| MadSci Network: Physics |
First of all, you're correct that not all bodies deform in a strictly linear fashion. However, it seems to be a reasonably good approximation for astronomical purposes, especially since a better model is not easy to come up with.
Your second point is also a very insightful one, but believe it or not, your intuition is wrong! The question of how the Moon migrates over time is really one of energy loss. Because the Earth's tidal bulges are moving each day (since the Earth rotates under the Moon and the bulge try to point at the Moon as best they can manage), there is heat being generated. It's very like kneading a piece of cold clay or flexing a paper clip. Since we're losing energy and since total angular momentum (Earth's spin angular momentum plus the Moon's orbital angular momentum) is not being changed, the Moon has to move outward. So what we really need to find out is how much energy is being lost per second. This lost rate due to tides depends on the difference between the Earth's rotation frequency (once per day) and the Moon's orbital period (once per month). So so far, you're perfectly right.
But there's more! In order to convert this energy loss rate into a rate of increase of the lunar orbit, we need to see where that energy is actually being taken from. If we study this, we see it comes from Earth's spin and the Moon's orbital energy. If we figure out how much energy can be lost per time at a given rate of recession for the Moon - remembering to conserve angular momentum - we see that this rate also depends on the difference of the frequencies. So when we equate the two energy loss rates to solve for the recession rate, the difference of the frequencies cancels out.
To summarize the above (which is a bit technical): the energy loss rate depends on the difference of frequencies, but that rate at which that energy loss can be used to increase the Moon's semi-major axis also depends on the difference of the frequencies. So it all comes out in a wash.
Well, almost in a wash. In the final expression for the recession rate we find that there's a term that depends on the sign of the difference between Earth's rotation rate and the Moon's orbital rate. So when Earth spins faster than the Moon orbits, the Moon moves away from us. If, however, the Moon orbited faster than we spin (so a month was less than a day), the Moon would be approaching us. And if the Moon were just at the right distance so that a month equalled a day, nothing would be happening at all. (Except that this is an unstable arrangement and any little bump away from this distance will grow and the Moon will start moving towards or away from the Earth.)
If you'll let me continue for a few more paragraphs...
I'm very suspicious of that paragraph which you quoted. For one this, it isn't at all obvious to me that one can go from the fact that tidal forces fall like one over the distance cubed (which is true!) to the recession rate of the Moon goes like one over the distance to the sixth. Not to be overly vain, but I actually work in this field and I didn't see the step. So I looked up a careful derivation of this in Murray and Dermott's book, Solar System Dynamics - an excellent text if you really want to see all the gory math. After a great deal of work, including some I've outlined above, they deduce that the Moon should be moving away from the Earth at a rate that goes like one over the distance to the 5.5. Close to what your author thought, but not quite. (It also depends on several other factors, including the way the Earth responds to tidal pulls and the masses of both bodies.)
So can we figure out where the Moon has been over the past 4.5 billion years? Not really. There are other effects which shape the Moon's behavior. As you alluded to, there's no real reason to think "k", the way Earth responds to tidal forces, is constant over all that time, either. The energy loss rate depends a lot on how much Earth tries to rest the movement of our tidal bulge. Since continents have formed and moved, this isn't going to be constant. In fact, we are probably in a very unusual time right now. Continents keep the ocean from being able to response to the tidal force as it would like, and so the oceans lose energy against the land masses all over the planet. But if you think back to pangea, when Earth's continents were basically all clumped together, there would be much less resistance to the oceans and energy would not be lost as quickly.
In short, I think that whatever you're reading is not considering all of the subtleties involved in this problem. (And it is an ongoing problem to understand where the Moon has been. I'm always thrilled to know that not all of the interesting questions have been answered!) You are quite right to have some skepticism on these points.
I hope that this has helped!
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Note added by MadSci Admin:
The following previous MadSci answer discusses some of
the details about the origin of the moon.
- - John Link, MadSci Admin
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