In the first episode of the Founding series on Apple TV, terrorists destroy the space elevator of the galactic empire. A great opportunity to talk about space elevators and figure out what could happen if such an elevator suddenly exploded (spoiler: not good).
People like to bring things out of the Earth’s atmosphere. Because of this, we have weather satellites, GPS satellites, and the telescope. James Webb. So far, however, the only option we have for launching something into space is to tie it to a device that provides controlled chemical explosions (we usually call it a “rocket”).
Don’t get me wrong, rockets are cool. However, it is also expensive and inefficient. Let’s figure out what it takes to lift a kilogram object into low Earth orbit. This is about 400 km above the surface of the planet – about the same place is the International Space Station. To do this, first, raise the object to a height of 400 km. However, if you just lift it up there, it won’t last long there. It will just fall back to Earth. Therefore, secondly, he will have to move quite quickly to stay in orbit.
Let me briefly remind you that in order to change the energy of a system by a certain amount, it is necessary to pour exactly that much energy into it – in physics this is called “work”. We can mathematically model different types of energy. Objects have kinetic energy due to their speed. If you increase the speed of an object, its kinetic energy will increase. Gravitational potential energy depends on the distance between the object and the Earth. By increasing the height of an object, we increase its gravitational potential energy.
Let’s say you want to use a rocket to increase an object’s gravitational potential energy (to get it to the right height) and also to increase its kinetic energy (to get it to the right speed). For this, speed will be more important than height – only 11% of the spent energy will turn into potential, and the rest will be kinetic.
The total energy that will need to be expended to put a kilogram object into orbit will be about 33 million J. For comparison, it takes about 10 J to lift a textbook from the floor and put it on the table. Much more energy is needed to enter orbit.
But in reality the problem is even more complicated. Chemical rockets need energy not only to put an object into orbit, but also to bring their own weight and the weight of fuel. And until the fuel burns out, it is just extra mass, which requires even more fuel to start. In real rockets, 85% of the mass can be fuel alone. This is completely inefficient.
What if, instead of launching an object on a chemical rocket, it could simply be lifted up on a cable that goes straight into space? This would be possible with a space elevator.
Space Elevator Basics
Let’s say we build a giant tower 400 km high. You can just take the elevator to the top floor and go into space. Just? Not really.
Firstly, it is impossible to build such a structure out of steel – its weight will simply crush its lower parts. In addition, the material will require too much.
But this is not the biggest problem either. Recall that high speed is required to be in orbit. If you were to stand on top of a 400-kilometer tower whose foundation rested on the equator, you would move as the planet rotates. It is similar to the movement of a person who is on the edge of a carousel. Since the Earth rotates in about one day (see “sidereal time” and “solar day”), its angular velocity is about 7.29 × 10-5 radians per second.
Angular velocity measures the speed of rotation, not movement in a straight line. If two people stand side by side on the edge of the carousel, they will have the same angular velocity. Let’s say it’s radians per second. However, if one of them is further from the center than the other, it will move faster. Let’s say one of them stands a meter from the center, and the other three meters. Then the first will move at a speed of 1 m/s, and the other – 3 m/s. The same with the rotation of the Earth – you can move far enough from the surface so that the rotation of the Earth gives you the necessary orbital speed.
Let’s go back to the man standing on top of the 400-kilometer tower. Is it far enough from the planet’s surface to remain in orbit? Given the rotation of the Earth, its angular velocity will be 2π radians per day. It may not seem like much – but at the equator, such a rotation leads to a speed of 465 m / s. This is 1674 km/h. But this is still not enough. The orbital speed required to avoid falling back at that altitude would be 7.7 km/s, or 27,720 km/h.
There is one more factor. As you move away from the Earth, the orbital speed decreases. As the distance from the surface increases from 400 to 800 km, the orbital velocity will drop from 7.7 km/s to 7.5 km/s. It seems to be a little – but you need to remember that the orbital radius matters, and not just the distance from the Earth’s surface. Theoretically, you could build a magical tower high enough that you could reach orbit from its roof – but its height should be 36,000 km. So you can forget about it.
However, there is more interesting and practical information. An orbit with a height of 36,000 km has a special name – geosynchronous. One revolution in such an orbit takes as much time as the Earth takes one revolution. If you place an object in this orbit above the equator, it will be above the same point on the Earth’s surface (then called the geostationary orbit). This is useful – you know where to look for this object. The geostationary orbit makes it easy to communicate with such objects, whether they are video satellites, weather satellites, or satellite cameras that need to keep track of the same point on the surface.
Let’s go back to the space elevator. If we can’t build a tower on the ground, we can hang a 36,000 km cable from an object in geostationary orbit. Op-la, and the space elevator is ready.
To do this, you need a large mass in orbit – a space station or a small asteroid. The mass must be large so that it is not pulled out of orbit by objects climbing up the cable.
Perhaps the problem with this design is obvious. Who needs a 36,000 km cable? A cable of this length, even made of the strongest material such as Kevlar, will have to be made too thick so that it does not break. And the thicker the cable, the more its lower part will weigh, and the thicker its upper part should be. This task seems to be in principle unsolvable. There is only one hope left – to find in the future some very light and strong material like carbon nanotubes. Perhaps we can build something similar, but not today.
What about a falling space cable?
In the first episode of “Foundations”, a group of people decide to set off explosions that separate the upper station of the space elevator from the cable. The cable falls to the surface of the planet, and causes serious damage.
What would a falling space cable look like in reality? Modeling this is not easy, but you can roughly guess. Let’s build a cable model consisting of 100 separate parts. Each of them at the first moment of time moves around the Earth with the same angular velocity as the planet. In a real cable, tension forces will act. For simplicity, we will model a cable, parts of which are only subject to the Earth’s gravity. Therefore, we can simply build motion models for each of the 100 parts.
Here’s what it will look like:
What’s going on here? Note that the lower part of the cable just falls to the ground, probably causing serious damage to it. In the model, the cable wraps around a third of the equator, but a full-fledged cable would wrap almost entirely around the Earth, since its equator is 40,000 km long.
Probably, some parts of the cable will not reach the Earth. They will start falling from a great height, and their speed will increase as they approach the surface. It is likely that these parts will simply accelerate strongly enough to enter a non-circular orbit around the Earth. For the inhabitants of the equator, this would be the best option – let this cable become space debris than fall on their heads.
Of course, some parts of the cable will pull on others, as a result, more and more cable will fall to the Earth, and at some point these forces will be large enough to break the cable.
It turns out that a space elevator is difficult to build, and this cable cannot be allowed to break and fall to Earth. Maybe it’s good that we are still in the rocket phase of space exploration.