I did this whole calculation as part of my PhD oral exam back in 1995. Let's see how much I remember. [img]/phpBB/images/smiles/icon_smile.gif[/img]
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On 2002-03-18 10:28, Gary Redmond wrote:
However the following is (astronomy book) conjecture:
"This has the additional effect of causing the Earth to rotate on its axis more slowly. Eventually the Earth will find itself tidally locked with the Moon. Both bodies will show only one face to each other."
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Um, well, it's not so much conjecture as extrapolation.
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The facts are:
The Moon does run slow (recession of nodes), however there is no evidence that it is "slowing".
The Earth does run slow (leap second), again there is no evidence that it is "slowing".
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Both not true. Think about it, if the moon is moving further away, it *has* to slow down. Kepler's third law.
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I'm not completely sure on this, but I don't believe that the gravitational mass can effect the rotation of the inertial mass.
If you locate the center of gravity of a mass (point around which all its weight is equally distributed) and then suspend that mass from that point it will be in perfect balance and will not rotate, or will continue to rotate indefinitely. The Earth's center of gravity suspends the Moon's center of gravity and vice versa. Thus the Earth cannot rotate the Moon, and the Moon cannot rotate the Earth.
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This is imprecisely stated. First of all, neither the earth nor the moon are spherically symmetric. It is the *torque* produced by the tidal bulges that drags
on the two bodies and changes the angular momentum of the joint system (torque = dL/dt). See below.
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Furthermore, tides are internal to a system so therefore they cannot change the angular momentum of the system.
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No. I think about it this way: the tides cause a deformation of the earth along the direction of the earth-moon axis. However, the earth does not have perfect elasticity, so as it rotates, the bulges are pulled ahead of the earth-moon axis. Therefore, if you look at the force the bulges exert on the moon (Newton's third law), that force is mis-aligned with said axis. Therefore, the torque (|t| = |r x F| = |r||F|sin(theta)) is non-zero, therefore the angular momentum of the system changes. This will manifest in two ways: the earth's rotation period will slow and the moon will shift into a higher orbit and revolve more slowly.
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"3) Since there is also a tide from the Sun (although smaller than the tide from the Moon) I presume that the Earth will become tidally locked to the Sun. How long will this take?"
Forever!
If the Earth were locked to the Sun, the Moon were locked to the Earth, and the Earth were locked to the Moon, the Earth Moon system would no-longer rotate the Moon would crash straight down and the Earth and Moon would become one mass:-)
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No. It's still rotating, it's just that from the point of view of an observer on the earth, the moon would *appear* to hover over the same spot on the earth. By your argument, geostationary satellites should come crashing down. When the earth/moon system tidally locks, the moon will be at the geosynchronous orbit, which will be about 12 times further out than current geosynchronous orbits, or half a million km. I remember that the asymptotic value for the final rotation period is about 42 (current) days. Douglas Adams would have been so proud. The moon is currently about 3.8e5 km, so if its drift away is linear (which it isn't), it would take about 3 billion years to drift that far. Actually, it will take much longer, since the drift rate will slow down as the moon drifts away and the torque decreases.
I haven't ever tried to calculate what will happen when the earth-moon system tidally locks with the sun. Safe to say it's not going to happen for a long, long time, but let me think about it...
My instinct is that over the long haul, the tiny tidal bulge will work towards locking the earth with the sun, which will cause the earth to move away (Neptune is demonstrably moving away from the sun, which is why Pluto and other KBOs are trapped in resonance orbits with Neptune) and slow down, which will in turn cause the moon to slow down and move away from the earth to keep up. Unless the moon is orbiting in the opposite sense, which I don't remember off the top of my head. Anyway, I suspect that eventually the moon will get stripped off the earth into its own solar orbit, but I haven't tried to caluclate that for sure.
I have to go to a meeting,
Don