Jacobi integral

Imagine you're playing with a spinning top. Have you noticed that as it spins faster, it becomes harder to knock it over? That's because it's spinning so fast that it stays upright, even though you're trying to tip it over.

The Jacobi integral is like a special trick that tells us whether objects (like planets or stars) will stay in a spinning orbit, or whether they'll eventually fly off into space. It's like a spinning top's "secret code" for staying upright.

The Jacobi integral is a measure of how much energy an object has in a spinning orbit. If this energy is too high, the object will escape from its orbit and fly away. If the energy is too low, the object will spiral inward towards the center of gravity and eventually crash into the object it was orbiting. But if the energy is just right, the object will stay in a stable, circular orbit without spiraling out of control or crashing.

So, scientists use the Jacobi integral to figure out how objects in space will interact with each other, and whether they'll stay in a stable orbit or not. It's a bit like a "magic formula" that helps us understand the way things in space move and behave.