Probability of Mercury's orbital instability over 5 Gyr under general-relativistic and tidal perturbations
Lowered from the drafter's 66: the old 71 graded the broad question of Solar-System stability, while this sharpened problem already has a rare-event method (QDMC) and an advection–diffusion model that reproduce the N-body statistics, so a path is visible and the remaining work is controlled computation plus a justified reduced model. That places it in the 30–49 band, near the ringdown (40) and CCSN (50) problems. Leverage 20 and application value 6 are unchanged.
The problem
Is the Solar System stable? Estimate the small chance that Mercury is thrown off course within five billion years.
- 1Mercury's orbit turns slowly
- 2Resonance with Jupiter can stretch it
- 3Einstein's gravity reduces the risk
- 4Chaos plus code errors blur the odds
- 5Need a controlled probability estimate
The question. Compute the probability that Mercury's eccentricity grows enough to lead to collision or ejection during the next 5 billion years, and how it depends on the strength of general-relativistic (GR) precession, tidal dissipation in Mercury and the Sun, and the initial conditions within observational uncertainties.
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Relativistic two-body dynamics and waveform modelsTheory · host
- Then
- Probability of Mercury's orbital instability over 5 Gyr under general-relativistic and tidal perturbationsThe problem
State of the art
Laskar & Gastineau (Nature 459, 817, 2009; doi:10.1038/nature08096) integrated 2,501 solutions and found about 1% probability of Mercury's eccentricity reaching high values over 5 Gyr. Abbot et al. (arXiv:2106.09091) used rare-event sampling to estimate the probability of a Mercury instability within 2 Gyr as about 10⁻⁴, at 100 times lower cost than direct simulation. Abbot et al. (arXiv:2306.11870) showed that a purely diffusive model of g₁ underpredicts the instability probability by factors of 3–10⁴ on timescales below 5 Gyr and built a subdiffusive phenomenological model tuned to N-body statistics over 1–40 Gyr. Brown & Rein (arXiv:2303.05567) ran 1,280 N-body integrations over 12.5 Gyr with time-varying GR precession: GR lowers the 5-Gyr destabilisation probability by a factor of about 60, an advection–diffusion (Fokker–Planck) model reproduces the instability times with and without GR, and there is no critical GR precession frequency below which the system becomes markedly less stable. Time-step-induced numerical chaos (Hernandez, Hadden & Makino 2022) may inflate instability fractions unless small time steps are used. The estimate is not settled across integrators and models, and no resolution is reported.
Realistic applications
The result would establish the long-term stability of the Solar System, supports architecture studies for exoplanetary systems with similar secular resonances and benchmarks N-body integrators and rare-event methods. No practical application.
What counts as solved
Resolved when two independent codes with demonstrated convergence give agreeing instability probabilities (with uncertainties) as functions of time to 5 Gyr, and the dependence on GR and tidal effects is explained by a validated reduced model.
How it gets verified
ReproductionA training run, experiment or computation rerun by someone else.
Two independent N-body codes with demonstrated convergence must give agreeing instability probabilities, with uncertainties, as functions of time out to 5 Gyr, and the role of GR and tides must be explained by a validated reduced model. The spec fixes the initial-condition ensemble size, integrator and timestep tests, and the instability criterion; the runs are repeatable by others on stated compute.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
Theory of secular chaos and large-scale computation with rare-event sampling.
Bottlenecks
ranked by votes- #1ComputeResolving rare early instabilities
Instability probabilities of 10⁻⁴ require rare-event methods and symplectic integrators with controlled time-step error.
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Progress
- Jun 20, 2023Subdiffusive modelsource
Abbot et al. show that a diffusive model underpredicts short-time instabilities and replace it with a subdiffusive model of g₁.
- Mar 9, 2023Advection–diffusion model with varying GRsource
Brown & Rein show GR lowers the 5-Gyr instability probability about 60-fold and find no critical GR precession rate.
- Jun 16, 2021Rare-event samplingsource
Abbot et al. estimate about 10⁻⁴ probability of instability within 2 Gyr.
- Jun 11, 2009Collisional trajectories foundsource
Laskar & Gastineau find about 1% probability of Mercury's instability in 5 Gyr.