Let's imagine, you've been riding along in
your spaceship and you just need somewhere to park it because you're going out of town. Where do you put it? You actually don't have that many options. Most places you leave it, it's going to start
drifting away. You come back in a few years and it's a million
miles away. But there are a few excellent parking spots
for your ship. These are called the Lagrange points. The Lagrange points are places where the forces
acting on an object are perfectly balanced. Assume we have an object that orbits the Sun
along with the earth. We're going to use a rotating coordinate system. This rotates at the same speed as the Earth
moves. So in this coordinate system, the Earth doesn't
move. Now let's look at all the forces that are
at work here. To account for the fact that this coordinate
system is rotating, we have to add in a centrifugal force that pushes out away from the Sun. Then we add in the gravitational force of
the Sun, and the gravitational force of the Earth. This graph shows us the effective potential
energy and it tells us how objects will move. An object whose orbit starts here will slowly
drift towards the Sun. An object starting here will slowly drift
towards the Earth. And an object starting here will slowly drift
away into the outer solar system. But an object won't move, if the surface is
flat. The surface is flat at five points called
the Lagrange points. The first three points were discovered by
Euler. The remaining two were discovered by Lagrange. The last two Lagrange points are stable. This means that if an object is slightly off
from the point, its orbit will shift slightly but not too much. The first three points are unstable. This means that if an object is just slightly
off from the point, it will just move farther and farther away. Since orbits near the L4 and L5 points are
stable, objects naturally go there and stay there. All of the major planets have their own Lagrange
points. And the most massive planet Jupiter has thousands
of asteroids clustered around its L4 and L5 points. These are called the trojans. By convention, these asteroids are called
after heroes of the Trojan war. The L4 asteroids are named after the Greek
warriors: Achilles, Odysseus, Agamemnon. The L5 asteroids are named the Trojan side:
Priam, Paris, etc. NASA has been parking spaceships at Lagrange
points for years. There's even room for your spaceship, as long
as you can fly it there. For more astronomical videos, please click
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