Newsgroups: sci.astro,alt.sci.planetary
From: Henry Spencer <henry@zoo.toronto.edu>
Subject: Re: Formation of Earth's Moon Likely Caused by Huge Planetary 
	Collision
Date: Mon, 4 Aug 1997 18:40:56 GMT

In article <5ro2dg$8a2$1@Jupiter.Mcs.Net>, Jorn Barger <jorn@MCS.COM> wrote:
>>Tidal effects have a strong tendency to circularize orbits.  Even a moon
>>that starts out in a highly elliptical orbit will end up in a circular
>>orbit, unless it's small and far out or a recent acquisition.
>
>I never heard this-- can you talk me thru it?

The quick-handwaving level is easy.  The moon has tidal bulges.  Assume
it's tidally locked to the planet, making one rotation per orbit (that
process is really quick).  If it's in a slightly elliptical orbit, there
are two complications.

One is that the strength of tidal forces is an inverse-cube function of
distance (it's one power stronger than the gravitational force because it
varies with the gradient of the gravitational force).  So the tidal
effects are stronger, and the bulges significantly higher, at periapsis
than at apoapsis.

The other is that the moon has to rotate at a uniform rate, while an
elliptical orbit doesn't go around the planet at a uniform rate.  So
the tidal bulges move back and forth over the surface a bit, staying
(more or less) with the orbital motion even though it doesn't quite
match the rotation.

Both of these effects dissipate energy, because the shifting bulges are
constantly flexing the rock (ice, water, whatever) of the moon.  In fact,
this is where the heat for Io's volcanoes comes from.  That energy has to
come from *somewhere*.  It comes from orbital kinetic energy, with some
suitable shuffling of orbital momentum so angular momentum is conserved.
(The reason why Io is still volcanic after all this time is that it's in
a resonance with two of the other Galilean satellites, which pump up the
eccentricity of its orbit as fast as tidal effects smooth it down.)

Exactly how all this turns into circularization of the orbit is a little
trickier to follow in detail, but consider the second effect, rotation
versus orbital motion.  At periapsis, where the moon is moving fastest,
rotation lags behind orbital motion.  The moon is not rotating fast enough
to keep the tidal bulges lined up with the planet.  The nearside bulge
pulls ahead of the planet, the farside one falls behind.  (If this sounds
like the wrong way around, you need to draw a picture -- this description
is correct.)  Since the two bulges are at different distances from the
planet, the gravitational pull on the nearside (leading) one is stronger.
Since the bulge is leading the moon slightly, and the pull is toward the
center of the planet, the pull is not quite along the planet-moon axis.
So there is a small backward force on the bulge, decelerating the moon.
The effect of deceleration at periapsis is lower apoapsis -- that is,
circularization of the orbit.

The effect is reversed at apoapsis, but tidal forces are weaker there.
In fact, it turns out that such unbalanced forces on or from tidal bulges
scale with the inverse *sixth power* of distance.  So the forces at
periapsis completely dominate the situation.

When the moon is also raising tidal bulges in the planet, the situation
gets distinctly more complicated, of course...

>I understand-- I think-- how the moon's tides cause the moon to show
>the same face to the earth.  But is it the earth's tides that affect
>the moon's orbit, or its own tides???

Mostly Earth's tides.  The dominant effect on the Moon's orbit, after all
is said and done, is the way it is being raised (and Earth's rotation
slowed) by the tides the Moon raises in the Earth.

>Wouldn't this also be the explanation of planetary rings, then?

In the case of rings, they tend to be circular for a more prosaic reason:
collisions between ring particles circularize their orbits very quickly.
(The same collisions are very effective at getting rid of out-of-plane
motions, which is why Saturn's rings are perhaps 100 *meters* thick
despite being 270,000 *kilometers* across -- a sheet of ordinary paper
would have to be over twice the size of a football stadium to have the
same width/thickness ratio.)
--
Committees do harm merely by existing.             |       Henry Spencer
                           -- Freeman Dyson        |   henry@zoo.toronto.edu