How Spacecraft Move in Orbit

BEFORE getting to the business of discussing the orbital aspects of modern spacecraft missions later in this chapter, there are a few fundamentals about the orbital motion of a spacecraft that we need to discuss, and a few popular misconceptions about it that need to be put to rest.

The first of these fundamentals is how a spacecraft remains in orbit around Earth, effectively forever, without having to fire rockets to sustain the motion. The answer lies in understanding that the spacecraft, like a stone falling down a deep well, is in a state of continual free-fall. Clearly, we do not expect the stone's motion to be assisted by rockets; it just falls unaided in the gravity field until it impacts the water at the bottom of the well. Free-fall in a gravity field is also the key to understanding the spacecraft's motion, although in this case it is perhaps not so apparent. And, of course, the spacecraft operator hopes that, in the process, it does not impact the ground like the stone!

To help with this discussion, we turn to a device that has become known as Newton's cannon, after its originator Isaac Newton. He first introduced the idea around 1680 in A Treatise of the System of the World, which he wrote as a popularization of his great work the Principia (see Chapter 1). Newton produced a diagram of his cannon in his treatise similar to that in Figure 2.1. To start with we have to imagine an impossibly high mountain, let's say 200 km (124 miles) high, for the sake of argument—a real challenge to the climbing fraternity. Not only is it a long way to the top, but when you get there you are effectively in the vacuum of space. Then you have to envisage dragging all the materials necessary to the summit to build a large cannon there that is capable of firing projectiles at a range of barrel speeds. This is also illustrated rather unimaginatively in Figure 2.1.

The cannon crew, presumably all dressed in space suits, now begins the serious business of firing cannonballs at the unsuspecting population below. You can see that if the crew fires a cannonball out of the gun with a barrel speed of, say, 2 km/sec (1.24 miles/sec), then it will do as you expect it to -

G. Swinerd, How Spacecraft Fly: Spaceflight Without Formulae, DOI: 10.1007/978-0-387-76572-3_2, © Praxis Publishing, Ltd. 2008

Newton's cannon

Newton's cannon

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Figure 2.1: Newton's cannon—a "thought experiment" devised by Isaac Newton to explain the nature of orbital motion.

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Figure 2.1: Newton's cannon—a "thought experiment" devised by Isaac Newton to explain the nature of orbital motion.

that is, take a curved path and impact the ground some distance away at point A (Figure 2.1). If the barrel speed is then ramped up to, say, 6 km/sec (3.73 miles/sec), the cannonball becomes intercontinental and travels a whole lot further to point B before impacting Earth's surface. However, something really interesting happens when the crew further increase the barrel speed to around 8 km/sec (5 miles/sec). Now the cannonball's path curves toward Earth's surface once again, but the curvature of the trajectory is matched by the curvature of Earth, and the ball continues to fall toward Earth without actually making contact! The projectile has now entered a circular orbit around Earth (Figure 2.1), and the cannon crew had better watch out as the ball will whizz past the gun about 90 minutes later, after making a complete orbit of Earth. Like the stone, the ball is now in a state of free-fall, and will continue to orbit Earth indefinitely.

To reinforce this idea, we can look at the same situation but consider Earth's curvature. At the mountain summit, the curvature is such that Earth's surface falls away below a truly flat horizontal plane by about 5 m (16 ft) in approximately every 8 km (5 miles) traveled over the ground. You may also recall that 5 m is roughly the distance fallen by Newton's apple in 1 second (Chapter 1). If you fire a cannonball from the gun at an initially horizontal speed of around 8 km/sec, it too will fall 5 m in the first second of flight, thus matching Earth's curvature. Therefore, no ground impact results, and you have orbital motion. To be more precise, for the cannonball to enter a circular orbit, it must be fired at a speed of 7.78 km/sec (4.84 miles/sec) from the summit cannon. For those of you who like more familiar units, this is about 28,000 km per hour (17,400 mph), which is a typical speed for a space shuttle in low orbit.

Newton's other orbital trajectories (Chapter 1) can also be produced using the summit cannon. For example, if we further increase the barrel speed to around, say, 9 km/sec (5.59 miles/sec), this has the effect of raising the height of the ball's trajectory on the other side of the globe, producing an elliptical trajectory (Figure 2.1). Since this is a closed trajectory, the cannonball will come back to haunt the cannon crew, about 23/4 hours after the projectile is fired, in this case. Note that the ball always returns to the low point on the orbit, the summit cannon, which is referred to as the orbit perigee. The high point, on the other side of Earth from the mountain, is called the orbit apogee. These are rather strange terms, but as the topic of orbit dynamics has been with us for so many years, a lot of wonderful terminology has come to us from history, as we will see later. Getting back to our cannon, further ramping up the barrel speed will result in higher and higher apogees, giving more and more elongated ellipses. Eventually, the apogee height will effectively reach infinity, an extremely long way away, and then the trajectory becomes an open parabola (Figure 2.1).

If you ask the cannon crew to check the barrel speed, the crew members will tell you that the parabolic trajectory occurred at around 11 km/sec (6.84 miles/sec). If you also recall the discussion about the parabola in Chapter 1, it is the trajectory that results in escape from Earth's gravity with the minimum energy given to the cannonball. The ball flashes out of the cannon at huge speed, but this energy is consumed by the gravity field as it climbs away from Earth, and when it reaches infinity it effectively has no energy left to go anywhere, that is, it has zero speed. Any further increase in the barrel speed of the cannon will result in the ball's trajectory being a hyperbola (Figure 2.1). If you recall, this gives the ball sufficient energy to escape Earth's gravity, with some left over to give it a constant speed once it has reached a great distance from Earth.

While Newton's cannon is helpful in revealing the nature of orbital motion, as you have probably guessed it does not have much to do with the realities of launching current spacecraft into orbit. This is done using launch vehicles, and we shall discuss in Chapter 5 how these are related to Newton's cannon.

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