The inner Solar System is chaotic, and the only meaningful statement about its long-term stability is statistical. Building on Laskar and Gastineau's 2009 discovery that about 1% of 2,501 five-Gyr integrations of the full Solar System lead, through secular resonant pumping of Mercury's eccentricity, to close encounters, collisions, or ejections of the inner planets — and that in the vicinity of one such destabilized trajectory roughly 25% of perturbed variants end with a collision of Mercury, Venus, or Mars with the Earth — the open problem is to convert these sparse ensemble results into a calibrated risk assessment: the probability of inner-planet destabilization and of Earth-impact (or Earth ejection) within the Sun's remaining ~5 Gyr main-sequence lifetime under the full physically admissible uncertainty of the present Solar System state, the conditional distribution of outcomes given destabilization, and the robustness of these probabilities to physical and numerical modeling choices. Later work (Zeeze 2015; the Abbot-group ensembles and rare-event-sampling program 2021–2024; instanton and quasi-integral analyses of the destabilization mechanism; passing-star simulations) has tightened the Mercury-instability statistics to O(10^-2) at 5 Gyr and exposed large sensitivities to integrator coordinates and stellar passages, but no study has yet sampled the full initial-condition uncertainty compatible with present knowledge or recomputed the conditional post-destabilization Earth-collision distribution, so the problem remains open.

Background

Whether the Solar System is dynamically stable is one of the oldest open questions in astronomy. The eight planets interact chaotically: two trajectories of the full equations of motion that start microscopically apart diverge exponentially, with an e-folding (Lyapunov) time of only about 5 million years for the inner planets, so the future cannot be predicted deterministically beyond ~100 Myr and must be treated statistically, through ensembles of integrations. Because the orbital periods are short compared with the timescale on which the orbital ellipses slowly rotate and deform, long-term evolution is often studied with secular (orbit-averaged) equations, in which the fast orbital motion is averaged out and the eccentricities and perihelia diffuse slowly under the planets' mutual torques. The dominant instability of the present Solar System involves Mercury: the precession rate of Mercury's perihelion (the secular frequency g1g_1) lies close enough to Jupiter's (g5g_5) that the g1−g5g_1-g_5 secular resonance — a near-commensurability of the two perihelion precession frequencies that can exchange angular momentum between the orbits — can raise Mercury's orbital eccentricity to ~0.86, the level at which Mercury's aphelion reaches Venus's orbit and the orbits cross, so that close encounters, collisions, or ejections of the inner planets become dynamically possible. General relativity is decisive here: it adds 0.43 arcsec/yr to Mercury's perihelion precession and moves the system away from the resonance; in a purely Newtonian model, more than half of all trajectories destabilize within 5 Gyr, against roughly 1% with relativistic precession included (Laskar 2008, Icarus). Long direct integrations use symplectic integrators — geometric numerical schemes that preserve the Hamiltonian structure over billions of years — augmented to handle close encounters. Laskar (2008) estimated from 1,000 secular integrations that the probability of Mercury reaching very high eccentricity within 5 Gyr is ~1%. Laskar and Gastineau (2009, Nature) then ran 2,501 five-Gyr integrations of the full, non-averaged Solar System (including the Moon and general relativity), started from initial conditions compatible with present knowledge of the planetary state: about 1% of solutions produce a large increase of Mercury's eccentricity leading to collisions of Mercury with Venus or the Sun; and in one of these solutions the Mercury excursion triggers, through angular-momentum exchange with the giant planets, a complete destabilization of all four terrestrial planets roughly 3.34 Gyr from now. Around that destabilizing trajectory, 201 additional slightly perturbed solutions were integrated: 5 ended with the ejection of Mars, 1 with a collision of Mercury with the Earth, 29 with a collision of Mars with the Earth, and 18 with a collision of Venus with the Earth — that is, roughly 25% of the perturbed variants end in a planetary collision with the Earth (Laskar 2012, 'Is the Solar System stable?', a review recounting the experiment). These results established that collisional trajectories of Mercury, Mars, and Venus with the Earth exist within the Sun's remaining main-sequence lifetime, but left the actual risk quantitatively undetermined: the 2,501-member sample perturbs only a sliver of the high-dimensional uncertainty of the present state — the solutions differ by only 0.38 mm in the initial value of Mercury's semi-major axis — the destabilizing trajectory sampled is essentially unique in the ensemble, and the probabilities are conditional on specific physical and numerical modeling choices. Beyond direct ensembles, rare-event sampling schemes and reduced secular or stochastic models have been used to converge Mercury-instability statistics at lower computational cost, provided they remain accurate through close encounters (Abbot et al. 2021, 2023).

Problem Statement

The common objective is a calibrated, model-robust statistical answer to the question Newton posed and Laskar & Gastineau (2009) made quantitative: what is the probability that the inner Solar System undergoes destabilization within the Sun's remaining ~5 Gyr main-sequence lifetime, given the full physically admissible uncertainty of the present Solar System state, and, conditional on such destabilization, the probability that Mercury, Venus, or Mars collides with the Earth or that a terrestrial planet is ejected? Here "physically admissible uncertainty" follows the criterion of Laskar & Gastineau (2009): initial states in agreement with present knowledge of the parameters of the Solar System. Since in a chaotic system every probability is defined only relative to a chosen initial-state measure, the relevant measure here is a stated uncertainty model (ephemeris covariance and treatment of neglected perturbations) within that criterion. This comprises three linked components, each of which supports the single risk assessment and cannot be detached from it without invalidating the others: (1) the marginal probability that chaotic diffusion drives Mercury's eccentricity through the g1−g5g_1-g_5 secular resonance to the point of inner-planet destabilization, sampled over the full high-dimensional uncertainty of the present state within that criterion (ephemeris covariances and neglected perturbations such as asteroids and passing field stars), not merely over millimetric offsets of one nominal trajectory — Laskar & Gastineau's own 2,501-member ensemble explored only a 0.38 mm spread in Mercury's initial semi-major axis; (2) the conditional probability distribution of outcomes given destabilization — what fraction of destabilizations end in Mercury–Venus or Mercury–Sun collisions, ejections, or collisions with the Earth — and whether rare destabilizing pathways like the one sampled by Laskar & Gastineau's 201-run neighborhood study are representative of the full set of destabilizing pathways in phase space; and (3) the sensitivity of both the marginal and conditional probabilities to physical and numerical modeling choices (for example general relativity, the Moon's contribution, tides, planetary internal structure, asteroid and stellar perturbations, integrator coordinates and time step near close encounters — an illustrative, non-exhaustive list of demonstrated sensitivities). The target is converged numerical estimates, with confidence intervals under the uncertainty model within that criterion, of the marginal probability of inner-planet destabilization (the onset event of component (1)) and of the conditional outcome distribution given destabilization (component (2)); the probability of a Mercury, Venus, or Mars collision with the Earth, or of a terrestrial-planet ejection, is the product of that marginal probability and the corresponding conditional outcome fraction; each demonstrated sensitivity listed in (3) is carried as an explicit term of the uncertainty budget rather than required to vanish.

References

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