Lightsaber physics

My roommate and I are watching The Empire Strikes Back. Remember the scene where Yoda sends Luke out into the wilderness, and he fights the apparition of Darth Vader? Luke chops the ghost-Vader's head off, and he falls to the ground, lightsaber still on and functioning. A number of questions arise.

Firstly, does the lightsaber have some sort of shutdown mechanism, by which it turns off once it leaves the user's hand? I vaguely recall scenes where the saber has been flung from some Jedi's hand, followed by a shot of an impotent handle clanking around on the ground. But in this case, Vader fell with the lightsaber still in his hand. Was it still on? Assuming it was, did it damage whatever part of his body it fell on, or did it start disintegrating the soil, or both?

For the sake of this post, let's assume that it stayed on (even after eventually leaving ghost-Vader's hand). Even if Vader did land on the lightsaber, it still would have come in contact with the soil, and my bet is that Dagobah's gravity would draw the lightsaber downward. Now, the question is, would the destructive power of the lightsaber's lightblade, combined with the force of gravity, draw said lightsaber inexorably downward, toward the planet's core (assuming, of course, that it wouldn't melt somewhere in the mantle)? I mean, gravity is a powerful force, but it doesn't pull us through the crust and into the bowels of the planet. This all comes down, I suppose, to what the nature of the lightsaber's power is. Does it automatically destroy everything in its path? Is there a limit to what it can reduce to atoms? If anyone is a physics major/Star Wars buff, please enlighten me.