Theme 3.3 Kepler Kepler s Basic Assumptions Kepler wanted to use the extensive data sets provided by Tycho to work out the shapes and relative sizes of all the planetary orbits. He had to make two simplifying assumptions. First, he assumed that the Sun was at the centre of the solar system; in other words, he accepted the Copernican notion. Secondly, he assumed that the Earth's orbit around the Sun is a circle. This simplified his further calculations of the orbits of the other planets. It can be justified by the fact that the Sun looks roughly the same size at all times of year so our orbit must be nearly circular in fact. Questions to Address If you think about it there are three obvious questions you could ask about the nature of the orbits followed by planets around the Sun. The answers to each of these led to one of Kepler's famous three laws. The first question is to inquire about the shape of the orbits of planets around the Sun. He assumed, as we know, that the Earth's orbit was circular but the others might not be, and he wanted to determine those shapes. Secondly, he considered the motion of an individual planet. Does it move at constant speed around its orbit, or speed up and slow down? This led to his second law. Third, he wanted to intercompare the planets. They all have different periods: some move around the Sun more rapidly than others, but he wanted to know what determined those periods, how they were related to the distance of the planets from the Sun. These led as we said to his three famous laws of planetary motion. Kepler s First Law Kepler discovered that the planets follow paths that are elliptical in shape. Notice by the way that this word has nothing to do with eclipse or the ecliptic: an ellipse is a geometrical form that we'll explore in a subsequent panel. He also discovered that the Sun lies at one focus of the elliptical path followed by any planet: again, we'll see what that means subsequently. Perhaps the simplest way to understand what an ellipse is, is to draw one. This could be done using the diagram shown on the left of this panel. Take a piece of string and tie it into a loop. Stretch the loop around two pins in a board and then use a pencil as shown to describe a curve, keeping the string taut at all times. Notice that this means the distance from one pin out to the curve and back to the other pin is the same for every point on the curve. That's indeed what defines an ellipse. The two points where the pins are stuck are each called a focus; the plural is foci, the two foci of the ellipse. On the right-hand side we see the same representation again, and below it we see a representation of a planetary orbit where you see that the sun sits at one of the foci of the ellipse, with a planet moving around and around. At the bottom, I have a link for you where you can explore this notion of drawing ellipses.
Ellipses can be thought of as flattened circles, and indeed if you put the two pins in the board very close together you'll effectively be drawing a circle. A circle is in fact really an ellipse, one that is not flattened. There's a parallel here: you can think of rectangles and squares. A square is a rectangle; all other rectangles could be described as flattened squares. In the same sense, an ellipse is a flattened circle. As noted, every ellipse has two foci, plus an attribute called the eccentricity. (Don't worry about its mathematical definition.) An eccentricity of zero means that the ellipse is not flattened at all. In fact it's a circle, and so the two pins were superimposed on each other on the board as you draw the ellipse. The larger the value of the eccentricity, which can approach 1, the flatter the ellipse; so you see in the diagram on the right hand side a very flattened ellipse with an eccentricity of 0.96. The Earth's orbital eccentricity (which you don't have to memorize) is 0.017, close to zero, and this tells us -- as Kepler had assumed -- that the earth's orbit is close to a circle around the Sun. From his analysis of Tycho's data, Kepler was to be able to work out not just the shapes of the orbits but also their relative sizes. So for example, Venus is closer to the Sun, being only about 72 % as far away from the sun as the Earth is, whereas Mars is about half again as far. The Sun plays an important role, sitting at one focus of the ellipse. It allows us to understand the motion of the planet around the Sun. Look at the diagram at the right for example where you see a planet approaching from the right, following the blue arrow. The strong gravitational influence of the Sun pulls the planet around the corner (so to speak) and sends it flying back out again to the right, along the direction of the orange arrow -- so gravity explains why the planet makes that U-turn as it nears the sun. (By the way, Kepler did not know about gravity, and could only speculate about what might be causing the motions of the planets at all.) So we understand why it is that the planet makes a U-turn around the sun in close proximity to it. It is the effect of gravity. But you might ask: why does the planet make another U-turn out at the other end of its orbit? Look at the diagram here for example. Why does the planet turn around and head back, if there's no second sun sitting out at the other focus of the ellipse? The answer is that it's again the gravity of the sun that's responsible. This is most easily understood if we turn the diagram on its side, as shown. Look at the planet moving away from the sun along the direction of the blue arrow. Think about the gravitational effect of the sun. As the planet moves away, it feels a force in the downward direction in this diagram because of the gravity of the sun, and will (like a stone falling back to earth) turn around and come back down. So once again the sun is what determines the orbital motion of the planet, even though it sits at only one of the foci of the ellipse. Kepler s Second Law
So Kepler was able to determine the shapes of the orbits of the other planets: Mercury, Venus, Mars, Jupiter and Saturn. (In Kepler's day, we had not yet discovered Uranus and Neptune.) He then turned his attention to the question of the speeds with which individual planets move. Do they speed up and slow down? And the answer is Yes: no individual planet moves at constant speed around its orbit. But Kepler did more than make that statement. He actually discovered a law that describes this motion. I'm going to describe the imaginative way in which Kepler expressed his second law. We'll see in a few panels' time that there's a much more straightforward physical expression of that law that we use now, but for the moment let's admire the imaginative way in which he addressed this question. The diagram shown here shows a planet moving around and around the sun. Imagine drawing a line from that planet to the sun and then seeing how much area it sweeps out as it moves, so for example, the planet moving from X to Y sweeps out an area shown by the green shading. Later, as it moves from A to B, it sweeps out the area shown by the blue shading. It was with respect to those swept-out areas that Kepler was able to describe his second law, the law of areas. Look at the diagram on the right here. As the planet moves from C to D, it sweeps out segment I, a tall skinny orange triangle as shown. Later, as it moves from A to B, it sweeps out segment II, again shaded in orange. What Kepler discovered is that if these two areas are the same then it takes the same time to move from C to D as it does from A to B. Since segment I is long and skinny, it must move rather slowly from C to D because that's a small distance. Since segment II is short and stubby it must move rather quickly from A to B because that's a longer distance and the time taken for the two segments is the same. In other words, when a planet is farther away from the Sun it moves slowly. When it's closer to the Sun it moves quickly. As noted, the orbits of the planets are indeed elliptical, and their speeds vary -- but not very much! The orbits are not much removed from circles. Halley's Comet, though, is an extreme example of an object that obeys the same physical laws (unknown to Kepler: he did not know about the orbits of comets when he was making these analyses). Halley's Comet moves very far out into the solar system out, indeed, beyond the orbit of Neptune, as shown in this figure. When it's out there, it's moving very slowly and it spends decades out in the outer parts of the solar system. As it comes in towards the sun it picks up speed and moves very quickly, giving us a brief glimpse of it once every 76 years. At the bottom of this page you'll find a link to a simulation that allows you to play with objects moving in elliptical orbits around the Sun, and you can change the eccentricity of the shape of the orbit and so on and test out these various laws to your satisfaction. What we've said, then, is that as planets or comets move in closer to the sun, they speed up in their motion, and then slow down as they move farther away. Does this sound familiar? It should, because this is in fact a statement of the conservation of angular momentum which we encountered earlier. (Remember the figure skater who draws in her arms to spin faster? It's the same physical principle.) As the object
moves closer to the Sun, it picks up speed because of the conservation of angular momentum, and that is in fact what Kepler's second law is equivalent to. Interestingly, we can also draw insights from another conservation law: the conservation of energy, which we also encountered earlier. Far away from the sun, the plane has a lot of potential energy. As it falls in it picks up speed -- kinetic energy! -- as it loses that potential energy. It moves quickly around the corner of the Sun, climbs back away, losing speed and kinetic energy but regaining potential energy. The total energy remains the same all the time. That's the conservation law, but it allows you to understand why it is that the planet moves so quickly as it passes the Sun. This thought about the conservation of energy has another important implication. Think about an object falling in towards the Sun from a great distance away. As it loses potential energy and gains kinetic energy, it moves faster and faster, but unless it runs directly into the Sun (which will bring it to a halt) it will be moving quickly enough in close proximity to the Sun that it has enough energy to climb away from the Sun again and to get out from the same large distance from which it began! (This is analogous to a swinging pendulum.) In other words, the object will not and cannot take up a new close-in orbit. It will not be captured by the gravity of the central object, which is a very common misunderstanding. This actually matters a great deal in the context of space craft and space travel. Imagine we want to put a space probe into orbit around Jupiter, so that we can take good close-up images of the planet. First we launch it on its way, and now we can imagine the spacecraft hurtling through the solar system in the direction of Jupiter. As it nears Jupiter, it speeds up under the influence of Jupiter's gravity -- and if we don't do anything, it will simply pass on by the planet, lose speed as it continues on its way but climb as far away from Jupiter as it was to begin with. It will not be captured! We actually have to slow it down a lot using rocket engines as it nears Jupiter so that it will go into a planned new orbit around the planet itself. What this means, among other things, is that we have to carry extra fuel with us for this purpose, and that reduces the amount of payload we can have on board the spacecraft. Kepler s Third Law For his third law, Kepler wanted to investigate how the orbital period and the speed of motion of a planet depended on its distance away from the sun. You might be surprised to discover that it's very straightforward to work out the orbital periods of the different planets, and they were known long before Kepler's time. Here's a panel that makes the process quite straightforward. Suppose we look out at night and discover that Mars is overhead as shown in the first diagram. A year later, we look out at night -- and there's no Mars to be seen. It's because Mars is in fact now on the far side of the sun. (We've gone around the sun once; Mars has made about half of its orbit.) Two years later, though, we see Mars again overhead at night, as shown in the third panel. The conclusion you draw is that Mars has an orbital period of about two years (and we can refine that quite precisely).
As you have just seen, it's quite straightforward to work out the orbital periods of the various planets, and we know, therefore, for example, that Jupiter takes almost 12 Earth years to go once around the Sun. But why does it take so long? Well, its orbital path is longer: since it's farther away from the sun, it has more territory to cover. (Even if it were moving at the same speed as the Earth, it would take longer to make one full orbit.) But what is the relationship? Are they moving faster or slower the farther out they are? This is what Kepler wanted to understand. What is the relationship between distance and orbital period? Just for interest, here are Kepler's numbers. (No need to memorize these by the way. And I remind you that one astronomical unit is the average distance between the Earth and the Sun, so we've expressed the planetary distances here in that convenient unit. Notice that Jupiter is a little more than five times the distance of the Earth from the Sun, and has an orbital period of almost 12 years, as I said.) What is the relationship between these sets of numbers? Perhaps amazingly, it took Kepler more than a decade of analysis to work out what is a fairly straightforward relationship mathematically. (A modern physics student would do so in a matter of minutes because we have better analytic tools -- and know, in a sense, what to look for. ) What Kepler discovered (and you do not have to memorize this relationship) is that the square of the period of a planet is proportional to its distance cubed. Let's see what this means arithmetically. Consider Jupiter, take its period of 11.86 years and square that (multiply those two numbers together). Take its distance from the sun in astronomical units and cube that (multiply that together three times as shown) -- and you get the same number! This holds true for all the planets. There's a very simple relationship between the orbital period and the distance in this mathematical sense. The discovery of Kepler's third law is profoundly important. It tells us that there's a fundamental relationship between orbital period and distance, and presumably therefore some physical law that drives that. We'll discover later that Newton showed it was because of the effects of gravity. Moreover this implies a predictability. If we find a new object orbiting the Sun, we can work out its distance by determining its orbital period -- or vice versa -- and that means we can launch a spacecraft into orbit at a certain distance, know what speed we'll need or have at that distance away from the Sun. There's a mathematical regularity and predictability to the entire system. In the end, Kepler had a complete understanding of what you might describe as a scale model of the solar system. He knew the shapes and relative sizes of all the planetary orbits out to Saturn, as shown here, and he knew the relationship between orbital period and relative distance to the Sun. It's interesting to remember that to him the planets were merely moving dots of light. He had no telescopes, no actual images, and no idea of their true physical nature and structure. He also did not know the actual true size of the solar system. He knew that Jupiter was roughly
five times the distance from the Sun that the earth is, but he had no idea what those distances were (except on the basis of very crude arguments that depended on historical speculations of various sorts). Kepler the Mystic Kepler is rightly admired and remembered for the derivation of the three laws that describe the orbits of the planets around the Sun. It must be remembered, however, that scientists of his day were a strange mix of modern scientific thinking and rather mystical views of the universe. Let's consider two ways in which this mysticism is displayed by Kepler. First of all, he asked the interesting question of why there were exactly six planets: Mercury, Venus, Earth, Mars, Jupiter and Saturn. Well, of course, the answer is we know that there are more! -- Uranus and Neptune we are not discovered until after Kepler's lifetime. But he came up with an explanation of why there should be exactly six. Kepler knew from his study of geometry that there are only five regular polyhedra. (A regular polyhedron is a solid that has all of its faces exactly the same. So for example, a cube has six identical faces, each of which is a square. An octahedron has eight identical faces, each of which is an equilateral triangle.) He now imagined making one of each of these solids but of different sizes, and then packing them one inside another, the smaller ones inside the bigger ones, but in different orders. The relationship of these would then give rise to a set of spacings that could (he thought) be related to the spacing of the planets, and by varying the orders he tested different possible packings. The result of this thinking is shown in the wood cut reproduced here. In the top panel, you see that there's a large cube that contains within it a tetrahedron. The bottom panel shows the innermost parts of the packing. Kepler was sure that this explained the spacing and number of the planets, and was very proud of this discovery. Indeed, he was prouder of this than he was of the discovery of the three laws that described the orbital motion of the planets around the sun! I've provided you with a link to a brief representation by an actor portraying Kepler visualizing this fantastic breakthrough (as he himself thought). A second aspect of Kepler's thinking takes us back to the time of Pythagoras, who studied tightly stretched strings. He knew that plucking or hitting these strings would create musical notes of different pitches, and it soon became apparent that these notes are caused by the vibrations of the strings, with the higher frequencies (the more rapid vibrations) producing higher-pitched notes. You might like to see this exemplified, so I've provided a link to a view of real vibrating guitar strings as seen from within the guitar and where you'll see the strings themselves vibrating as they are plucked. Just as the strings of a guitar produce notes of different pitches depending on how rapidly they're vibrating back and forth, Kepler reasoned that the planets moving around and around the Sun were exhibiting periodic, regular motions of different frequencies -- Mercury going around the Sun very rapidly at high frequency, Jupiter more slowly, etc. He reasoned that this would cause music of the
spheres -- some sort of unearthly unhearable music created by the planets and their motions; and because the motions of the planets were not steady but variable in speed, there would be a gradual change in the pitches as each planet exhibited its orbit. These are shown here. Galileo, who was a real scientist, had absolutely no patience for this interpretation, by the way. Setting aside those mystic contributions, it's nevertheless clear that Kepler made enormous profound contributions to our understanding of the solar system -- but there are still two essential items missing. The first is that we don't have any real proof yet. Kepler had assumed that the Sun was the centre of the solar system, and on that basis he derived some pleasingly simple behavioral laws; but that doesn't prove that the Earth and the planets are orbiting the Sun. We need such a proof before we're absolutely convinced. Secondly, there's no physical explanation of why the planets would move the way they do. What are the laws of nature that govern the motions? The answers to these two questions came from first Galileo and then Newton.