## 1. IntroductionThe recent works (Laskar The evolution of the Earth-Moon system is far from being a new
subject. Evidence of the loss of the Earth's angular momentum has long
been observed through paleogeological clocks (for a review, see
Williams, 1989), while the present deceleration of the lunar mean
motion can be directly measured by Lunar Laser Ranging (Dickey On the other hand, the future of the evolution of the Earth's
obliquity has already been explored by Ward (1982) which showed, using
a simple model with isolated resonances, that the precession frequency
is expected to cross planetary secular resonances in the future, which
could allow the obliquity to increase up to 60 degrees. The reality is
much more severe, as, from the extended work (Laskar Indeed, using Laskar's method of frequency map analysis over more
than 250 000 numerical integrations of the Earth's obliquity for
various values of the precession constant, it was possible to obtain a
clear picture of the global dynamics of the Earth's obliquity (Laskar
and Robutel, 1993) (Fig.1). Each point of the graph represents one
value of the couple (obliquity, precession constant), the precession
constant being a quantity proportional to the speed of rotation (see
formula (1) in Sect. 2). One dot in the non-hatched zone corresponds
to a stable position, where the obliquity suffers only small (nearly
quasiperiodic) variations around its mean value, whereas one point in
the hatched zone corresponds to a chaotic behavior so that the hatched
area delimits a region of resonances overlap where the Earth can
wander horizontally. The present Earth is located in a stable region
( = 23.44 degrees, = 54.93
arcsec/year), and the present variations of the obliquity are limited
to degrees around its mean value (Laskar
Fig. 1 can be considered as a snapshot of the dynamics of the Earth's obliquity, constructed in a conservative framework, over a relatively short time scale on which the dissipation due to tidal interaction or core-mantle coupling is not yet visible. However, this picture already allows to forecast the future and past evolutions of the Earth's obliquity on much longer time scales, of several billions of years, when the various dissipative effects can no longer be neglected. Indeed, the consequence of this dissipation is to slow down the rotation of the Earth, so that an initial point of the graph is slowly brought down to lower values of the precession constant. This suggests that the Earth's spin has smoothly evolved since the formation or capture of the Moon. Our aim in the present work is to give a precise view of the future evolution of the Earth's obliquity, and more specifically, to describe quantitatively its path in the chaotic zone. The main limitation on a precise evolution of the Earth's rotational state is as much the crudeness and uncertainty of dissipative models as it is the values of their parameters, such as the amplitude of the tidal dissipation, and even more, the viscosity of the outer core. The choice of the dissipative model does not seem to be fundamental here, essentially because the Earth's speed of rotation is not subject to large changes: different models would not lead to very different variations, especially when compared to the ones produced by planetary perturbations. Besides, in order to overcome difficulties arising from the uncertainty of the parameters, we will rely on the geological observations of the length of the day (Williams, 1989). This will allow us to obtain a set of plausible values which correspond to these observations, and to fix the time scale of the evolution along the way. Section 2 is devoted to determine the averaged equations of secular rotational dynamics with planetary perturbations. In Sect. 3 we present the chosen models for estimating the additional dissipative contributions to these equations, while some limits for the coefficients of dissipation will be obtained in Sect. 4. Then we discuss in Sect. 5 the history of the Earth's obliquity proposed by Williams (1993). Finally, we present in Sect. 6 the results of a set of 500 numerical simulations of the future evolution of the Earth for the next 5 Gyr, using Laskar's method of integration of the solar system (Laskar, 1988, 1994a), and starting with very close initial conditions. Indeed, the chaotic nature of the motion prevents us from computing a single orbit, and only a statistical approach becomes meaningful for this problem. Before concluding, we discuss some alternatives to the results in relation to the coefficients of dissipation. © European Southern Observatory (ESO) 1997 Online publication: July 3, 1998 |