What Happens If a Space Elevator Breaks

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.

www.wired.com

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