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

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6. The next five billion years evolution of the Earth-Moon system

As we have managed to set up some limitations on the possible values of the tidal dissipation and the viscosity of the outer core, by using the available geological observations of the past evolution of the Earth, we are now ready to study its future over its expected lifetime, i.e. about 5 Gyr.

6.1. with the present tidal coefficient and no core-mantle friction

Using Laskar's theory of the solar system, we simultaneously integrate over 5 Gyr the motion of all the planets (Pluto is not taken into account) and the angular momenta of the Earth and the Moon with a 250 yr time step.

The equations for the planetary orbital motion used here are the averaged equations which were previously used by Laskar for the demonstration of the chaotic behavior of the solar system. They include the Newtonian interactions of the 8 major planets of the solar system (Pluto is neglected), and relativistic and Lunar corrections (Laskar, 1985, 1989, 1990). The numerical solution of these averaged equations showed excellent agreement when compared over 4400 years with the numerical ephemeris DE102 (Newhall et al., 1983, Laskar, 1986), and over 3 millions years with the numerical integration performed by Quinn, Tremaine and Duncan (Quinn et al., 1991, Laskar et al., 1992). Similar agreement was observed with subsequent numerical integration by Sussman and Wisdom (1992).

This system of equations was obtained with dedicated computer algebra and contains about 50000 monomial terms of the form [FORMULA] (Laskar, 1985). It was first constructed in a very extensive way, containing all terms up to second order with respect to the masses, and up to 5th degree in eccentricity and inclination, which led to 153824 terms, and then truncated to improve the efficiency of the integration, without significant loss of precision (Laskar, 1994). The numerical evaluation of this simplified system is very efficient, since fewer than 6000 monomials need to be evaluated because of symmetries. Numerical integration is carried out using an Adams method (PECE) of order 12 and with a 250-year stepsize. The integration error was measured by integrating the equations back and forth over 10 Myr. It amounts to [FORMULA] after [FORMULA] years (40 000 steps), and behaves like [FORMULA]. Ignoring the chaotic behavior of the orbits, this would give a numerical error of only [FORMULA] after 10 Gyr.

It is clear that because of the chaotic dynamics with a characteristic time of 5 Myr, the orbital solution loses its accuracy beyond 100 Myr. This is not important since we do not look for the exact solution, but for what happens when the system enters the chaotic zone; the fact that this zone may not be at the exact location has not much importance.

We have chosen to take the near present value of 600 seconds for [FORMULA] and no core-mantle friction. In order to have a statistical view of the possible behaviors, the whole system is simultaneously integrated over 5 Gyr for 500 different initial orientations with obliquities very close to [FORMULA]: 10 initial phases [FORMULA] separated by [FORMULA] rad and 50 initial obliquities separated by [FORMULA] degree.

Thus we have performed a frequency analysis (see Laskar, 1993) on the precession frequency and also plotted the minimum and maximum reached obliquity each 10.26 Myr. The whole computation, for such an experiment, took about 13 days on a IBM-RS6000/390.

It is quite obvious that we cannot display all the various solutions, and we just selected two examples of the possible evolution of the Earth (Fig.4a-b) which are representative of the whole experiment. The two curves plotted in Figs. 4a and 4b have initial obliquities differing by 36 [FORMULA] as. We see that the obliquity enters the chaotic region at about [FORMULA] 1.5 Gyr and that it can go from [FORMULA] to values close to [FORMULA] as was the case in the conservative framework. When superimposed on Fig. 1, those graphs show possible paths of the evolving obliquity through the different zones of the global dynamics.

[FIGURE] Fig. 4a. Example of possible evolution of the Earth's obliquity for 5 Gyr in the future, for [FORMULA] s. The background of the figure is the same one as in Fig. 1, and is a global view of the stability of the obliquity, obtained by means of frequency map analysis (see Laskar and Robutel, 1993). The precession constant (on the left) is plotted against the obliquity: the two bold curves correspond to the minimum and maximum values reached by the obliquity. The right y-axis gives the corresponding time for the motion. The non-hatched zone corresponds to very regular regions, and we actually observe that in these regions, the motion suffers only small (and regular) variations. The hatched parts are the regions of strong chaotic behavior. Indeed, in the present simulation, as soon as the orbit enters this chaotic zone, very strong variations of the obliquity are observed, and very high values, close to 90 degrees, are reached.
[FIGURE] Fig. 4b. Same as Fig. 4a, but with a difference of [FORMULA] degree in the initial obliquity.

The computed speed of rotation of the Earth after 5 Gyr is about 0.42 [FORMULA]. Provided that [FORMULA] kg m-3 (Hinderer et al., 1990) and that [FORMULA], one can easily check that condition [FORMULA] of Sect. 3 has not been violated.

The 500 different paths obtained in this manner allow us to get a fairly good idea of the probability for the obliquity to attain some given threshold once the chaotic zone entered. For instance, we have found that 342 maximum obliquities have exceeded [FORMULA] at least once, hence a probability [FORMULA] (see Fig. 5).

[FIGURE] Fig. 5. Probability P for the maximum obliquity [FORMULA] to exceed a given value [FORMULA] for the Earth with [FORMULA]. This was performed over 500 orbits with very close initial conditions followed over 5 Gyr.

6.2. some alternatives

[FORMULA] [FORMULA] seconds is close to the present measured value of the dissipation coefficient, and is in agreement with the observations at -500 Myr (Fig. 3a), but this leads to a lunar collision at about 1.2 Gyr in the past. For this reason, we also considered for [FORMULA] the smaller value of 200 seconds which is close to the lowest value compatible with these geological observations (Fig. 3a). As previously, for [FORMULA] seconds, we followed the evolution of 500 obliquities. As the dissipation is three times weaker, the Earth reaches the chaotic zone on a much longer time, after about 4.5 Gyr, and after 5 Gyr it has spent only about 500 Myr in this chaotic zone; the probability of reaching a given high value of obliquity is then lower than in the previous case of [FORMULA] seconds for which the same situation lasted 3.5 Gyr (see Fig. 6), and we have [FORMULA], which nevertheless is not a small value. If we continue the integrations over 6 Gyr, which is still a possible future lifetime for the Earth, we obtain for [FORMULA] the much higher value of [FORMULA] (Fig. 6). We carried on the computation till 8 Gyr in order to look at the evolution of this probability, and we also plotted in Fig. 6 the corresponding curves for 7 and 8 Gyr. Then, the set of the four curves shows that the longer the Earth remains in the chaotic zone, the higher are the probabilities for the maximum obliquity to reach any value (the possible maximum hardly exceeding [FORMULA] after 8 Gyr).

[FIGURE] Fig. 6. Probability P for the maximum obliquity [FORMULA] to exceed a given value [FORMULA] for the Earth with [FORMULA] after 5, 6, 7 and 8 Gyr.

We can thus conclude that for any value of the tidal dissipation compatible with the geological observations depicted in Fig. 3a, a very large obliquity in the future of the Earth is a highly probable event.

Finally, we notice that all curves present a falldown at about [FORMULA] and a step till a second falldown to 0 close to [FORMULA]. This can be understood by the fact that, as is shown in Laskar et al. (1993b), the chaotic zone is divided into two regions of strong overlap of secular resonances. In each of these regions, the diffusion of the orbits is rapid, but the connection between these two boxes is more difficult. As soon as a given orbit enters the second box, related to high values of the obliquity, it will rapidly describe it entirely, so we observe in this case a jump in the maximum value reached by the obliquity.

[FORMULA] One would like to consider some very larger coefficients [FORMULA] or [FORMULA] in order to accelerate the effect of the dissipation and to shorten a lot the time of integration by the way. For example, Touma and Wisdom (1994) set a tidal effect about 4000 times stronger than the present value in their study of the past evolution of the Earth's obliquity. We have integrated the system with three different values: [FORMULA], and [FORMULA] seconds, the last one roughly corresponding to what Touma and Wisdom took. The equivalent despinning of the Earth is then respectively achieved after 100 Myr, 10 Myr and 1 Myr instead of 5 Gyr.

The results clearly show that the dynamics are altered as much as the time scale of braking is reduced (see Figs. 7a-c). In the first case, we have found [FORMULA]. In the second one, the obliquity remains confined below [FORMULA]. Finally, secular resonances have a faint effect in the last case, the obliquity never exceeding [FORMULA]. For both last cases, the 500 initial conditions nearly give the same evolution.

[FIGURE] Fig. 7a. as in Fig. 4ab, the precession constant (on the left) is plotted against the obliquity: the two bold curves correspond to the minimum and maximum values reached by the obliquity. Example of evolution for the obliquity of the Earth with [FORMULA].

[FIGURE] Fig. 7b. Example of evolution for the obliquity of the Earth with [FORMULA].

[FIGURE] Fig. 7c. Example of evolution for the obliquity of the Earth with [FORMULA].

It is then clear that such a strategy has to be excluded: the time scale of action of the dissipation must be of the same order as the true one.

[FORMULA] We have chosen to neglect the core-mantle friction because the tidal effects are traditionally considered as the only substantial effects. We could also have undertaken some integrations with non-zero viscosities in order to see the core-mantle friction play an important role, the obliquity being slowly decreasing. As was discussed in Sect. 4, such a scenario is plausible. In any case, the rate of change in obliquity would be relatively small according to relation [FORMULA]. As the Earth enters the chaotic zone before [FORMULA] 4.5 Gyr and undergoes strong variations due to the planetary perturbations, the choice of the couple [FORMULA] has finally not very much consequence on the very long term evolution of the spin, provided the observations and the time scale of dissipation are respected, according to Fig. 3.

Nevertheless, it must be stressed that for very high viscosities, the CMF could dominate the evolution of the obliquity before the 5 Gyr term. With the highest possible value [FORMULA] m2 s-1 suggested by the palaeo-observations, [FORMULA] reaches only [FORMULA] /Gyr when [FORMULA]. Then, such a situation appears to be extreme; this supports our studies with [FORMULA].

[FORMULA] The formulation of the tidal dissipation used in these integrations corresponds to a model for which the function Q is inversely proportional to the speed of rotation. Consequently, as suggested by MacDonald's computations (MacDonald, 1964) with a constant geometrical phase lag and the present tidal dissipation rate, the choice of the model for which Q is constant would lead to a faster despinning in the future, and the chaotic zone would be attained sooner.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998