Paper Information

Citation: Sussman, G. J., & Wisdom, J. (1992). Chaotic Evolution of the Solar System. Science, 257(5066), 56-62. https://doi.org/10.1126/science.257.5066.56

Publication: Science 1992

Additional Resources:

What kind of paper is this?

This is a computational/numerical methods paper that combines algorithm development with scientific discovery. The core contribution is both methodological (demonstrating the effectiveness of symplectic mapping for long-term orbital integration) and empirical (providing the first direct numerical confirmation that the entire Solar System exhibits chaotic dynamics). The work validates theoretical predictions through unprecedented computational experiments spanning nearly 100 million years of simulated planetary motion.

What is the motivation?

The authors aimed to address the fundamental question of the long-term stability and predictability of the Solar System. Prior work had limitations:

  • Sussman & Wisdom (1988): Found chaos in Pluto’s orbit but did not integrate the full system.
  • Laskar (1989): Found evidence for chaos in the whole system (excluding Pluto) but relied on analytically averaged equations, which are perturbative and truncated.

A direct integration of the full system without averaging approximations was required to validate these findings and determine whether the observed chaos was an artifact of the approximation methods or a genuine property of the planetary dynamics.

What is the novelty here?

The study represents the first direct, full-system integration spanning nearly 100 million years. Key innovations included:

  • Symplectic Mapping: Application of the Wisdom-Holman mapping method, which allows for much larger time steps (e.g., 7.2 days) compared to multistep methods (which require ~100 steps/orbit) while maintaining long-term energy conservation and numerical stability.
  • Custom Hardware: Use of the Supercomputer Toolkit, a multiprocessor system optimized for ODEs, where each processor was 3x faster than the entire previous generation “Digital Orrery”.
  • Direct Numerical Validation: Unlike Laskar’s secular perturbation theory approach, this work directly integrates Newton’s equations without analytical approximations, providing independent verification of chaotic behavior.

What experiments were performed?

  • 100 Myr Integration: Eight separate integrations of the entire planetary system were run for ~100 million years (reversed time) with slightly different initial conditions to measure exponential divergence of trajectories.
  • Validation:
    • Compared a 3-million-year segment against the high-precision integration by Quinn, Tremaine, and Duncan (QTD).
    • Compared results with Laskar’s secular resonance angle calculations to verify consistency with the perturbative approach.
  • Subsystem Analysis: Additional integrations of just the Jovian planets (outer system) and massless Pluto particles were performed to isolate the source of chaos and determine which subsystems contribute to the overall chaotic behavior.

What were the outcomes and conclusions drawn?

  • System-wide Chaos: The Solar System is chaotic with an exponential divergence timescale (Lyapunov time) of approximately 4 million years, meaning that initial condition uncertainties grow by a factor of $e$ roughly every 4 million years.
  • Jovian Chaos: The Jovian planets (Jupiter, Saturn, Uranus, Neptune) are independently chaotic, though the timescale varies (3-30 Myr) depending on step-size and initial conditions.
  • Pluto’s Robust Chaos: Pluto’s chaotic motion (10-20 Myr timescale) is robust and persists even when the Jovian planets are artificially forced into quasiperiodic motion.
  • Mechanism: While specific secular resonance angles (like Laskar’s $\sigma_1$ and $\sigma_2$) were observed to alternate between libration and circulation, the authors did not definitively identify resonance overlap as the sole mechanism driving the chaos.
  • Predictability Horizon: The chaotic nature fundamentally limits our ability to predict planetary positions beyond roughly 100 million years into the past or future, regardless of improvements in observational precision or computational power.

Reproducibility Details

Data

  • Initial Conditions: Derived from JPL Ephemeris DE102, matching the setup used by Quinn, Tremaine, and Duncan (QTD) for direct comparison.
  • Masses: Planetary masses consistent with DE102.

Algorithms

Integrator: Symplectic n-body mapping (Wisdom & Holman method)

The Hamiltonian is split into Keplerian motion and planetary interactions:

$$H = H_{\text{Kepler}} + H_{\text{Interaction}}$$

  • Structure: Evolves Kepler Hamiltonian for half-step, followed by interaction kicks, ending with half-step Kepler evolution. This splitting preserves the symplectic structure.
  • Step Size: 7.2 days (chosen to align with QTD output timestamps for validation).
  • Precision: Pseudo-quadruple precision (128-bit effectively) for position accumulation to minimize roundoff errors, though retrospectively deemed unnecessary for this problem.

Validation Integrator: Traditional linear multistep Störmer predictor (used for 22 Myr validation checks).

Models

  • Hamiltonian: The system is modeled using a split Hamiltonian approach separating two-body Keplerian motion from perturbative interactions.
  • General Relativity: Modeled using the potential approximation of Nobili and Roxburgh (1986) rather than full post-Newtonian corrections, to maintain integrability of the Keplerian part.
  • Earth-Moon: Treated via a quadrupole approximation similar to QTD, representing the Earth-Moon system as a single body with appropriate mass distribution.

Evaluation

The primary metric for chaos was the Lyapunov Exponent, estimated via the divergence of nearby trajectories with slightly perturbed initial conditions.

MetricValueNotes
Divergence Timescale (Full System)~4 MyrDominated by inner planets initially
Divergence Timescale (Pluto)10-20 MyrConsistent across methods
Eccentricity Error (vs QTD)$10^{-5}$ to $10^{-6}$Excellent agreement over 3 Myr

Hardware

  • System: Supercomputer Toolkit (MIT/Hewlett-Packard collaboration)
  • Configuration: 8-processor configuration used for parallel integrations
  • Performance: ~30 years of Solar System evolution per second per processor
  • Total Compute: ~1000 hours of Toolkit time for the main experiments

Citation

@article{sussmanChaoticEvolutionSolar1992,
  title = {Chaotic {{Evolution}} of the {{Solar System}}},
  author = {Sussman, Gerald J. and Wisdom, Jack},
  journal = {Science},
  volume = {257},
  number = {5066},
  pages = {56--62},
  year = {1992},
  month = {jul},
  doi = {10.1126/science.257.5066.56},
  abstract = {The evolution of the entire planetary system has been numerically integrated for a time span of nearly 100 million years. This calculation confirms that the evolution of the solar system as a whole is chaotic, with a time scale of exponential divergence of about 4 million years. Additional numerical experiments indicate that the Jovian planet subsystem is chaotic, although some small variations in the model can yield quasiperiodic motion. The motion of Pluto is independently and robustly chaotic.}
}