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

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4. Some limits to the coefficients of dissipation

Goldreich (1966), Mignard (1979, 1980, 1981) and Touma and Wisdom (1994) have studied the past evolution of the Earth-Moon system taking the distance of the Moon to the Earth as the independent variable instead of the time. This is because if one takes the present value [FORMULA] 638 seconds which fits the observed receding of the Moon for which Dickey et al. (1994) give 3.82 cm per year, it is found that the capture or formation of the Moon occurred at about -1.2 Gyr which contradicts most of paleogeological observations (see for example Piper, 1978, Lambeck, 1980).

So [FORMULA] has clearly been smaller in the past, probably because of the changes in the continental distribution (Krohn and Sündermann, 1978) and in the oceanic loading during glaciations. We only need for our purpose to consider some acceptable average value anyway.

With a review of experiments in laboratories and observations of fluctuations of the Earth's spin, Lumb and Aldridge (1991) give a range of possible values for the effective viscosity [FORMULA] going from [FORMULA] up to [FORMULA] m2 s-1. Although the influence of the friction is proportional to [FORMULA], the uncertainty about the effect of internal friction still remains a serious problem. Rochester has chosen to set the upper limit to 10 m2 s-1 which comes from observations of the fluctuations of the Earth's nutation (Toomre, 1974). Actually this last value changes the spin very little. On the opposite, Williams asserts that [FORMULA] should be much larger. This comes out from his assessment of observations of sediments and fossils which suggests that the history of the obliquity would have been very different from the consensual one, starting from [FORMULA] and going down to the present [FORMULA] with a drastic falldown at about -630 Myr (Williams, 1993).

A simple way to determine some strong constraints, provided that Rochester's model is assumed always valid, is to look for the couples [FORMULA] which give, with the present model, an evolution of the length of the day (LOD) similar to the ones given by the observations of the Earth's ground over the last two billion years (Williams, 1989).

The data gathered by Williams (1989) are of distinct origins and they give some quite different rates of increasing of the LOD, especially for the first hundreds Myr. We have decided to take as references two of these rates. The first one is based on the maximum values of the LOD at about -500 Myr which corresponds to a 20.33 hour rotation period. This gives the ratio [FORMULA]. Such a choice is supported by the sake of getting upper bounds for the dissipation coefficients. The second rate is based on the -2 Gyr data and the observation of Elatina formation at -650 Myr. It corresponds to a 19 hour rotation period at -2 Gyr which gives [FORMULA] This one is suggested by the fact that most ancient observations match better with the Moon's orbital history and corresponds to a low rate of braking.

We have computed the speed of the Earth's rotation at -500 Myr and -2 Gyr for 22 values of [FORMULA], going from 0 to 630 seconds by steps of 30 seconds, and for 25 values of [FORMULA], going from 0 to [FORMULA] m2 s-1 by cubic steps.

The following physical parameters of the present Earth appearing in the previous equations have been taken from Lambeck (1988):

[EQUATION]

and [FORMULA] (Legros, private communication). All other parameters and initial conditions are taken from Laskar (1986).

The results are shown in Figs. 3a-b where [FORMULA] is given for a wide range of [FORMULA]. In the computation at -2 Gyr, [FORMULA] only goes up to 360 seconds because higher values have led to the collision with the Moon before this date. In Fig. 3 we have superimposed the curves corresponding to the ratios [FORMULA] and [FORMULA]. For the case -500 Myr, we have also drawn the curve of [FORMULA] which corresponds to the lower observed rate of deceleration, so that the strip formed by both curves represents the global uncertainty on the couple [FORMULA].

[FIGURE] Fig. 3. a Percentage of the ratio of the speed of rotation at -500 Myr over the present one for various values of the tidal delay [FORMULA] and the viscosity [FORMULA]. The two bold lines delimit an acceptable range, in agreement with the observations from sediments and fossils. b Same percentage at -2 000 Myr. The bold line corresponds to the observation of Williams (1989).

It is clear that 600 seconds is an upper limit for average [FORMULA]. The observations at -2 Gyr set this limit at about 200 seconds. Moreover, the upper limit for average [FORMULA] (with no tidal effects) is about 7400 m2 s-1, and we see that the core-mantle friction has no significant effect below several m2 s-1. The observations at -2 Gyr lead to [FORMULA] m2 s-1.

It would be interesting to know the lower limit of [FORMULA], which should be the value of the mantle alone because most of the fluctuations comes from the changes in oceanic loading. It is possible to give an rough estimate to it, knowing that the energy dissipated in the oceans accounts for about 90 or 95% of the total (Zschau, 1978), (Cazenave, 1983), (Mignard, 1983), (Lambeck, 1988). In this case, the lowest [FORMULA] would equal 30 or 60 seconds, hence a largest [FORMULA] of about 600 or 800 m2 s-1 if one relies on the -2 Gyr observations, and about 4400 or 4700 m2 s-1 for the -500 Myr ones.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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