The Discrepancy Between Earth And Moon Measures

We have seen above that tidal friction is causing the spin rate of the Earth to fall, and to compensate for that the Moon is receding from us. Direct measurements are made of two quite different things: the rotation rate of the planet (using historical eclipses for the long-term changes, and ultra precise radio astronomical techniques and so on for the short-term changes), and the distance of the Moon (through laser ranging). You might expect the results obtained from these distinct measurements to be in agreement, but that is not the case: there is a marked discrepancy between them. Why is this?

Let us go back to thinking in terms of the length-of-day (LoD) because it is the easiest parameter to understand. We have said that the LoD is increasing by around 1.7 milliseconds per century. So, as I write in the year 2001 the LoD is 1.7 milliseconds longer than it was in 1901, and in the year 2101 it will be (we anticipate) close to 3.4 milliseconds longer than it was back in 1901. The LoD is a quantity we can measure directly.

The rate at which the Moon is receding is also measured directly (from the retro-reflectors left on the lunar surface; see Figure 6-1), and that inch-and-a-half per year can be converted into the equivalent increase in the LoD that would result over a century. But when we do this, the answer is 2.3 milliseconds, and not the 1.7 milliseconds we might have expected. How does the discrepancy of 0.6 milliseconds arise? Some other process must be counteracting a part of the slowdown due to the tidal drag imposed by the Moon.

The answer to this puzzle seems to be related to the Ice Age cycle. At first sight it might appear that, as sketched earlier, the melting of the vast ice packs at latitudes beyond 40 degrees (which occurred around 10 millennia ago) would lead to a simple expansion of the rock and soil that had been compressed beneath them. Such an expansion would increase the average distance of these landmasses from the spin axis of the Earth, and so the spin rate would fall just as when the ice-skater spreads her arms. But it is not quite that straightforward.

It seems that since the ice burden melted (one can hardly call it polar ice because it covered about a quarter of the globe) the shape of the Earth as a whole has been changing due to the migration of the liquid water so released. Rotating objects are not spherical, but oblate; that is, they are flattened slightly, the distance pole to pole through the middle being less than that measured crosswise in the equatorial plane. Since the termination of the last glacial period it appears that the Earth has become a little less flattened (that is, it has become closer to spherical), with oceanic water moving away from the tropics and towards the poles. This means that the water involved is nearer to our spin axis, and so possesses less angular momentum. Overall, this boosts the planet's spin rate very slightly. The enhancement caused by shape change is equivalent to 0.6 milliseconds (per day per century). Subtracting that from the tidal drag imposed by the Moon and Sun, the value of 2.3 milliseconds, the overall change measured directly is 1.7 milliseconds.

I have just slipped something else in there. I mentioned the Sun imposing a tidal drag, as it surely does. Although the Moon is the major cause of the tides as such, it is the solar influence that produces the difference between the heights of spring and neap tides. Because of the Sun's effect, the angular momentum of the Earth—Moon system is not conserved precisely: the system is not completely isolated. This, though, is a minor complication. In fact, you can probably imagine what is happening here. The tidal friction due to the Sun produces a change that must be taken up by the orbital angular momentum of the Earth, and in consequence the mean Earth—Sun separation increases a little. But the change involved is minute, compared to what is happening in the Earth-Moon system.

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