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Those questions that we asked in the last lecture about

Â how that material got inside Jupiter?

Â How that non hydrogen and helium material got inside?

Â Really bring up the big questions of how does Jupiter or

Â a giant planet like Jupiter form which in

Â fact brings up the big question of how do planets for in general?

Â We're going to digress for a few lectures now and

Â talk about planet formation in general, getting back to a couple of different

Â ideas on how Jupiter itself could have formed.

Â And we'll expand on these ideas of planetary formation as we go talk about

Â the small bodies in the solar system in the next unit.

Â If you knew nothing about the planets in the solar system and just looked at them

Â for the first time, one of the things that would strike you is that the planets,

Â all eight of them, circle around the Sun in merely a very flat disc.

Â There's a little bit of variation but almost none.

Â What's more, most of the planets also themselves rotate in that same direction.

Â Not all of them, Venus goes the wrong way, it's got the issues with the Sun.

Â Uranus and Neptune are more tilted on their sides, the reasons for

Â that are less clear.

Â But the other planets, like the Earth have their north poles nearly straight up,

Â north poles meaning where they rotate in the same direction as the Earth.

Â All of that rotation in one direction and

Â all of that flatness of where the planets are clearly has a cause.

Â As far as Emmanuel Conte and even earlier, the idea was proposed that the planets and

Â the Sun itself formed a collapsing cloud of gas and dust.

Â You can look up in the sky and see these clouds of gas and dust.

Â They collapse down, and if they're spinning just a little bit.

Â They're huge, they're spinning a little bit.

Â As they get smaller, and smaller they start to spin faster, and faster.

Â The analogy that everyone always use is of an ice skater who has his hands out, and

Â he's spinning slowly.

Â And then he pulls himself in, and gets in that really fast spin.

Â These clouds do the same thing, and as they spin faster and

Â faster some of the material goes into the center star.

Â But there's so much spinning that a lot of it flattens out into a very think disc

Â going around that star.

Â It doesn't make it into the star itself.

Â As I said, that ideas has been around for a long time.

Â These days we actually see them.

Â Okay, okay.

Â They don't look like as good as this.

Â This is an artist conception of what they look like, but

Â here I'll show you the real one in a minute.

Â But let's get the impression on what's going on first.

Â There's a star here in the center.

Â This is our young star that's just in the process of forming, and

Â this disc around it, you can sort of see that there's a cavity there in the middle

Â of the star has blasted away.

Â And materials also as we'll see, falling onto the star.

Â But out in here, is where all the planets are being formed.

Â Let's look at a real picture now from the Hubble Space Telescope.

Â This is a famous picture from the Hubble Space Telescope of an object called HH30.

Â The HH stands for Herbig Harrel object, which is not important, but

Â you might look at this and say, it's not that spectacular, but you're wrong.

Â This, I remember when this image first came out.

Â It was just a stunning picture to be able to see a disc like that.

Â But let me tell you what you're actually seeing,

Â because it's not clear when you look at it what's really going on.

Â The disc, this disk that's being formed is actually,

Â you're looking at it exactly edge on and the disc is, if I could draw it.

Â If you could see the disc.

Â The disc would be sort of like this.

Â 5:03

These are not so bad.

Â Try to pay attention.

Â If you can't follow along just sort of understand what's going on here.

Â But it's actually sort of interesting that we can in some pretty simple first

Â principles, understand a process by which you could take dust and grow big planet.

Â You might think at first that, okay, you have dust that sticks together.

Â How are you going to go from these tiniest little grains of dust up to giant planet

Â planets like Jupiter or even smaller planets like the Earth?

Â And it turns out that the process is amenable to our mathematical thought.

Â Well, let me say.

Â Here's the way I'm going to explain it for now.

Â When we go into the next unit and talk about the small bodies,

Â we'll talk about the fact that this may not actually be the way it works.

Â There might be some complicated,

Â interesting ways in which planets form in very different ways.

Â But for now, let's do this one.

Â Okay, imagine that you are a chunk of material in this nebula.

Â You're a little bit bigger than the other stuff around you.

Â And there's all this dust around you.

Â And the dust is moving.

Â We're going to consider you stationary.

Â Everything is going around the Sun.

Â In orbit around the Sun.

Â So, if I'm sitting on top of this big chunk,

Â everything else has slightly different velocities around it.

Â So we'll call the velocity of these other things, we'll call them V, and

Â I'm going to use V infinity because I'm going to mean the velocity that

Â they have when they're infinitely far away from this thing.

Â They don't really have to be infinitely far away, that's okay.

Â So, if I'm sitting in a sea of dust and

Â the sea of dust has a density, I'm going to call it a number density.

Â You might think of density as kilograms per meter cube,

Â they're grams per centimeter cube.

Â But now we do ask do number density which means number of

Â dust grains per meter cube.

Â And each of this dust grains again has this velocity,

Â something like the infinity the question is, how often do the dust grains hit this?

Â The answer is a very simple one.

Â This has a radius, R.

Â 7:32

This is equal to number of impacts per second.

Â Let's double check to make sure the units make sense here.

Â We have meters squared, we have meters per second squared,

Â we have number per meter cubed.

Â Notice the meter cube cancels out the meter and the meter squared,

Â we're left with number per second.

Â Number of impacts per second by this very simple formula and it makes perfect sense.

Â As the impacts occur, the object gets bigger, it's cross section becomes bigger.

Â More impacts occur, it grows faster, but not by much.

Â If this were the only process going on in the nebula,

Â the growth of objects would take, essentially, forever.

Â But there's one extra thing, or two extra things, depending on how you think of it,

Â that helped immensely.

Â Let's leave this formula over here.

Â And let's talk about one more process that happens.

Â This is the process that's called Gravitational focusing.

Â And, it's a very simple process to think about.

Â The point is that if you have this object here in the middle, as some mass,

Â that has some radius, and the dust particles are moving along.

Â And the dust particle,

Â if it's here and going along at the right Inside the target area, it'll hit.

Â But a dust particle moving like here,

Â will be deflected by the gravity of this object and might hit it too.

Â So we need to account for the gravity of this central object, because it now has,

Â if objects all the way from here to here impact,

Â it now has a much bigger cross section than just this radius, right here.

Â Turns out to be, again,

Â a very simple calculation to make by considering just two things.

Â One is conservation of energy.

Â Two is conservation of angular momentum.

Â Let's think about a particle that's just barely going to impact.

Â This one right here just hits the one that's just barely going to impact,

Â just hits the backside, right there.

Â Anything closer is going to hit, anything farther is not going to hit.

Â So that's going to be our impact parameter, we'll call it.

Â That's the term we use in physics, usually given by the letter b.

Â I have no idea why.

Â The impact parameter is the new radius that will give us the cross-section.

Â How do we calculate that?

Â Well, it will also depend on what this V infinity is.

Â You can imagine that if V infinity is very small,

Â it's moving along very slowly, it's going to be easy to have it impact.

Â If it's moving along very fast, it'll barely be deflected.

Â So we need to have the V infinity there too.

Â This is all we need to know.

Â And the reason is, because that we can say that the energy at this point and

Â this point, the energies are the same.

Â The energy here is purely kinetic energy.

Â If this is far enough away, that's why we say infinity.

Â If this is far enough away the gravitational attraction between these two

Â is so small that it doesn't add any energy.

Â So, sufficiently for our way there's only kinetic energy.

Â Kinetic energy is one-half MV squared, V infinity squared.

Â Energy is conserved so at impact, we're going to have kinetic energy.

Â We'll call it the impact squared.

Â 10:39

But the other thing we have is potential energy due to the interaction of these two

Â and that's going to be minus G big M is the mass of this guy little m, I didn't

Â even mention that, but figured it out this has a mass of little m, GMm over R.

Â So we have kinetic energy, kinetic energy, potential energy, conserved easy enough.

Â One other thing that we have is angular momentum conserved,

Â angular momentum is the rotation angular momentum about this central object.

Â Angular momentum is a perpendicular distance times the velocity so

Â the perpendicular distance to this object is again b,

Â right here the velocity is V infinity.

Â So angular momentum of this guy even though he's way out here,

Â the angular momentum is V infinity times b and the angular momentum by the time

Â he reaches here perpendicular velocity is, well, he's right there so that's easy.

Â It's V impact, times R.

Â This is just orbital mechanics, this is the exact same thing we did when we were

Â figuring out how to get to Mars many lectures ago.

Â Because until the moment that it impacts,

Â this thing is just sort of in orbit around this object.

Â We can just do a little manipulation and now we can figure this out.

Â We can say, we don't care about the impact so we can say the V impact

Â is V infinity times b over R.

Â We can then ignore this term from now on and

Â we're going to substitute back into this big equation here.

Â And we'll, say one-halve MV infinity

Â squared equals one-half little m Vi

Â is V infinity squared times b squared

Â over R squared and then minus G Mm over R.

Â Notice that all these terms have the little m in them, and they all cancel out,

Â so it actually doesn't matter what the mass of this object is here, so

Â we're going to cancel that out here.

Â Not entirely true, I really should have done this in a center of mass.

Â So I'm assuming that this object is much more massive than this object is

Â how it essentially is working here.

Â And remember what we're trying to do, we're eventually trying to solve for

Â b squared, because we're going to get the cross-section where we

Â use the cross-section here was pi R squared.

Â We're going to use the new cross section which will be pi b squared.

Â So, we're solving for b squared.

Â So, let's quickly solve for it.

Â We're going to multiply everything through by

Â these terms that are on the b squared side.

Â Bring this over to this side and I'm going to take the b over to this side and

Â I will write is as b squared equals this term just becomes R squared.

Â 14:00

B squared is now going to replace R squared in this term here,

Â to say how frequently the impacts occur.

Â And you can see, that it's going to depend not only on R squared,

Â as R gets bigger, it's going to depend on the escape velocity,

Â which of course depends on how much mass the object has, as the object gets bigger.

Â The escape velocity increases, but the most important term is this V infinity.

Â It will end up depending on how fast the other particles around are going.

Â If V infinity becomes very small,

Â if the objects are moving very slowly with respect to each other, you'd think gee,

Â they're never going to impact, no big deal.

Â But no, if they're moving very slowly with respect to each other,

Â the only influence they have is of gravity.

Â V infinity becomes zero, b squared becomes infinity.

Â And the fact that there's a V infinity squared on the bottom here and

Â a V infinity on the top here means that the overall impact

Â rate scales as 1 over V infinity.

Â So if the velocities get small the impacts go up dramatically.

Â Why would the velocities get small?

Â Let's do one other process, we had gravitational focus,

Â the other process that's important is called Dynamical friction.

Â It has nothing to do with friction, so it's kind of a weird term.

Â But what it really means is, if you have a bunch of objects around.

Â Let's say, you have some big ones, massive ones, and

Â then you have a bunch of small ones, little ones.

Â And they are moving around through space,

Â gravitationally interacting with each other.

Â An interesting thing is going to happen.

Â First, if we start them out all at the same velocity, going around the Sun,

Â they're all going at the same velocity.

Â What's going to happen is that the big ones are going to slow down.

Â They're not going to slow down going around the Sun.

Â They're all going around the Sun.

Â But they're going to slow down their relative motions to very small, relative

Â velocities, while the small ones will end up with very high relative motions.

Â The process is similar, though not exactly the same to what you could imagine.

Â What if you had a bunch of bouncy balls.

Â Let's say some big beach ball size ones like this.

Â And then a bunch of little super balls.

Â And you put them inside of a big box and

Â you initially start them all moving at the same velocity.

Â Well, the small ones are going to hit the big ones and start go faster and faster

Â every time whenever a big one hits a small one it comes off on very fast speed.

Â Every time a small one hits a big one it slows the big one down by just

Â a little bit eventually the big ones will have very similar speeds,

Â while the little ones will be moving around really quickly.

Â There are a couple of ways of thinking about this,

Â the other way you could think about it is called equipartition of energy,

Â the particles end up having similar amounts of kinetic energy.

Â Kinetic energy is one-half MV squared.

Â If M is really big, you better have a very small v.

Â If M is small, you have a very big V.

Â The other way of thinking about it is this.

Â If you had a single object sitting here.

Â And a sea of small bodies was coming around going this way.

Â Well, as they go by, the ones that don't hit are deflected like this.

Â And so, in front of the object, the object is moving this way.

Â In front of the object, there's a uniform sea of particles.

Â Behind the object, there's a little bit more density of objects right behind it

Â because they've been deflected this way.

Â A little bit less out here.

Â And that little bit extra density gives a little bit of a tug here and

Â slows this body down.

Â While these evolved and sped up a little bit.

Â Yeah, there are a lot of different ways of looking at it.

Â But the important point is, in a sea of particles, small ones will end up

Â going fast with respect to each other, and with respect to the large ones.

Â And the large ones will get increasingly slowed down.

Â The larger they are, the slower they will get with respect to each other,

Â their relative velocities will be very small.

Â What does that mean?

Â That means that in this initial protoplanetary disc,

Â in this initial disc of gas and dust.

Â Particle start to impact each other.

Â They start to stick.

Â They starts to be a little bit of gravitational focusing and

Â then there's feedback.

Â The particles starts to get bigger particles that

Â objects starts to get bigger.

Â Have more gravitational focusing.

Â They get even bigger still, then they start to slow down with respect to

Â all of the objects that are around there.

Â And maybe there's another one over here doing the exact same thing.

Â Suddenly, these objects have zero velocity with respect to each other.

Â Forget about all these little objects going around here, but

Â think about these big ones now.

Â These big ones now have almost no velocity with respect to each other and so

Â their gravitational cross section,

Â their gravitational focusing cross section becomes huge.

Â And they all merge.

Â This is a process that we call Runaway Growth, and it's a significantly faster

Â process than you would get by simply saying how many objects are going to

Â hit this or even how gravitationally focused are you going to be?

Â It's that combination of gravitational focusing and

Â Dynamical friction, which leads to this process of runaway growth.

Â This process of Runaway growth can continue until

Â everything in a region of the disk is combined into one single object.

Â We'll talk about what those single objects are in the next lecture.

Â