Where does the energy come from?

Since the moon is more stretched at the point of perigee, I would think that is where the shape of its orbit is affected the most. The stretching will cause internal heat of course, but that would also tend to make the point on the moon's surface closest to the planet want to speed up, and the farther point to slow down (being different distances from the center). The moon itself is roughly circular, so the two points are proportionally distanced from the center. The gravitational force from the planet is non-proportional, however, due to the inverse square law. So gravity being unproportionally stronger on the near side of the moon, I would think it would make the moon as a whole want to speed up more than it would want to slow down. So as a result the moon speeds up at perigee, matching the energy imparted as internal heat due to friction (assuming that energy is conserved). So the perigee will be pushed away from the planet until it eventually matches the apogee, and the orbit is circular. How's that for a thought experiment? Of course I'm just speculating.

Energy is stored in the moon's being in an orbit around the planet. You can see this by imagining an asteroid being captured by a planet to become a moon of the planet. The asteroid comes into the planet's gravity well and its motion relative to the planet adds a lot of energy to the new system of planet and moon. This energy is stored in their revolution around each other.

The tidal forces stretching and shrinking decrease the system's (planet and moon) angular momentum, and hence decreasing the energy stored by the revolution.

Whether the moon's orbit will become more circular or whether it will just become a smaller ellipse is an interesting question but in either case, energy will be converted from the angular momentum of the orbiting moon into heat. I think if more heat is generated in the planet than in the moon, then the planet will slow down and the moon will orbit farther away.

Obviously the two objects are being stretched from a gravitational force. The motion is elliptical because the moon wants to move in a straight line, but it gets pulled back by the planet. Our moon falls back about a half inch each year.

However the heat generated is friction when the objects stretch and particles rub together. The further the two objects, the less the pull, hence less friction.

1/x^2, where x is the distance between objects relates to the gravitational pull the objects have on each other.

friction is the source of the heat as the moon gets closer to the planet the planets gravity is trying to pull the moon apart pulling it into an 'A' semitrailer shape then as it moves away its own gravity pulls it back into a sphere for an example look at Io Jupiter's closest moon. now depending on the moons composition it would either be ripped apart or it would move into a circular Orbit

"What is the energy source that is winding down?"

I don't understand

as far as I know the moon is growing further and further away. there must have been enough potential kinetic energy from its velocity and the impact that created our moon to continually force it away from us.

http://www.youtube.com/watch?v=ibV4MdN5wo0&#t=0m18...

there might even be enough energy in the tides to accelerate this by warping and elongating the orbit of the moon with imbalanced gravity (tides)

other than that I'd figure the energy would be absorbed as heat within the moons core. It does have a very minor magnetic field.

http://upload.wikimedia.org/wikipedia/commons/3/37...

notice how the field is now split between far side and near side instead of north and south.

still kinda confused as to where you're headed with this.

Gravitational potential energy. Yes, gravity is the source of that energy, and it's not eternal because it's decreased every time it's converted into other forms of energy.

That energy represents lost kinetic energy in the moon's motion.

It will slow in it's rotation until it is, (like our moon), 'tidally locked',

and get closer to it's primary.

friction of the moon caused by the stretching and shrinking, not an expenditure of energy

carbohydrates