from
https://www.wired.com/story/what-happens-if-a-space-elevator-breaks/

What Happens If a Space Elevator Breaks

These structures are a sci-fi solution to the problem of getting objects
into orbit without a rocket—but you don’t want to be under one if the
cable snaps.

space elevator in foundation
PHOTOGRAPH: TCD/PROD.DB/APPLE TV+/ALAMY

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.

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.

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.)

Here's what it would look like:

PLAY/PAUSE BUTTON
VIDEO: RHETT ALLAIN

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.

Rhett Allain is an associate professor of physics at Southeastern
Louisiana University. He enjoys teaching and talking about physics.
Sometimes he takes things apart and can't put them back together.
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