The Lyapunov Time of the Planets

Plate 81 speaks for itself: “Beyond about 5 million years, even the planets can't be predicted.”

The Physics

Laskar's integrations showed the inner Solar System is chaotic: tiny errors in the planets' positions grow exponentially with a Lyapunov time of about 5 million years.

δ(t) ≈ δ₀ e^(t / t_L)

• δ: separation between two nearby solutions, m
• δ₀: initial separation, m
• t: time, s (quoted in Myr)
• t_L: Lyapunov time ≈ 5 Myr, s

A 15 m error in Earth's position: ~150 m by 10 Myr, and ~150 million km by 100 Myr.

Sources

• J. Laskar, Nature 338, 237 (1989).
• J. Laskar, Icarus 88, 266 (1990).

On the poster

Plate 081, “The Lyapunov Time of the Planets”: the pixel-art plate with its slogan “Beyond about 5 million years, even the planets can't be predicted.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.