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Re: [Phys-L] our second, at least, moon

This story involves some important basic physics, which the
popular press hasn't covered very well.

To a first approximation, the earth by itself cannot capture
anything into an orbit. That's because of conservation of
energy and momentum.

A lot of introductory science books -- and even science
/standards/ -- say that
balls spontaneously roll downhill, and more generally
things spontaneously settle into low-energy states,
which are stable ...
but this is just not true. Energy considerations tell us
that a ball will roll down one hill and then roll up the
other hill on the far side of the valley. If you want
it to roll down and /stay/ down, you need some form of
dissipation. All questions of spontaneity, stability,
and reversibility depend on *entropy* not energy. In
/some/ cases decreasing energy can be used as a proxy
for increasing entropy, but this is not a general rule.

Considering the earth by itself, the gravitational field
is a static central force. An object that is not already
in a closed orbit will follow a hyperbolic trajectory.
It will fall into the earth's potential well, whiz past,
and then immediate climb up the well and fly away.
There are theorems that say a static potential gives
rise to a force of constraint, which means you can't
do work against it.

This gets more complicated and more interesting because
the earth already has a good-sized moon. The earth/moon
system is essentially a double planet. This is a big
deal, because we now have a /moving/ potential. You
can't do work against a static potential, but you
certainly can against a moving potential. Water is
held in a ladle by forces of constraint, but you can
ladle water from a lower reservoir to a higher one,
doing work against the water, using a time-varying

In astronautics this is sometimes called the slingshot
effect, although I don't find that to be a particularly
descriptive name. You could equally well call it the
tennis-racket effect or a hundred other things.

I haven't looked at the details, but there is basically
only one way it could have happened: The newcomer fell
into the moon's gravitational potential, but when it
was ready to climb back up and fly away, the potential
was different, because the moon had moved. So the
newcomer lost energy during the encounter.

The window for pulling off this feat is verrry small.

The resulting orbit is not stable. It is only a matter
of time before the reverse process occurs and the
object gains enough energy to fly away, never to be
seen again.

Interplanetary gas and dust provides "some" friction,
which in principle "could" lead to permanent capture,
but that is too weak to be relevant on the timescales
we are considering.

Students find this a bit counterintuitive, because their
intuitions are mostly based on experience with systems
that have lots of friction ... and based on static (not
dynamic) gravitational fields.


Here's a somewhat-practical example that involves some
of the same principles: The carnival "swinger game".

The game is essentially unwinnable. Physics says if
the ball misses the pin outbound, it will miss again
inbound. If there were a lot more friction, or if
the pivot supporting the pendulum were moving, then
you would have a chance.

Tossing the ball (so the supporting chain goes
slack) is not allowed.