In brief
The Earth isn't a perfect sphere. It's actually a bit fat around the middle, like a squash. This 'oblateness' means that gravity isn't the same everywhere. When a satellite passes over the equator, the extra mass of the Earth pulls on it a little harder. Over hundreds of orbits, these tiny tugs add up. If you don't account for them in your math, your satellite will be miles away from where you thought it was. Then there is the Moon. Even though it's far away, its gravity is strong enough to pull satellites out of their lanes. It's a constant tug-of-war between the Earth, the Moon, and the satellite.
The atmosphere is a moving target
The biggest challenge is the air itself. Even a few hundred miles up, there are still a few stray molecules of gas. When a satellite hits them, it slows down just a tiny bit. This is called atmospheric drag. The problem is that the atmosphere isn't still. It grows and shrinks depending on what the Sun is doing. When the Sun is active, it heats up the upper atmosphere, causing it to puff out. This means there is more 'wind' for the satellite to fly through, which slows it down faster. Scientists use something called the NRLMSISE-00 model to try and guess how thick the air will be on any given day. It's basically a weather report for the edge of space.
Xenon: The fuel of the future
To fight this drag and stay in orbit, many modern satellites use ion thrusters filled with xenon gas. These engines are great because they can fire for a long time without using much fuel. This is vital for debris remediation satellites—the ones sent up specifically to move old junk. These satellites have to do a lot of maneuvering. They have to match the speed of the junk, grab it, and then push it down. Every time they fire their engine, it changes their 'orbital elements.' This is just a list of numbers that describes their orbit. After every burn, they have to recalculate everything to make sure they are still on the right path.
Why it is so hard to be right
Predicting a re-entry window is like trying to guess where a leaf will land after falling from a tree in a hurricane. You might know the general area, but the exact spot is a mystery until the very end. This is why you often hear news reports saying a satellite will fall 'somewhere between Friday and Sunday.' As the satellite gets lower, the drag gets stronger, and the math gets harder. The engineers have to constantly refine their models to account for 'non-conservative forces.' These are things that take energy away from the satellite, like the friction of the air.
Doesn't it make you feel small to realize how much the Sun controls what happens in our own sky?
The goal of the game
The reason we put so much work into this is safety. We want to make sure that when a rocket stage or an old payload falls, it doesn't land on someone's house. By using accurate ephemerides, we can predict a safe 're-entry window.' This lets us steer the satellite toward a remote part of the ocean. It also helps avoid collisions while the satellite is still in orbit. If we know exactly where two things are going to be, we can make sure they don't try to occupy the same space at the same time. It's a high-stakes game of orbital chess that never ends.
So, the next time you see a 'shooting star,' remember it might just be a very well-calculated piece of math finally coming home.