Specifically, they found that Mercury has an approximately 1% chance of colliding with the Sun or Venus in the Sun’s remaining 5 Gyr lifetime. Yet in a dramatic suite of numerical orbit integrations requiring 8 million CPU hours, Laskar and Mickaël Gastineau of the Paris Observatory found in 2009 that dynamical instabilities, while possible, are rare. In 1989, Jacques Laskar demonstrated that the Lyapunov timescale for the terrestrial planets was only a few million years (Myr). The model provides a simple theoretical framework unfortunately, it suffers from an important tension. This parameter corresponds to the time between steps in the random walk. Every chaotic dynamical system has a characteristic timescale over which predictability is lost, called the Lyapunov timescale. The simplest picture then follows the orbits of the terrestrial planets (Mercury, Venus, Earth, and Mars) and models the chaos as driving a random walk in their eccentricities and inclinations-until the orbits become so elliptical that they go unstable. But the practical question of how long such small effects would need to build up to cause dynamical instabilities had to await the advent of computers.Ī recent insight is that, loosely speaking, the orbits of the outer, more massive giant planets remain well behaved over the age of the Solar System. A century later, Henri Poincaré discovered that our Solar System is chaotic, demonstrating that these neglected, higher-order terms cannot be ignored, and collisions may eventually occur (for more on the problem’s history, see Research News: The Final Piece in the Solar System-Stability Puzzle?). In the 1780s, Pierre-Simon Laplace and Joseph-Louis Lagrange thought they had proved the eternal stability of the Solar System by finding an approximate solution after expanding the expression for the planets’ average gravitational effect on one another to lowest order in the orbits’ small eccentricities and inclinations.
Do these tiny gravitational tugs then simply average out, or can their effects build up and lead to instabilities and planetary collisions over such long timescales? Planetary systems like our own live for roughly 10 billion years (10 Gyr) before their central star runs out of nuclear fuel. The catch is that the relevant timescales are astronomical. The gravitational perturbations the planets exert on one another are at least 1000 times smaller than the dominant central force from the Sun. At face value, the problem seems trivial. Shortly after discovering the law of gravity, Isaac Newton wondered whether it would allow our Solar System to remain stable. BernardH/ CC BY-SA 3.0/Wikimedia Commons Figure 1: A Markus-Lyapunov fractal makes use of the approach devised by Alekesandr Lyapunov (1857–1918) for characterizing the continuous evolution of a dynamical system from order into chaos.
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