Stability of the Solar System
The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have been stable historically, and will be in the short term, their weak gravitational effects on one another can add up in unpredictable ways. For this reason (among others) the Solar System is stated to be chaotic, and even the most precise long-term models for the orbital motion of the Solar System are not valid over more than a few tens of millions of years.
The Solar System is stable in human terms, in that none of the planets will collide with each other or be ejected from the system in the next few billion years, and the Earth's orbit will be relatively stable.
Since Newton's law of gravitation (1687), mathematicians and astronomers (such as Laplace, Lagrange, Gauss, Poincaré, Kolmogorov, Vladimir Arnold and Jürgen Moser) have searched for evidence for the stability of the planetary motions, and this quest led to many mathematical developments, and several successive 'proofs' of stability for the Solar System.
Overview and challenges 
The orbits of the planets are open to long-term variations, and modeling the Solar System is subject to the n-body problem.
Resonance happens when any two periods have a simple numerical ratio. The most fundamental period for an object in the Solar System is its orbital period, and orbital resonances pervade the Solar System. In 1867, the American astronomer Daniel Kirkwood noticed that asteroids in the asteroid belt are not randomly distributed. There were distinct gaps in the belt at locations that corresponded to resonances with Jupiter. For example, there were no asteroids at the 3:1 resonance – a distance of 2.5 AU – or at the 2:1 resonance at 3.3 AU (AU is the astronomical unit, or essentially the distance from sun to earth).
Another common form of resonance in the Solar System is spin–orbit resonance, where the period of spin (the time it takes the orbit to rotate once about its axis) has a simple numerical relationship with its orbital period. An example is our own Moon, which is in a 1:1 spin–orbit resonance that keeps the far side of the Moon away from the Earth.
The planets' orbits are chaotic over longer timescales, such that the whole Solar System possesses a Lyapunov time in the range of 2–230 million years. In all cases this means that the position of a planet along its orbit ultimately becomes impossible to predict with any certainty (so, for example, the timing of winter and summer become uncertain), but in some cases the orbits themselves may change dramatically. Such chaos manifests most strongly as changes in eccentricity, with some planets' orbits becoming significantly more—or less—elliptical.
In calculation, the unknowns include asteroids, the solar quadrupole moment, mass loss from the Sun through radiation and solar wind, and drag of solar wind on planetary magnetospheres, galactic tidal forces, the fractional effect, and effects from passing stars.
Neptune–Pluto resonance 
The Neptune–Pluto system lies in a 3:2 orbital resonance. C. J. Cohen and E. C. Hubbard at the Naval Surface Warfare Center Dahlgren Division discovered this in 1965. Although the resonance itself will remain stable in the short term, it becomes impossible to predict the position of Pluto with any degree of accuracy, as the uncertainty in the position grows by a factor e with each Lyapunov time, which for Pluto is 10–20 million years (the Lyapunov time) into the future. Thus, on the time scale of hundreds of millions of years Pluto's orbital phase becomes un-knowable, even if Pluto's orbit appear perfectly stable on 10 Gyr time scales (Ito and Tanikawa 2002, MNRAS).
Jovian moon resonance 
Jupiter's moon Io has an orbital period of 1.769 days, nearly half that of the next satellite Europa (3.551 days). They are said to be in a 2:1 orbit-orbit resonance. This particular resonance has important consequences because Europa's gravity perturbs the orbit of Io. As Io moves closer to Jupiter and then further away in the course of a orbit, it experiences significant tidal stresses resulting in active volcanoes, which Voyager observed. Europa is also in a 2:1 resonance with the next satellite Ganymede.
Mercury–Jupiter 1:1 resonance 
The planet Mercury is especially susceptible to Jupiter's influence because of a small celestial coincidence: Mercury's perihelion, the point where it gets closest to the Sun, precesses at a rate of about 1.5 degrees every 1000 years, and Jupiter's perihelion precesses only a little slower. One day, the two may fall into sync, at which time Jupiter's constant gravitational tugs could accumulate and pull Mercury off course. This could eject it from the Solar System altogether or send it on a collision course with Venus or Earth.
Asteroid influence 
Chaos from geological processes 
Another example is Earth's axial tilt which, due to friction raised within Earth's mantle by tidal interactions with the Moon (see below), will be rendered chaotic at some point between 1.5 and 4.5 billion years from now.
Project LONGSTOP (Long-term Gravitational Study of the Outer Planets) was a 1982 international consortium of Solar System dynamicists led by Archie Roy. It involved creation of a model on a supercomputer, integrating the orbits of (only) the outer planets. Its results revealed several curious exchanges of energy between the outer planets, but no signs of gross instability.
Digital Orrery 
Another project involved constructing the Digital Orrery by Gerry Sussman and his MIT group in 1988. The group used a supercomputer to integrate the orbits of the outer planets over 845 million years (some 20 per cent of the age of the Solar System). In 1988, Sussman and Wisdom found data using the Orrery which revealed that Pluto's orbit shows signs of chaos, due in part to its peculiar resonance with Neptune.
If Pluto's orbit is chaotic, then technically the whole Solar System is chaotic, because each body, even one as small as Pluto, affects the others to some extent through gravitational interactions.
Laskar #1 
In 1989, Jacques Laskar of the Bureau des Longitudes in Paris published the results of his numerical integration of the Solar System over 200 million years. These were not the full equations of motion, but rather averaged equations along the lines of those used by Laplace. Laskar's work showed that the Earth's orbit (as well as the orbits of all the inner planets) is chaotic and that an error as small as 15 metres in measuring the position of the Earth today would make it impossible to predict where the Earth would be in its orbit in just over 100 million years' time.
Laskar & Gastineau 
Jacques Laskar and his colleague Mickaël Gastineau in 2009 took a more thorough approach by directly simulating 2500 possible futures. Each of the 2500 cases has slightly different initial conditions: Mercury's position varies by about 1 metre between one simulation and the next. In 20 cases, Mercury goes into a dangerous orbit and often ends up colliding with Venus or plunging into the Sun. Moving in such a warped orbit, Mercury's gravity is more likely to shake other planets out of their settled paths: in one simulated case its perturbations send Mars heading towards Earth.
See also 
- Clearing the neighbourhood
- Future of the Earth
- Resonant trans-Neptunian object
- Risks to civilization, humans and planet Earth
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