## A triumph for Newtonian gravity

As children of our age, we find it natural to think of the planets as cousins of the Earth: remote and taciturn, perhaps, but cousins nevertheless. To visit them is not a trip lightly undertaken, but we and our robots have done it. Men have walked on the Moon; live television pictures from Mars, Jupiter, Saturn, Uranus, and Neptune have graced millions of television screens around the world; and we know now that there are no little green men on Mars (although little green bacteria are not completely ruled out).

Among all the exotic discoveries have been some very familiar sights: ice, dust storms, weather, lightning, erosion, rift valleys, even volcanos. Against this background, it may be hard for us to understand how special and mysterious the planets were to the ancients. Looking like bright stars, but moving against the background of "fixed" stars, they inspired awe and worship. The Greeks and Romans associated gods with them, and they played nearly as large a part in astrology as did the Moon.

It was a giant step forward for ancient astronomers, culminating in the great Greek astronomer Ptolemy, to show that what was then known about planetary motions could be described by a set of circular motions superimposed on one another. These were called cycles and epicycles. It was an even greater leap for Copernicus to argue that everything looks simpler if the main circular orbits go around the Sun instead of the Earth. Not only was this simpler, but it was also a revelation. If the Earth and the planets all circle the Sun, and if the Earth is simply in the third orbit out, then probably the planets are not stars at all, and the Earth might be a planet, too.

This probability became a virtual certainty when Galileo trained his first telescope on the night sky. Not only did he discover that Jupiter had moons, just like the Earth, only more of them, he also saw craters and mountains on the Moon, indicating that it was rocky like the Earth; and he saw the phases of Venus, the shadow that creeps across Venus as it does the Moon as these bodies change their position relative to the Sun. This meant that Venus was a body whose size could be measured: it was not a mere point-like star. When Kepler (whom we met in Chapter 2) showed, by extraordinarily painstaking calculations, that the planetary orbits were actually ellipses, the modern description of the motion of the planets was essentially complete.

But Galileo's study of motion on Earth soon raised an even bigger problem. Since bodies travel in straight lines as a rule, and since the planets do not, what agency forces them to stay in their orbits? Newton saw this problem clearly, and he had the courage to say that it needed no extra-terrestrial solution: gravity, the same gravity that makes apples fall from trees, also makes the planets fall toward the Sun. But what was the law of gravity? What rule enables one to calculate the acceleration of the planets?

In this chapter: applied to the Solar System, Newton's new theory of gravity explained all the available data, and continued to do so for 200 years. What is more, early physicists understood that the theory made two curious but apparently unobservable predictions: that some stars could be so compact that light could not escape from them, and that light would change direction on passing near the Sun. Einstein returned the attention of astronomers to these ideas, and now both black holes and gravitational lenses are commonplace.

>This name is pronounced "Tolemy". His full name was Claudius Ptolomaus, and he lived in Alexandria during the second century AD. Little else is known of him.

>The image behind the text on this page is from a beautiful photograph of the Moon taken by the Portugese amateur astronomer A Cidadao on 1 March, 1999. Used with permission.

Figure 4.1. Both the Moon (left) and Mars (right) appear to be desolate, uninhabited deserts. But Mars appears to have experienced erosion, probably by flowing water, at some time in its past. Left image courtesy nasa; right courtesy nasa, the Jet Propulsion Laboratory (jpl), and Malin Space Science Systems.

Figure 4.1. Both the Moon (left) and Mars (right) appear to be desolate, uninhabited deserts. But Mars appears to have experienced erosion, probably by flowing water, at some time in its past. Left image courtesy nasa; right courtesy nasa, the Jet Propulsion Laboratory (jpl), and Malin Space Science Systems.

In this section: we learn how knowledge of the Moon's distance, which was available to Newton, makes the law of gravity that he invented very plausible.

How to invent Newton's law for the acceleration of gravity

Let us look first at the nearest "planet": the Moon. If the laws of mechanics postulated by Newton are to apply in the heavens as well, then we should be able to deduce from the motion of the Moon what the force on it is. Suppose that the Moon is in a circular orbit. This is a good first approximation, but we shall have to return to the question of elliptical orbits later. We have seen in Chapter 3 that, for circular motion at speed V and radius R, the acceleration a is V2/R. We can eliminate V in terms of the radius of the orbit R and the period P, because the speed is just the distance traveled (circumference of the orbit) 2nR divided by the time taken, P. This means that the acceleration is (2n/P)2R towards the Earth.

Now, Newton knew the distance R to the Moon; even the Greeks had a value for it, by measuring its parallax from different points on the Earth, as shown in Figure 4.2. Since Newton also knew the period P of the Moon's orbit, he could work out its acceleration.

If we consult Table 4.1 for the modern values of these numbers, we can calculate from Equation 3.1 on page 19 or Equation 3.3 on page 22 that the acceleration of the

Figure 4.2. The direction to the Moon is different at different places on the Earth; this is called the Moon's parallax. One can use the parallax to determine the Moon's distance D. Suppose the Moon is directly overhead at a point E on the Earth, and suppose one measures its position in the sky from point G at the same moment, obtaining that it lies at an angle d down from the vertical there. The point G is known to be more northerly on the Earth than E, by a latitude angle a. Simple geometry then says then that the angle between E and G as seen from the Moon is ¡3 = d - a, and the angle of the triangle opposite the desired distance D is 180° - d. By the law of sines for triangles, we have sin(180° - d)/D = sin ¡3/R, where R is the radius of the Earth. This can be solved for D. Ptolemy performed essentially this measurement of d, and knew a and R reasonably accurately. He deduced that the Moon's distance was 59 times the Earth's radius. The right value is just over 60.

Figure 4.2. The direction to the Moon is different at different places on the Earth; this is called the Moon's parallax. One can use the parallax to determine the Moon's distance D. Suppose the Moon is directly overhead at a point E on the Earth, and suppose one measures its position in the sky from point G at the same moment, obtaining that it lies at an angle d down from the vertical there. The point G is known to be more northerly on the Earth than E, by a latitude angle a. Simple geometry then says then that the angle between E and G as seen from the Moon is ¡3 = d - a, and the angle of the triangle opposite the desired distance D is 180° - d. By the law of sines for triangles, we have sin(180° - d)/D = sin ¡3/R, where R is the radius of the Earth. This can be solved for D. Ptolemy performed essentially this measurement of d, and knew a and R reasonably accurately. He deduced that the Moon's distance was 59 times the Earth's radius. The right value is just over 60.

Table 4.1. Data for the Moon.

Average distance Period P2/R3 Average Mass from the Earth, R (km) of orbit, P (s) (s2 km-3) speed (km s-1) (kg)

Moon is 0.0027m s-2. Newton reasoned that if this was due to the Earth's gravity, then (like any gravitational acceleration) it could not depend on the Moon's mass, so it could depend only on how far the Moon was from the Earth. In particular, he guessed that it might depend only on how far it was from the center of the Earth. Compared with the acceleration of gravity on the Earth's surface, 9.8 m s-2, that of the Moon is smaller. How much smaller? The ratio of the two accelerations, 9.8/0.0027, is 3600. The ratio of the radius R of the Moon's orbit, 384000 km, to the radius of the Earth itself, 6380 km, is 60.

Clearly, the ratio of the accelerations is the square of the ratio of the distances taken in the opposite sense: the acceleration produced by the Earth is inversely proportional to the square of the distance from the Earth's center.

We have already seen in Equation 2.3 on page 13 that the simplest form of such an inverse-square law of gravity that obeys the equivalence principle is

Fgrav =

GMiM2

where M1 in this case is the mass of the Earth and M2 that of the Moon.

One can imagine how Newton might have reacted to this result: such a simple relation cannot be coincidence! Surely it must also apply to the planets in their orbits around the Sun. But Newton knew that the orbits of the planets were ellipses, not simple circles. Could that also be a consequence of the inverse-square law? That is what we turn to next.

The orbits of the planets described by Newton's law of gravity

In Table 4.2 on the following page I have listed the main properties of the planets and their orbits. Here we encounter for the first time the astronomer's unit of distance in the Solar System, the astronomical unit, denoted AU. It is defined as the average distance of the Earth from the Sun, 1.496 x 1010 m. In these units, distances become easier to comprehend: 2 AU is just twice the radius of the Earth's orbit, but what is 3 x 1010m? (It is just 2 AU again.)

Now look at column 4, where I have calculated the ratio of the square of the period to the cube of the average distance from the Sun. Kepler had noticed that these values were remarkably similar for all the known planets (out to Saturn), and we now call this Kepler's third law. (He didn't know the absolute distances between planets very well, but could deduce their ratios from observations, and that was enough to deduce the constancy of this number.)

Newton recognized that this strange relation provides the crucial evidence that the gravitational force does indeed fall off as the square of the distance. Again we consider an idealized circular orbit, for simplicity. From Equation 3.2 on page 19, the quantity in column 4 is, in terms of the acceleration a and the radius R,

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