What's the math that powers rockets? How does it help us get them to space? And how do we use that math to put a satellite or person in orbit around the Earth? Let's find out.
The Mathematics of Getting to Space
When people think about going to space, they usually think about going up. And that’s certainly true, but it’s only part of the story. It’s sort of hard to define exactly where the atmosphere ends and outer space begins (since the atmosphere gradually falls off as you go up in altitude), but one popular choice is the so-called “Karman line” at a height of 100 km (or around 62 miles) above sea level. A lot of people are surprised to find that space begins only 100 km up … since that’s really not that far. But the problem with getting there is that it’s “uphill” the whole way, which means you have to fight gravity the whole way.
A rocket traveling at 8 km/s completes one orbit every 90 minutes.
But getting up that high is only half the battle of getting into orbit around the Earth. Because if you fly a spacecraft 100 km straight up and then turn off the engines, it will simply come right back down to the ground (this is called a sub-orbital flight). If your goal is to get a satellite into orbit around the Earth or to deliver a person to the International Space Station, the rocket doesn’t just need to get into space, it needs to stay there. And that means it needs to end up flying sideways really, really fast—around 8 km/s or almost 18,000 miles per hour!
How fast is that? Well, a rocket or satellite traveling at 8 km/s completes one orbit every 90 minutes. Which is amazingly fast considering it takes 5 hours to fly across the United States in an airplane. For comparison, a rocket in orbit crosses the US in about 10 minutes.
The Mathematics of Orbiting the Earth
But why does a rocket or satellite or space station need to be moving sideways so fast to stay in orbit? The answer is mainly geometry (and a healthy dose of physics). As you know, the Earth is roughly spherical. While it’s possible to go around the Earth (or anything else) in an elliptical orbit (which looks like a squashed circle), we’re going to think about the simple case of a perfectly circular orbit. If you think about it, you’ll see that a rocket going around the spherical Earth in a circular orbit some height above the ground will stay at that height above the ground the entire orbit. This is kind of obvious, but it really is the key to understanding the mathematics of being in orbit.
Orbits come down to geometry and traveling sideways really fast.
To see how this works, imagine standing at the edge of a tall cliff overlooking the ocean. If you drop a ball, the ball will fall straight down into the water. If you throw the ball with a bit of sideways speed, the ball will travel in a parabolic arc and land a bit further away from the cliff. Now imagine throwing the ball harder and harder with more sideways speed. Each increase in horizontal speed means the ball lands in the water farther from the cliff than before. If you throw the ball hard enough (and we’re talking really hard), something weird happens: the amount the ball falls towards the Earth is exactly matched by the amount the spherical Earth curves away from the ball. The net result is that the height of the ball above the water doesn't change—and it will just keep going and going.
Keep in mind that even though it doesn't hit the ground, the ball is actually falling towards the Earth the whole time—it simply never gets closer to the ground since its curved trajectory matches the curvature of the Earth. In other words, the ball is in orbit. As I said, orbits come down to geometry and traveling sideways really fast. Of course, you can’t actually get a ball into orbit by throwing it off a cliff like this since air molecules in Earth’s atmosphere will slow it down and eventually make it fall to the ground. Which is exactly why rockets also have to travel upwards into space before they can orbit the Earth.
The Rocket Equation
Now that we know what it means to get a satellite into orbit, let’s think about how we get it there. In other words, let’s think about what determines how big a rocket needs to be to lift a satellite into space and get it moving sideways fast enough to orbit the Earth. To begin with, let’s contemplate what we have to do to put a person (and their toothbrush) or a satellite into orbit. The answer is that we need to attach a rocket underneath this payload that has enough fuel and power to lift the required mass into orbit. But, the rocket we just attached to the payload also has some mass (mostly its fuel), which means we need another rocket under the first that has enough fuel and power to lift it. But, this second rocket we just attached also has some mass (again, mostly its fuel), so we once again need another rocket to lift it! And on, and on, and on. Even if a rocket's payload is small, it needs a lot of fuel to lift it … and it needs fuel to lift the fuel … and so on. As I said earlier, space is only 100 km away, but it’s 100 km straight up … which makes it hard to get to.
Space is only 100 km away, but it’s 100 km straight up … which makes it hard to get to.
There’s an equation that summarizes this whole situation and tells us roughly how much fuel is needed to lift a given amount of mass into orbit by a particular rocket. It’s called, logically, the rocket equation. We’re not going to go into all the details of this equation, but the gist is that it tells engineers how to calculate the speed gained by a rocket as it burns its fuel. In particular, the equation says that the speed increase is proportional to the logarithm of the initial mass of the rocket (including the rocket itself, the payload, and all of its fuel) divided by the final mass of the rocket (once all the fuel is burned). This ultimately tells us that adding more and more fuel to a rocket offers diminishing returns in terms of speed gained since, as we’ve seen, all of that fuel requires even more fuel. Which is exactly why rockets have to be such enormous, magnificent, and beautiful machines.
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