In these notes we will look at what the evidence is which underlies some beliefs which we have had drilled into us since early childhood:
First of all you should realize that all of the beliefs listed in the previous paragraph are ingredients of a theory. When I say this I don't mean that they are "just a theory," that they are just dreamed up and could be wrong as easily as right. What I do mean is that they are inferred from observations, and are ingredients of a picture of the Universe which has been built up to explain those observations. They are not observed directly -- until quite recently there was no way for human beings, bound to the surface of the earth, directly to see the earth moving around the sun or spinning on its axis. The tricky part is this: whenever you have a limited set of observations (that is, always), there will be the possibility of coming up with two or more theories which all explain those observations. The process of deciding which one (if any) is correct is delicate, and it always involves making more observations in hopes that all but one of the competing theories will fail to explain the new observations. On the other hand, you are always free to revise and improve a theory as new observations come to light, so if your old theory doesn't explain the new observations, you can try to come up with a modification of it which does.
There are two or three things which any good theory must do. The first is, of course, to explain some set of observations. Theories about how many angels can dance on the head of a pin generally fail do this. You should realize that no theory explains everything. There are always some observations out on the fringes which the theory doesn't explain -- either it has nothing to say about them, or no one has yet worked out what it has to say about them, or it does make a prediction and the prediction is wrong. But the theory should have a domain, and preferably a known domain, in which its predictions are reliable. The difficulties on the fringes are always there, and sometimes they point to future modifications to the theory. The second thing a good theory must do is predict the results of observations which have not yet been made. "That's the way God made it" is a theory that can explain any observation, but because it can explain anything it can predict nothing. That makes it useless as a theory. The more predictions a theory makes which turn out to be correct when the observations are made, the more confidence we have that the theory is correct, and the more reluctant we will be to discard it in favor of a new theory. The theory of the earth moving around the sun, etc., makes a huge number of predictions which agree very well with observations. The last thing which we sort of want from a theory is simplicity or elegance. This isn't really a criterion -- we never really throw away a perfectly good theory because it's ugly -- but experience has shown that really good theories usually have a conceptual elegance about them. This experience is summed up in a rule called Occam's razor, which can be stated (this is one of many ways), "given two theories which both explain some set of observations, the simpler one is more likely to be correct." Not that the simpler one is always correct -- people's ideas of simplicity differ, and so people may even disagree as to which one is simpler -- but if one is clearly simpler, don't bet the rent money on the other one.
Two final comments before we move on to the task at hand. First, authority is not evidence in favor of (or against) a theory. Your third-grade teacher may have said that it's so, but why does your third-grade teacher believe it? Eventually, the evidence has to come from observations. Second, when trying to choose between two competing theories you must be careful not to discard one too quickly! If both theories agree with the existing observations, you need to keep both as possibilities, even if you're leaning toward one. Either theory might be correct, and you don't want to throw away the right one. Sometimes you just have to live with the uncertainty of not knowing which one to believe. Don't worry, eventually the issue will be settled. And new ones will come up. That's science.
Now let's look at the observations which lead us to believe that the earth rotates about its axis, and that it and the planets revolve around the sun. We will also need to show why this theory is accepted and its competitors are not. (This will not be in anything like historical order. For that, refer to the textbook.)
Start by looking up at the sky on a clear, moonless night. If you look at the stars for a while your eye will start to pick out patterns. Now go away and come back in an hour or so. You will see the same patterns of stars, but not in the same place as before. The stars will have kept the same positions relative to one another, but they will have moved as a unit. If you put out a camera with the lens open, and leave it for several hours, you'll get a picture which shows the tracks which the stars followed across the sky during the time the camera lens was open. What you see is a nest of concentric circles, with (if you are in the Northern Hemisphere) the star Polaris near the center [see Figure 1.8 on page 9 of the textbook]. We now have two observational facts to explain: first, the stars seem to move across the sky along circular tracks which all have the same center; second, as the stars do this they keep the same positions relative to one another.
[To see this demonstrated, try running the program nitesky1.ctb. Each click of the mouse button advances time by about 4 minutes. You can also get the program to run as a time lapse, and you can get the stars to leave trails behind them as they would on the film of a camera whose lens is left open.]
There are basically two pictures of the structure of the Universe that can account for these two facts. One is the theory that the stars actually do move -- and they must be connected to one another so that they all move together. Aristotle believed this theory; he thought of the stars as being points of light on a big sphere which rotates around the Earth. The other theory is that the stars only seem to move -- actually, they are fixed while the Earth spins slowly (once per day) around its axis. This gives you the illusion, as you are carried along by the rotating Earth, that the stars seem to circle the opposite way in the sky.
Both of these theories account for the observed motion of the stars across the sky, and we can also extend them to include the sun. In the first theory, we assume that the sun also rotates around the earth once per day. In fact, the sun is observed to cross the sky in the same direction the stars do (roughly east to west), and so we might even assume that the sun is attached to the same sphere that the stars are attached to. In the other theory, we assume that the sun and the stars are stationary and the earth rotates, making it look like the sun and stars are moving across the sky. Note that this latter theory would be in big trouble if the sun didn't cross the sky in the same direction as the stars!
With the (meager) evidence that we have at hand so far, there is no way to decide which of these two theories, if either, is right. You might prefer one because it seems more elegant, or because it agrees with your ideas of how the universe should be. Or you might believe one because somebody with some authority tells you which one to believe. But these are weak arguments. After all, Aristotle believed his theory because it agreed with his ideas of what the universe should be like (the Earth at the center of everything, and the heavens built up of perfect circles and spheres). Almost everybody else up until the sixteenth century believed Aristotle's theory because Aristotle was so famous he just had to be right. And now everybody believes that he was wrong. So we need some more observations to decide between the two theories.
Another interesting observation comes if you travel. If you go far enough south, you will notice that the point in the sky which the stars appear to circle around is lower in the sky than at home. Similarly, as you go north that point appears higher and higher in the sky until you reach a (very cold) place where it appears directly overhead.
[Most of this can be demonstrated by running the program nitesky1.ctb and changing the latitude. The screen distorts pretty badly if you get too close to the poles (latitude 90° or -90°), but otherwise you can look at the sky from anywhere in the northern or southern hemisphere. Latitudes south of the equator are given as negative numbers.]
The only explanation I can think of for this is to theorize that the earth is round, and that "up," the direction from your feet to your head when you are standing, is always away from the center of the earth. The diagram then shows how this works. For each position on the Earth's surface it shows the directions "straight up," "along the ground to the north," and the direction you have to look to see the star Polaris, which is very close to the point in the sky around which all the stars seem to revolve. Clearly this direction appears closer to "straight up" the farther north you are. (You should think about what happens when you travel past the north pole, or south past the equator.) Incidentally, the ancient Greek philosopher Erastosthenes had another argument for the earth being round, which the book explains on page 24.
It's interesting that the earth is round, but unfortunately this does not help us decide which of our two theories to believe. The sun and stars still seem to move across the sky the same way whether they actually move around an axis which comes vertically out of the earth's north pole, or whether they stand still and the earth rotates around that axis. It's only the relative motion of the earth and the stars that counts.
Now let's take a clock, which divides our 24-hour day evenly into hours, minutes, and seconds, and use it to get more detailed information about the motion of the sun and stars. This leads us to a striking observation: each star which rises on the eastern horizon or sets on the western horizon during the night does so about 4 minutes earlier each night! (To be more precise, the figure is 3 minutes and 55.9 seconds.) Thus if you see a star rising in the east at 9:00 p.m. today, then tomorrow it will rise at 8:56, the next day at 8:52, the next at 8:48, and so on. After 15 days it will rise at 8:00, after 30 days at 7:00. After 90 days, it will rise 6 hours earlier than it did today, after 180 days it will be 12 hours earlier. After 360 days it will rise 24 hours earlier than it does tonight, or in other words it will be rising -- for the 361st time - at the same time on the clock as it rises tonight. This is all figured using the estimate of 4 minutes for the nightly advance; if we use the more precise number of 3 minutes 55.9 seconds, we find that the stars come back to rising at the same time after 365 (and a quarter) days. This is remarkable! The motion of the stars across the sky is a little bit faster than that of the sun (the sun takes 24 hours to go once around the sky, but the stars make the same trip in 4 minutes less time), and the difference is related to the length of the year (in one year, the stars circle the sky exactly once more than the sun does).
The first theory can easily accomodate this new fact. All we need to do is abandon our preliminary idea that the sun is attached to the same sphere as the stars, and assume instead that the stars are on one sphere and the sun on another. The two spheres rotate at slightly different rates. This addition to the theory doesn't explain why there is that peculiar relation between the rates at which the star sphere and the sun sphere rotate; rather it could accomodate any rates for the two spheres. Pending some clever explanation, it is necessary to just call it a coincidence. This isn't very satisfying, but it could be true.
The second theory has a somewhat more difficult time handling the new information. We can no longer assume that both the sun and the stars are stationary while the earth turns on its axis under them. We can assume that either the stars or the sun remains stationary, but then we must assume that the other moves. Somehow. There are many ways of tweaking this theory to get it to incorporate the new information. We could assume that the sun is stationary and that the stars circle slowly around the earth, or that the stars are stationary and the sun circles slowly around the earth, etc. Thus this one theory splits into many. One of the many, however, gives a particularly elegant explanation of the relative rates of rotation of the sun and stars. This is the only one I will continue with, since of all the theories that have the earth rotating about its axis, it seems to be the best (explaining, as it does, something that the other variations fail to explain). It is the theory that the earth rotates about its axis, once per day, and at the same time revolves around the sun once per year. This makes at least some sense, because the year very obviously has to do with the sun's motion (or with the relation between the earth and sun): one year is the time it takes for the point at which the sun rises to go from its southernmost point to its northernmost point and back. [Why this happens has to do with the fact that the earth's axis of rotation is not parallel to the plane in which it revolves around the sun, but that is a side issue to the topic we're working on in these notes, and so I won't discuss it any further. Suffice it to say that the year is determined by the relation between the earth and the sun, and the stars have nothing to do with it.] Now let's see how this theory explains the relation between the motion of the sun and that of the stars.
To keep things as simple as possible, let us pretend that (a) the earth orbits the sun in exactly the same time that it takes to spin six times about its axis, and (b) the axis about which the earth spins is perpendicular to the plane of its orbit around the sun. The diagram shows the earth rotating counterclockwise about its axis while it revolves, also counterclockwise, around the sun. There are two observers shown on the earth. To start with, the earth is in position 1. It is noon for one of the observers (that is, he sees the sun directly overhead). It is midnight for the other observer, who is located halfway around the world from the first; let us suppose that there is a star conveniently located directly above her head, and very far away. As time passes, the earth makes its way to position 2, rotating about its axis all the while. At this point it is sunset for the first observer (he sees the sun on the horizon), and it is sunrise for the second observer (she also sees the sun on the horizon). When the earth gets to position 3, the first observer now sees that star directly over his head. Time passes, the earth continues to work its way around the sun and turn about its axis, the first observer sees the sun rise while the second sees it set, and later still the earth reaches position 4. This is the point at which the second observer again sees the star directly overhead. That star has now completed one circuit around the sky. But it is not yet noon for the first observer, because his head is not yet pointing toward the sun! It is not until the earth turns a bit farther, and progresses to position 5, that the first observer sees the sun directly overhead again.
[To continue the saga, run the program sidereal.ctb. It should not be hard to convince yourself that the first observer will see noon return for the fifth time (not counting the one when we started) when the earth gets all the way around to position 1 again, and that this will be the sixth time that the second observer sees the star return to the overhead position.]
Another way of saying what we have just seen is that it takes slightly less time for the stars to circle the sky than it takes for the sun to circle the sky. In our example, the time from one noon to the next, which is called the solar day, is one-fifth of a year. The time it takes the stars to complete one circuit of the sky, which is called the sidereal day, is only one-sixth of a year. For the real earth, the numbers work like this: The earth completes an orbit of the sun in 365.26 days (that is, 365.26 solar) days. In this time, the stars circle the sky exactly once more than the sun does, so we go through 366.26 sidereal days. Thus the sidereal day is 365.26/366.26 of a solar day. The solar day is 24 hours, or 24×60×60 = 86,400 seconds, so the sidereal day is (365.26/366.26)×86400 = 86164.1 seconds, which is shorter than 24 hours by 235.9 seconds, or -- voila -- 3 minutes 55.9 seconds!
[Now a question for the reader: how would the story change if the earth were revolving clockwise around the sun instead of counterclockwise?]
At this point neither theory has much of an edge on the other by reason of elegance or simplicity, although the second does manage a nice explanation of one observation for which the first can only manage a pretty lame one. But this is probably not enough to convince anyone to discard the first theory. Not yet.
Another thing you notice if you watch the stars very carefully, keeping notes of their positions from night to night, is that not all the stars really keep the same positions relative to one another. There are a few (five that are visible to the naked eye) which drift relative to the other stars, and relative to each other. That is, they circle the sky like the rest of the stars do, except that they do not come back to exactly the same spot after one sidereal day, as the vast majority of stars do. If you record what spots in the sky these misbehaving stars are at after each sidereal day, or equivalently where they are relative to the other stars, then these spots are the things that drift. The drift, incidentally, is always close to the path that the sun takes across the sky. The ancient Greeks called each of these five stars a "wanderer" or, in Greek, planetes. This has worn down to become our word "planet".
[To see this behavior, run the program nitesky3.ctb. Each click of the mouse advances time by one sidereal day, so that the stars (other than the planets) have all returned to the same place in the sky. You can also have the program show where the sun is in the sky -- remember that the program only looks at the sky once a day! -- and you can change your latitude, your altitude (which means how many degrees above the horizon you are facing, not how high you are off the ground!), and your azimuth (how many degrees east of north you are facing). Try looking at an azimuth of between -120° and -180°, or between 60° and 180°.]
It is easy to incorporate these planets into both theories. In the first theory, you assume that each planet is rotating on its own sphere at its own rate. In the second, you assume that each planet is another body which, like the earth, revolves around the sun. In both theories you have to put in the rate at which each body rotates around the earth or sun, as the case may be.
The planets, however, execute some pretty bizarre maneuvers relative to the stars. The one called "Mars", for example, usually drifts eastward relative to the other stars, but at regular intervals it stops and drifts westward for a few weeks, then resumes its eastward drift. Each planet also changes in brightness, being brightest when it is drifting westward. Explaining this behavior is quite a job for either of our theories.
To get the first theory to handle this behavior, a major modification is needed. Instead of assuming that the planets circle around the earth, you must assume that each planet revolves on a circle whose center revolves around the earth. (The circle on a circle is called an epicycle.) If you choose correctly the speeds with which the planet moves around the epicycle and the center of the epicycle moves around the earth, you can get the planet to drift eastward most of the time but sometimes westward. [A good illustration is Figure 2.9 on page 33 of the textbook. To get a better one, run the program epicycle.ctb] This also explains why the planets appear brightest when they are moving westward: at these times they are closest to the earth, and so they will appear brighter at these times even if their intrinsic brightness never changes. Unfortunately, it is very difficult to get precise agreement between where the theory says the planets should be and where the planets really are. In order to get this agreement to the level of precision of the observations available in his day, Ptolemy (circa 140 A.D.) had to do things like putting the planets on epicycles whose centers move on epicycles whose centers move on (etc.) -- up to a total of over 80 circles -- and to allow each of the basic circles to be centered on some point other than the earth. And more precise observations are available today. Clearly this theory is in trouble with Occam. In order to match observations we have to make the theory more and more complicated, and we really get very little out of the theory other than what we have put in.
The second theory has a somewhat easier time of it than the first. In fact, having the earth and the planets all orbiting around the sun immediately allows the planets periodically to drift backward across the sky: if Mars is farther from the sun than earth is, and moves more slowly along its orbit, then it appears to move westward across the sky as the earth passes it. This happens when Earth and Mars are closest together, which accounts for the fact that Mars appears brightest at these times. [One place to look for an illustration is Figure 2.8 on page 34 of the textbook. To see it animated, run the program retrogrd.ctb.] Unfortunately, detailed agreement between the predictions of the theory and actual observations is still hard to get. To do as well as Ptolemy's complicated theory does, you would still have to make the planets travel on epicycles on epicycles on ... whose centers circle around the sun. Thus this theory rapidly gets almost as complicated as the other one. But there turns out to be one small modification to this theory which allows us to get much better agreement with the observations than we could get from Ptolemy's theory, and still keeps the theory fairly simple. We do this by abandoning the idea that the planets always have to follow circular orbits. If we assume instead that the planets, and the earth, follow elliptical orbits instead of circular ones, then we can get excellent, detailed agreement between the theory and the observations.
At this point, the second theory has a definite edge over the first in simplicity and elegance, and it also agrees somewhat better with the observations. By now we should be leaning heavily toward it. But there is more to come.
If we assume that the earth and the planets follow elliptical orbits, we still have to propose what the sizes and shapes of those orbits are, and how the earth and planets move along them. Of course, we do this in such a way as to get the best possible agreement between our theory and the observations. To describe the results, I first need to describe the geometry of an ellipse. To create an ellipse, first choose two points (which we will call the foci of the ellipse), and choose some length which is longer than the distance between the foci. A point is on the ellipse if the sum of the distances from that point to the two foci is equal to that chosen length. (If you happen to have chosen the two foci to be at the same place, then your ellipse is really just a circle, and the chosen length is the circle's diameter. So we say that a circle is a special case of an ellipse.) Now for some more terms -- you should refer to the diagram. The line going from one end of the ellipse, through the foci, to the other end is called the "major axis", and it is the longest line you can fit inside the ellipse. The midpoint of the major axis, which is halfway between the foci, is called the "center" of the ellipse. The line perpendicular to the major axis through the center, from one side of the ellipse to the other, is called the "minor axis". Finally, the semimajor axis is just half the length of the major axis, and the semiminor axis would then be half the length of the minor axis. Now for the results. To get the best agreement between the theory and the observations, we must assume that:
A really good theory gives you more than you put into it, and one place where our theory starts to do this arises from an objection that a defender of the other theory might raise. The objection is this. If the earth really orbits around the sun, then in the course of a year the relative positions of the stars in the sky should change, because we would be looking at them from different vantage points at different times of the year. For instance, the diagram shows two stars and the earth in its orbit. When the earth is closest to the two stars, they would appear widely separated in the sky. That is, you would have to turn your head through a rather large angle to go from looking at one of them to looking at the other. Six months later, that angle would be smaller, and the stars would appear closer together in the sky. Another six months later and they would again appear far apart. Why, the geocentrist might ask, is this not observed?
The answer is that it is observed, it's just that the change in the angle of separation between any pair of stars is tiny. In fact it is always less than one second of arc, which is one sixtieth of one minute, which is one sixtieth of one degree. For comparison, the moon takes up an angle of about half a degree in the sky. You can't notice a shift in angle of less than a second of arc without good instruments. With those instruments, however, you can not only detect it but measure it, and from such measurements you can eventually determine how far away some of the stars are.
So far the second theory wins over the first on all counts. It agrees more precisely with the observations they were both constructed to explain, it is far more elegant, and it gives some results beyond what we put into it. We certainly expect that it is true -- or at least closer to true than the other. But the thing which really settles the issue is the way it ties in with a separate theory, which was constructed to do different things and which is extremely well verified by experiments (up to limits which are themselves quite well understood). That theory is an explanation of how terrestrial objects move, and how their motion is affected by forces. It was proposed by Isaac Newton in the late 1600's and consists of three statements:
First let's pretend that the planets follow circular orbits -- even though they are actually elliptical, the orbits are pretty close to being circles -- and ask what force could possibly account for Kepler's third law, the one that says that R³/T² is the same for all planets. According to Newton (the second law), a force is required to keep an object moving in a circle, since it is always changing its direction of motion. Quantitatively (and if you haven't had a physics course somewhere, please just accept this), if you're going to make an object of mass m go around a circle of radius R at constant speed, so that it takes a time T to go once around, then somehow you have to exert a force on it which is directed toward the center of the circle and is proportional to mR/T². If you tie a rock to a string and whirl it around your head, it is the tension in the string that provides that force; if a car goes around a curve, it is the friction between the tires and the road; when you go over the top on a Ferris wheel, it is the earth's gravity. For a planet circling around the sun, it seems natural to assume that the sun exerts a gravitational attraction on the planet. This would be a force toward the center of the circle (where the sun is), as we need. The magnitude we need is proportional to mR/T², where m is now the mass of the planet, R the radius of its orbit, and T its period. To get Kepler's third law to come out, the T² will have to be proportional to R³, and so the force must then be proportional to m/R² -- proportional to the mass of the object being attracted, and inversely proportional to the square of the distance between the two objects. Finally, we pull in Newton's third law, which says that if the sun is pulling on the planet, then the planet is also pulling on the sun with an equal force. Since the force must be proportional to the mass of the object being attracted, and both objects are being attracted to each other, then the force must be proportional to both masses. The upshot is this: in order to make the planets follow circular orbits at rates which obey Kepler's third law, the sun must exert a gravitational attraction on each planet which is equal to GMm/R², where M is the sun's mass, m is the planet's mass, and R is the distance between them. G is a constant of proportionality which must be measured in the laboratory (a difficult measurement, which wasn't actually accomplished until a hundred years after Newton did this calculation.).
Now we know what force the sun must exert to keep the planets in circular orbits; does this force law also allow elliptical orbits? The answer takes some involved mathematics (junior year physics), but it is "yes". It allows both circular and elliptical orbits, and also parabolic and hyperbolic orbits (which are followed by some comets!). For planets on elliptical orbits, the calculation shows that all three of Kepler's laws would be satisfied, with a couple of small modifications. The calculation does not quite predict that the sun would be at one focus of the ellipse, but rather that the center of mass is at the focus. This is a tiny difference if the sun is very much more massive than the planet. The equal-areas law comes out (it is conservation of angular momentum). The third law appears in a more specific form, namely that R³/T² is proportional to G(M+m). This depends on the mass m of the planet, so it is not exactly the same for all planets, but again if the mass of the sun is much larger than the masses of the planets, then the differences are tiny. This is a tremendous bonus -- it gives us a way to measure the sun's mass! All we need to do is observe the motion of the planets and infer each planet's period and its distance from the sun, and then the mass of the sun is given to very good precision by the formula. Similarly, whenever we see a (relatively) small object orbiting a larger object (e.g., a moon orbiting a planet), we can determine the mass of the object being orbited the same way.
If we want to believe that Newton's laws account for the motion of the planets, then we have to accept that Kepler's laws are not exact. In fact, there are some tiny discrepancies between Kepler's laws and the actual motion of the planets across the sky, and the modifications above due to Newton's laws make the discrepancy even smaller. We do even better if we do the Newton's-law calculation correctly, including not only the gravitational attraction between the sun and the planet, but also the gravitational attraction of all the other planets. This makes the agreement with observations all but perfect. (The little remaining discrepancy was eventually taken care of by a new theory -- Einstein's general relativity.)
By now the geocentric theory is just about dead. It explains the motion of the sun, planets, and stars across the sky to pretty good (but not very good) accuracy, but at the cost of tremendous complexity -- and it does very little else. The other theory, that the earth rotates about its axis and that it and the planets revolve around the sun under the influence of gravitational forces according to Newton's laws, does much more and does it more simply. It explains the observed motions of the sun, planets, and stars across the sky, which is what we set out to do, with extraordinary precision. It also explains the motion of comets, suggests that the sun is much more massive than the earth and the planets, and gives us a way to measure its mass. Finally, it ties in with a huge body of experimental evidence that Newton's laws correctly describe the motion of objects on earth. The theory nowadays has no serious competition.