Minimize the Cost

Everyone wants to save a buck. In this chapter and the next, I diverge from the ordinary-life lesson and talk about how rocket scientists try to save time and money.

When a shuttle astronaut drinks a sixteen-ounce bottle of water it costs about $10,000.

Why is it so expensive? There are two reasons. The first is the poor design of the shuttle itself, which has unfortunately made space travel more expensive (not to mention more dangerous) than it has to be. (More on this point later.)

The second reason has to do with the laws of orbital mechanics. To get that bottle of water into orbit (along with the astronauts, their life-support system, and everything else), it has to be accelerated to a speed of five miles per second. After the rocket engine burns out, the bottle will be in free fall—it will fall toward Earth, but it is moving so fast that Earth's surface curves away from the bottle at the same rate that the bottle falls toward the ground. These two effects, the falling of the bottle and the curving of Earth's surface, cancel out at five miles per second. To be placed in a circular orbit, the bottle must be moving parallel to Earth's surface at this incredible speed.

To make this concept of circular orbit clearer, let's do a thought experiment. Imagine driving your car off a cliff over the Grand Canyon. (And please remember that this is only a thought experiment.) If you drive at sixty miles per hour, your car will continue at that speed after your wheels leave the road and you start falling into the canyon. (We are neglecting the effect of air drag, which makes the problem a bit more complicated but doesn't change the basic concept.) How long it takes you to hit the ground does not depend on your speed (of sixty mph) but only on the constant acceleration (i.e., the pull) of gravity and your height above the canyon floor. If you drove at 120 mph, you'd still hit the ground at the same time, but you'd be twice as far down range. The faster you go, the farther down range you will travel, always hitting the ground at the same time. This analysis starts to break down when you travel at very high speeds, because you cover so much down-range distance that the curvature of Earth starts to matter. At five miles per second, the surface of Earth curves downward at exactly the same rate that you accelerate downward from the pull of gravity. Your car never hits the ground because the ground is falling below you due to the curvature of our planet. While you and your car fall together, you feel the sensation of falling. Because your car falls with you, you seem to be floating inside the car. You feel you will crash into the ground, which is exactly how astronauts in orbit feel: You are experiencing weightlessness. (It takes even astronauts a while to learn to ignore the sensation that they are falling to their deaths. After a few days, the vomiting stops and it can become a pleasant experience.) In this thought experiment, we have ignored the effect of drag, which would slow you down and cause your orbit to decay. To circumvent the orbit decay problem, spacecraft have to be at least one hundred miles above Earth's surface where the atmospheric drag is negligible.

So weightlessness is not the absence of gravity at all—weightlessness is falling under the pull of gravity. If you want the sensation to last indefinitely (and who doesn't?), then you have to be traveling at the fantastically fast speed of five miles per second. (This is technically known as the "circular speed" for low-Earth orbit.) At this speed, a spacecraft will zip past the entire United States in under eight minutes; in ten minutes it will cross the Atlantic Ocean.

To get the spacecraft into orbit, it is placed in the payload bay of a launch vehicle such as a Delta, an Atlas, or the shuttle. The launch vehicle is ignited and launched straight up into the air. As it ascends vertically to greater and greater heights, it starts to tilt

Chapter 35 Minimize the Cost over to one side. This tipping is intentional. The rocket continues to tip while gaining altitude until it is no longer traveling vertically, but instead is traveling horizontally. When the rocket reaches a height of one hundred miles it achieves a speed of five miles per second and is traveling parallel to Earth's surface. The engine cuts off, and the astronauts experience falling without ever hitting the ground (i.e., weightlessness).

The tipping rate of the launch vehicle is determined by a control scheme called "the steering law." If the tipping occurs too fast, it will cost more pounds of propellant to get into orbit. If it occurs too slowly, it will also cost more.

Rocket scientists use a type of mathematics called "calculus of variations" to find the cheapest way to get into orbit. According to the theory, the steering law that takes the minimum time to reach orbit is best because the rocket burns the least amount of propellant.

When rocket scientists apply this technique to the shuttle, the result is that it costs $10,000 to give an astronaut a drink of water!

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