## On the Shoulders of Giants

Newton was born on Christmas Day 1642, just 23 years after Kepler had published the last of his three laws—and what a gift to the world! Newton has been described in cliched terms by numerous biographers: "the foremost scientific intellect of all time''; "the father of modern science''; and so on. However, when applied to Newton, these glowing epithets are arguably fully justified. Newton made major contributions to many areas of scientific activity, including optics and light, mathematics, dynamics, gravitation, and theoretical astronomy. He himself, however, summarized his contribution to science by stating, "If I have been able to see further, it was only because I stood on the shoulders of giants,'' referring to some of those giants discussed in the preceding sections of this chapter.

Newton's quest to understand the world began with his undergraduate career at Trinity College, Cambridge University, at the age of 18. However, his time at Cambridge was interrupted in the summer of 1665, when the university was closed down by an outbreak of plague. Newton then returned to his birthplace, the isolated village of Woolsthorpe in Lincolnshire, where in an amazingly productive period of 2 years he revolutionized science.

In summary, during this period he devised his law of gravitation and his laws of motion. Combining these, he was able to formulate the equations that governed the motion of the planets around the Sun. He then realized that these equations could not be solved using the methods then available. However, he was not a person to let such a small detail inhibit his efforts, and so he set about inventing a new branch of mathematics, called calculus, to remove the barrier. All of these accomplishments have had a lasting impact upon science and engineering to the present day, and any one of them would be considered a major intellectual achievement. For them all to have come from one individual in such a short period of time is extraordinary.

It is worth pausing a few moments to consider in more detail each of the steps that comprised Newton's achievement. Perhaps the thing most people associate with Newton is his law of gravitation, along with the story of the mythical apple that is supposed to have fallen on his head and given spontaneous birth to the idea. It is likely, however, that the formulation of his understanding of gravity took a little more time and effort. The formal statement of Newton's law of gravitation is given below, and as can be seen it is expressed once again as what might be called a word equation:

Newton's Law of Universal Gravitation - the force of gravity between two bodies is directly proportional to the product of their masses, and inversely proportional to the square of their distance apart.

However, it can be easily understood in simple terms. The phrase "directly proportional to the product of their masses'' simply means that the force of gravity between two large objects—say two planets, or two stars—is large, and indeed will govern the way these celestial bodies move with respect to each other. On the other hand, the force of gravity between two small objects will be tiny. For example, if you place a couple of balls on a pool table, you expect them to remain firmly attached to the surface, since they are attracted to the large mass which is the Earth beneath the table. It is only the structural strength of the table that is preventing them from responding to the force by whizzing off toward Earth's center. At the same time, we do not expect them to move across the table toward each other, since the force of gravity between them is so small as to be effectively zero. The game of pool would be somewhat different if it were otherwise!

The way the force of gravity varies with distance, as described above, is sometimes referred to as the inverse square law. This describes how the force between two bodies diminishes as they move further apart. If you think of two objects a particular distance apart—strictly this distance is measured between their centers—then the force of gravity between them will have a particular strength. When we move them apart so that the distance between them is doubled, the inverse square law says that the force is one fourth of what it was before. To get this, we take 2 from "twice the distance,'' square it to give 4, and then take the inverse to give us the one fourth. In the same way, we can move the bodies 10 times further apart, and the same argument tells us that the force of gravity is reduced by a factor of 1/100.

There is some debate among scientific historians about how Newton settled upon the inverse square law for gravitation. Some believe he was influenced by his studies of the way light behaved; he discovered by experiment that the intensity of light falling upon a surface decreased in proportion to the inverse square of the distance between the source of light and the surface. However, more likely he proposed the inverse square law since it was consistent with Kepler's third law of planetary motion, which can be shown by the use of some simple mathematics that can be done literally on the back of an envelope.

Coming back to Newton's apple, we can explore some of Newton's thinking during his brief but prolific period of exile in the Lincolnshire countryside. Having thought about his law of gravitation in a universal context, Newton's observation of the fall of an apple from a tree engendered universal questions in his mind such as, Why doesn't the moon also fall to the ground? To answer this one, we can compare the motion of the apple with that of the moon.

Taking the apple first, when it is released from the tree it responds to the force of gravity by accelerating toward the ground. It starts from rest up in the branches of the tree and builds up speed until impact with the ground. If we were able somehow to measure this impact speed and the time of fall, we would be able to calculate its acceleration. For example, if the height of the tree was such that the apple took 1 second to hit the ground, we would find that its impact speed was about 10 meters per second (32 feet per second). In this case, the distance fallen can be estimated as about 5 meters (16 feet)— quite a tall apple tree. If the ground were not in the way, the apple would continue to accelerate toward Earth's center, gaining 10 meters per second in speed for every second of the fall. This acceleration due to gravity at Earth's surface of 10 meters per second per second is usually expressed as 10 m/sec/ sec or 10 m/sec2 (32 feet/sec2).

Newton was the first to realize that the moon must also respond to the force of gravity in the same way. However, the moon is around 60 times more distant from Earth's center than his apple. Applying his law of gravitation, he estimated that the acceleration of the moon toward the Earth will be much less than that of the apple by a factor of 1/3600—that is, the inverse of 60 squared. In its distant orbit then, the moon will fall toward Earth with an acceleration of approximately 10/3600 meters per second per second, or about 3 millimeters/sec2 (1/10o feet/sec2). With this small acceleration downward, it is easy to estimate that in 1 second the moon falls a small distance toward Earth of about 1.5 millimeters—much less than the 5 meters fallen by the apple. However, it is instructive to consider what happens to the moon's motion during the period of a minute, as then the numbers are a little easier to grasp. Because of the coincidence that the moon is 60 times further away from Earth's center than the apple, and that there are 60 seconds in a minute, the mathematics tell us that the moon falls the same distance in 1 minute as the apple falls in 1 second—about 5 m. However, at the same time the moon has a relatively high speed along its orbit so that in 60 seconds it moves horizontally approximately 61,100 meters (200,500 feet). Figure 1.8 shows that the combination of these horizontal and vertical motions result in the near-circular orbital path that we observe, so that although the moon does actually fall continually toward Earth, fortunately it never reaches the ground!

To understand the motion of bodies, such as the apple and the moon, in this way, Newton had to devise not only his universal law of gravitation, but also his three laws of motion, which are stated as follows:

Newton 1 - A body will continue in a state of rest, or of uniform speed in a straight line, unless compelled to change this state by forces acting upon it.

Newton 2 - The rate of change of momentum of a body is proportional to the force acting upon it, and is in the same direction as the force.

Newton 3 - To every action there is an equal and opposite reaction.

moon 60 seconds later without gravity moon 60 seconds later without gravity

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Figure 1.8: Newton applied his universal law of gravitation to the Moon, as well as to his apple, to show how the Moon orbits Earth.

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Figure 1.8: Newton applied his universal law of gravitation to the Moon, as well as to his apple, to show how the Moon orbits Earth.

It is important to note that these laws, written in the form of words here, have their most powerful manifestation when expressed in mathematics. They revolutionized 17th century science, and indeed still dominate engineering science today. Newton's contribution is summed up by noting that 21st century engineers still use the mathematical expression of these laws to design buildings, bridges, cars, airplanes, and indeed spacecraft. I would love to stand with Isaac Newton at the end of a modern airport runway as a jumbo jet is taking off, and tell him that he is responsible for this apparently impossible apparition of 350 metric tonnes of predominately metal soaring into the sky—an amazing legacy!

In terms of our understanding of the solar system, Newton's revolution came about when he combined his law of gravitation with his laws of motion to produce equations that described the motion of the planets around the Sun. As described earlier, the solutions of these equations were obtained only after Newton had devised a new branch of mathematics. But once this was done, Newton rediscovered Kepler's three laws of planetary motion in his mathematics, thus giving a theoretical basis to Kepler's empirical work completed almost half a century before. However, Newton found not only Kepler's work in his new formulism but lots more. His mathematics were saying that objects moving in a gravity field, for example, a planet moving around the Sun, or a spacecraft moving around a planet, were not confined to elliptical paths. The shape of the path of such an object could also be that of a circle, a parabola, or a hyperbola. Most people are familiar with circles and ellipses, but what about the parabolic and hyperbolic shapes? These four shapes are referred to as conic sections,

because you can get them by sectioning, or slicing, a cone as illustrated in Figures 1.9 and 1.10.

In Figure 1.9a, we can see that if we slice the cone horizontally we get a circle, and if we slice it slightly obliquely we get an ellipse. Another trajectory shape is obtained by slicing the cone in such a way that the plane of the slice is parallel to the side of the cone, as in Figure 1.9b. In this case we get a parabola. The parabolic trajectory is not a closed one—as are the circle and ellipse—so we have to imagine that the cone is infinitely big, and not truncated at the base as illustrated. An example of an object on a parabolic trajectory is a comet that falls with initially zero speed from infinity—or effectively from a great distance—and is swung around the Sun and ends up heading back to the same place, arriving again at infinity with zero speed. Perhaps this type of trajectory can best be describes as a celestial U-turn.

The final trajectory shape that Newton found in his mathematics is the hyperbola, which is obtained by slicing our cone vertically, as shown in Figure 1.9c. This trajectory is again an open one, stretching to infinity, so we have to imagine that our cone is very large. An example of this type of orbit is a spacecraft swinging by a planet. The spacecraft approaches from great distance, initially traveling at constant speed relative to the planet in a straight line, as the gravitational influence of the planet is tiny. However, as the spacecraft closes in, the gravitational force increases and the trajectory is deflected. The gravity of the planet swings the spacecraft's path around so that the vehicle leaves the planet in a new direction, traveling in a straight path again at the same planet-relative speed once it has reached great distance. As we will see later in Chapter 4, this type of swing-by trajectory is commonly used by engineers designing interplanetary spacecraft missions. The hyperbola is distinguished from the parabola by the deflection angle; in a parabolic trajectory the object is deflected by 180 degrees by its encounter

Figure 1.10: The light cast by a table lamp shows the hyperbola-shaped, swing-by trajectory of a spacecraft.

with the planet, whereas the hyperbolic trajectory is deflected by an amount less than this.

Surprisingly, the hyperbola is also a fairly common sight in everyday life. All you need is a lamp with a circular shade, which projects a cone of light both upward and downward, producing a circle of light on the ceiling above and on the table below. If, however, the lamp is placed next to a wall, then the cone of light is effectively sliced vertically, as in Figure 1.9c, to produce the shape of a hyperbolic trajectory on the wall. A photograph of such a shape is shown in Figure 1.10a, and how it relates to the orbital trajectory is shown in Figure 1.10b. If you find yourself on a dinner date in a restaurant with this common type of wall lighting, you could use your knowledge of celestial mechanics to break the ice. (On the other hand, your guest may just think that you are a rather sad person who needs to get out more!)

Newton himself thought of his scientific investigations as a small contribution toward revealing the fundamental laws of the universe that were "written" by God, its designer. Thinking about his discoveries in this context, it is rather strange that we live in a universe where the shapes of gravitational trajectories are related to slices of a cone!

The final episode in this story of Newton's achievement is also surprising. Having "solved the universe'' in this way, Newton then failed to communicate his work to anyone! Meanwhile, unknown to him, contemporary scientists— principally Robert Hooke and Edmund Halley (of Halley's Comet fame)— were struggling with the problem of planetary motion in the coffee houses of London. Finally in 1684, Halley visited Newton in Cambridge, hoping to gain some insight into the riddle. When Halley posed the question about the shape of gravitational trajectories about the Sun, Newton revealed that he had already solved the problem, but had characteristically misplaced it. It was ultimately Halley who encouraged Newton to write his landmark work, the Philosophiae Naturalis Principia Mathematica, requiring 2 years of hard work to complete. In this rather circuitous manner, Newton was finally recognized as being one of the greatest scientific thinkers of all time.

The only other individual described is this way is perhaps Albert Einstein, whose scientific genius was unleashed upon the world at the beginning of the 20th century.

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