Stardate 20030925.1248

(Captain's log): Sean writes:

I read with interest your post about the Gallileo mission and I had a question for you about orbital mechanics. I realize that you can use a planetary body to alter and accelerate the orbit of a satellite, but what I don't understand is HOW? if a satellite enters the gravity well of a planet and speeds up, doesn't it then have to exit the well, slowing it back down? my high school physics teacher off handedly said that it was caused by conservation of angular momentum, but he didn't have time right then to explain. now that I have been out of school longer than I was in it, I can't exactly go back to Los Angeles and ask Mr. Fisk again. Could you explain how it works?

Energy is always conserved.

(Actually, according to relativity, mass-energy is conserved, because mass and energy are the same thing. According to Quantum Mechanics, mass-energy is conserved in the long run but a certain amount of cheating is permitted, as long as it doesn't get out of hand. But those things don't affect Galilean Mechanics. [This message has been brought to you by the "Don't nitpick" division of USS Clueless.])

And forces always balance. As I sit here typing, I am pulled down into my chair by Earth's gravity, but at the same time Earth is being pulled towards me by my gravity.

Any object can be subjected to many forces at once in different directions with different magnitudes, which can therefore be represented as vectors. If the vector sum of the forces has a non-zero magnitude, then that object will accelerate in the direction of the vector sum, with the acceleration being proportional to the magnitude of the vector sum.

If I stand up and jump, I will move up and then down. The force from my legs makes me accelerate upwards, and then as my feet leave the ground gravity will make me accelerate downwards. While I'm pushing with my legs, the vector sum of forces points upward because my legs push harder than gravity, but after my feet leave the ground the vector sum of the forces points downward, because my legs exert no force and gravity predominates.

But the Earth moves in the opposite direction. When I jump, I'm moving my body, but I'm also shoving the Earth away, and the Earth moves when I do so. And just as I move upwards, stop, and then again move downwards, the Earth moves away from me, stops, and moves towards me again until we make contact and both stop moving (because my legs once again exert force).

Of course, the Earth doesn't move very far. The gravitational force in question is the same for both in absolute terms, but not proportionally. Earth applies about a kilonewton of force to me via gravity, and I apply about a kilonewton to the Earth in precisely the opposite direction. While my legs are shoving, they generate a force of several kilonewtons on both. (The standard unit of force in the MKS system is the newton, and of course it is named in honor of Isaac Newton. By definition, one newton of force causes a kilogram to accelerate one meter per second per second.)

The force on me and on the Earth varies during the period in which I'm jumping, but at any instant is the same for both (but in exactly opposite directions). But our masses are not even remotely the same, and though a force of that magnitude can cause considerable acceleration in a mass like me, it causes a lot less acceleration to something the size of the Earth, whose mass is more than 22 orders of magnitude larger. Theory tells us that the Earth moves when I jump, but the resulting motion is far too small to measure. (In fact, it will move a distance far smaller than the diameter of a hydrogen atom.)

Nonetheless, it does actually move.

Let's consider what happened when Voyager 2 passed Jupiter. There was gravitational force between them, too. And Voyager 2's orbital path took it behind Jupiter in Jupiter's orbit around the sun. That's critical.

Every object in the universe exerts gravitational force on every other object at all times. Voyager 2 is interacting gravitationally with Jupiter even now, but the magnitude of the force isn't significant. For purposes of this discussion, it was only really important during a period of a few weeks when Voyager 2 was closest to Jupiter.

There's the old saw about the irresistible force and the immovable object and what happens when the irresistible force is applied to the immovable object. (The question turns out to be nonsense. It's logically impossible for both to exist in the same universe, so it's logically impossible for them to ever meet. Therefore it makes no sense to discuss what would happen if they did.) In our universe it turns out that every force is irresistible and no object is immovable. Any object, no matter how massive, will respond to any force, no matter how small. The response may be miniscule, but it isn't zero.

During its closest approach, Voyager 2 and Jupiter were gravitationally attracted to one another. Because Voyager 2 was behind Jupiter, the force vector on Jupiter was in the direction opposite Jupiter's orbital motion. Jupiter responded to that force by losing orbital energy.

There's a curious thing about how orbits work which seems counterintuitive: all other things being equal, a lower energy orbit has a faster orbital velocity. Kinetic energy is proportional to the square of the velocity, so a