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ASTR 160: Frontiers and Controversies in Astrophysics
Lecture 4
 Discovering Exoplanets: Hot Jupiters
Overview
The formation of planets is discussed with a special emphasis on the bodies in the Solar System. Planetary differences between the celestial bodies in the Inner and Outer Solar System are observed. Professor Bailyn explains how the outlook of our Solar System can predict what other star systems may look like. It is demonstrated how momentum equations are applied in astronomers’ search for exoplanets. Planet velocities are discussed and compared in relation to a planet’s mass. Finally, the Doppler shift is introduced and students learn how it is used to measure the velocity of distant objects, such as galaxies and planets.
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htmlFrontiers and Controversies in AstrophysicsASTR 160  Lecture 4  Discovering Exoplanets: Hot JupitersChapter 1. Theory of Planetary Formation [00:00:00]Professor Charles Bailyn: We were talking last time about the objects in the Solar System. And we’d gone through kind of two of the three stages of the scientific method as it’s applied to observational science rather than experimental science. And the first thing was just making observations, finding a bunch of things. And so I gave you a little slide show depicting some of the objects in our Solar System. And after observations the next thing to do is classification, and we did some of that too, and I divided all these objects that had been discovered into six categories. And then, once you’ve done that, once you have some categories that kind of make sense, then the next thing is to interpret these results and to try and explain where these categories come from, how they arise, and actually figure something out. And I’ve described that as interpretation, and I want to offer you a little bit of interpretation about what we’ve found about the Solar System. Now, I only want to explain some of the classifications. This is actually a common thing to do. When you find a whole bunch of different things and you’ve got twelve classes and three subclasses and two exceptions, you kind of want to explain the big features first and then worry about the little things later. This is commonly done. So, what I’m going to do is I’m going to talk about only the inner terrestrial planets. You’ll recall that these are small, rocky things in relatively short orbit, and contrast them with the outer planets, the Jovian, the Jupiterlike planets, which are large and have not only rocks, but also lots of ice and gas. And these things are in wider orbit, but the orbits of both of these are, basically, more or less circular. Not precisely circular, they’re actually elliptical, but quite close, and they’re all in the same plane. That is to say, they’re all going around the same way. There’s nothing that’s going this way instead of this way. So, they’re circular and coplanar. And let’s try for an explanation of those particular features. Okay, so this would be something like a Theory of Planetary Formation: how the planets formed, why they get that way. This word, “theory,” is a serious problem. This is one of the foremost examples of a word that means something different when scientists use it than when normal people use it. In scientific parlance, a theory is something which has a lot of support, which explains a lot of observed or experimental fact. In everyday life, of course, a theory means a wild guess. So, there’s a pretty stark difference between those two definitions. This gets our friends the evolutionary biologists, in all kinds of trouble because they keep talking about the theory of evolution. And a certain segment of society interprets that as the wild guess of evolution and this creates various kinds of difficulties. The problem is, there isn’t another word that one can readily use for what the science definition of this–you could use “paradigm,” you could use “scenario.” These are kind of ugly sorts of words, so I think we’re stuck with “theory.” But I mean this in the sense of something that explains a lot of facts, rather than in the sense of wild guess. Okay, so here we go: Theory of Planetary Formation. So, the idea is that the planets form from a disk of material around the Sun. So, the Sun and the planets are created out of a collapsing cloud. The cloud collapses, some parts of the cloud are rotating, that prevents them from collapsing. And so, you have a situation, after a while, where you have a sort of starlike thing in the middle, and then a kind of disk of stuff around it. So, this is a side view, and in the top view, all these things are in orbit here. And this disk consists of basically the same material as the Sun, which is to say, lots of gas, by which, I mean hydrogen and helium in particular. Some ice, of course, when it’s in a star it’s all melted. But these are elements like carbon, nitrogen, oxygen, that go into making ices, and a little bit of heavier elements that could, if you put them all together, form dust and rocks, and things like that. A little heavier element: things like silicon and iron. So, that’s what the Sun consists of, and so does this disk orbiting around it. Then what happens? In the disk, things gradually stick together. So rather than having individual atoms or molecules, the molecules and atoms sort of run into each other, form dust grains, the dust grains run–or ice crystals or whatever, those run into each other, form bigger things. And you gradually make bigger things as the little things collide with each other and stick. So in the disk, things gradually stick together and become tiny little objects, which are sometimes given the name “planetesimals.” And then the “planetesimals” bump into each other and stick. These stick together until you’ve got a situation where in each region, each orbit, each distance from the Sun, you’ve coalesced everything into one large object. And so you end up with one big object in each region, by which I mean distance from the Sun, distance from the star. And this explains the orbits, because since these large objects have been created by running a lot of small objects together, the ellipticity of any of the elliptical part of the orbit of any of these things tend to cancel out. Because one of these objects will be elliptical in one direction; another will be elliptical in another direction. If you put them all together they’ll–that orbit will tend to be circular. Similarly, some of the objects will be going up out of the plane of the Solar System, some will be going down, but you’ll run them together and they’ll all end up with kind of similar circular coplanar orbits. So makes approximately–that’s the approximate sign [~]–circular, coplanar orbit. So, that’s good because that’s one of the things we’re trying to explain. And there is an expected difference between how this works out in the inner parts of the Solar System and how it works out in the outer parts of the Solar System. In the Inner Solar System the ice and gas in the “planetesimals” evaporates and does not become part of the planet that gets formed. And so the planets are only the rocky parts. They actually hold onto a little bit of the ice and gas, but not very much. Whereas, in the Outer Solar System, the ice is frozen, and so it behaves just like rocks. And so planets have rocks and ice. This means that they’re substantially more massive. And if they’re sufficiently massive, then they have enough gravity to hold onto the gas also–onto gas as well. It’s also true that in the Outer Solar System there’s more volume, there’s more stuff, and so there’s more stuff to build the planets out of in the first place. And so, this nicely explains the difference between the Inner Solar System and the Outer Solar System. You build these things up out of little chunks. But in the Inner Solar System the temperature is high enough that the chunks of ice evaporate, and so you can’t build them up out of that. And so you get much, much smaller things made almost entirely out of rocks, in contrast to the Outer Solar System where you have enough ice, you build much bigger planets, and in some cases you hang on to a lot of the gas as well. So, this is interesting because it makes a prediction about how other Solar Systems ought to look. Namely, that this difference between inner planets and outer planets ought to exist everywhere. That planetary systems–there should be a general feature, because there’s nothing of what I’ve said so far that’s unique to the Sun. So planetary systems should have inner rocky planets and outer Jupiterlike planets, sort of gasplusice sorts of planets, much bigger. And the dividing line between these two kinds of planets is determined by temperature. Because there will be some temperature where those ice things melt and therefore you don’t expect it to be at the same distance away from a star. You expect it to be at the place where the temperature is the same. Therefore, if you have a very bright star, hot star, you’ll have inner planets out further from that star, innertype terrestrial planets will go out further. And if you have a very dim faint star, which doesn’t generate as much heat, the gas planets will–the dividing line between the gas planets and the terrestrial planets will be much closer in. So the dividing line is determined by temperature, hence, by the luminosity–the amount of energy given off of the star. So now, this tells us what we’re supposed to do next. Namely, go out and find a whole bunch of other Solar Systems and verify this prediction. Namely, that if you’ve got a really bright, hot star, you ought to have rocks out fairly far in the Solar System, whereas, if you have a dim star, you’ll have Jupiterlike things coming in much closer. And so that’s what I want to talk about now, is how people went about doing this and what the results were. And I’ll tell you the punch line in advance, which is that this totally doesn’t work. But it’s an obvious prediction from our theory of planet formation that came about by an examination of what was going on in our own Solar System. Chapter 2. Observing Moving Exoplanets and Stars [00:12:04]Okay. So, observing exoplanets. How do you find these things? And you’ll recall we started down this track in the last lecture. The key point here is that stars move too. It’s not just the planet going around a star, it’s the planet and the star both go around the center of mass of the system and the stars move, too. You can’t see the planets independently. And there was a little equation–the velocity of the star times the mass of the star is equal to the velocity of the planet times the mass of the planet. This is basically an equation of momentum. The distance of the star to the center of mass, times its mass, is equal to the distance of the planet to the center of mass, times the mass of the planet. This is kind of a definition of the center of mass. And sometimes, you don’t want to deal with the individual velocities or distances, you want to deal with the total velocity or distances. And so that’s just defined where if you want to talk about the total that’s obviously justD_{star} plus D_{planet}. Similarly, for the total velocity and the total mass, and that’s just defining terms. And then, in order to relate these distances to things that have shown up in our equations and Kepler’s Third Law, it’s true that the maximum value of the total distance–that is to say, the distance between these things can vary, because they can be in elliptical orbits. Sometimes they’re closer than others. And if you take the maximum distance between them–that’s the maximum of D_{total}–that’s a, that’s the semimajor axis of the orbit. And now, for nearly circular orbits, as the planets turn out to be, then D_{total} is always the same, because if it’s circular the distance between them doesn’t vary. So then, D_{total} is more or less equal to the semimajor axis because it’s always the same, therefore it’s always near its own maximum. And then, you can also say something interesting about the velocity. The velocity–what’s the definition of velocity? Velocity is miles per hour, or something like that. So, it’s distance per time. And let’s take a time period of one orbital period and ask the question, how far does something go in one orbital period? Well, it goes all the way around its orbit. And you may recall from high school geometry that if you know the radius of a circle, you also know its circumference. The circumference is the distance it would have to travel, that’s 2π times the radius. This is the basic fact from geometry and so that’s 2π times the distance, in this case, the semimajor axis. But this is only true for nearly circular orbits and the reason is that in highly elliptical orbits, the velocity changes by a substantial amount. It moves much faster when it’s closer. And so you can only really define what the overall velocity of the thing is, if you’ve got a nearly circular orbit. But in that case, as is true for planets, this 2πa over P gives you a value for velocity. Now, so we have another little equation here, V equals 2πa / P. This is an important one, so you’ll want to remember that. And I should say, which kind of Vs and as, remember up here there is V_{star}, V_{planet}, V_{total}, all these different kinds of things. What do I actually mean by that? And it can mean any of them, but it has to be consistent. So if you’re dealing with the velocity of the star then a is equal to D_{star}. Remember these are all nearly circular orbits so D isn’t going to change. And if you’ve–if you’re dealing with V of the planet, then a is equal to D of the planet, and that is approximately equal toD_{total}. Because the mass of the planet is so low that almost all the motion in the system comes from the planet. So, you can also deal with V_{total} _{,} which is equal to V_{planet}. And so, all these things go together. But if you’re worried about the velocity of the star, you have to be careful, because it’s not a of the orbit as a whole, it’s just a tiny piece of the orbit that involves the motion of the star. Okay? Let’s do an example. How fast does the Earth move? V = (2πa) / P. Well, we know a is equal to one Astronomical Unit. P is equal to one year for the Earth’s orbit, and so the velocity of the Earth is 2π Astronomical Units per year. Pretty straightforward but not very informative, because we don’t have a feeling for measuring velocities in Astronomical Units per year. There’s a joke in–a kind of physics joke that you convert all velocities into furlongs per fortnight just to be annoying. But let’s not do that; let’s convert it instead into meters per second, because then we have the hope of understanding what’s going on. V is equal to 2π. One Astronomical Unit is 1.5 times 10 to the 11 meters. And a year is 3 x 10^{7} seconds. π / 3 = 1. 2 x 1.5 = 3. 10^{11} / 10^{7}. 11  7 = 4. So, this is 10^{4} meters per second, that’s 30 kilometers per second. So, we move right along as we go around the Sun. If you did this for Jupiter, plugged in the various values for Jupiter–I won’t actually do that calculation, you can do it on your own–you discover that Jupiter moves about half as fast as the Earth. So, it goes around 15 kilometers per second, that’s 1.5 x 10^{4} meters per second. So now, we can ask the question, “How fast does the Sun move in response to the orbit of these planets?” So how fast is the solar motion induced by Jupiter? Okay, and now we go back to this momentum equation. Velocity of Jupiter times the mass of Jupiter is equal to the velocity of the Sun times the mass of the Sun. What we want to know is the velocity of the Sun. And so that is equal to the velocity of Jupiter, which we just calculated, times the mass of Jupiter, divided by the mass of the Sun. 1.5 x 10^{4}, that’s the velocity of Jupiter. Mass of Jupiter, as it happens–I think I wrote this down last time, 2 x 10^{27} kilograms. Mass of the Sun, 2 x 10^{30}. Twos cancel, obviously. We get 1.5 x 10^{4}. Times 10^{27}, that’s 10^{31}. 10^{31} / 10^{30}=10^{1}. Is equal to 15 meters per second, not kilometers now, meters. And so the Sun–Jupiter moves 15 kilometers per second, the Sun moves 15 meters per second. That makes perfect sense, because the Sun is 1,000 times more massive than Jupiter, so it has to be going 1,000 times more slowly. So instead of moving at some number of kilometers per second, it’s moving at some number of meters per second. This, it turns out, can be detected with modern equipment. “Detectable,” let’s say, in distant stars. You can see things that move by 15 meters a second. We’ll come back to how that’s done in a minute. How about the Earth? Solar motion due to Earth. Now, you might think is going to be bigger, because the velocity of Earth is bigger than the velocity of Jupiter. But, of course, the mass is much, much smaller. So we have M_{earth}V_{earth} is equal to M_{sun} V_{sun}, where V_{sun} now means the motion induced by Earth. This number is bigger than it was for Jupiter, but this number’s a whole lot smaller. And so the overall effect is that the V_{sun} is going to be smaller, V_{earth}M_{earth} / M_{sun}. That’s 3 x 10^{4}, that’s the velocity. Mass of the Earth, as it turns out, is 6 x 10^{24} meters per second. The Sun, down here at 2 x 10^{30}, same Sun. 6 / 2 = 3. 3 times 3 is 10, so we get (10^{1} x 10^{4} x 10^{24}) / 10^{30}. One–five–29 over 30. That’s 10^{1} = 1 / 10 of a meter per second, or 10 centimeters per second. So that’s much, much slower than the Sun moves in response to Jupiter. Why? Because the Earth is so much less massive. So, 15 meters a second for–as the result of Jupiter, only 10 centimeters a second, a tenth of a meter per second, as a result of Earth. And with current technology, things that slow are not detectable, yet in other–around–in other stars. So, we have a situation, and this is what was happening about ten years ago, where instruments had been developed that could, in principle, see the reflex motion of stars due to planets like Jupiter, but weren’t yet capable of seeing the motion of stars due to planets like Earth. But, we expect that Solar Systems ought to have planets like Jupiter, and so people went out to try and look. Chapter 3. Doppler Shift [00:23:45]All right, how do you find–How do you observe these things? And now, if you’ve taken high school physics, you will recall, perhaps, something called the Doppler Shift. This is the key. And this is a way of measuring velocity and it turns out oddly enough that velocities are some of the most easy and straightforward things to measure in astronomical objects because you can determine them by the Doppler Shift. And so just to remind you or to inform you, if you haven’t seen this before, and there is some help sheets and things that you can look at about this too. Light is characterized by its wavelength, which is usually given the Greek letter Lambda [λ]. And light that is something like 4 x 10^{7} meters. A wavelength has units of length. This looks blue to us. Light that is–let’s color code this for your convenience, 5 x 10^{7}meters looks green, 7 x 10^{7} meters kind of looks red. Longer wavelengths are what we call “infrared.” And shorter wavelengths that we can’t see are called “ultraviolet.” And so, ultraviolet up here. And if you get really, really long–if you have, like, meter wavelengths–that’s radio waves, out here. And if you have really short wavelengths, those are xrays and gamma rays. So, all of these kinds of radiation are basically the same thing, called “electromagnetic radiation”; again, there’s a help sheet. These are all electromagnetic radiation, and what kind of radiation it is depends on the wavelength. And the key to the Doppler Shift is that the observed wavelength changes, depending on the relative motion of the thing emitting the light in the observer. Motion of source and observer. In particular, if they’re moving toward each other, then the wavelength gets shorter, and if they’re moving away from each other, the wavelength gets longer. Do you feel an equation coming on? Because obviously this is going to need to be quantified, right? How much shorter? How much longer? But before we do that, let me just point out that this motion towards is sometimes called a “blueshift” because it makes–it pushes the light from the red end of the visual spectrum towards the blue and this kind of thing here, the motion away from each other is called the “redshift.” And let me show you why this is true before I write down the equation. Let’s see, it’s just–let me get out of that and try this one instead. It’s just a property of how waves look, so look what happens. If the thing is stationary there, in the middle, that’s emitting the waves, then the waves propagate equally in all directions, and both observers see the same distance between successive waves. That is to say, the same wavelength. And you can see that there. Then, when the thing is moving in some direction, each successive wave is emitted a little bit closer to one observer, and a little further away from the other observer. And so, because the waves are emitted at different places, the wave fronts here–I’ll wait until this cycle goes through again. The wave fronts for this observer are closer to each other and the wavelengths looks shorter. So, when the thing is coming towards you that’s emitting the wave, it looks shorter to the observer it’s going towards. Whereas, for this guy, the waves–each successive wave is emitted a little bit further away. And so the wave fronts are further away from each other when they pass, and then the wavelength becomes longer. So, that’s the kind of conceptual thing that’s going on. And the key thing is that the velocity that’s relevant here is velocity toward and away from you. If the thing is going sideways, it doesn’t make any difference. And so, it’s not actually velocity that you observe by looking at the Doppler Shift. It’s radial velocity, which is the technical term for–is the thing coming towards you or moving away from you. And how fast is it coming towards you and how fast is it moving away from you? Okay, let me turn this off here. So here’s the equation for that–let’s see here. Lambda is the wavelength–I’ll explain all these terms in a minute. And this is important. And the terms mean the following things: this is radial velocity [V_{r}], and it’s positive when it’s going away from you. It’s negative when it’s going towards you. And it’s zero when it’s going sideways. And it’s a velocity, it’s in meters per second or whatever the appropriate units of–furlongs per fortnight, or whatever the appropriate units are. The only restriction on the units is, it has to be in the same units as the thing in the denominator, here. That’s C, that’s the speed of light. That’s 3 x 10^{8} meters per second. Maybe we should have you work it out in furlongs per fortnight. No, no, no, we won’t do that. But as long as the velocity here is expressed in the same terms as you express the speed of light, then the units will work out. This λ with the little zero at the bottom is the rest wavelength, so that’s the wavelength you would observe from whatever light source, electromagnetic radiation source you have, if nothing was moving. And Delta Lambda [Δλ], this is not Δ times λ. That’s one symbol, confusingly enough. Delta always means change; you may remember this from calculus if you’ve taken calculus. Delta always means change, so this is a change in velocity–sorry, change in the wavelength. And it is defined such that the observed wavelength is equal to the rest wavelength, plus the change in the wavelength induced by the radial velocity. So now, look how this works. If this side of the equation is negative–if it’s coming towards you–then this quantity is negative. That means this quantity is negative. That means this quantity is negative. That means the observed wavelength is shorter than the rest wavelength, which is exactly how it’s supposed to be. When we come towards you, it’s blueshifted, the wavelengths get shorter. Similarly, if this is going–if something’s going away from you then V_{r} is positive, this is positive, and you end up with a longer wavelength. Okay? All right, example: how fast do you have to go to turn a red light green? This is potentially useful should you ever be pulled over for running a red light. You can just say, “it looked green.” How fast to make a red light green? And let’s call green light 5 x 10^{7} meters. Red light is 7 x 10^{7} meters. So Δλ had better be equal to (7  5) x 10^{7}. That’s 2 x10^{7} again in meters. And we want this to be negative, because we want λ_{0} to be the red. That’s what it would be like if nothing was moving and we want λ_{observed} to be green. And λ_{observed} = λ_{0} + Δλ. And this had better be,  2 x 10^{7} meters. And so Δλ / λ _{0}, that’s ( 2 x 10^{7}) over (5 x 10^{7}).So that’s  ^{2}⁄_{5} is equal to the radial velocity over the speed of light. So, if you’re going at  ^{2}⁄_{5} the speed of light, then the red light looks green. Now, the minus just means you have to be moving toward that light. Now, two things about this, first of all, don’t use this as an excuse, because it’ll cost you much more in the ticket for going over the speed limit if you’re going at ^{2}⁄_{5} of the speed of light. Second of all, be wary of this a little bit because there is, in fact, a change to the equation that happens when you’re going close to the speed of light, and we’ll talk about that when we get to relativity. And so this is just an example of how this works out. But when you’re dealing with the motions of stars ‒ 15 meters per second, 10 centimeters a second–you’re nowhere near the speed of light, and so the equation that I wrote down is actually fine. So, what do you expect to see when you’re looking at a star, which has a planet in orbit around it? Looking at a star in some kind of orbit–so here’s the radial velocity as a function of time. And the key thing about orbits is that the radial velocity changes, because first the thing’s coming towards you then later in its orbit it turns around goes the other way. Then it comes back and it comes towards you, turns around and goes the other way. So, the radial velocity will change from positive to negative and back as the object first comes towards you, then away from you. And so, it’ll look like this if you make a whole bunch of observations of this. It turns out that for circular orbits, this is a sine wave. And if you were to observe this, you could observe directly from such a plot, if you made repeated observations of the radial velocity of a star, or some other thing in orbit, you could observe two things. First of all, you would immediately be able to tell what the orbital period is. That’s the amount of time–this is a time axis–it takes for the object to come back to the same place in the orbit for a second time. Second of all, you could tell something about what the velocity is. The amplitude of this sine wave is something to do with the overall velocity, because that’s the maximum velocity it has coming towards you. But you have to be a little careful here, because that’s only true if the object–if the orbit is edgeon. Let me explain what I mean by that, we’ll come back to this later. If the orbit’s going this way, then it never comes towards you or goes away from you–it’s always going sideways. If, on the other hand, the orbit’s going this way, then first it comes towards you, then it goes away from you. And if it’s somewhere in between, you only see part of the motion of the orbit in terms of radial velocity. So, this amplitude is V if the orbit is edgeon. If not, V is going to be more than that, because you’re only seeing part of the motion. That’s a detail we’ll come back to later. So, this is what you expect to see if there’s a planet going around the star, and if you have enough sensitivity in your measurements of the Doppler Shift to be able to actually see that motion. Chapter 4. 51 Pegasus and “Hot Jupiters” [00:37:47]Okay, so here’s what they saw. This is a star called 51 Pegasus, in the Constellation of Pegasus. It is a “solar analog,” socalled, by which they mean, it’s about as much like the Sun as they can–as you can find if you go out and look at other stars. So it’s very, very much like the Sun. This is a radial velocity curve. I have taken the axis off for a reason I’ll point out later. This is radial velocity versus time, and this is exactly what you expect to see. It goes up; it comes down–very good news. This is exactly what you expect to see for a planet. There are two problems here, as it turns out. Problem number one is the Xaxis, because it turns out, the amount of time it takes you to go one orbit around for this object is a little more than four days. The shortest orbital period of a planet in our own Solar System is that of Mercury, which is eightyeight days, so this thing is way closer to the star than anything in our solar system. Xaxis problem is that P is equal to around four days. Problem number two is the Yaxis. And now let me put some units on to this thing. I’ve done a very bad thing of showing you a graph with no units, but it was just to prolong the suspense here. V_{R}. Zero. So, that’s going sideways. This is 50 meters a second. This is negative 50 meters a second. And so first it–here it’s coming towards you, there it’s going away from you. And so, the amplitude of this thing, and therefore the velocity of the star, is something like 50 meters a second. Now, it’s less obvious why that’s a problem, but it is. And let me show you why. Let me do the equation–do the equations a little bit. Okay, so this is a solar analog. What is the semimajor axis of the planets orbit? Axis of planet. We know P is equal to–let’s see 4 / 365.24 is equal to 1 / 100 is equal to 10^{2} in years. That’s four days over a year. And M is equal to 1 solar mass because it’s a solar analog. So, a^{3} = P^{2}. M = (10^{2})^{2} M is equal–one. 10^{4}– So a is equal to 10 to the–well let’s–we got to do this right. (100 x 10^{6})^{⅓}. 5^{3} = 125. So the cube root of 100 is 5. 5 x 10^{2} Astronomical Units. Or let’s put it in real units here. 5 x 10^{2}. An Astronomical Unit is 1.5 x 10^{11} meters. 5 x 1.5 is like 7 x 10^{9} meters. So that’s how–that’s the semimajor axis–and if you compare it, if you go look at the lists of planets in our own Solar System, what you’ll discover is that’s way closer than any of the planets in our own Solar System–way closer to the star than Mercury is. I haven’t used the Yaxis yet. Now I’m about to, because what I’m going to do is, I’m going to take–I’m going to figure out the velocity that this planet is moving at. That’s 2 π a / P. You wrote that down a little while ago; a, I just figured out. So, 2 times π times 7 times 10^{9}. P is 1/100 of a year. So, that’s 10^{2}, times 3 x 10^{7}, which is the number of seconds in a year. π / 3 = 1. 2 x 7 = 15. 10 ^{9} ‒ 10^{2} x 10^{7}. 10^{5} – So, this is 15 x 10^{4}, or 1.5 x 10^{5}. So this is thing is going–the planet now, this is V_{total}, which is approximately equal toV_{planet}, is going at 150 kilometers a second. Much faster than the Earth is going around; well, that makes perfect sense. It’s in closer–got to move faster to stay in its orbit. And so, this is 150 kilometers a second if you prefer those units. Now, we know the velocity of the planet, we know the mass of the star, we know the velocity of the star and so we can figure out the mass of the planet. Here’s how it works. M_{p}V_{p} = M_{star}V_{star}. V_{planet}, we’ve just figured out, is 1.5 x 10^{5}. Actually, let’s leave it as 15 x 10^{4}, that’ll turn out to be easier for the arithmetic. The mass of the stars, the same as the mass of the Sun, 2 x 10^{30}. The velocity of the star we just observed. We saw it moving back and forth, it’s 50 kilometers a second, 5 x 10^{1}. Okay, so now–and this is multiplied by the mass of the planet. So, the mass of the planet is equal to 2 x 10^{30}, times (5 x 10^{1}) / (15 x 10^{4}). 5 / 15 is a third, so this is a third times 10^{3}, times the mass of the Sun. 10^{3} times the mass of the Sun, that’s the mass of Jupiter. So, this is equal to ⅓ of the mass of Jupiter–which is, by the way, bigger than the mass of Saturn or any other object in our own Solar System. Okay, so that’s catastrophic, right? That’s a hopeless disaster, because the first part of the lecture and the second part of the lecture have entirely contradicted each other. Because in the first part of the lecture I gave you a whole song and dance, and you all wrote it down, and it sounded believable at the time. How inner planets were going to be these small rocky things. And now the very first planet we go out and find turns out to be a very close planet that’s quite massive. And this is impossible according to this nice little theory of planetary formation that I promulgated to you earlier in the lecture. And so, the very first planet that was observed turned out to be a screwup. Worse than that, there were soon dozens more like it discovered. These are given the name “Hot Jupiters,” and you can see what the problem is. If you figure out what the surface temperature of these things ought to be, it’s over 1,000 degrees. There’s no way you could have gas or ice on such a thing, it would totally melt. And there aren’t enough rocks in the whole rocky elements–silicon and iron, and so forth–in the whole of the planetary system of our own Solar System to add up to ⅓ of a Jupiter, even if you put them all together. So how is this done? What is going on here? It seems clear that one of the two things must be true. Either this thing isn’t a planet and there’s some other explanation for this attractive bunch of data here, or something has gone seriously wrong in our understanding of how planets form. And stay tuned, we’ll talk about that next time. [end of transcript] Back to Top 
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