The sun is 320,000 times as massive as the earth, but only 400 times
as far from the moon as the earth is. Therefore the force of the sun's
gravity on the moon should be twice as big as the earth's and the moon
should go around the sun instead of around the earth. Since it
doesn't, Newton's theory of gravity must be wrong!" What's the matter
with this reasoning?

The attractive force of gravity is proportional to the mass of the two
objects in question and to the square of distance.

Mass of the earth:   about 6 x 10^24 kg
Earth-moon distance: about 384,000 km
Mass of the sun:     about 2 x 10^30 kg
Sun-moon distance:   about 150,000,000 km

The sun's gravitational pull on the moon is proportional to 2 x 10^30
/ (1.5 x 10^8)^2 = about 8.8 x 10^13

The earth's gravitational pull on the moon is proportional to 6 x
10^24 / (3.84 x 10^5) = about 4 x 10^13

Sure enough, the sun pulls on the moon harder than the earth does! 
Sounds crazy, doesn't it?

Consider the earth by itself - it would orbit the sun pretty much like
it does now.  Consider the moon by itself - it would orbit the sun
pretty much like it does now.  We say that the moon orbits the earth
because it's convenient.  But it's more true to say that the moon
orbits the earth and the sun.  The earth and the sun orbit the center
of our galaxy.  The galaxy and the sun don't so much orbit as go
pretty straight (there's a mind-boggling amount of mass in a galaxy -
hard to change its direction!)

If you trace the path the moon makes in space, it's a spiral around
the sun.  The earth also orbits in a spiral because of the affect the
moon's gravity has on our planet.  Since the earth is rather more
massive than the sun, the moon moves more than the earth does in
response to the gravitational attraction the two share, but both
bodies orbit the sun.

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Request for Answer Clarification by sousme-ga on 06 Nov 2002 12:23 PST

How come the moon isn't pulled away from the earth then?

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Clarification of Answer by haversian-ga on 07 Nov 2002 10:46 PST

As I said, both the moon and the earth orbit the sun.  Since they are
at (about) the same distance and moving at (about) the same angular
velocity, they have (about) the same orbit.

One way to think about it is to smash the earth and the moon together
into one bigger mass situated at the current center of mass of the
earth-moon system.  This new planet would orbit the same way the
center of mass of the earth-moon system orbits currently.  Now that
you know how the center of mass moves, you can put the earth and moon
back where they belong, revolving around that point as it in turn
revolves around the sun.

Just as the gravity of the sun doesn't tear the earth apart (the
eastern and western hemispheres are tied together with gravity, just
like the earth and moon are), it doesn't tear the earth-moon system
apart.

If this isn't making sense, do you want me to look for some basic
orbital mechanics sites to point you to that might have helpful
drawings?

Hello haversian,

A useful web reference that may help you understand the issue of
gravitational forces can be found at

http://hyperphysics.phy-astr.gsu.edu/hbase/forces/isq.html#isqg

It all relates to the "inverse square law". 

regards

rcd-ga

You may be interested to know that the path of the Moon's orbit is
everywhere concave toward the Sun.  Sometimes when people draw it as a
sinusoid (more or less) superimposed upon the nearly circular orbit of
the Earth around the Sun, it is sketched as convex to the sun inside
the Earth's orbit and concave toward the Sun outside.  In fact, it is
nowhere convex inward.

So in that sense the moon does orbit the Sun with only a small
perturbation by the presence of the Earth.