astr-160: Frontiers and Controversies in Astrophysics
Lecture 1 - Introduction [January 16, 2007]
Chapter 1. Introduction [00:00:00]
Professor Charles Bailyn: Some things about this course. This is a course for non-scientists. That portion of the enrollment policies is not a suggestion. I really don't want science majors in this class. If you are a science major, I'm going to notice because that's one of the things that appears on the class list; what your major is. So, don't take the course if you're a science major. Let me point out that freshmen don't have major, so it doesn't matter if you intend to be a science major if you're a freshman. If you are a science major I recommend Astro 210, which is being given this term. I have a little handout on all the different introductory astronomy courses at the front of the room if you're interested.
Let's see, it is also true that this course is kind of intended for non-science majors who have a certain basic high school level comfort with tenth-grade science and math. If you're extremely phobic about these kinds of things, I would say that Astronomy 120, while it has a similar level of math, has a somewhat shallower learning curve and a somewhat deeper safety net. So, if you're the kind of person who breaks into a sweat when somebody writes down an equal sign, check out 120. Let's see, but that's not the biggest difference between this class and 120. I think the biggest difference is what the class is trying to do. Astronomy 120 and also 110 and other courses in our department, and elsewhere in the university, are basically survey courses. Most introductory science courses are survey courses. They cover a fairly wide subject matter.
This is--isn't that, what this course is supposed to do is we're going to talk about three particular topics in very considerable detail. Enough detail so that by the end of our discussion you'll understand what's going on in current research in this topic. And by current, I don't mean this decade, I mean this week. Astronomy is currently in a stage of very rapid advancement, and one of the things that's happened every time I've taught this course in the past, is that at some point during the semester someone will publish some piece of research which changes some aspect of the curriculum. I'll come in waving some paper and everything will be changed, and I can't guarantee that of course, because I can't predict the future but it's happened every time in the past. So, we really are trying to get you all the way out to the frontiers of the subject.
I think this is actually a better approach for non-science majors, because after all, we live in the Internet age. If you want to find out a bunch of facts about some scientific topic you could go online and go to Wikipedia or wherever, look these facts up. That's not a big problem. The problem comes when there are two sets of facts which directly contradict each other. This happens quite frequently in scientific topics these days, particularly those with kind of political or moral overtones, and you get facts that directly contradict each other. What are you supposed to do about that? What I'm hoping is that by talking about situations in which the facts at the moment really aren't known yet, you can develop some skill in interpreting these kinds of contradictory facts for yourself. If you don't do that, then the only alternative is to listen to the experts argue with each other and vote for whoever argues the loudest or looks the best when they're doing it, or has a degree from Harvard or whatever it is. You guys can do better than that. So, the hope is that by practicing this kind of skill of evaluating science when the answer isn't fully understood, that you can develop skills that will stand you in good stead when you run into scientific controversies in a political context or a legal context, or just as ordinary citizens in the course of your lives.
It also happens that the three particular topics I think are of some real interest and importance in themselves. And I'll get to the three topics again in just a moment here. Let me point out that this kind of approach has a downside to it and this has been pointed out repeatedly on course evaluations. Because we're dealing with stuff which ultimately the answers are not yet understood there's no textbook. There can't be a textbook. We haven't figured out what to put in the textbooks yet. And the problem with that is that that makes the lectures very important because that's the only information you're going to get. There's a whole bunch of online readings and stuff but they tend to have a point of view, and so it's really the lectures that are the basis of the course. The problem with that is that I've chosen to give this course at the ungodly early hour of 9:30 in the morning, and you guys are going to have to show up and so here's the deal. I'll make a deal with you: Your job is to get to class by 9:30 in the morning. My job is to keep you awake once you're here, and so if we both succeed in cooperating in this sense we'll probably be okay. But seriously, if you're anticipating regular difficulties in getting to class this is not actually a great class to take because there's no backup in the form of a textbook.
Chapter 2. Topics of the Course [00:05:38]
All right, the particular topics that are under discussion, I've listed them here in green. The first of them are extra-solar planets, by which I mean planets around stars other than the Sun. It's well known that there are many, many of these planets. All you have to do is watch Star Trek or something like that and you'll find many, many examples and this has been a staple of science fiction for quite a long time. Oddly enough, until ten years ago, there was absolutely no evidence for this. We assumed that, because the stars are normal stars and there are many other stars--the Sun is a normal star and there are many other stars like the Sun that there must be many planets of the same kind as the planets in our own Solar System circling around all these other stars. But until 1995 there was not one bit of evidence to support that idea. Since 1995, this has become a huge growth industry and research, and we now know of literally hundreds of planets, all of them discovered in the last ten years. So, this is a situation in which, what ten years ago was science fiction, has become science fact and we're very rapidly trying to figure out exactly what kinds of planets these are, whether there are Earth-like planets out there, and that has some bearing on what the science fiction people refer to when they say as, "life as we know it." And so, that's currently one of the hottest topics in astronomy.
The next topic is going to be black holes, and this is a similar situation. Fifteen, twenty years ago black holes were sort of poised precariously on the boundary between theoretical physics and science fiction. A boundary that is more porous than you might believe. But again, in the past fifteen years or so this has been converted into a standard topic in observational astronomy. There are dozens, probably hundreds of objects we can point to in the sky and say, "yes those things are black holes." And so now, the current topic of research is do these things that we are pretty sure are black holes actually behave in the incredibly bizarre, science-fictiony manner that the theoretical physicists have been talking about for the past thirty or forty years. So, to what extent are these very exotic behaviors actually manifested in real life?
Finally, I want to talk a little bit about cosmology. Cosmology is the study of the Universe as a whole. That's too big a topic to go into in depth, so I've picked one piece of it. The piece I've picked is the existence, which was discovered in the late 1990s, of something called "dark energy." Dark energy is an all-pervading anti-gravity; it's a repulsive force that turns out to occupy essentially all of the Universe, and 75% or more of the entire mass energy of the Universe turns out to be in the form of this mysterious dark energy. The evidence for this comes largely, but not entirely, from observations of a certain kind of supernova. And so what I'm going to focus on is the observations of the supernovae and how they demonstrated that, in fact, all ordinary matter and energy and so forth is a tiny fraction of what's actually going on in the Universe, and what's really happening out there is something we totally don't begin to understand. So, that will be the third topic of the course.
These topics have something in common. All of them involve observing something that you can't actually see directly. We don't see these planets directly because they're too faint and too far away. We don't see black holes directly. By the definition of black hole you can't see these things directly. And of course, dark energy, by its very name, is also undetectable. So, how do we know that these things are there? The answer is we know that they're there because of their influence on other objects that we can see, and in particular, their gravitational influence on other objects that we can see. And so, what binds these three topics together, are first of all, the fact that the observational techniques to discover them are actually quite similar to each other. And second, that they all involve different manifestations of gravity. And so, we'll be talking in the first part of the course about Newtonian gravity. In the second part of the course when we get to black holes, that's relativistic gravity, general relativity, Newton's--Einstein's theory which supplanted Newton's theory. And then by the time we get to dark energy, it may not even be correctly described by Einstein's work, and we may be in the area of whole new kinds of physics that the theorists haven't even thought about yet. So, there will be a progression to more and more sophisticated theories of gravity underlying these observations.
There's another feature that these topics have in common, and that is that they can be understood in some detail without particularly sophisticated mathematics. Now, let me pause here and say some things about math. Astronomy is a mathematical topic. There will be math in this course, there ought to be math in any astronomy course or it isn't really an astronomy course, it's just a slide show. Now, the math in this course has been kept at a deliberately low level. That is to say, the kind of math we'll be doing is stuff you did in ninth and tenth grade. Introductory high school algebra, high school geometry, I think we take the sine of an angle a couple of times, but it's the one case it cancels out almost immediately, so don't let that scare you. It's the kind of thing that you all did on the math SATs and since you're all sitting in this room you must have done okay.
Having said that, I have discovered that saying that is misleading. And the reason it's misleading is cast your mind back to ninth grade; ninth grade math is hard. Remember? In particular, word problems are hard. You remember word problems. This is where you drive from here to Cleveland and you fill your tank up with gas, and the gas costs so much per gallon, and the question is what is your shoe size or something. [laughter] The way one approaches that is through a kind of common sense approach which involves the fact that many of us have been in a car, driving from City A to City B, perhaps not Cleveland, but somewhere else, and so you have a kind of intuition to fall back on. When you do math problems that are logically the same, but apply to astrophysical systems, for which you have absolutely no common sense to back you up, then you have to reason purely from the internal logic of the problem and that's hard to do. It's a skill that can be learned; it's a skill that's worth learning; it's a skill that I'm sure many of you already have to a large extent, but it isn't an easy thing. So, the fact that the level of the math is low doesn't mean that the problems are easy. We do have a lot of help mechanisms, which I'll describe perhaps on Thursday, to keep you up to speed if you start having trouble with these things.
Chapter 3. Course Requirements [00:12:57]
So, I should say something about course requirements here. Let's see, we have sections in this class. The sections are not just problem solving sections, these are actually required. The fact that we're dealing in topics for which the answer isn't fully known means that one can actually have discussion sections unlike many science courses, so we're going to do that. And so, the structure of the course is like a history course. Two lectures a week plus required section, and so 10% of your grade comes from sections. A large fraction of that is just showing up, but there will also be something in terms of saying something intelligent once you get there. That's 10% of the course; 30% of the course is problem sets. We will hand these things out once a week. The first problem set will show up on Thursday, and if you have any question about whether this course is appropriate for you, the right thing to do is to look at that problem set and ask yourself is this reasonable. I will say that students on their evaluations have pointed out that it does--the course does get harder. It's not that the math gets more complicated, but the situations get more complicated. So, if you have serious trouble with the first problem set that's probably a warning sign. As I say, that will be handed out on Thursday. These things come about once a week; it's 30% of the grade. I'll say more about problem sets later on Thursday.
Thirty percent comes from two midterm exams. The way we do this is the one where you get the better score counts 20%. The one that you get the worst score counts 10%. So, that gives you a little bit of a break. And then there will be the Final exam, that's the last 30% of the class. There's also an optional paper. If you choose to do that, that will count 15% of your grade, and what it will do is it will de-weight whichever the worst of your 30% parts of your grade are back down to 15%. So, if you're a word person rather than a number person, you get this opportunity to augment your score and de-weight some other part of the class in which you may have done less well.
All of this stuff is on the classes server [Yale's online course tool]. I should say that the syllabus that I've put out here is just a direct copy off of what's on the classes server, so feel free to take that. But all the information, and actually more information is online. Let me pause now and ask whether there are questions about the course and the course procedures. Yes?
Student: This may be a silly question, but I saw on the web that right below the times listed for this course was a "to be determined" or some sort of notation that could indicate that there is another of this class at a different time?
Professor Charles Bailyn: No, no this class is going to meet now. I'll have to check and see what you were thinking of, but it may be that what that was referring to was section times, and actually this is something that I haven't mentioned. Sections are required. They're all going to be on Mondays. We're going to have a wide range of times, all of them on Mondays from 12:30 until I think 8:00 at night. But you do have to sign up for a section. Let me also say, I've mentioned here, I don't think this is--; actually, looking at the number of people here, I think we're going to be able to accommodate everyone, including juniors and seniors. But I did set it up in such a way that freshmen and sophomores get first crack. The way that's going to work is the online sectioning form opens up on Monday and juniors and seniors won't be allowed to officially register for the class until Tuesday. So, the freshmen and sophomores get to fill up the sections first. My guess is, again, looking at the number of people here today that we won't have any problem, and that if you're a junior or a senior you'll get in just fine. So, we'll be picking sections through what is now the standard online sectioning thing, which is going to open for business next Monday. I'll check the website and see if that's actually what you meant, but it may have been something else. Other questions?
Chapter 4. Planetary Orbits [00:21:03]
Let me, in general, encourage you to ask questions. I know that that's hard to do in a big lecture setting, but we have an advantage over other courses, particularly science courses. We're not trying to prepare you for the astronomy part of the MCATs, so we don't have to cover a specific syllabus. We're not even trying to follow a textbook. And so we have a little more leeway than is ordinarily true to ask questions and go in weird directions, so please feel free to do that. I reserve the right to put a question off into the future or into discussion section or something, but do by all means ask. We have some freedom of action. Yes?
Student: Is it possible to take an early final?
Professor Charles Bailyn: An early final? Let me think about that. I prefer to avoid it because then I have to invent another final. The problem with that is trying to make them come out even. I will say this, that if I do an early final, I'm probably going to err on the side of making it hard. But it's very hard to make them come out even, but let me think about that. Other questions? Yes.
Student: In discussion sections, is it just going to be like discussing things or is it going to be working on the problem sets?
Professor Charles Bailyn: It's going to be some--So, the question is, "What are the discussions sections going to be like?" Are there going to be discussion of the problem sets or is it going to sort of general discussion of the course material? The answer is both. I think there will be both, in any given discussion section, there will probably both be an opportunity to talk about the previous problem set and to clarify things about the next problem set, and also some kind of activity that sort of extends and advances what we've been talking about in class. So, I'm hoping to do some of both. If we veer too much in either one direction that's probably not a good thing. There will be other ways of getting help as well, if you start to have trouble on the problem sets or in the course generally. I'll talk about those a little bit on Thursday. Yes sir?
Student: How are problem sets graded?
Professor Charles Bailyn: How are problem sets graded? Very carefully. Let's see, I think we'll probably--it'll probably be on a kind of zero to twenty-point schedule. But let me say this about the problem sets. There are going to be two kinds of things on the problem sets. One are kind of quantitative problems which have a right answer. Those are relatively easy to grade on some kind of a point scale; you give partial credit and so forth. But we will also--because this is a course that's not only about the specific of this topic but also about science in general, we're also going to have things that look kind of like essay questions on the problem sets. Those are a little harder to grade in this way, but we've got to grade them in the same way so that we can add the points up. And I'll talk a little bit more about how those are graded. I will say one thing; one thing that we do is we make sure that each problem or essay is graded by one T.A. or by myself, so that we don't have different people--so that if you're in a section it's not like your--all the problem sets for that section are all graded by your section leader and some other section leader grades all the other problems, because that leads to imbalances of various kinds. So, we assign each problem to a specific person for the whole class. It's basically a zero through twenty scale, although what that means varies depending on what kind of a problem it is. I'll say a little bit more about that.
I will also say there is a rather detailed lateness policy that's linked to the classes server, please read that. We're going to stick to it. And one of the features of that is that there will be answer sheets. Problem sets are typically due Thursday, there will be an answer sheet up the following Tuesday, so if you don't get it done by five days after it's due, you're toast because the answers are posted. Other questions?
Great, let's start. This is very cool. All right, this is going to be all kinds of fun. Planets, planets around other stars, but planets in general. So, let's start by talking a little bit about orbits, planetary orbits. You probably know some of this story, originally in the old days, people used to think that the Earth was the center of the Universe. So, the Earth was at the middle and planets went around them in circles. That's not much of a circle [drawing on overhead], but you know what I mean. And so, everything was circles around the Earth. And that's what planets did, where planets also included to their way of thinking, the Sun and the Moon as well, and so you had these circles around the Earth. This is what's called the geocentric model; Earth at the middle. It's associated with the name of a Greek astronomer named Ptolemy. The problem with this model is very simple. Namely, that if you actually go out and observe where the planets, and the Sun, and the Moon are night after night after night it doesn't work very well. So, this doesn't fit the observations. Doesn't fit observations.
So, they said, all right well maybe that doesn't work all that well, so what we'll do is instead of imagining that the planets are on circles around the Earth, we'll imagine that there are circles on circles around the Earth, and the planets go on those. So, you add a kind of extra circle here, so the circle goes around the Earth and the planet goes around on that circle. These circles were called epicycles. So, add epicycles. And what happened is they would add an epicycle and then they'd go out and observe some more, and in particular, the Arab astronomers a thousand years ago. A thousand years ago the center of all science was in the Arab countries; they gave us all their--all our star names by the way are in Arabic, so are mathematical techniques such as algebra; it all comes from the Arabs. They knew what they were doing back then when the Europeans were kind of in squalor. And they made these great observations, and every time they made more observations it turned out it didn't fit. So, they had to add more epicycles. So then, they added one here, and one here, and so on until you had circles, and circles, and circles, and circles in order to explain the observations. So, add epicycles repeatedly. And this is kind of unsatisfying because it's not a good thing where every time you get more or better observations you have to revise and extend your theory.
That's not such a great theory. In fact, the word epicycles has now become a kind of a swear word in the scientific community, meaning a sort of theory that has become so complex it's just ridiculous and you don't want to believe it anymore. So, someone will come up with some really seemingly sophisticated but very complicated theory and if you don't like that you just go that's just epicycles, forget about it. So, this has become a little bit of a swear word, and it was unsatisfactory at the time. Now, let me pause for a moment and confess that the story I've just told you, which is the standard story about Ptolemaic epicycles is, well, it has what I think Colbert would refer to as "truthiness." It's a commonly told story that people like to believe, but if you talk to the historians of science this isn't actually how it happened. And, in fact, this idea of circles on circles, on circles that isn't the way epicycles worked, they had circles and they did get more complicated every time they fit the observations, but not by adding more and more circles. They would move the circles side to side, they would have things going at variable speeds around the circle, all sorts of things but this little picture that I've just drawn here has a kind of "truthiness" to it. I would say that this is a general issue with the way scientists describe how science works.
We tell these nice anecdotes and we put them in the textbooks too; in the little bars that go down the side of the textbook, where you get the head and shoulder shot of the famous dead white male scientist and so forth. And then we tell these stories. And the historians of science hate this because it isn't actually what happened. Nevertheless, we persist in telling these stories, and I've been thinking about why that is. I think the way to think about this is what these stories are, are fables. And like any fable, the point is not that the story is true. The point is that it vividly illustrates a moral, which tells you how to behave or how not to behave and they're useful for that reason. You'll recall the famous fable of the ant and the grasshopper. Grasshopper sings and plays and dances all summer long. The ant is very industrious, piles up food, doesn't have any fun. But then in the winter, the grasshopper starves and the ant does fine. If an entomologist were to come along and say but that's not how ants and grasshoppers behave, you would correctly say that he's missed the whole point. And the point is that it's just a nice story which illustrates certain kinds of behaviors and whether they're good or bad. So, here's what I'm going to do; I'm going to tell these stories, but I'm going to label them fables and I'm going to point out the morals explicitly. And the optional paper is going to be: go and take any one of these things and find out what really happened and comment somewhat on the implications of the real story for science.
I should say that the biggest of these fables is probably the one about Galileo and the Catholic Church, where the Catholic Church oppresses the pioneering scientist and the scientist stands firm against this huge impersonal bureaucracy, and the establishment trying to squelch them and so forth. The truth of that is actually very subtle and very interesting and I can't go into it now, among other things because I'm not a historian of science, I'm not the best person to talk about it, but check that out sometime.
Anyway, for this particular--this is the fable of the Ptolemaic epicycles and the moral is that simple theories are better. And you particularly don't like theories which get more and more complicated, the better and better your data become. I should say that the word simple in there turns out to have a technical meaning if you take a statistics course. What I mean by simple is something that has relatively few free parameters. I'll just leave that at that. You can go talk to the statisticians about it. So, if your theory is getting overwhelmed by epicycles, then you'd better go out and come up with some other better theory. And so, people tried to do that, and the first step along the way was, of course, Copernicus.
Copernicus, as you probably recall, decides that the geocentric model is wrong, things ought to be heliocentric; the Sun in the middle. So, you put the Sun in the middle and everything, including the Earth, goes in circles around the Sun. This was revolutionary, and in fact, the title of the book he published was De Revolutionibus Orbium Coelestium, which means "of the revolutions" in the sense of "revolving of the celestial spheres." The use of that word revolution is one of the things that pushed the word revolution to its current meaning, meaning overthrowing authority in some ways. Originally, it just meant to revolve but this was so revolutionary that people started to use the word in the other way. This wasn't actually as great a theory as you might think, because it still needed epicycles. Not as big, not as many, but it didn't get rid of the problem with epicycles. And that didn't work itself out until a generation or two later when Kepler came along.
Kepler was a famous astronomer and he had in his possession, because he stole them, the best naked-eye results that had ever been obtained of the motions of the planets, in particular, Mars. He described these motions in Three Laws of Planetary Motion. You can look them all up in a textbook. In other kinds of courses we would have you memorize these things; I'm not going to do that. The key point here is that these are not circles; they're ellipses around the Sun. That, it turns out, gives you a model for planetary orbits which, when you take better and better data, doesn't change. They're still ellipses; you don't need little ellipses on top of these ellipses to explain everything that's going on. So, this now has excellent descriptive power. It really describes what's going on, and when you make further observations, it still describes what's going on. It does not have any explanatory power in the sense that if you say, "why ellipses?" Kepler had no idea. That's just the way God made it. So, it's not in any particular way an explanation. For the explanation you have to wait another generation or two until we get Newton.
Chapter 5. From Newton's Laws of Motion to the Theory of Everything [00:31:32]
Newton writes down three laws of his own, but these are now three laws of motion, not planetary motion in particular. And again, one could write these down and memorize them and learn them, and that would be a good thing. Let me write down one of them, the Second Law, looks like this: F = ma, force equals mass times acceleration. And I write this one down simply to point out that that equation is the entire intellectual content of Introductory Physics for physics majors. If you go take Physics 180 this is all that they do and they spend the whole time. It turns out you don't actually want acceleration, that doesn't tell you what you want to know. What you want to know is the trajectory, where the object is as a function of time. Those of you who have taken some calculus may recall that if you take the acceleration, and you take an integral twice, you'd come up with the position as a function of time. So here's what--so in the next thirty seconds I'm going to explain Physics 180 to you. You substitute in some kind of a force, you divide by mass, you take two integrals, and that gives you the trajectory of the thing. That's all you need to know. Technically, of course, it's quite hard, but conceptually pretty straight-forward.
One of the things that Newton did with this equation was he took a particular force, namely the force of gravity, which he also wrote down a Law of Gravity. That tells you for any given situation what the force due to gravity is, substituted it in here, and figured out what the motions of the planets ought to be. And it turns out that he could derive Kepler's Laws. He derives Kepler's Laws. Very nice. Now, of course, in order to do this he has to invent calculus, so it takes a little while. He was a great genius but even so, inventing calculus from scratch, not something you want to attempt at home. And that was basically the start of both modern science and modern mathematics.
So, this marks the start of science in the following sense--that Newton has to make a couple assumptions along the way, sort of deep assumptions about how the world works. One is that the Universe is governed by laws, and in fact, by universal laws. What I mean by universal, in this sense, is that they apply everywhere; that the same law of gravity that resulted in the top of my pen falling to the floor over there also is responsible for the orbits of the planets and the motions of the stars. This was a new idea. It's very familiar to us by now, but the idea that the planets ought to behave according to the same rules as stuff down here on Earth was a whole new concept. The other piece of the new concept is that these laws are mathematical in nature. This is why science is hard, because it's hard for human beings. I think it's something to do with the way our brains are wired, to accept that this is true. It's very easy to imagine a world in which that's not true. Go read any fantasy novel. Any fantasy novel has a situation where the hero or the villain, by virtue of their strength of character influences the events around them. So that is a rule governed by laws, perhaps, that are not mathematical in nature, but depend on the moral character of the individuals involved. Every human culture has such stories including our own. It's very hard to get away from it, and the idea that there's just this sort of mathematical structure and that your moral stature has no bearing on what's going to happen is kind of hard to accept. Fortunately, people turn out to be pretty good at math, so we can actually solve these problems and move forward. These two ideas were revolutionary and they are the basis pretty much of all science.
So then Newton's laws get elaborated on for several centuries. By the end of the nineteenth century things are starting to come apart a little bit. There are now problems that show up with Newtonian physics. It's been a big success on the whole but there are now problems. And in the early twentieth century what happens is two new laws of physics are invented. These are the given the names quantum mechanics and general relativity. And the situation with these is they don't overturn Newton's laws, they extend them. It turns out that in the kinds of situations that Newton was looking at, both quantum mechanics and general relativity, reduced down to Newton's law. So, you have a situation where here are Newton's laws, Ns Laws, of which Kepler's laws are a tiny subset. And then general relativity; I'm drawing a kind of Venn diagram here, is here, relativity, occupying Newton's laws but that's some other stuff. Quantum mechanics looks kind of like this; extends in a different direction. Let me make these axes-specific. I don't like Venn diagrams when they don't tell you what you're actually plotting. This is mass, so heavy things are when relativity kicks in. This is size, and so small things are when quantum mechanics kicks in.
But you can see the problem. We've got two big theories. You really want those theories to be encompassed by one yet bigger theory. And that is the current goal of theoretical physics, to try and find the one great theory that encompasses both quantum mechanics and general relativity, which contradict each other in various awkward ways, particularly in this region up here. This is called the Theory of Everything, or TOE. And the best current guess as to what kind of a theory that will be is that it will be some kind of string theory. I won't go into string theories now, you can go read many popular books on this; it's very exciting. There is currently no string theory that really works out all that well but the people who are studying this kind of thing like to believe that that's going to work out sometime in the future. This is good.
Chapter 6. The Newtonian Modification of Kepler's Third Law [00:38:10]
We've gone about forty minutes from the start of science to the Theory of Everything, so we're done. Everything else is a detail and so the whole rest of the course is filling in details. The first of which--so let's start on the details. The first of which, I want to go back and catch one of Kepler's Laws. And I want to write down the Newtonian Modification of Kepler's Third Law. That is an equation that looks like this:
a3 = GMP2/4 π2
We're going to circle this in red. This is something you're going to want to memorize. This, it turns out to be, a basis of a large fraction of what we're going to do in this course. So, let me explain the symbols; a is the semi-major axis of an elliptical orbit. Remember these orbits are going to be ellipses; here's an ellipse. The long side is the major axis; the short side is the minor axis. Half the major axis is the semi-major axis, so this is a right here. P is the orbital period, how long it takes the planet or whatever orbiting object you've got to go around one orbit. M is the total mass of the two things in orbit around each other, of the orbiting bodies. And the existence of that M is why this is Newton's modification. In Kepler's law, it was always planets going around the Sun, so the mass was always the same; the mass was that of the Sun and so it cancelled out. In general, you can use the same equation to deal with things orbiting the Earth or things orbiting the Moon as long as you put in the right mass there. G is a constant of nature, the gravitational constant, and it equals some value depending on what units you use. And we'll come back to that later. Four is 4, π is this obscure number from elementary mathematics 3.14159 whatever the heck it is. And you can punch it in on your calculator or whatever. So, you can use this equation to find things out.
Now, these numbers tend to be awkward to work with. The mass of the Sun is some huge number of kilograms, G is a very awkward number, π is always a mess. But let me show you a trick. Consider the Earth's orbit around the Sun. The semi-major axis of the Earth's orbit is a very common unit in astronomy, and it's called an Astronomical Unit. It's a unit of length, or AU. The mass of the Sun, mass of the Earth plus the Sun is mostly the mass of the Sun; of Sun, is called the solar mass obviously, and it's given this symbol M with a little circle with a dot inside, that's the symbol for the Sun. What's the orbital period of the Earth? A year, thank you very much. Period of Earth--one year. That's what a year means; it takes a year for the Earth to go around the Sun. So, it must be the case that one Astronomical Unit cubed, is equal to Gtimes the mass of the Sun, times one year squared, that's P2 over π2.
Now, let me show you a trick. Take the general equation, it's a useful trick, and divide by the specific equation. So a3 =P2GM/ π2 and we're going to divide that by 1 AU3 equals one year squared, G mass of the Sun over 4π 2. We can do this because these two things are equal so we're dividing both sides of the equation by the same amount. G cancels, 4π 2cancels; that's very nice. We end up with a over 1 AU3 equals P over one year squared, M over the solar mass. This is just saying that quantity is a in units of an Astronomical Unit. This quantity is P in units of a year. If this is two years then this number will come out to 2, and this is M in units of the mass of the Sun. So, you can say a3 = P2M, providing you're dealing in units of the mass of the Sun, units of one year, and units of an AU. So, this is now much easier to work with. You've got rid of all kinds of terrible things, so let me give you the first numerical example of the course. This will be the last thing we do today, namely, the orbit of Jupiter.
Turns out the distance from Jupiter to the Sun is about five times the distance of the Earth to the Sun. So, a of Jupiter is approximately five times a, a of Earth; a of Earth you'll recall is this 1 AU so this is about 5 AU. So, how does this equation work out? You get 53 equals P2M, M is the mass of the Sun, 1 solar mass. And since Jupiter is going around the Sun that's equal to 1. So, you have 53, 5 times 5 is 25, 25 times 5 is 125, so you end up with 125 equals P2, so you can answer the question now. What is the orbital period of Jupiter in years? Obviously, that's going to equal the square root of 125. Here's another trick. What's the square root of 125? Quickly? Good, more decimals? You could type it into your calculator though and find out, but let me make a suggestion. Don't take the square root of 125; take the square root of 121 instead. What's the square root of 121? 11. Much easier, right? And notice this, a of Jupiter is approximately five, so 53 is approximately 125, and it's just as good to say 121 is equal to the square root--the square of the period, and Pequals 11 years. That's the orbital period of Jupiter.
All right, so now, I'm aware that many of you are shopping the course today and may not be back for future lectures. And so, I want for those people who have decided against this that they'll do something far more worthwhile with their time, I want to leave you with something you can carry through your life from your brief experience with Astronomy 160. And that is the following piece of advice: Don't take the square root of 125, take the square root of 121. It's much easier. This is what the business people call thinking outside the box. Don't do the stupid hard thing. Do the thing that is just as good but requires some thought first in order to make it easy. So, I will leave you with that, the rest of you I'll see you on Thursday morning.
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