I’ve noticed some of my most-read posts aren’t the heavy
duty quantum mechanics but the more bread-and-butter physics questions. I got
invovled with the moon last month because I wanted to make a point about the
quadrupole component of the magnetic field in the Stern Gerlach experiment, but
then the moon topic took on a life of its own. First I got messed up with the
reflectance, and then I said I was going to explain the tides. I guess now it’s
time to pay the piper.
The easy thing to explain about the tides is why there are
two of them, I’ve already shown the picture that explains it completely. It’s
the distortional component of the moon’s gravity that causes the tides, not the
direct force, and it’s easy to see that the nature of the distortional force is
what we call the quadrupole field.
The tough question turns out to be: what is the effect on
the earth-moon system of the tidal friction? It’s not too hard to convince
yourself that it ought to slow down the rotational speed of the earth. But what
about the moon’s orbital speed? If the earth is slowing down, shouldn’t the
moon speed up? Otherwise how would angular momentum be conserved?
It turns out this question drove me crazy for about two
days. I thought I had it all figured out: the moon speeds up to conserve
angular momentum, and therefore it is dragged down into a lower orbit. Why
lower? Because the velocity of a satellite in a low orbit is greater than the
velocity of the same satellite in a high orbit. Then I looked it up on
Wikipedia, and it was backwards. They had it going into a high orbit to
conserve angular momentum!
I have to say I never like to be on the wrong side of
Wikipedia when it comes to questions like this. The people who write articles
like this one
aren’t just pulling that stuff out of
their ass. So where was my mistake?
It’s actually hard to believe, as near as I can figure out,
this is how it works. If you somehow get up into orbit and push on the moon to
speed it up, it actually slows down. No fooling. Let’s say you set off an
atomic bomb on the leeward side of the moon, to boost its orbital speed. The
effect of that one impulse is to throw the moon into an elliptical orbit. But
if you keep on continuously setting off bombs, until you’ve gone one full
orbit, the effect of the ellipticity (is that a word?) has to cancel itself out.
The net result is you’ve got yourself back into a circular orbit, but it’s higher up than the one you started with.
But being a higher orbit, it’s also a slower
orbit. (Yes, that’s how it works.) By pushing on the moon to speed it up,
you’ve ended up slowing it down. You’ve driven it to a higher orbit, but it’s
velocity is less than when you started out.
(It’s velocity is lower, but its angular momentum is
actually higher. That’s because angular momentum is the product of velocity and
radius, and the increase in radius is more than compensates for the decrease in
velocity.)
There’s still one thing missing from this explanation, and
that is a mechanism. It’s fine to say that the moon speeds up to conserve
angular momentum, but what makes it speed up? I have to admit I had an awful
time getting this question all straightened out, but to give myself credit, I
had the mechanism right all along. It’s the shift in the tidal bulge. The
picture we showed earlier showed how the bulge would look if the earth and moon
were just sitting there. In fact, the earth is spinning fast, dragging the
bulge along with it, and so the moon is effectively trailing behind the bulge.
I’ve drawn the picture here showing the bulge at a 45 degree angle to the earth-moon axis: this isn’t
the physical case, it’s the ideal case of a uniform ocean with a natural
frequency exactly equal to the driving frequency of the earth-moon system. In
other words, it's an ocean at resonance, cresting to the maximum possible power
and dissipation. You can see that the gravitational effect of the bulges is to
drag the moon along, tending to speed it up to synchronize it with the earth.
The leading bulge, being closer to the moon, is more effective than the trailing bulge. The perverse response where the moon actually slows down in response to being
dragged along is another thing altogether.
1 comment:
Excellent article! You’ve answered this question phenomenally.
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