Program 11 - "Kepler's Laws of Planetary Motion"

 

Music This means something.I know it.It must mean something.Circles, I'm so obsessed with circles.I don't know what I'm thinking.At night I dream about circles and I want to eat Cheeriosand Dorios and Spaghetti-o's and circles, circles, circles, what am I thinking...Oh, oh, I ruined my circle.Oh, oh, no.Wait, it's just bent, a flattened circle.What a concept.Maybe it's not ruined, maybe it's just bent.

 

MusicSilico: "We are back, with the Nature of Physical Science.The telecourse that helps you to focus with no hocus or pocus."This is Program 11, Lesson 2.3, "Kepler's Laws of Planetary Motion."Before we're done with this program we will have learnedof the important contributions of Gilbertand Bacon to the river of knowledge.We will learn Kepler's three laws of Planetary motion and usethem to describe the orbits of the planets.We will learn the anatomy of the ellipse, and we will learnabout the significance and the implications of Kepler's laws.Here are the objectives for today's lesson.These objectives are also in the Study Guide at the beginning of the lesson.Before you begin to study the lesson, take a few minutesto read the objectives and the study questions for the lesson.Look for key words and ideas as you read.Use the Study Guide and follow it as you watch the program.

 

Focussing on the Learning Objectives will help youto study and understand the important concepts.Compare the objectives with the study questions for this lessonto be sure that you have the concepts under control.Kepler's laws of planetary motion mark a really important turningpoint in the transition from geocentrism to heliocentrism.They provide the first quantitative connectionbetween the planets, including earth as one of the planets.But even more important, they mark a time when the importantquestions of the times were changing.By this time in the middle of the 16th century there were manyintellectuals who favored the simplicity of a heliocentricsystem, but they were unwilling to throw out the comfortablegeocentrism of Ptolemy without good evidence in support of a new system.The focus of the fundamental questions shifted from whichsystem it "really" was to what kinds of suppositions would berequired if we were going to justify a heliocentric system.In other words, what kinds of things would we have to change?And what kinds of things would we have to know aboutif we were going to assume there was a heliocentric system.So, it's kind of difficult to characterize this, but you know,we, the common thing we learn is that by this time people stillhad not accepted the heliocentric theory at all.That's really not quite true.What I mean is that it mattered less to keepthe old theories just for the sake of keeping them.Back in the old days, of course, it was important to keep the old theories.But the evidence for and against both systems, both theheliocentric and the geocentric system, was being reworked,and most of the thinkers of the time would have easily acceptedheliocentrism if it was feasible.

 

The problem was it wasn't really feasible.Because the nagging thing that remained was the problem of motion.That is, it's cause and effects.You see, even if you accepted heliocentrism, simply basedupon Brahe's data or other evidence, there's still theproblem of what makes the planets go around in curvedpaths regardless of the shapes and the speeds of the planets.You still have to consider motion.Remember that Aristotle's cosmology linked us all to the prime mover.So, the question is or I should say the question in Kepler's timewas is the Copernican system really a systemor is it simply another device to calculate?The Copernican system did make astronomical calculationseasier, but it was seen by most mathematicians around the endof the 16th century as nothing more than another mechanical device.To nonscientists in Kepler's time, like to most nonscientiststoday, who could care about the mathematical simplicityor the absence of it, they found no serious objectionsto the geocentric theory and were quite unwilling to exchange it.I mean, why do you want to change a good theoryunless there's some good reason to change it?That means in the same way that you don't want to keep a theoryjust to keep it, you also don't want to change it just to just to change it.So, around this time there were two other individuals,both of whom were English scientists whose ideasadded to this growing river of knowledge.William Gilbert was of Tycho's generationand Francis Bacon was a contemporary of Kepler.

 

So, I want to take a minute to give you some background on these two guys.William Gilbert lived from 1544 to 1603, roughly the same as Tycho Brahe.Brahe was born in 1544 and died in 1601.Gilbert was the Royal physician to Elizabeth I in England whoat that time was managing England's rise to world power status.Gilbert published a book called, "De Magnete," which means"the magnet" or on magnetism, in which he summarizedeverything that was known to date on the properties of electricity and magnetism.There wasn't much known in those times about either one of these things.Mostly what was known about magnetism was that there werethese certain kinds of naturally occurring minerals or rocksor stones, if you like, called lodestone which seemed to alignthemselves with the earth's magnetic field.They'd been used for compasses for many years,both in the Mediterranean and in the Far East.The concept that most intrigued Gilbert about this whole thingwas the ability of a magnet to attract another magnetthrough empty space, apparently without anything intervening.This will turn out to be an important idea as we'll see for Kepler.Among other things, Gilbert's work was a treatiseon lodestones and specifically their use in navigation.In order to predict, for example, how the magnet would behaveor how the compass would behave, in different partsof the earth, Gilbert took a big piece of lodestone and carvedit into a spherical shape and then used that as a model, a modelof earth to predict where the compass needle would pointand how it would behave in different locationson a spherical earth, not a flat earth, but a spherical earth.So he took this piece of lodestone and he held magnets in variousplaces and he watched how they lined up and he used that in the treatise on navigation.

 

So now recall that although the geocentric theory was stillin favor, the flat earth idea had been lost forever.This had happened in the early part of the 16th century whenMagellan's fleet had sailed around the world earlier in the century,15, 0, 15, 0, 0 something or other.So, exactly what this idea of magnetism has to dowith the planetary motions, we'll get to in a little while.When you think about this, can you see a connection now?What connections could there possibly be between twomagnets attracting each other and the motion of the planets.It might be easier for us today to figure this out thanit was for Gilbert and people in his time.So, even without that connection, which we'll establish when weget to Kepler's laws, Gilbert made a very important statementwhich went far in defining the new scientific paradigm that was about to bloom.In fact in the preface to "De Magnete" he wrote a quotewhich I'll read to you, which basically set the tonefor the scientific revolution, which he helped to precipitate.This is the preface:He says, "To you, alone true philosophers, ingenuous menwho not only in books but in things themselves look for knowledge,have I dedicated these foundations of magnetic science--a new style of philosophizing." You see what he's doing here, don't you?He's basically saying, "Remember that Augustine and all theseother people who'd said look only internally."What Gilbert is doing is saying, "Ingenuous men who look notonly in books, but in things themselves."In other words, check reality.No matter how good your ideas are, it's the thingsthemselves that give you knowledge.Along these same lines, was another Englishman namedFrancis Bacon.

Bacon was a contemporary of Kepler born in 1561, died in 1626.Francis Bacon was one of those interesting Englisharistocratic characters who didn't do much under Queen Elizabeth I.He was kind of young, but he found his place when Elizabeth'ssuccessor, James I came in to rule, and worked his way upthrough the aristocratic ranks and into the bureaucracyand eventually in 1621 became the Lord Chancellor which is afairly high position in the King's Cabinet.But unfortunately, in the same year that he accepted thatCabinet post, he plead guilty to accepting bribesand sort of went into a forced retirement.It's more like get out of my sight and get out of the castle sort of thing.So his career as a statesman was very tainted.But his philosophical musings helped to spurn the growthof experimental science in England where the scientific revolutionreally would occur with Newton a hundred years later.His contributions to the river of knowledge basically werein his inductive approach to experimental science which was later refined by Galileo.He put forth this idea and an essay titled, "The New Atlantis."The New Atlantis was a society he proposed in the New Atlantis was a utopian society.

 

You know what utopia is, right?Utopia is where everybody is happy and has everything they want.But the New Atlantis was a utopian society based upon scientific principles.That's a scary thought, isn't it?A society based on science.Whew.This was a completely new idea in Bacon's time and it really didhelp to define what would become the modern scientific principles.I mean, after all, if you're going to have a society foundedon scientific principles, you need at some point to say what those principles are.I used the word earlier about Bacon's inductive approach.We'll study induction later and compare it with deduction,but for now we need to understand that induction is a processwhere you go from the specific to the general.But again, we will study this down the road, but it's a goodidea to think a little bit ahead on that.You might want to look this up in the dictionary.

 

Look up the word "induction" in the dictionary,and at the same time when you're done looking up induction, look up the word, "deduction."And, so what do you think?You do this, can you write a short essay comparingand contrasting these two terms, and maybe even using anexample of the difference between the two.So the idea with induction as far as the specifics of the generalis that if you experiment, this is the key word, "experiment,"that experimentation will allow you to gather factsand according to Bacon, from those facts you should be ableto ascertain the underlying laws that cause the things to behave that way in your experiment.This is contrary, of course, to a deductive method in which youstart with first principles and you derive or you predictthe behavior of something based upon what you assume to be theories.This is the way this has been done all the way through the Greek classical period.Is to have, first you invent the theories and you make predictions.So, the other thing that Bacon contributed here might be,you might call this actually our modern concept of researchand development or even trial and error in problem solving.

 

Bacon made statement which I think points very nicely to this.He said, "Truth comes out of error more easily than it comes out of confusion."In other words, it's better to make an experiment and come outwith wrong results than it is to not do theexperiment at all and be totally confused about it.So, again, the idea of induction raises its head in here.And now we're ready to turn our attention to Kepler's laws themselves.

 

The laws are most simply stated in their modern form,largely because Kepler did not state them clearly.In fact, it's really difficult even to locate the lawsin his writings, which ramble on about harmony and justificationfor abandoning the Ptolemaic system, and defense of Copernicus, and so on.I mentioned before, I think in the last program, that Kepler was not a particularly good writer.In fact, to illustrate the style of writing and also the obscurityof the principles, I want to read you Kepler's laws in his own words.Pay attention to this.Here's what Kepler wrote:Ready for this?"...equal diurnal arcs on one and the same eccentric are nottraversed with equal velocities, but that these are differenttimes in equal parts of the eccentric are to each other asthe distances from the Sun, the source of the motion;and on the other hand, that the times being supposed equal, as,for instance, one natural day in each case, the true diurnal arcscorresponding to them in a single eccentric orbit are inverselyproportional to the two distances from the Sun.It has likewise been shown by me that the orbit of a planet iselliptical, and the Sun, the source of motion, is in one of the foci of this ellipse."Silico: "What did you say, could you repeat that please?"Yeah, that's funny.So, everybody all together, repeat what Kepler said.What did he say?I can't hear you.

 

 

OK, come on, seriously, this excerpt contains Kepler's firstlaw and even parts of the second law.And you might want to look at this quote in the Study Guideafter you've learned the laws and their meanings to see if can figure out what he really says.But can you see from this why the work didn't cause much stir?This, in fact, is from De Harmonice Mundi which isThe Harmony of the World which Kepler published in 1619and this is actually a restatement of the laws in a clearer formthan he had originally stated in his 1609 work in The New Astronomy.You may remember The New Astronomy based on causationand the work of art and the really long thing.So, I want to point out to you, please don't feel like you need to memorize this.But I wanted to point out to you that Kepler's style of writingwas so obscure that it really took Newton's genius, and in fact,it's a tribute to Newton's genius, as if he needed more tribute,that he was able to find in this statement one pieceof his solution to the gravitational puzzle.And he was actually able to figure out what Kepler was sayingand put it for us in our modern form so that we know what it really says.There is one of the things you'll notice about this passage when you study it.Kepler says it twice, very explicitly.That the sun is the source of planetary motion.

 

The sun is the source of planetary motion.This is something completely new with Kepler.And it's so hidden in this statement that it's no wonder that Church officials didn't see it.I mean, who can read through this stuff and pick out Kepler saying that?Remember, in Aristotle's thing, the cause of motion, is the prime mover.So, here Kepler is bringing this heliocentric thing backand saying it's the sun that's the cause of motion.Well, I'll spare for you at this point Kepler's wording of the third law, for now.But the third law was Kepler's favorite, and we'll come backto it after we examine all the other laws in a little bit more detail.

 

OK, so now let me state the first law for you in modern terms.The first law basically says that the planets, including earth,revolve around the sun in elliptical orbits.The sun is at the focus of one ellipse, I should say,one of the focuses of the ellipse, and the other focus is empty.I'll say that again.The planets, including earth, revolve around the sun in elliptical orbits.The sun is at one focus of the ellipse, the other is empty.Not circular orbits, not spherical rotations, but elliptical orbits.This is such a simple statement that it's really amazing that ittook so long and it was so difficult for someone to come up with this idea and to produce it.In the picture of the ellipse, here, we see that what this means isthat sometimes the planet is closer to the sun than others, and sometimes it's further away.You also notice in here that although the second focusof the ellipse is a geometrical point of symmetry, it has no physical reality.Only one of the focuses of the ellipse has anything to do with that.It will be useful at this point, I think, to take a closer lookat the anatomy and the properties of the ellipse.The ellipse is a conic section, like the circle.We saw in the last program how the circle and the ellipse arerelated in terms of slicing or sectioning a right circular cone.Now, we want to consider the properties of the ellipse from a slightly different perspective.Simply from the perspective of the flat figure that we call the ellipse.

 

OK, so the circle can be defined as the set of all points equidistant from a single point.Remember that the circle and the ellipse are in the same familyof curves called the conic sections.So, in other words, all the points on the circle are the same distance from the center.The center is a single point.That distance is called the radius of the circle.So, what about the ellipse?The ellipse can be defined as the set of points equidistant from two points.Each of the two points is called a focus, and I'll show youin a couple minutes how you can draw this and we'll look at the anatomy a little closer.Each focus in the ellipse plays the role for the ellipse that the center plays for the circle.So we might sort of make an analogy like this:The ellipse is to the circle as the rectangle is to the square.What does that mean?Something you can write about or think about, if you want to.The ellipse is to the circle as the rectangle is to the square.

 

Well we want to look at the anatomy of the ellipse,but it's not necessary for us to completely dissect it as a mathematician might.Our interests are simply in seeing what the terms mean and how itaffects the shape of the ellipse, and how we describe the ellipse.It's important for us simply to see how it's described and how itscharacterized as well as how it's constructed and some of its properties.

 

OK, so in general there are several terms we need to know about the ellipse.The first of these is the focus.There are two focuses, or foci, of the ellipse.The further apart the two foci, the more squashed the ellipse appears.The two foci are highly symmetrical, meaning that they're mirror images of each other.In fact you can cut the ellipse in half either way and you'll still have a mirror image.Numerically, we talk about the focus as the distancefrom the center of the ellipse to one focus.So you might see that this plays the same role in the ellipse that the radius plays for the circle.There's a difference, though.Unlike a circle which has a single radius, each ellipse has both a long axis and a short axis.The axis is the length of any line which cuts the ellipse in half.OK, so by definition, cutting it in half means that any axis willpass through the centerpoint of the ellipse.The semi major axis is one half of the length of the ellipse.It's the distance from the center to the furthest point on the ellipse from the center.We'll see this in the picture in a minute.The semi minor axis is one half of the width of the ellipse,or the distance from the center to the closest point on the ellipse.

 

OK, now we've seen how the ellipse is described,let's take a look at the construction of the ellipse.But before we do that, let me remind you once again,you do not have to memorize and reiterate all these facts about the ellipse.The point is, and this is the point, that the ellipse is a very Pythagorean figure.It's not a circle, but it's very Pythagorean because it hasmany interesting numerical and geometric properties.You see, in the time of Plato, the conic sections had not yet been described.

 

Nobody knew about the ellipse and the hyperbola and the parabola.And it was not really known in Plato's time howsimilar the ellipse and the circle really are.In fact, it wasn't until Euclid and Alexandrian Greeksin Alexandria that we see an in depth study of these curvingplane figures such as the ellipse and the hyperbola and so forth.So, except for the circle, the classical Greek mathematiciansconsidered mostly Polygon shapes.That is, things with flat side, not things with curved sides.It was simply too sophisticated for them at the time.So, it wasn't really too much of a stretch for Kepler to considersubstituting one geometric figure for another when the two wereclosely related as we saw in the last program.So, seeing that the ellipse has these properties it should helpyou to visualize the planetary motions and understand thembetter, and also will help you to understand what Newton didwith this when we get to that program a little bit later on.It's sort of like this, understanding how your cars'engines work might make you a better driver, even if you don'tunderstand the engine well enough to be able to takeit apart and put it back together again.It's the same with the ellipse.

 

OK, let's go to the ELMO now and I'll draw some ellipses and some figures for you.The first thing I want to do is not to draw an ellipse, but to draw a circle.A circle and an ellipse are related so what I've gothere on the ELMO is a mat pin and a string.Well, what I'm going to do here first is to draw a circle.So the circle, you may recall is defined as the centerof all points equidistant from a single point.So, I can fix that by making sort of a string compass here.So what I'm doing is just not stretching the string,but allowing it to extend to its full length.And following it along and what I get when I do this is a circle.Well, it's actually not a bad circle.String's a little too long to go around it, but that'sbecause I need it for the ellipses later on.So, on this circle, every point on the circle isthe same distance away from the center.That part of the circle, of course, we know by the word, radius.It's the radius of the circle.So what I'm going to do now is to put a new sheet of the paperon here and you remember that for an ellipse we don't need just asingle point, but rather we need two points.

 

So, I can make the second point with a second mat pin.I want to start these out with the pins fairly close together.So, the two pins now represent the two focuses or foci of the ellipse.I'll loop the string over them like this,and how you see that what happens is that because thestring is a fixed length, then the distance from the focusto the outside of the string is always the same.Right?It's the sum of this side plus this side.So, if I draw it here and I draw the ellipse like this.As I'm drawing this I want to remind you that you'rewelcome to try this at home.This not something that only professionals can do.And I think you'll get a much better appreciation for whatthe ellipse is and how it's constructed.So, what do we have here?You know, that looks like a circle to me.But part of the problem we've been havingwith this heliocentric, geocentric thing and with the planetaryorbits is distinguishing between the planetary orbits that are circles.Remember that all of the devices and so forth were put in thereto begin with to deal with the fact that circles are not quite circles.So, I can change now and make the ellipse a little more squashed,because all I have to do is move the focuses or the foci a little bit further apart like this.

 

Now, what do you think is going to happen to this new ellipse that I'm going to draw?Is it going to look like the old one?Is it going to be bigger, smaller?What's the difference?You see what's happening already, right?But now the same technique but with the foci further apart,the ellipse becomes squashed down this way, but it alsobecomes reduced in size this way.So the effect is to squash it.I think you can imagine what would happen if I was to movethese very far apart, like say over here.What do you guess this ellipse is going to look like?Now let's see.Suppose I draw this and sure enough still get the samebasic elliptical shape, sort of the flattened shape, but now younotice that the ellipse is very elongated.I can keep doing this.I can move them further and further apart and I can makeellipses that are more and more squashed.

 

Now that we've seen the construction of the ellipsewe can look at some of the other properties.Hopefully you're beginning to understand why these shapesheld so much fascination for the early mathematicians.Hopefully also, you're beginning to ask questions.These questions should be things, for example, like why do theseshapes have mathematical properties?What is it about the universe that causes shapes to have mathematical properties?And, is there really magic in these numbers or not?Well, let's take a look here.

 

Let's go back to the ELMO and I want to point out to you thatwhen I was constructing the ellipse what I was really doingwas using the property that the ellipse is really the focus,I should say that the center of all points is equidistant from two points.Those two points are called the focus, and what this reallymeans is that if I take a line and draw from one focusto the outside of the ellipse, let me move this a little bit,and then draw it back to the other ellipse.If I add up the length of this line and this line, the sum of those is a fixed number.Right, it has to do with the string, with the length of the string.But, if I go over here and take another point on the ellipse,and draw this line to the focus and draw this line to the focus,so if I add up the sum of this line and the length of this line,the sum is the same as the original line segment.In other words, every point on the ellipse has that same distance.I think we can take a look at this on the slide, a little bit betterdrawing and get a better sense of how this anatomy works.

 

OK, so, why is this so?Why does it work like this?You know what the answer to that is?I don't know either.The fact is that we know it works like this and I don't thinkanybody really has an explanation for whythe ellipse has these properties.

 

OK.The next one.The ellipse, remember.I mentioned earlier that the ellipse is a Pythagorean figure.Well, I didn't really say that I guess, but we know that circle'sa Pythagorean figure and ellipse is related to the circle,so what's the relationship between the ellipseand the circle as far as Pythagorean goes?This is really a pretty neat thing.

 

OK.Here on the ellipse.I'm going to draw the ellipse out like this.I'll kind of exaggerate this, elongated, so when we wantto consider the ellipse, we can sort of put a cross here on itwhere the center of the cross here represents the center of the ellipse.Once again, my drawing is not to scale and I'll showyou another better picture here in a minute.The length here is what we called before the semi major axis.This is semi major.It's called semi, means half, major means its the long, longer of the two axis.In this particular picture I'm going to label this with the letter "A."This length.The width of the ellipse is called the called the semi minor axis.OK.Semi minor.Again, minor meaning the smaller of the two distances.Semi meaning half of that.

 

OK.At the same time every ellipse has a focus.I'll put this focus over here and, of course, there is a matchingfocus on the other side and I mentioned earlier that thenumerical way of characterizing the focus is the distancefrom the center of the ellipse out to the focus.So, I'm going to label this "C."So what, you say.Well, you notice, if you take a look at this that the "C"and "B" are perpendicular to each other.They're at right angles.I drew this line at right angles.So what's the relationship between the semi majoraxis, the semi minor axis and the focus?Well, suppose I call semi minor axis "B," and thenconnect these two together like this.Remember the word, hypotenuse?I have a right triangle here, right?Pythagoras' right triangle.The relationship on this triangle is that C squaredplus B squared equals A squared.

 

If we go back to the screen, I think you can see the pictureof this a little bit clearer, little bit better drawing.The amazing thing about this is that the propertiesof the ellipse are such that, that Pythagorean triangle is true for all ellipses.In other words, for any ellipse, if you look at the focus,the semi minor axis make a right triangle out of them and drawthe hypothenuse, the length of the hypothenuse isexactly equal to the semi major axis.So, the ellipse is more than just a squashed circle.It's a circle that's squashed in a particular way.Remember the idea of the Pythagorean triples, right?So, why should it be that this particular figure that happensto be a slice of a cone that's related to a circle would havethis particular relationship between these three numbers,between the major axis, the minor axis and the focus.

 

 

OK.One more thing, actually two more things, but one of them hasto do now with the concept of how do we characterize how squashed an ellipse is.The measure of the degree of the flattening of the ellipse is called the eccentricity.It's a number between zero and one and is defined very simplyas the ratio between the focus and the semi major axis.Let me draw this on the ELMO.I've drawn sort of an elongated ellipse.I'm going to put the little cross in here again so we can find the center of the ellipse.If the focus out here is of length "F" or I guess I used the word,the letter "C" before so I'll stick with that.So that "C" represents the length of this line segment.The length of the semi major axis is this, which I called "A" before, right?So, we defined this thing called the eccentricity.OK, as simply the ratio of "C" over "A."

 

OK, I think the picture on the screen now has a little bit clearer picture of this.So I think if we switch back to that we can see this a little bit better.You can see that here that because of the eccentricityof the ellipse is always going to be a number between zero and one.For example, if the focus is zero, what does that mean?Well, if the focus is zero, that means that the point "F"corresponds with the center of the ellipse.When you divide zero by the semi major axis, what do you get?Well what do you get when you divide anything by zero?You get the number zero.So, if you have an ellipse which has the focus coincidingwith the center, then the semi major axis is the same as the radius of the circle.Does that make sense?So, another way to say this is that an ellipse with eccentricityof zero is a circle, or you might simply say that a circle isreally a special case of an ellipse which has zero eccentricity.

 

OK.Remember these two things are related, they're part of the same family.On the other hand, if the focus of the ellipse is the same lengthas the semi major axis, what does that mean?That means that the focus is now not within the bodyof the ellipse, but it's out on the edge of the ellipse.The ratio of the focus to the semi major axis is now equal to one.Any number divided by itself is equal to one.So, what we'd have now is simply a straight line where the focusand the semi major axis of the same length.The semi minor axis is now squashed.So in other words it's like taking a circle and squashing it,completely down to a straight line.So, from that perspective we can say that the straight lineor the straight line segment is really a special case of an ellipse with an eccentricity of one.

 

 

Now, this kind of funny if you think about it, isn't it?I mean, who would suspect that there's a relationship between a straight line and circle?In fact, a straight line is a straight line and a circle is acircle and the fact that both of these can be consideredto be special cases of an ellipse with varyingeccentricities is something pretty remarkable.I want to point out to you, by the way also, that the eccentricityfor a specific planetary orbit within out solarsystem is very, very close to zero.The most eccentric of all the orbits is thatof Mercury, and it's about .25 or something like that.The ratio of the focal length to the semi major axis.For most of the planets, like for Neptune, for example,the eccentricity is about .0007.Which means, that for all practical purposes, if you wereto look at the actual orbit of the planet, except for twoof the planets, you would not be able to tell whetheror not they were circles or ellipses.

 

OK.I've got one more, one more neat thing about the ellipse to showyou, and this is something that's not necessarily relatedto Kepler and the laws, but it's something that's interesting anyway.Let me go to the ELMO for now.I'll get it.OK.In the Metropolitan Museum in New York City they have a roomin the Art Gallery called a Whispering Gallery.The Whispering Gallery is an elliptical shaped room, and the way it works is this.On the floor of this room, first of all it's oval an shapedor elliptical shaped room, and on the floor of the roomthey have two tiles marked with dots.And what happens is that one person stands in, can standin one focus and whisper, very, very softly, and no matter howlarge the room is, a person standing at the other focus canhear it with perfect crystal clarity, as if they were standing right next to the person.How do you suppose this works?It's sort of magical property, isn't it?I mean any place else in the room, by the way, if person "A" isstanding here and person "B" is standing here, they hear perfectly.But if a person "B" moves to a different part of the room,say over here, then when person "A" whispers, person can hardly can hear at all.What's happening here is another property of the ellipse.It has to do with the relationship between the fixed lengthof the, the fixed distance, I should say, from the two foci.

 

So, here's what happens.A person stands here and whispers.The sound rays go out from, the sound waves go out from there in all directions.Sound waves are sort of broadcast out in all directions.Some of them are going to pass directly through point "B."And in normal circumstances, for example, if you're outsidetalking to someone, the waves that pass directly from thatperson to you are the ones that you see.What happens in the whispering gallery is this.This sound wave which bounces off of this side is reflectedin such a way that it passes through the point "B."In fact, because of this property of ellipses, it relatesto the equidistance, every straight line that passes through onefocus, if it's reflected off the other, off the wallof the ellipse, will be reflected back to the other focus,so that all these sound waves that go out in different directionslined up being focussed back to the point "B."So that basically it's, you see where the word, focus, comes from?I think if we go to the picture on the screen again, you can seethat the picture's a little bit clearer, quite a bit clearer, in fact.So, in the Whispering Gallery, the sound that's made by one personis reflected off the ellipse in such a way that all of the soundwaves are focussed back on to the person at "B," very muchin the same way that the sun's rays are focussed througha magnifying glass when you use it to burn something.You see where the origin of the word, focus, comes from?

 

OK.So that's the first law.That the planets move in elliptical orbits around the sunwith the sun at one focus, and Kepler was verycareful to include earth as one of the planets.Now we're ready to look at the second law.The second law is a little more complex, but it's a veryinteresting one and a very Pythagorean one,that Kepler discovered quite by accident.The second law basically just says that a planetsweeps out equal areas at equal times.The word, "sweeps out," needs a little elaboration, I think.Imagine a sand box where you put your arm down into the sandand sweep an arc like this with your arm.What you're going to do is create a triangular shaped areawith a rounded edge--an arc of a circle, if you like.The area of the triangle, of any triangle, is equalto one half of its base times its height.You can think of a triangle as half of a rectangle or half of a square.So, for example, if you have a square and you cut it in half,the triangle that results is fairly uniform in size.But suppose you have a very long rectangle, like a rectanglethat's only this wide, but really, really long.

 

OK.So you cut that rectangle in half.How do you compare the area of that triangle to the areaof the triangle formed by cutting the square in half?Well, you multiply one half times the base times the height.So, the point of this is that you can create a trianglewith similar areas either by having a triangle that's sortof a broad triangle, or having one that's very elongated.So, here's what happens.Although the planets move in a circular arc as they movearound the ellipse, for any short period of time the line is very nearly straight.But if you look close enough at a curved line, it will appear to be straight.So, if you take the amount of time that it takes a planet to travelthat certain distance and then connect it back to the sun,as you see in the picture, you see that because the planet movesfaster when it's closer to the sun, it movesthrough a larger angle in a given time.When it's further away from the sun, the planet moves slower.So, it moves a shorter distance along the arc, but because thebase of the triangle, which is the distance of the planetfrom the sun, is longer, then the area of the two triangles is the same.Kepler thought this was amazing when he discovered it.He stated this in the quote that I read for you earlier.That the planets move not only not in circles,but also they do not move at uniform speeds.In other words, they slow up and slow down and speedup at different points in the orbit.The fact that the planets move at different speeds at differenttimes is directly contrary to Plato's assertion that the speedof each planet must not only be circular, but must also be constant.

 

So, here in the second law is really a greater deviationfrom the Platonic perfection of the Pythagoreanperfection than the first law.Because, after all, the ellipse is still like a circle, right?Part of the family, but moving at a different speed along the ellipse.It is quite a departure from Plato's principle.This second law might take you a little bit of time to figureout, but if you look at the picture on the screen and also lookat the pictures in the text and in the Study Guide, I think you can get a sense of what this means.Now we can turn our attention to the third law.The third law is the most complex, I guess to understand and it's also Kepler's favorite.He was very amazed when he actually stumbled on to this law, after, he says, eleven years.I won't bore you with the quote where he actually states the third law.But he was very amazed.He had stated the first two laws in 1609 in the New Astronomy.But the third law in the De Harmonice Mundi, you couldreally sense Kepler's fascination with this particular law.It is the most Pythagorean.In fact, it's often called the harmonic law for that reason.There are several ways to say this, so what I want to dois give you several different ways to say this and we'll lookat a table of numbers where actually we can actually seethese Pythagorean or what Kepler thought were Pythagorean relationships.The third law simply states that the planet's period and itsaverage distance to the sun are related by the two-thirds power.Two thirds power.

 

Now those of you who are not mathematically inclined,these numbers may sound a little strange at two-thirds power,after all, it's not something we deal with on a daily basis.But, I'll talk you through some of these things and see if we can't get a sense of this.So, in this statement we use a couple of words.The period is the length of time for one revolution.That's the length of the planet's year.The average distance to the sun, the distance we're talkingabout is the average of the semi major and the semi minor axes.So this is like imagining that the planet's actually on a circle, but it's really not.There are many different ways to state this third law, but they are all equivalent.So let me go through some of these with you.Another way to say it is that square of the time isproportional to the cube of the distance, where, again, the timeis the time for one period, one revolution, and the distance is the average distance.

 

Another way to say this is that the ratio of the time squaredto the distance cubed is the same for all the planets, except one.The one that it's not same for is the moon.This was kind of amazing to Kepler, too, because, of course,if all the planets have the same number and the moon doesn't, what does that mean?Oh, especially if I was to point out to you that the earth hasthe same number as the rest of the planets.When I say the same number, what I mean that its ratio is the same for all the planets.So that's another way to say it.

 

By the way, another way to say this is the period and theaverage distance are related by the two-thirds power.That's what I said in the first place.An easy memory device to think of this and you will need to knowthis law is to think of the word, Times Square, because that willassociate the word "time" with the word "squared," so whenyou're trying to remember whether it's time squaredor time cubed, once you get the Times Square part, you're half way home.So, the square of the time is proportional to the cube of the distance.So, the table that I put on the screen now contains the orbitalnumbers for some of the planets of our solarsystem, all of which revolve around the sun.I didn't put the moon on this chart because the moon does have adifferent number and that's actually a different type of object.You understand, of course, that the moon goes around the earthrather than around the sun, so the moon has a different setof constants in Kepler's third law than any of the other planets do.In this table "T" is the period in earth years, "D" is the distancefrom the sun in astronomical units.The astronomical unit is a distance unit basedon the earth's average distance from the sun.It's about equal to 93 million miles or 150 million kilometers.When you look at this table you notice something interesting here.

 

First of all I want to point out that if you look at earth,the numbers for earth, that the numbers both, all turn out to be "1."This is because the period of earth is one year,the distance is one astronomical unit.But you notice that for each of the planets the numbers"T squared" and "D cubed" are almost identical.This is saying that the ratio of "T squared"to "D cubed" is very nearly one for all of the planets.The numbers used are modern determinations and aresomewhat more accurate than those available in Kepler's time.So his numbers didn't work out even quite as well as ours do.But, look at the numbers here. Let's take these down in order.So, for planet Mercury its period is .24.That means it takes about a quarter of a year, or about threemonths for Mercury to make one orbit around the sun.Its distance from the sun is about .39 astronomical units.That means that it's about 40 percent closer to the sun than the earth is.So, .24 squared is .059; .39 squared is, .39 cubed, I'm sorry, is .059.In the right hand column "T squared" is equals "D cubed" for the planet Mercury.

 

Here we go to planet Venus.The period is .62.That means it has a length of year, it's about 60 percent of that of earth.Its distance is .72.It's about 72 percent closer to the sun..62 squared is .38; .72 cubed is .37.Not perfect agreement, but, .38 and .37 are certainly close.For the earth, of course, all the numbers are one.As we go down the table here, I won't read all the numbersfor everything, but you notice for Mars,the numbers "T squared" is 3.53; "D cubed" is 3.58.Jupiter.Notice, by the way, how much further Jupiter isaway than any of the other planets.Like Mars, for example, is only one and a half timesearth's distance from the sun.Jupiter is almost five and a half times.

 

Jupiter is quite a bit further away than Mars so, you would thinkthat it might be some sort of a different thing going on here,but Jupiter's period--almost twelve years--twelve years for one revolution.Jupiter's distance about five and a half, 5.31 times thatof the earth, but look again at "T squared" 142; "D cubed" 141.You see the same thing for Saturn.Again.In both of these cases the numbers did not exactly match up.But 142 versus 141.You look down the two columns and you match the two numbers.So, what we're seeing here is something which I have to pointout that even Kepler had no explanation for.It's simply a Pythagorean occurrence or harmonicrelationship between the planets.Why do you suppose we call this harmonic?Remember back in the earlier program we talkedabout harmony as being the ratios of two lengths of pipe and small whole numbers.Well, here we see that the ratio of these two numbers when we usethe measurements and astronomical units and years,the ratio of these two numbers is almost exactly one in each case.So, I'll leave you with the third law to work on yourown with one small question.Is the fact that these numbers are not exactly the same, in otherwords, the ratio is not exactly one, is this a significantcontradiction to Kepler's laws or to the paradigm?We talked about, or I asked you about

significantcontradictions back in Program 2.

 

So, here we have a case where the numbers are very close.Are they close enough to have convinced you that this ratio holds?They certainly were close enough to convince Kepler.And I should point out that his numbers were not even this close.So, you might have to go back now and think about each of the three laws.You especially want to be able to describe the three laws,and certainly for the third law you want to understand what wemean by this ratio between the square of the time and the cube of the distance.

 

Well now I want to turn our attention to the significance of Kepler's laws.As we noted above, the laws are significant because theyoverthrew the circular paradigm or at least replaced the circularparadigm with the elliptical paradigm.That's the sense in which we usually think of Kepler's laws,overthrowing the circular paradigm.But, in terms of the continuity of ideas here in our river of time,just how radical was it to break the circular paradigm?In fact, was the paradigm really broken?Or did Kepler just bend it a little bit?These are things for you to think about.Kepler spent much of his effort in writing these various booksarguing that he was not really breaking anything at all.

 

You know how sometimes when you need to do something,you need to do something a different way, you can't do it,if it goes against your morals and you have to sort of justify it?Well, Kepler was doing this.It's like, "Well, I could do this, it's only a little white lie,it's only bending the paradigm, it's not really breaking it."In fact, his writings rambled on and on and on about this.They contained logical arguments, very similar to thoseof Ptolemy, but he goes back to Aristarchus and talks about,remember how Aristarchus said this and thisand heliocentrism, and all that kind of stuff.But, he takes the same arguments that Ptolemy used and in manycases reaches the opposite conclusions.He also spends quite a bit of time justifying the Copernican systemand pointing out that Copernicus was not a badguy, and that a lot of the stuff makes sense.It was interesting, not only in terms of Kepler, but in general.It was interesting that the more data you have,the easier it is to argue away inconsistencies.Well, let's put it this way.When Brahe's data came in and it showed that the Ptolemaicsystem was wrong, it showed the Copernican system was wrong,it showed that the Tychonic system was wrong, what are you going to do?Who are you going to call?What's the problem?Right?The problem is, of course, that you've got a system that's been2500 years which is no good for anything.It doesn't work.So, when you're faced with a crisis like this, like Kepler was,it's much, much easier to argue away the objections to the moving earth.

 

I started out the program by saying that the ideas in Kepler's times,the questions pertaining to what sorts of suppositions do wehave to make if we're going to assume that there's a heliocentric system.What Kepler really did was to force that decision by pointingout that the heliocentric system is much, much simpler,it works much, much better, and in fact, we wind up with onlyseven eccentrics or orbits instead of the 30 some that Copernicus had used.So, the laws do definitely support the Copernican theory,but is that the only way in which they were really Copernican?We kind of think it is right?Because remember, the Copernican system, although it washeliocentric, still used or still insisted on perfect circles.They still required devices like the equant and the eccentric.Kepler's orbits, on the other hand, had no epicycles.They had no spheres within spheres.They had no other devices moving or stationary.It took a total of one slightly flattened sphere per planet.So, is it really Copernican?I don't really think so.I think it's heliocentric, but I don't really think it'sCopernican, because Copernicus was stuck in the circular paradigm.This is really the key.

 

Overall, in Kepler's scheme the elliptical orbits made for amuch simpler understanding of the planetary motions.And more importantly, allowed a much easier and more precisemethod of calculating their future whereabouts.Although interestingly enough Kepler didn't use them.Well, I'll get to that later.So, here's now some more of the problems and some more of the significance.In the Scholastic Philosophy which was inheritedfrom Aristotle, remember, put togetherby St. Thomas Aquinas, motion had a very special place.The prime mover in Aristotle's view wasnecessary to keep the planets moving.In Kepler's scheme the planets don't need this anymore.Right?Or what I should say, I guess, is that the planets need a reasonto keep on moving different from the prime mover because nolonger is this cosmic, cosmological connection thatAristotle had between the motions.Not only do we need a way to keep the planets moving, assumingthat Kepler's laws really do describe the motion of the planets.Not only do we need to keep them moving, but we also needa way to explain how the elliptical orbits could remain stable.In fact, this was one of the great criticisms of Kepler's work,even in Newton's time, is how can you possibly have anoff-centered nonsymmetrical figurelike an ellipse which will provide a stable orbit.You'd think that just like if you try to roll a ball that'selliptical in shape it would sort of bounce around in an odd fashion.

 

People thought the planetary orbits would do the same thing.The next thing in the significance here is theidea of the concept of a central force.You'll hear this term many, many times when we get into talking about Newton.But, the idea of a central force really begins with Kepler.The central force concept is an idea, the concept that there's aforce of some kind exerted by the sun which continually actson the planets to keep them moving in closed stablepaths from one orbit to the next.In other words, what keeps the planets going around in theseelliptical orbits without getting off-centerwithout falling out of the orbits, and so on?It was apparent to Kepler that this force wasdirected toward the focus of the ellipse.In other words to the sun.But, he couldn't describe the nature of the force, nor couldhe prove that this force was directed toward the sun.

 

I mentioned earlier that Kepler in two times in describing thefirst law talked about the sun as the source of the motion.Well, any good theory, whether it's a heliocentric or geocentricor some unheard of theory needs to have some sortof an explanation for why things happen the way they happen, even if it's wrong. Remember what Bacon said, "It is better to be in errorthan to be not understanding."Well, Kepler had read Gilbert's treatise on magnetism, De Magnete.So, Kepler immediately saw a correspondence herebetween the fact that a magnet can attractanother magnet through empty space.It provided the perfect mechanism for the sun attracting the planets.So, Kepler speculated that the force that held theplanets in orbit was a magnetic force.He noticed from Gilbert that magnets can exert forceson each other through empty space, so simply asked himselfif magnets can do it through empty space, why can'tplanets do it through empty space as well.If magnets can do it, there's no reason to expect thatthe planets might do otherwise.

 

OK.Another significance here is that although Kepler had brokenwith the concept of the perfect circles, the sacred geometryof the universe is still not violated.What this means is simply that although the planetary motionsare no longer circular, they are describable in geometric terms,even if those terms are slightly different from what the ancients thought.It's Pythagorean because it's harmonious.It's Platonic because the ellipse is almost a circle.So the circular paradigm is only bent, it's not really broken in the strict sense.It's also, of course, Euclidian.Remember, Euclid was the great father of geometry.It's Euclidian because it's a conic section, a family of spheres,of shapes, I'm sorry, of which the circle is a member, but it's not the only member.It's the exemplary member.It's the perfect member.So, having a figure that fits all three of these criteria,it's Pythagorean, it's Platonic and Euclidian,doesn't seem to really break the paradigm that much.

 

OK.Couple more things here on the significance.The mathematical relationships that Kepler discovered madeone further very important statement concerning thisheliocentric geocentric controversy.You see, Kepler's critics could argue or had argued,in fact, that the ellipses were just one more devicerather than a new cosmology.After all Ptolemy had put these devices in so it's easy to saythat the ellipses were simply a devicefor calculating, but still did not represent the reality.But, from Kepler's third law when we compared the "Ds squared"and the "Ts cubed," we find that all the planets have the samenumber which represents that ratio.That's all of the planets, including earth.

 

So, here in the third law, in these Pythagorean numbers was theproof that the earth was a planet just like the rest, or at leastit certainly had something in common.You might say that the proof was in the Pythagorean pudding.We should also note, by the way, that number again representingthe third law ratio is different for only oneheavenly object, that, of course, is the moon.(Cough.)Excuse me.So, the last significant feature of Kepler's laws is that this was,for the first time, the first mathematical law whichactually linked the motions of all the planets together.Aristotle tried to do this, but in his cosmology he had claimedthat the motions were linked, but it was aqualitative model, not a quantitative one.You recall that Ptolemy gave up on the concept of linkingthe motions because he found it unnecessaryin order to calculate the motions.Well, the third law in Kepler's eye links them right back togetheragain, but in a heliocentric framework, not a geocentric one.You see, prior to Kepler's formulation of the laws,mathematics was used for calculations only.You could set out to do a particular calculation.You could get a result, but there was nothing in the mathematicsfor recognizing relationships between one thing and another.In fact, Kepler's laws were the first general numericalrelationship anywhere in the physical sciences.

 

OK, finally as far as the significance goes, and this maybe one of the more significant of all. As far as Kepler's influences on Newton a half century later,it was necessity for an explanation of some kindfor the relationships Keplerdiscovered that stimulatedNewton's curiosity and helpedhim to consider the motionsof the planets in the same terms as he considered the motion of the apple.In other words, why is it that Kepler's laws behave the way they do?Right?An explanation is needed.Kepler's magnetism explanation wasn't very good.

 

Well, in this program we have summarized the influencesof Gilbert on Kepler's work and we looked briefly at Francis Baconwhose preference for experiments would go on the drive Galileo'sinvestigations and basically define our scientific method.We also saw that Kepler's laws of motion advanced the heliocentric view, but with a difference.Had the planets moving in an elliptical rather than circularorbits with the sun at the focus, rather than at the center,sweeping out equal areas in equal times, and all having the samerelationship between period and distance.We also learned about the properties of the ellipsein order to reinforce our Pythagorean concept of the nature of this conic section.

 

Well, here we are again.Another program in Nature of Physical Science under our belts.Don't forget to send in your program responses.I guess, you know, I guess that's it for today,so remember, when it comes to science, get physical.Silico: "May I ask a question before we go?Was it Gilbert and Bacon who wrote those silly operettas?"Silly operettas?No, that was Gilbert and Sullivan.You crazy?Come on boot up your timeline, you'll see theylived two hundred years apart.That's a good question, but that's one you could have looked upfor yourself, and you know, part of what you're supposed to learnas a computer in this course is that you need.....Music