How did they find the secret space lab in Captain Marvel?

OK, not everything is perfect with a piece of state space. Imagine that you wanted to trace the motion of a hidden laboratory orbiting the Earth. What would it look like? Honestly, it wouldn’t be that easy. In the example of the oscillating spring, it is in one dimension. This means that there is only one value for the position (the x value) and one value for the speed (the x speed). But real life is in 3D. The actual position would be a 3D vector (with three values: x, y and z). Also, the velocity would be a 3D vector with components in the x, y, and z directions. That’s six values. You would need six coordinates to completely plot the state space for an orbiting object. Good luck trying to draw a 6D object.

Even if you assumed that an object had a flat orbit in the xy plane, it would still be two coordinates for position and two for speed – a 4D plot. Oh but I’ll make one for you anyway. One way to make this work is to create two state space plots – one for the x vs. v_X and one for y against v_Yes. I didn’t want it to be boring, so it’s for a non-circular orbit around the Earth.

Of course, you are still waiting for the answer to the all-important question: would this work? Could you give the state vectors for an orbiting space lab and then find it six years later? May be.

Let’s say you know the EXACT position and speed of an object at an EXACT time. If you know all the forces acting on this object, then yes – you can use the initial conditions (position and speed) and find the position and speed at any time in the future. But what if you don’t know all the forces and interactions? If there are forces that you don’t take into account (like air drag), the speed and position will change from what you expect. Even a small interaction can make a big difference on a six-year timescale.

In fact, objects like the International Space Station have very small interactions with the Earth’s atmosphere. Even at an altitude of 400 km (like the ISS) there is a tiny bit of air. As the space station moves through this tiny amount of air, there is a pushing force backward that decreases speed (and spoils your state space plot). Wait! It’s even worse. The amount of air drag on the space station changes over time as the atmosphere expands and contracts with changes in weather. It is therefore almost impossible to know how the state vectors of an orbiting object will change over time.