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Astron. Astrophys. 318, 975-989 (1997)

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1. Introduction

The recent works (Laskar et al., 1993a-b) and (Laskar and Robutel, 1993) emphasized the sensitivity of the obliquity [FORMULA] of a planet to the planetary perturbations. Indeed, secular resonances between the precession motion of the rotation axis of a planet and the slow secular motion of its orbit due to planetary perturbations can result in large chaotic variations of its obliquity. In the case of the Earth, the presence of the Moon changes the Earth's precessing frequency by a large amount, and thus keeps it in a stable region, far from the large chaotic zone which results from secular resonances overlap (Laskar et al., 1993b). But due to tidal dissipation, the Moon is slowly receding from the Earth and the Earth's rotation is slowing down. Ultimately, the Earth will reach the large chaotic zone due to planetary perturbations, and its obliquity will no longer be stable. The object of the present work is to provide a quantitative description of the long term evolution of the Earth's obliquity, in the future, but also in the past.

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 et al., 1994) with great precision. Nevertheless, accurate quantitative estimates of the length of the day over the age of the Earth are still lacking, and it is still a difficult question to know the precise origins of these evolutions. Apart from the early work of (Darwin, 1880), the major works on the past history of the Earth-Moon system are due to MacDonald (1964), Goldreich (1966) and Mignard (1979, 1980, 1981). They found the same trends in the variations of the Earth's spin and the lunar orbit due to the effects of the tides raised on the Earth by the Sun and the Moon. Adding the planetary perturbations to the Earth's orbit, Touma and Wisdom (1994) recently confirmed those past variations, the rates of which nevertheless remain uncertain because the coefficient of tidal dissipation is not well known. On the other hand, none of these studies have taken into account the action of the friction between the core and the mantle of the Earth as has been done in studies of Venus' obliquity (see for example Goldreich and Peale, 1970, Lago and Cazenave, 1979, Dobrovolskis, 1980, Yoder, 1995). However, as is pointed out by Williams (1993), this effect could be of great importance, providing the spin with an obliquity decreasing with the time. The controversy about the efficiency of the core-mantle friction arises from the fact that the possible effective viscosities of the outer core cover a very large range of values (Lumb and Aldridge, 1991), and in the scenario presented by Williams (1993) for the past evolution of the Earth's obliquity, a very high value of this viscosity is assumed in order to obtain a past obliquity of the Earth reaching 70 degrees one billion years ago.

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 et al. 1993b) and (Laskar and Robutel, 1993), we know that as the Moon recedes from the Earth, as soon as the obliquity reaches the first important planetary secular resonance, it will enter a very large chaotic zone, with the possibility of attaining very high obliquities up to nearly 90 degrees.

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 ([FORMULA] = 23.44 degrees, [FORMULA] = 54.93 arcsec/year), and the present variations of the obliquity are limited to [FORMULA] degrees around its mean value (Laskar et al., 1993a).

[FIGURE] Fig. 1. The zone of large scale chaotic behavior for the obliquity of the Earth for a wide range of precession constant [FORMULA] (in arcseconds per year). The non-hatched region corresponds to the stable orbits, where the variations of the obliquity are moderate, while the hatched zone is the chaotic zone. The chaotic behavior is estimated by the diffusion rate of the precession frequency measured for each initial condition [FORMULA] via numerical frequency map analysis over 36 Myr. In the large chaotic zones, the chaotic diffusion will occur on horizontal lines ([FORMULA] is fixed), and the obliquity of the planet can explore horizontally all the hatched zone. With the Moon, one can consider the present situation of the Earth to be represented approximately by the point with coordinates [FORMULA], which is in the middle of a large zone of regular motion. Without the Moon, for spin periods ranging from about 12h to 48h, the obliquity of the Earth would undergo very large chaotic variations ranging from nearly [FORMULA] to about [FORMULA]. This figure summarizes the results of about 250 000 numerical integrations of the Earth's obliquity variations under perturbations due to the whole solar system for various initial conditions over 36 Myr. (Laskar and Robutel, 1993, Laskar et al. 1993b).

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.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998