Is it possible to compute the past climate of the Earth at the time of dinosaurs? This question was answered by Jacques Laskar during his lecture entitled “Astronomical calibration of the Geological Time Scales” at the **workshop “Mathematical models and methods for Planet Earth”** at Istituto Nazionale di Alta Matematica (INdAM) on May 27-29.

Laskar explained that Louis-Joseph Lagrange was the first to suggest a link between the past climates of the Earth and the variations of the parameters characterizing the Earth’s elliptical orbit: the latter concern the changes in the major axis, in the eccentricity, in the obliquity of the Earth’s axis, and in the precession of the Earth’s axis. These parameters undergo periodic oscillations, now called the Milankovitch cycles, with different periods between 20,000 and 40,000 years. These oscillations essentially come from the attraction exerted on the Earth by the other planets of the solar system. They have some influence on the climate, for instance when the eccentricity of the ellipse changes. The oscillations of the Earth’s axis also influence the climate: when the axis is more slanted the poles receive more Sun in summer but the polar ice spreads more in winter. And scientists compare the measurements of ice cores and sedimentary records, showing the correlation between the past climates and the computed oscillations of the parameters of the Earth’s orbit.

But how do we compute these oscillations? We use series expansions to approximate the motion of the Earth when taking into account the attraction of the other planets of the solar system. But these series are divergent as was shown by Poincaré. Hence, the series can only provide precise information over a limited period of time. Jacques Laskar showed in 1989 that the inner planets of the solar system are chaotic and confirmed this later in 2009 with 2500 simulations in parallel of the solar system. One characteristic of chaos is sensitivity to initial conditions, which means that errors grow exponentially in time. Hence, inevitably, the errors of any simulation will grow so much that we can no more learn anything reliable from the simulation. The whole question is then: how fast grow these errors?

This is measured by the Lyapunov time, which we will define here as the time before we lose one digit of precision—that is, the error is multiplied by 10. When modeling the planets, this Lyapunov time is 10 million years. The extinction of the dinosaurs took place 65 millions years ago, and simulating the solar system over 70 millions years we lose 7 digits precision. This is a lot, but it is still tractable, and it is relatively easy to prove that the Earth is “stable” at this time horizon. But when we speak of the influence of the parameters of the Earth’s orbit on the climate, we need more precision. It does not suffice to include in the simulations all planets of the solar system as well as the moon and the mean effect of the asteroid belt. The largest asteroids have to be considered individually, and some of them play a role. The two largest are Ceres and Vesta. Both are highly chaotic, with Lyapunov times of 28,900 years and 14,282 years, respectively. These asteroids are sufficiently large to have an influence on the orbit of the Earth, and there are other chaotic asteroids in the asteroid belt. Imagine: for each million years, the errors coming from Ceres are multiplied by 10^34 and those from Vesta by 10^70! In a paper “Strong chaos induced by close encounters with Ceres and Vesta” published in 2011 in Astronomy and Astrophysics, Jacques Laskar and his co-authors showed that we hit the wall and cannot obtain any reliable information past 60 million years. Hence, we may deduce the climate at the time of dinosaurs from geological observations, but there is no hope to compute it through backwards integration of the solar system.

Christiane Rousseau