| Kepler the Mystic.
Laws in Threes:
It is an amusing coincidence that many of the famous laws in physics come in sets of three - Newton's laws, Kirchoff's laws, the laws of thermodynamics, and Kepler's laws among them. (We will encounter all of these, except the laws of thermodynamics, as the course progresses.) What are Kepler's laws, and how did he derive them?
A Philosophical Starting-Point, and a Bold AssumptionTo start with, Kepler immediately abandoned the geocentric models. He was sufficiently persuaded by the simplicity of the Copernican model to accept it as a starting point, notwithstanding Tycho's dying wishes. He wanted to make sense out of all the data, but how? The answer is that he assumed that the Earth's orbit is nearly a perfect circle, precisely centered on the Sun. This was a very bold (and possibly dangerous) assumption which turns out to be nearly correct. That he was correct is very fortunate in two quite different senses. The first consideration is that if the orbit were not nearly a circle, the Earth's climate might be very variable (since we would sometimes be much closer to the sun than average, and sometimes much farther away). We might not be able to survive on such a world, and there would have been no Kepler to make the assumption! So in this sense Kepler's assumption had an element of inevitable correctness. The second more prosaic point is that the correctness of the simplifying assumption meant that Kepler could carry out straightforward graphical calculations to work out the orbits of the other planets, some of which move in more obviously non-circular orbits. It is fun, although ultimately fruitless, to speculate about how things might have developed if the situation were otherwise. It is conceivable, after all, that life forms might adapt to the extremes of climate to be expected on a planet which orbits a star in a distinctly non-circular way. On such an Earth, a Kepler-like creature would have had a much harder time deducing the true nature of the Solar System from observations like Tycho's. (Success would eventually have come anyway, however, since technological developments like radar now allow us to determine the distances of all the planets and the grand layout of the Solar System quite unambiguously, with no need of simplifying assumptions.)
Some Simple Surveying.Kepler's approach was very clever. He knew that the Earth goes around the sun every 365.25 days, and was able to work out the correct numbers for the other planets as well. (For instance, Mars goes once around the sun every 687 days - its orbital period is about 1.88 Earth years. Determining this number is actually fairly straightforward. The details of how we do so are not important.) Suppose that we look at Mars on 1 January 1570. We don't know how far away it is, but we know what direction it lies in. If we observe it again exactly 687 days later, in mid-November of 1571, we know that Mars has made one complete orbit of the sun and has returned to exactly where it was on 1 Jan 1570. But we (on the Earth) have not returned to our original location, since only 1.88 Earth years have passed! Consequently, we are looking at Mars from a different location in our own orbit, and (with the simplifying assumption that our orbit is a circle) we can use the two lines of sight to work out the unique spot where Mars actually is on both of those dates. Here is an analogy which may help. Suppose you look up every night at precisely 5:00 P.M. and see a plane directly overhead (say, one travelling from Toronto to Montreal). You might be curious to know how high up it is, but have no direct way of judging that. What you could do, however, is travel 5 miles North (say), and at exactly 5:00 P.M. the next day make note of the angle at which you see the plane. You know it is directly above your house at that instant, since that is where it is every day at that time, and now your second line of sight allows you to pin down its location precisely - provided you make the measurement at the right moment! Kepler did nothing more than this. The planet Mars is like the airplane, coming back to the same spot every 687 days. But we are on the moving Earth, and after 687 days we see Mars from a different point of view. This allows us to nail down its true location. Obviously this gives you only one point on Mars's orbit, but you want lots more. That was the beauty of Tycho's impressively modern scientific approach: he provided Kepler with many precise observations of Mars, taken over many years. By combining them in pairs separated by intervals of 687 days, Kepler was able to map out the full shape of Mars's orbit.
Just a Scale Model.It is important to realize that what Kepler got out was merely a scale model. Since he did not know the distance from the Earth to the sun in real units (like miles or kilometers), he was able to figure out the size of Mars's orbit only relative to that of the Earth. What Kepler discovered is shown on page 202 of your text. (Look in particular at the orbits of Mars and the Earth.) You can see that Mars's orbit is not as circular as that of the Earth, and that the Sun is not at its very centre. On average, Mars is about 1.5 times as far away from the Sun as the Earth is, but its variations in distance are quite striking. In truth, Kepler was amazingly lucky! Some of Tycho's best and most complete observations were of Mars, and it turns out to have a distinctly non-circular orbit -- something which need not have been the case. (Mars might have had an orbit just as circular as our own.) His finding, of course, provided ample stimulation for further thought and interpretation. Nor did he stop with Mars: he went on to analyse Tycho's observations for all of the visible planets (Mercury, Venus, Mars, Jupiter and Saturn).
The Obvious Questions, and Three Laws.If you were Kepler, given the data and techniques we have described, what questions would you have asked yourself? I don't think it requires much imagination to see that there are exactly four fundamental questions. Of these, the fourth one could not be answered in his time, while the others lead to his famous three laws. Here are the questions: 1 What are the shapes of the orbits? 2 If you look at any particular planet, does it move around the Sun at constant speed, or is it speeding up and slowing down from time to time? If the latter, what determines its speed at any given moment? 3 If you compare any one planet with another, do they move at the same speed? Or are some of the planets particularly speedy, while others move more lethargically? What determines the speed with which a particular planet moves around its orbit? You might expect that to depend on how close the planets are to the sun -- does it? If so, is there some law that describes this dependence? 4 How big is the Solar System? How far is the Earth, or any other planet, from the sun in really meaningful units like miles or kilometres? I hope you will agree with me that there is nothing more that one could have hoped to extract from Tycho's data. In fact it took literally decades for Kepler to complete the job of answering the first three questions!
K-I: The Shapes of the Orbits.After a lot of experimentation and thought, Kepler finally derived his famous first law: K-I: The orbit of each planet around the sun is an ellipse, with the sun at one of the foci of the ellipse. Clearly this law will be a bit opaque to you unless you have a clear idea of what an ellipse is! There is a nice description of this on pages 72-73 of your text (and I would encourage you to try out the simulations and exercises on the CD which came with your textbook). Please read the text and become familiar with the terminology there. But above all note the important philosophical breakthrough: the assumption of motion in perfect circles was gone forever! It is worth emphasising that Kepler's conclusions were only possible because of the very precise and numerous observations made by Tycho Brahe. In effect, Kepler was trying to see if the old model (circles within circles) could successfully predict where the planets would be at various times. In one example, when he applied the old model to work out where Mars ought to have been on a certain date, he discovered that Brahe's observation was not in agreement with the prediction. The discrepancy was only about an eighth of a degree - about a quarter of the diameter of the full moon! But he was so sure that Brahe's observations were correct that he boldly took this as firm evidence that there must be something wrong. Another person might simply have concluded that Brahe had made a small mistake. One of the ways we characterise an ellipse is by calculating its eccentricity, which is a measure of how far it differs from a perfect circle. (I remind you that a circle is merely an ellipse in which the two foci coincide; it has an eccentricity of exactly zero. See the figure at the bottom of page 72 of the text.) Indeed, in common terminology, someone who is "eccentric" [like an astronomer?] is unconventional, out of the ordinary. So, too, ellipses can be thought of as 'circles gone astray.' The larger the value of the eccentricity, the "flatter" (less circular) the ellipse is. Something like Halley's comet orbits the sun in a very elongated ellipse, with an eccentricity bigger than 0.95. To see what that means, have a look at a representative sketch of the orbit of a comet, shown on page 379 of the text. Halley's Comet is much more eccentric even than that! By contrast to Comet Halley, the eccentricies of the planets orbiting the sun are very small, and the orbits are not easily distinguished from circles. For instance, the eccentricity of Mars's orbit is only 0.09, and its orbit looks pretty much like a circle unless one makes very careful measurements -- which Brahe did, and which Kepler interpreted! We now know that the Earth's orbit has an eccentricity of 0.017, much less than that of Mars and in confirmation of Kepler's working assumption of the near-circularity of its orbit.
K-II: No Planet Moves at Constant Speed.Knowing where each planet was at different times in its orbit, Kepler was able to ask a very sensible question: does each planet move at some steady unchanging speed as it orbits the sun? A bit of analysis led Kepler to the correct conclusion that the planets do not orbit the sun at constant speed. Indeed, it is fairly easy to determine that a given planet moves faster when it is relatively close to the sun, and slower when it is farther away. The problem for Kepler was to figure out the empirical law which describes this effect. In the end, he discovered what came to be known as the ``Law of Areas'' - his Second Law, which states: K-II: A given planet sweeps out equal areas in equal times as it orbits the sun. The discussion on page 73 of your text makes clear what this implies: a given planet must be moving rather slowly when it is far away from the sun, and faster when it is close in. This explains, for instance, the behaviour of Halley's comet, which spends many years out near the orbit of Neptune, moving quite slowly; then it comes in near the Sun and absolutely races past it (which is why we see it so fleetingly). I should emphasise that this law is obeyed by all the planets, but that the areas covered by the different planets in a given time interval are different. For instance, the area swept out by Mars in an interval of fifty days is not the same as the area swept out by the Earth in fifty days. However, the area swept out by the Earth in one interval of fifty days is that same as that swept out by the Earth in any other interval of fifty days. In other words, Law K-II does not say anything about intercomparisons between planets. Incidentally, the derivation of Kepler's Second Law from Tycho's data strikes me as a particularly remarkable achievement. Would you, I wonder, have thought of considering the numbers in this way?
K-III: No Two Planets Move at the Same Speed.So far Kepler had worked out two things: (i) the shapes and relative sizes of the planetary orbits; and (ii) the variation of speed displayed by an individual planet as it moves along its orbit around the sun. He published these findings, in the form of his first two laws, about ten years before he found and published his third law - the so-called ``harmonic law.'' That law came finally as a consequence of his trying to work out the answer to an obvious question: is there a simple relationship between the distance a planet is from the sun and the time it takes to go around the sun? [Digression: in what follows, to avoid speaking in terms of million of kilometers all the time, we introduce a new unit: the astronomical unit, which is the average distance of the Earth from the Sun, about 150 million km.] Consider three possibilities: 1 Suppose all the planets move at the same speed. (They might, for instance, all move at 30 km/sec, which happens to be how fast the Earth moves around the sun, although Kepler had no way of knowing that number.) In that case, the planets farther out will take longer to go around the sun than the Earth does simply because they have a greater distance to cover. Their orbital paths are obviously much longer. 2 On the other hand, if the outer planets move more quickly than the inner ones, then their orbital periods could be the same, or even shorter. 3 Finally, if the outer planets move more slowly than the inner ones, then their periods will be very much longer indeed. Kepler was able quickly to work out that the last of these is the case. Consider Jupiter as an example. It is about 5 astronomical units from the sun. (Because of its elliptical orbit, of course, Jupiter's distance from the sun is not quite constant, so we use the average value). Thus Jupiter's orbital path around the sun is about 5 times as long in total distance as the Earth's orbital path. But Jupiter takes more than 11 of our years to go once around the sun, which must mean that it travels more slowly than we do - indeed you can see that it must be moving at a bit less than half our speed. Kepler was quick to see this, but slow to work out the empirical law which quantifies the effect. After a lot of fiddling about with the numbers, mostly by trial and error, he concluded correctly that: K-III: The square of the period of a planet orbiting the sun is proportional to the cube of its mean (average) distance from the sun Try this out on Jupiter: it is 5 astronomical units from the sun, so its distance cubed is 5 x 5 x 5 = 125. It takes 11 years to go around, and the square of that quantity is 11 x 11 = 121. These numbers are pretty much the same (but not precisely, since the ``11'' and ``5'' I quoted you are only approximate values. The figure on page 73 of your text shows the actual good agreement for the planets). Incidentally, it is a little sobering to realise that it would take a modern scientist about five seconds to work out Kepler's third law, given a simple tabulation of planetary distances and orbital periods. [For those who know the terminology: simply make a log-log plot of the data. The linear distribution which you see implies a power-law relationship, with an exponent given by the slope of the line. Problem solved!]
Kepler the Mystic.During his lifetime, and indeed for decades thereafter, Kepler's laws were purely ``phenomenological'' -- that is, they described how things moved, but gave no explanation as to why. An understanding of the physical reasons for these three laws did not arise until the development of the science of mechanics and the notion of the law of gravitation, both the product of Isaac Newton's great mind. But Kepler did at least speculate that the sun was somehow responsible for creating the "driving force" that made the planets move around in their orbits, and in this sense he was looking for cause and effect as a modern scientist would. That having been said, it must also be noted that Kepler was very much a mystic and totally different from the modern stereotypical model of a scientist as pure dispassionate rationalist (not that modern scientists are really like that anyway!). There are two amusing examples of this: 1 The Spacing of the Planets: Kepler was puzzled to know why there were exactly six planets, and no more. (Of course, we now know that there are more. In his day, only Mercury, Venus, Earth, Mars, Jupiter and Saturn were known.) He developed the notion that the spacing of the planets was determined by a quasi-numerological relationship which described how the so-called "regular" polyhedra could be packed into one another. The regular polyhedra are bodies which are solid but which have all of their faces exactly alike - for instance, a cube has six identical faces, each of which is a square. They also have the property that all the vertices, or points, are equidistant from the centre, so that they can be nicely enclosed in a sphere with all the points just touching the inner surface. A tetrahedron is a regular solid: it has four faces, each of which is an equilateral triangle. (You may have encountered these in the form of small 'pyramidal' milk containers in restaurants, each containing a single cream serving for a cup of coffee.) An Egyptian pyramid, by contrast, is not a perfect polyhedron, since it has a square base but triangular faces. Mathematicians had long since proven that there were only five perfect polyhedra - the cube, the tetrahedron, the octahedron, the icosahedron, and the dodecahedron. Kepler described putting these one inside another as follows: "The Earth's orbit is the measure of all things; circumscribe around it a dodecahedron, and the circle containing this will be Mars. Circumscribe around Mars a tetrahedron, and the circle containing this will be Jupiter. Circumscribe around Jupiter a cube, and the circle containing this will be Saturn. Now inscribe within the Earth an icosahedron, and the circle contained within it will be Venus. Inscribe within Venus an octahedron, and the circle contained within it will be Mercury. You now have the reason for the number of planets." Of course this is now seen as pseudo-scientific nonsense, but Kepler was prouder of this, and of the models he built, than he was of his laws!
2 The Music of the Spheres.Since the time of the ancient Greeks, it has been known that music has a mathematical basis. The more rapidly a string vibrates, for instance, the higher the pitch of the note produced. One string vibrating exactly twice as fast as another identical one sounds exactly an octave higher; and so on. This simple physics is quite correct, and the basis of our understanding of the way in which instruments produce musical tones of very complex form and quality. Kepler reasoned that the inner planets, like Mercury, go around the sun very frequently in the same time as the outer planets do so only a few times. He elaborated the notion that these motions gave rise to musical notes, unheard by humans but nonetheless real, which he called "the music of the spheres." (This term was not in fact new to Kepler, but seems to have originated with Pythagoras in ancient Greece.) The Earth itself was thought to play only two notes, the "mi" and "re" of a classical scale, because of the fact that we live in a state of vice, corruption, and mi se re re (Latin for 'misery'). Needless to say, this is not now considered serious science. Previous chapter:Next chapter
0: Physics 015: The Course Notes, Fall 2004 1: Opening Remarks: Setting the Scene. 2: The Science of Astronomy: 3: The Importance of Scale: A First Conservation Law. 4: The Dominance of Gravity. 5: Looking Up: 6: The Seasons: 7: The Spin of the Earth: Another Conservation Law. 8: The Earth: Shape, Size, and State of Rotation. 9: The Moon: Shape, Size, Nature. 10: The Relative Distances and Sizes of the Sun and Moon: 11: Further Considerations: Planets and Stars. 12: The Moving Earth: 13: Stellar Parallax: The Astronomical Chicken 14: Greek Cosmology: 15: Stonehenge: 16: The Pyramids: 17: Copernicus Suggests a Heliocentric Cosmology: 18: Tycho Brahe, the Master Observer: 19: Kepler the Mystic. 20: Galileo Provides the Proof: 21: Light: Introductory Remarks. 22: Light as a Wave: 23: Light as Particles. 24: Full Spectrum of Light: 25: Interpreting the Emitted Light: 26: Kirchhoff's Laws and Stellar Spectra. 27: Understanding Kirchhoff's Laws. 28: The Doppler Effect: 29: Astronomical Telescopes: 30: The Great Observatories: 31: Making the Most of Optical Astronomy: 32: Adaptive Optics: Beating the Sky. 33: Radio Astronomy: 34: Observing at Other Wavelengths: 35: Isaac Newton's Physics: 36: Newtonian Gravity Explains It All: 37: Weight: 38: The Success of Newtonian Gravity: 39: The Ultimate Failure of Newtonian Gravity: 40: Tsunamis and Tides: 41: The Organization of the Solar System: 42: Solar System Formation: 43: The Age of the Solar System: 44: Planetary Structure: The Earth. 45: Solar System Leftovers: 46: The Vulnerability of the Earth: 47: Venus: 48: Mars: 49: The Search for Martian Life: 50: Physics 015 - Parallel Readings.
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(Monday, 17 May, 2021.)