In the first episode of the Foundation series on Apple TV, we see a terrorist try to destroy the space elevator used by the Galactic Empire. This seems like a great chance to talk about the physics of space elevators and to consider what would happen if one exploded. (Hint: It wouldn't be good.)
People like to put stuff beyond the Earth's atmosphere: It allows us to have weather satellites, a space station, GPS satellites, and even the James Webb Space Telescope. But right now, our only option for getting stuff into space is to strap it to a controlled chemical explosion that we usually call "a rocket."
Don't get me wrong, rockets are cool, but they are also expensive and inefficient. Let's consider what it takes to get a 1-kilogram object into low Earth orbit (LEO). This is around 400 kilometers above the surface of the Earth, about where the International Space Station is. In order to get this object into orbit, you need to accomplish two things. First, you need to lift it up 400 kilometers. But if you only increased the object’s altitude, it wouldn't be in space for long. It would just fall back to Earth. So, second, in order to keep this thing in LEO, it has to move—really fast.
Just a quick refresher on energy: It turns out that the amount of energy we put into a system (we call it work) is equal to the change in energy in that system. We can mathematically model different types of energy. Kinetic energy is the energy an object has due to its velocity. So if you increase an object’s velocity, it will increase in kinetic energy. Gravitational potential energy depends on the distance between the object and the Earth. This means that increasing an object’s altitude increases the gravitational potential energy.
So let's say you want to use a rocket to increase the object’s gravitational potential energy (to raise it to the right altitude) and also increase its kinetic energy (to get it up to speed). Getting into orbit is more about speed than height. Only 11 percent of the energy would be in the gravitational potential energy. The rest would be kinetic.
The total energy to get just that 1-kilogram object into orbit would be about 33 million joules. For comparison, if you pick up a textbook from the floor and put it on a table, that takes about 10 joules. It would take a lot more energy to get into orbit.
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But the problem is actually even more difficult than that. With chemical rockets, they don't just need energy to get that 1-kilogram object into orbit—the rockets also need to carry their fuel for the journey to LEO. Until they burn this fuel, it's essentially just extra mass for the payload, which means they need to launch with even more fuel. For many real-life rockets, up to 85 percent of the total mass can just be fuel. That's super inefficient.
So what if, instead of launching atop a chemical rocket, your object could just ride up on a cable that reaches all the way into space? That's what would happen with a space elevator.
Space Elevator Basics
Suppose you built a giant tower that is 400 kilometers tall. You could ride an elevator up to the top and then you would be in space. Simple, right? No, actually it's not.
First, you couldn't easily build a structure like this out of steel; the weight would likely compress and collapse the lower parts of the tower. Also, it would require massive amounts of material.
But that's not the biggest problem—there's still the issue with speed. (Remember, you need to move really fast to get into orbit.) If you were standing on the top of a 400-kilometer tower with the base somewhere on the Earth's equator, you would indeed be moving, because the planet is rotating—this is just like the motion of a person on the outside of a spinning merry-go-round. Since the Earth rotates about once a day (there's a difference between sidereal and synodic rotations), it has an angular velocity of 7.29 x 10-5 radians per second.
Angular velocity is different than linear velocity. It’s a measure of rotational speed instead of what we normally think of as velocity—movement in a straight line. (Radians are a unit of measurement to use with rotations, instead of degrees.)
If two people are standing on a merry-go-round as it spins, they will both have the same angular velocity. (Let's say it's 1 radian per second.) However, the person that is farther from the center of rotation will be moving faster. Let's say one person is 1 meter from the center and the other person is 3 meters from the center. Their speeds will be 1 m/s and 3 m/s respectively. This same thing works with a rotating Earth. It's possible to get far enough away such that the Earth's rotation gives you the required orbital velocity to stay in orbit around the planet.
So let’s go back to our example of a person standing on the top of a 400-kilometer tower. Are they far enough away from Earth that they can stay in orbit? For one complete rotation of the Earth, their angular velocity would be 2π radians per day. That might not seem very fast, but at the equator this rotation gives you a speed of 465 meters per second. That's over 1,000 miles per hour. However, it's still not enough. The orbital velocity (the velocity needed to stay in orbit) at that altitude is 7.7 kilometers per second, or over 17,000 miles per hour.
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Actually, there's another factor: As you increase your distance from the Earth, the orbital velocity also decreases. If you go from an altitude of 400 to 800 kilometers above the surface of the Earth, the orbital speed decreases from 7.7 km/s to 7.5 km/s. That doesn't seem like a large difference, but remember, it's really the orbital radius that matters and not just the height above the surface of the Earth. Theoretically, you could build a magical tower that was high enough that you could just step off of it and be in orbit—but it would have to be 36,000 kilometers tall. That’s not going to happen.
Here is something that's very cool and more practical: An orbit at the altitude of 36,000 kilometers has a special name. It's called a geosynchronous orbit, meaning that the time it takes an object to complete one orbit is exactly the same time it takes the Earth to rotate. If you put this object in an orbit directly over the equator, it will appear in the same location in the sky relative to the surface of the Earth. (Then it's called a geostationary orbit.) That's useful, because you know exactly where to find it. A geostationary orbit makes it easier to communicate with objects like TV or weather satellites, or for satellite cameras that need to stay focused on the same part of the Earth.
OK, back to the space elevator. If we can’t build a tower from the ground up, we can hang a 36,000-kilometer cable from an object that’s in a geostationary orbit. Boom: That's the space elevator.
To get this to work, you would need a large mass in orbit—either a space station or a small asteroid. The mass has to be large so that it doesn't get pulled out of orbit every time something climbs up the cable.
But perhaps now you can see the problem with a space elevator. Who wants to make a 36,000-kilometer-long cable? For a cable that long, even the strongest material, like kevlar, would have to be super thick to prevent it from breaking. Of course, thicker cables means more weight hanging down below, and that means the higher parts of the cable have to be even thicker to support the cable below. It's a compounding problem that seems essentially impossible. The only hope for the future of space elevator construction is to figure out how to use some super strong and lightweight material like carbon nanotubes. Perhaps we will make this work someday, but that day is not today.
What About a Falling Elevator Cable?
In the first episode of Foundation, some people decide to set off explosives that separate the space elevator’s top station from the rest of the cable. The cable falls to the surface of the planet and does some real damage down there.
What would a falling space elevator cable look like in real life? It's not that simple to model, but we can make a rough guess. Let's model the cable as being made up of 100 individual pieces. Each piece starts in a motion around the Earth, but with the same angular velocity as the Earth. (So, not in orbit.) In an actual space elevator cable, there would be some tension forces between pieces. But just for simplicity, in the model each piece will only have the gravitational force from the interaction with the Earth. Now I can just model the motion of these individual 100 parts of the cable to see what happens. (It's actually not too difficult to do this with some code in Python—but I'll skip all of that.)
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Here's what it would look like:
So, what's going on? Notice that the lower part of the cable just falls to Earth and probably causes some severe destruction. In this model, it wraps about a third of the way around the equator, even though its full length would almost make it all the way around the Earth, which has a circumference of 40,000 kilometers.
But some of the parts of the cable might not even hit the surface. If the pieces start high enough, their velocity will increase as they get closer to the surface. It's possible that the pieces will speed up enough to put them in a non-circular orbit around the Earth. If you are living on the equator, that's a good thing. Better to have that debris in space than falling on your head, right?
Of course, if the cable is still intact, then each piece would be pulling on other nearby pieces. This would cause more of the cable to crash into Earth. But at some point, the forces in the cable would become so strong that it would just break apart. You would still end up with space debris.
So not only is building a space elevator very difficult, but you really don’t want the cable to snap and fall. Maybe it’s a good thing that we are still in the rocket phase of space exploration.
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