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In regard to Mark Sylvester's case (i): the stake through the
spheres contributes a force not present in the gravitational
problem. I would expect this force to be important as it has the
same magnitude as the gravitational attraction. Why is this case
relevant to the discussion?
I strongly disagree with the proposition
that the effects of oceans, rotations, and revolutions can be added
later. They can be meaningfully added only if the stake is
simultaneously removed. In other words, only if we solve the problem
at hand.
His case (ii) of the spheres falling freely toward each other is(further clarifications deleted)
just the case we are discussing. The Earth and the Moon *are*
falling freely toward one another....
Sorry to consume bandwidth, but I did not say that the tides are due
*only* to the gravitational gradient....
By 'nailed down' I meant that 'some point' on earth and one on the
moon was fixed, and it doesn't have to be the center of mass.
Imagine a huge framework with a stake to it through the earth and
one through the moon. For convenience of discussion, imagine the
stake through the earth passing along the earth's polar axis. Also,
for convnience, do this at a time when the earth is above the
equator. Now imagine that both earth and moon have some elasticity,
that they can be deformed by gravitational forces. Keep both earth
and moon at rest.
Under these conditions the tidal forces (gradient of the
gravitational force) on each piece of each body, are such that both
bodies are distorted from their initial spherical shape into, what
is the term, oblate spheroids. Anyway, slightly elongated along the
axis joining the two bodies. With respect to the polar axis (the
stake, remember) there are two bulges on the earth, one toward the
moon, one on the other side. This fact of there being *two* bulges
will be the case even with motion, rotation, or fussing about the
center of mass. Now, of course, if you want to get quantitative
about the exact relative size of the bulges, you'll want to talk
about centers of mass. Then you'll unstake the planets and let them
rotate, which will introduce centripetal effects which will further
modify the bulges. Drag forces will ensure that the bulges lag, so
they no longer are directly along the earth-moon line, etc. etc.