A Coordinate System for Dynamical Instabilities in Hierarchical Systems in REBOUND

Technical Summary: A Coordinate System for Dynamical Instabilities in Hierarchical Systems in REBOUND

Problem Statement
The $N$-body problem remains a central challenge in classical mechanics, particularly for simulating planetary systems over gigayear timescales. While Wisdom-Holman (WH) hybrid integrators have become the gold standard for long-term integration by splitting the Hamiltonian into a dominant Keplerian term and a perturbative interaction term, their efficacy relies heavily on the choice of coordinate system.

• Jacobi coordinates offer high accuracy by solving the two-body problem exactly but are inflexible; they fail when orbital ordering changes due to dynamical instabilities.
• Democratic Heliocentric Coordinates (DHC) allow for orbit crossings and are the standard for hybrid integrators (e.g., MERCURY, SyMBA, TRACE) but perform poorly in systems lacking a single dominant central mass, such as wide binary systems where a planetary system orbits one component of a binary star.

In wide binary systems, DHC fails to maintain the necessary hierarchy for efficient integration, while Jacobi coordinates cannot handle the close encounters and orbital instabilities common in these environments. There is a need for a coordinate system that supports close encounters between any pair of bodies (planets, host star, and companion star) while retaining the speed benefits of WH methods.

Methodology
The authors implement a coordinate system derived by J. E. Chambers et al. (2002), termed "Wide Binary Coordinates" (WB), into the TRACE hybrid integrator within the open-source REBOUND package.

1. Coordinate Transformation:
   The WB system defines new canonical coordinates ($\vec{Q}$) and momenta ($\vec{P}$) that separate the barycentric motion of the central star, the relative motion of the companion star, and the relative motion of planets with respect to the host star. This allows the Hamiltonian to be split into three components:

   • $H_{Jump}$: Barycentric motion of the central star.
   • $H_{Interaction}$: Planet-planet and planet-companion interactions.
   • $H_{Kepler}$: Keplerian motion of planets and the companion around the host star.
     In the absence of a companion, these coordinates reduce to standard DHC.

2. Integration Scheme:
   The authors adapt the time-reversible switching scheme of TRACE to the WB coordinates. The integrator proceeds in substeps (leapfrog style) advancing $H_{Interaction}$, $H_{Jump}$, and $H_{Kepler}$.

   • Switching Logic: When close encounters occur (planet-planet, planet-star, or planet-companion), the integrator dynamically shifts terms from $H_{Interaction}$ into $H_{Kepler}$ to ensure the Keplerian term remains dominant.
   • Encounter Handling:
     • If a planet approaches the host star or the companion star within a critical radius (default: 0.1 Hill radii for the companion), the system reverts to inertial coordinates and uses a traditional integrator (Bulirsch-Stoer or IAS15) to solve the full equations of motion.
     • If planets approach each other, specific interaction terms are moved into $H_{Kepler}$ and solved using a Bulirsch-Stoer integrator, while the rest of the system continues with the WH solver.

3. Benchmarking:
   The performance of TRACE in WB coordinates was benchmarked against:

   • IAS15: A high-precision, adaptive timestep integrator (serving as the ground truth).
   • TRACE in DHC: The standard implementation for comparison.
   • WHFast in Jacobi coordinates: Used for comparison in stable regimes.

Key Results

• Parameter Space Analysis: A survey of 10,000 simulations varying companion mass ($1 M_J$ to $3 M_\odot$) and semimajor axis (50–1000 AU) revealed that WB coordinates significantly outperform DHC for larger, more distant companions. The authors provide a power-law guideline (Equation 11) to determine when WB coordinates are necessary:
  $$\log_{10}\left(\frac{M}{M_\odot}\right) \approx 0.9 \log_{10}\left(\frac{a}{1 \text{ AU}}\right) - 4.67$$
  WB coordinates perform competitively with WHFast in Jacobi coordinates across the entire explored parameter space.

• Planet-Planet Scattering: In simulations involving three Jupiter-mass planets and a 0.5 $M_\odot$ companion, TRACE in DHC produced statistically incorrect results (overpredicting semimajor axes and inclinations, underpredicting eccentricities) compared to IAS15. TRACE in WB coordinates recovered results statistically similar to IAS15 while achieving a 9× speedup.

• Stellar Flybys: In tests simulating stellar flybys on a Neptune-like planet, TRACE in DHC failed to reproduce analytic predictions for energy, eccentricity, and inclination changes. TRACE in WB coordinates matched both analytic predictions and IAS15 results, offering a 2× speedup over IAS15.

• ZLK Oscillations: For von Zeipel-Lidov-Kozai (ZLK) cycles, fixed-timestep integrators often struggle with the extreme eccentricities reached during high-eccentricity phases.

  • In DHC, the valid parameter space for TRACE to match IAS15 accuracy was extremely limited.
  • In WB coordinates, TRACE matched IAS15 accuracy for binary inclinations up to $70^\circ$ ($e_{max} \sim 0.9$).
  • In a specific test case involving an inner Earth-like planet (forcing a small global timestep), TRACE in WB coordinates was 20× faster than IAS15 while maintaining accuracy, whereas DHC failed to reproduce the correct oscillation period.

Significance and Guidelines
The paper claims that the implementation of WB coordinates in TRACE addresses a specific regime where traditional hybrid methods fail: wide binary systems prone to dynamical instabilities and close encounters. The authors emphasize that for large ensembles of chaotic systems in binary stars, this method may be the only viable integration approach that balances accuracy with computational feasibility.

The authors provide the following guidelines for selecting integrators in REBOUND:

1. IAS15: Use when computation time is not a constraint or exact trajectories are required.
2. WHFast: Use when computation time is a concern and no close encounters are expected.
3. TRACE: Use when computation time is a concern, close encounters are expected, and qualitatively correct trajectories are sufficient.
   • Coordinate Choice: For systems around Sun-like stars, use Equation 11 to decide between DHC and WB. WB is strongly preferred for true stellar companions.

The work demonstrates that TRACE in WB coordinates offers sufficient accuracy for studying dynamical instabilities, planet formation, and stellar flybys in binary systems without the prohibitive computational costs associated with high-precision adaptive integrators.