Those questions that we asked in the last lecture about how that material got inside Jupiter? How that non hydrogen and helium material got inside? Really bring up the big questions of how does Jupiter or a giant planet like Jupiter form which in fact brings up the big question of how do planets for in general? We're going to digress for a few lectures now and talk about planet formation in general, getting back to a couple of different ideas on how Jupiter itself could have formed. And we'll expand on these ideas of planetary formation as we go talk about the small bodies in the solar system in the next unit. If you knew nothing about the planets in the solar system and just looked at them for the first time, one of the things that would strike you is that the planets, all eight of them, circle around the Sun in merely a very flat disc. There's a little bit of variation but almost none. What's more, most of the planets also themselves rotate in that same direction. Not all of them, Venus goes the wrong way, it's got the issues with the Sun. Uranus and Neptune are more tilted on their sides, the reasons for that are less clear. But the other planets, like the Earth have their north poles nearly straight up, north poles meaning where they rotate in the same direction as the Earth. All of that rotation in one direction and all of that flatness of where the planets are clearly has a cause. As far as Immanuel Kant and even earlier, the idea was proposed that the planets and the Sun itself formed a collapsing cloud of gas and dust. You can look up in the sky and see these clouds of gas and dust. They collapse down, and if they're spinning just a little bit. They're huge, they're spinning a little bit. As they get smaller, and smaller they start to spin faster, and faster. The analogy that everyone always use is of an ice skater who has his hands out, and he's spinning slowly. And then he pulls himself in, and gets in that really fast spin. These clouds do the same thing, and as they spin faster and faster some of the material goes into the center star. But there's so much spinning that a lot of it flattens out into a very think disc going around that star. It doesn't make it into the star itself. As I said, that ideas has been around for a long time. These days we actually see them. Okay, okay. They don't look like as good as this. This is an artist conception of what they look like, but here I'll show you the real one in a minute. But let's get the impression on what's going on first. There's a star here in the center. This is our young star that's just in the process of forming, and this disc around it, you can sort of see that there's a cavity there in the middle of the star has blasted away. And materials also as we'll see, falling onto the star. But out in here, is where all the planets are being formed. Let's look at a real picture now from the Hubble Space Telescope. This is a famous picture from the Hubble Space Telescope of an object called HH30. The HH stands for Herbig Haro object, which is not important, but you might look at this and say, it's not that spectacular, but you're wrong. This, I remember when this image first came out. It was just a stunning picture to be able to see a disc like that. But let me tell you what you're actually seeing, because it's not clear when you look at it what's really going on. The disc, this disk that's being formed is actually, you're looking at it exactly edge on and the disc is, if I could draw it. If you could see the disc. The disc would be sort of like this. The star is in the middle. You don't see the star. The star is being blocked by all the dust in the disc. What you see out through here, is the star light that's scattered in the upper parts of the disc. So I really should draw the disc coming out more like this. The star light scatters in the very upper parts and the very lower parts and you see just a tiny bit of that star light but mostly, you see the disc by the absence of light down here in the middle. We found another object like this a few years later and we called it The Hamburger, and you can sort of see why. The one that we found actually looked more like a hamburger. The other thing you see are these funny columns coming out this way and going out this way. These are jets of material that are going out of the poles of the star as that star is forming magnetic fields are twisting up and around and material is being shot out. And you can actually if you take pictures of this and other objects like this you can sometimes see knots of material in it and you can take a picture a year later and they moved out little bits at a time. It's just spectacular that we can see these discs in the process of forming. So we know we have these discs, we have these discs of dust. We can see the dust blocking out the star. We know that we have these big gas and dust clouds that collapse down to form these discs. The question to ask yourself then is, where do the planets come from? This is a process,in it's simplest terms, the mathematics were worked out more than 40 years ago. I'm going to go through this a little bit. Now, I know some of you will tune out every time I write down equations. These are not so bad. Try to pay attention. If you can't follow along just sort of understand what's going on here. But it's actually sort of interesting that we can in some pretty simple first principles, understand a process by which you could take dust and grow big planet. You might think at first that, okay, you have dust that sticks together. How are you going to go from these tiniest little grains of dust up to giant planet planets like Jupiter or even smaller planets like the Earth? And it turns out that the process is amenable to our mathematical thought. Well, let me say. Here's the way I'm going to explain it for now. When we go into the next unit and talk about the small bodies, we'll talk about the fact that this may not actually be the way it works. There might be some complicated, interesting ways in which planets form in very different ways. But for now, let's do this one. Okay, imagine that you are a chunk of material in this nebula. You're a little bit bigger than the other stuff around you. And there's all this dust around you. And the dust is moving. We're going to consider you stationary. Everything is going around the Sun. In orbit around the Sun. So, if I'm sitting on top of this big chunk, everything else has slightly different velocities around it. So we'll call the velocity of these other things, we'll call them V, and I'm going to use V infinity because I'm going to mean the velocity that they have when they're infinitely far away from this thing. They don't really have to be infinitely far away, that's okay. So, if I'm sitting in a sea of dust and the sea of dust has a density, I'm going to call it a number density. You might think of density as kilograms per meter cube, they're grams per centimeter cube. But now we do ask do number density which means number of dust grains per meter cube. And each of this dust grains again has this velocity, something like the infinity the question is, how often do the dust grains hit this? The answer is a very simple one. This has a radius, R. It has a cross section, that means that the dust is moving along, and it sees a target. The cross section is the projected circumference. So pi, R, squared. The bigger the cross section, the more likely impact is going to occur. The bigger the velocity that these things have, the more likely an impact is going to hurt because more things will impact more frequently. And the larger the number density, the more frequently an impact will occur. This is equal to number of impacts per second. Let's double check to make sure the units make sense here. We have meters squared, we have meters per second squared, we have number per meter cubed. Notice the meter cube cancels out the meter and the meter squared, we're left with number per second. Number of impacts per second by this very simple formula and it makes perfect sense. As the impacts occur, the object gets bigger, it's cross section becomes bigger. More impacts occur, it grows faster, but not by much. If this were the only process going on in the nebula, the growth of objects would take, essentially, forever. But there's one extra thing, or two extra things, depending on how you think of it, that helped immensely. Let's leave this formula over here. And let's talk about one more process that happens. This is the process that's called Gravitational focusing. And, it's a very simple process to think about. The point is that if you have this object here in the middle, as some mass, that has some radius, and the dust particles are moving along. And the dust particle, if it's here and going along at the right Inside the target area, it'll hit. But a dust particle moving like here, will be deflected by the gravity of this object and might hit it too. So we need to account for the gravity of this central object, because it now has, if objects all the way from here to here impact, it now has a much bigger cross section than just this radius, right here. Turns out to be, again, a very simple calculation to make by considering just two things. One is conservation of energy. Two is conservation of angular momentum. Let's think about a particle that's just barely going to impact. This one right here just hits the one that's just barely going to impact, just hits the backside, right there. Anything closer is going to hit, anything farther is not going to hit. So that's going to be our impact parameter, we'll call it. That's the term we use in physics, usually given by the letter b. I have no idea why. The impact parameter is the new radius that will give us the cross-section. How do we calculate that? Well, it will also depend on what this V infinity is. You can imagine that if V infinity is very small, it's moving along very slowly, it's going to be easy to have it impact. If it's moving along very fast, it'll barely be deflected. So we need to have the V infinity there too. This is all we need to know. And the reason is, because that we can say that the energy at this point and this point, the energies are the same. The energy here is purely kinetic energy. If this is far enough away, that's why we say infinity. If this is far enough away the gravitational attraction between these two is so small that it doesn't add any energy. So, sufficiently for our way there's only kinetic energy. Kinetic energy is one-half MV squared, V infinity squared. Energy is conserved so at impact, we're going to have kinetic energy. We'll call it the impact squared. But the other thing we have is potential energy due to the interaction of these two and that's going to be minus G big M is the mass of this guy little m, I didn't even mention that, but figured it out this has a mass of little m, GMm over R. So we have kinetic energy, kinetic energy, potential energy, conserved easy enough. One other thing that we have is angular momentum conserved, angular momentum is the rotation angular momentum about this central object. Angular momentum is a perpendicular distance times the velocity so the perpendicular distance to this object is again b, right here the velocity is V infinity. So angular momentum of this guy even though he's way out here, the angular momentum is V infinity times b and the angular momentum by the time he reaches here perpendicular velocity is, well, he's right there so that's easy. It's V impact, times R. This is just orbital mechanics, this is the exact same thing we did when we were figuring out how to get to Mars many lectures ago. Because until the moment that it impacts, this thing is just sort of in orbit around this object. We can just do a little manipulation and now we can figure this out. We can say, we don't care about the impact so we can say the V impact is V infinity times b over R. We can then ignore this term from now on and we're going to substitute back into this big equation here. And we'll, say one-halve MV infinity squared equals one-half little m Vi is V infinity squared times b squared over R squared and then minus G Mm over R. Notice that all these terms have the little m in them, and they all cancel out, so it actually doesn't matter what the mass of this object is here, so we're going to cancel that out here. Not entirely true, I really should have done this in a center of mass. So I'm assuming that this object is much more massive than this object is how it essentially is working here. And remember what we're trying to do, we're eventually trying to solve for b squared, because we're going to get the cross-section where we use the cross-section here was pi R squared. We're going to use the new cross section which will be pi b squared. So, we're solving for b squared. So, let's quickly solve for it. We're going to multiply everything through by these terms that are on the b squared side. Bring this over to this side and I'm going to take the b over to this side and I will write is as b squared equals this term just becomes R squared. And this terms becomes plus, there's going to be a 2 GMR over V infinity squared. Or we can write this as V squared equals R squared times 1 times 2 GM over R times 1 over V infinity squared. This term turns out to be the escape velocity. If you're sitting on the surface of a body and you want to escape it's gravitational influence, this is the escape velocity of the object. You can work that out and so we could write this, rewrite this as b squared equals R squared times 1 plus V escape squared over V infinity squared. B squared is now going to replace R squared in this term here, to say how frequently the impacts occur. And you can see, that it's going to depend not only on R squared, as R gets bigger, it's going to depend on the escape velocity, which of course depends on how much mass the object has, as the object gets bigger. The escape velocity increases, but the most important term is this V infinity. It will end up depending on how fast the other particles around are going. If V infinity becomes very small, if the objects are moving very slowly with respect to each other, you'd think gee, they're never going to impact, no big deal. But no, if they're moving very slowly with respect to each other, the only influence they have is of gravity. V infinity becomes zero, b squared becomes infinity. And the fact that there's a V infinity squared on the bottom here and a V infinity on the top here means that the overall impact rate scales as 1 over V infinity. So if the velocities get small the impacts go up dramatically. Why would the velocities get small? Let's do one other process, we had gravitational focus, the other process that's important is called Dynamical friction. It has nothing to do with friction, so it's kind of a weird term. But what it really means is, if you have a bunch of objects around. Let's say, you have some big ones, massive ones, and then you have a bunch of small ones, little ones. And they are moving around through space, gravitationally interacting with each other. An interesting thing is going to happen. First, if we start them out all at the same velocity, going around the Sun, they're all going at the same velocity. What's going to happen is that the big ones are going to slow down. They're not going to slow down going around the Sun. They're all going around the Sun. But they're going to slow down their relative motions to very small, relative velocities, while the small ones will end up with very high relative motions. The process is similar, though not exactly the same to what you could imagine. What if you had a bunch of bouncy balls. Let's say some big beach ball size ones like this. And then a bunch of little super balls. And you put them inside of a big box and you initially start them all moving at the same velocity. Well, the small ones are going to hit the big ones and start go faster and faster every time whenever a big one hits a small one it comes off on very fast speed. Every time a small one hits a big one it slows the big one down by just a little bit eventually the big ones will have very similar speeds, while the little ones will be moving around really quickly. There are a couple of ways of thinking about this, the other way you could think about it is called equipartition of energy, the particles end up having similar amounts of kinetic energy. Kinetic energy is one-half MV squared. If M is really big, you better have a very small v. If M is small, you have a very big V. The other way of thinking about it is this. If you had a single object sitting here. And a sea of small bodies was coming around going this way. Well, as they go by, the ones that don't hit are deflected like this. And so, in front of the object, the object is moving this way. In front of the object, there's a uniform sea of particles. Behind the object, there's a little bit more density of objects right behind it because they've been deflected this way. A little bit less out here. And that little bit extra density gives a little bit of a tug here and slows this body down. While these evolved and sped up a little bit. Yeah, there are a lot of different ways of looking at it. But the important point is, in a sea of particles, small ones will end up going fast with respect to each other, and with respect to the large ones. And the large ones will get increasingly slowed down. The larger they are, the slower they will get with respect to each other, their relative velocities will be very small. What does that mean? That means that in this initial protoplanetary disc, in this initial disc of gas and dust. Particle start to impact each other. They start to stick. They starts to be a little bit of gravitational focusing and then there's feedback. The particles starts to get bigger particles that objects starts to get bigger. Have more gravitational focusing. They get even bigger still, then they start to slow down with respect to all of the objects that are around there. And maybe there's another one over here doing the exact same thing. Suddenly, these objects have zero velocity with respect to each other. Forget about all these little objects going around here, but think about these big ones now. These big ones now have almost no velocity with respect to each other and so their gravitational cross section, their gravitational focusing cross section becomes huge. And they all merge. This is a process that we call Runaway Growth, and it's a significantly faster process than you would get by simply saying how many objects are going to hit this or even how gravitationally focused are you going to be? It's that combination of gravitational focusing and Dynamical friction, which leads to this process of runaway growth. This process of Runaway growth can continue until everything in a region of the disk is combined into one single object. We'll talk about what those single objects are in the next lecture.
