$. In Modak \& Hamilton (2023), we proved that this DF cannot have been produced by Galactic tidal torques starting from any realistic DF that was not already superthermal. Here, we consider the other major dynamical effect on wide binaries: encounters with passing stars. We derive and solve the Fokker-Planck equation governing the evolution of binaries in semimajor axis and eccentricity under many weak, impulsive, penetrative stellar encounters. We show analytically that these encounters drive the eccentricity DF towards thermal on the same timescale as they drive the semimajor axes $a$ towards disruption, $t_\mathrm{ion} \sim 4\,\mathrm{Gyr}\,(a/10^4\,\mathrm{AU})^{-1}$. We conclude that the observed superthermal DF must derive from an even more superthermal (i.e. higher $α$) birth distribution. This requirement places strong constraints on the dominant binary formation channels. A testable prediction of our theory is that $α$ should be a monotonically decreasing function of binary age.

Motivation & Objective

• To determine whether stellar encounters can explain the observed superthermal eccentricity distribution (P(e) ∝ e^α, α ≈ 1.2) in wide binaries (a > 10³ AU).
• To model the combined evolution of semimajor axis and eccentricity under weak, impulsive, isotropic stellar encounters using a Fokker-Planck framework.
• To test whether dynamical evolution via encounters can transform a subthermal or thermal initial distribution into the observed superthermal state.
• To constrain viable formation mechanisms for wide binaries by requiring that the initial eccentricity distribution be more superthermal than observed.
• To derive a testable prediction: the power-law index α should decrease with binary age if the theory is correct.

Proposed method

• Derivation of a Fokker-Planck equation in (a, e²) space to describe the evolution of wide binary populations under weak, impulsive, penetrative, and isotropic stellar encounters.
• Solution of the Fokker-Planck equation to show that eccentricity distribution thermalizes at each semimajor axis a on the same timescale as semimajor axes evolve toward ionization.
• Use of the ionization timescale t_ion ≈ 4 Gyr × (a/10⁴ AU)⁻¹ as a key benchmark for the rate of both thermalization and hardening/softening.
• Analytical demonstration that any initial distribution, regardless of form, evolves toward thermal eccentricity distribution at fixed a under repeated encounters.
• Combination of this result with Paper I’s finding that Galactic tides cannot produce superthermal distributions from subthermal initial states.
• Application of the combined constraints to rule out formation mechanisms like cluster dissolution and favor turbulent fragmentation.

Experimental results

Research questions

• RQ1 Can stellar encounters alone produce the observed superthermal eccentricity distribution (α ≈ 1.2) in wide binaries from a non-superthermal initial state?
• RQ2 What is the relative evolution timescale of eccentricity thermalization and semimajor axis hardening/ionization due to stellar encounters?
• RQ3 Does the requirement of a more superthermal initial eccentricity distribution (α_initial > 1.2) rule out formation via cluster dissolution or other standard channels?
• RQ4 Can the observed α ≈ 1.2 eccentricity distribution be explained by dynamical evolution alone, or must it be primordial?
• RQ5 Is there a testable prediction linking the power-law index α to binary age, based on the model’s dynamics?

Key findings

• Stellar encounters drive the eccentricity distribution of wide binaries toward a thermal distribution (P(e) ∝ 2e) at each semimajor axis a, regardless of initial conditions.
• The thermalization of eccentricity and the ionization of binaries occur on the same timescale, approximately t_ion ≈ 4 Gyr × (a/10⁴ AU)⁻¹ for solar-mass binaries.
• The observed superthermal eccentricity distribution (α ≈ 1.2) cannot arise from dynamical evolution via stellar encounters or Galactic tides unless the initial distribution was already more superthermal (α_initial > 1.2).
• This requirement rules out formation via dissolution of young stellar clusters, which would produce a subthermal or thermal initial distribution.
• The theory favors formation via turbulent fragmentation, which naturally produces highly superthermal initial eccentricity distributions.
• A testable prediction is that the power-law index α should decrease monotonically with increasing binary age, providing a direct observational test for the model.