In the last few lectures we've seen how we can take dust, coagulate it, eventually have gravitational forces build up larger and larger planetesimals. And have those planetesimals eventually eat all of the material inside of its feeding zone, and become an isolated mass. And, we've seen that that isolated mass at the location of Jupiter, can be something like the mass of the core for which we're looking. The last step then in making something like Jupiter is adding all the rest of that mass, adding all of the hydrogen and helium that didn't come in as the solid core in the isolation mass. How does that happen? Well, well when you make something as big as 10 Earth masses at the location of Jupiter, you can also start to pull in some of the gas. Smaller than 10 Earth masses something even the size of the Earth, has a hard time holding on to gas because the gas doesn't behave the same way as solid particle. It's not like the gas comes in, hits Jupiter or the core of Jupiter and stays, the gas comes in, the gas comes by, it feels the gravitational attraction of Jupiter, but it also feels other forces. It feels gas pressure from the other gases around it, and that pressure is caused by the temperature of the gas. So you can imagine that a high temperature gas is not going to be able to be attracted to the planet. It will be attracted but it won't stay at the planet. This is one of the reasons that we have almost no hydrogen and helium around the earth today. The earth is too small to hold on to it. And, the temperatures are so high that whenever there is any in the atmosphere, it quickly escapes out the top of the atmosphere. This 10 earth mass core though is sufficiently large that it can hold onto a little bit of gas. It'll hold onto a shell, an envelope is the word that's usually used, an envelope of gas. A, a shell is not really the right word it's, it's a, it's a small amount although the size actually is quite large. And it's an envelope of gas in hydrostatic equilibrium. Hydrostatic equilibrium, as you remember, it balances out the gravitational pull of all the stuff here down at the bottom that's pulling everything in versus the pressure. In this case, these are relatively low pressure environments so something like the ideal gas law works very well. So really it's the temperature that matters. If the temperatures are high, you can support quite a bit of mass against the the pressure. If the temperatures are low, this envelope could perhaps collapse down upon itself. And this process of slowly accreting gas from the disc goes on for potentially millions and millions of years. A little bit more gas gets added on which means there's a little bit more mass around this entire planet, which means it can attract a little bit more gas. There's in a sense a feeding zone in, in sort of the same sense that we had the isolation zone for the solid terrestrial planets. Jupiter or a giant planet like this can't gravitationally attract things that are beyond its feeding zone, but it all, is adding in all this extra mass and its feeding zone is getting bigger and, if you remember, there's still a lot of gas left over in this feeding zone. There's a lot more gas than solid material in the disk, so there's the potential to get a lot of this gas. It doesn't get a lot of this gas because there's only a certain amount of material that you can put inside of this envelope in hydro static equilibrium, and that amount of material is determined by the total mass on the inside that it, that can pull it in and also the temperature of this envelope in hydro static equilibrium. Essentially, the temperature remains sufficiently high that this envelope is big and relatively thin. Something funny though happens as the mass of this proto-planet proto-Jupiter begins to get larger and larger. When the amount of gas approaches the same amount as solids, so maybe 10 earth masses of gas, 10 earth masses of solids. There's a sudden transition that occurs or a very quick transition that occurs where the temperature of this envelope, which is caused by the heating, caused by all this material come in, the temperature of this envelope is no longer warm enough to support this envelope. This extra mass that's now coming into the envelope, and so it has to collapse. It has to collapse onto the core this once big puffy envelope all that material is no longer in hydrostatic equilibrium, it is collapsing down in a, in a hydrodynamic collapse. What that also means is that there's no longer anything preventing other material from falling on to the planet, that's now growing. It can basically take material as fast as the disc can give it to it. And, if allowed, it would eat the entire gas out of its own feeding zone. This is sort of the hand wavy version of how it all works. Let me show you at least a, a figure out of a paper where this was simulated in great detail, and show you how the masses of the core, the mass of this, this envelope in the gas actually evolve overtime. Okay. It looks something like this from this paper. This is time down on the bottom in millions of years and the mass up here in terms of earth masses of material. The first thing that happens is within only a half a million years, only, that's pretty quick. Half a million years, this, this is the Z, this is the core. We'll call this the solid material. This material quickly builds up to its isolation mass. This is exactly the process that we talked about before, and it comes out to around 10 earth masses like we had before. And notice it stays pretty close to that for a long time. There, it has eaten all the material in its feeding zone, so it can't get any bigger. And then look what happens. This M X Y, X Y is everything except for the solids. X and y, x is usually hydrogen y is helium, z is everything else so massive hydrogen and helium is down here. And actually I think it's more interesting to look at the total mass that's up here on the top adding these two together. And you can see that this envelope, this is just the envelope growing slowly. The mass is getting bigger. So the, the ability to pull in an envelope gets a little bigger, but the temperatures are still sufficiently high that they can support this very large envelope as time goes on. And suddenly you get to a point where the core mass, and the gas mass are about the same. Right around here and things start to change dramatically. The, the gas mass starts to increase and then suddenly has this ridiculous runaway growth by a factor of 10 in almost no time. And that's when the collapse is happening onto that core, the, the envelope can longer be supported. The gravitational pull for the temperature of the envelope is trying to support it from collapsing and so the envelope collapses down and it pulls in any other gas around and the whole planetary mass, that's MP, increases dramatically. This is a nice process that could explain why we have something like a 10 earth mass core and a lot of material sitting on top of it. The time scales work out pretty well. We now know from looking at disks around stars and looking at how long they seem to eh, to last. That typical disks don't last any longer than 10 million years so the gas is gone after 10 million years but actually a lot of them only lasts for more like 3 million years and so, it's a little uncomfortably long, this process here in the middle takes a long time, but maybe that's okay. Maybe we happen to have a disk that lasted that long. In general, this process, I think, is a very nice, elegant solution to how you take a core that something like 10 earth masses and and add all this extra material on top of it. It does a pretty good job of describing something like Jupiter. It has some difficulties with some of the other planets that we'll talk about a little bit later, but before we do that, I want to talk about an entirely different mechanism that could potentially lead to something like Jupiter being formed. This last one that we just did was called the core instability model. Core instability because you form a core and then you have this hydrodynamic instability where all the stuff collapses onto you. The other model is called the disk instability. [SOUND] And that's a process by which the disc itself, the disc going around the star becomes unstable to gravitational collapse and instead of having this 10 million year long process of slowly adding material in, it happens essentially instantaneously. Okay. Let's figure out why that might happen. If we took a, looked at a little chunk of disc sitting right here, what keeps that from just gravitationally collapsing instantly? Well, there's some amount of material inside of here and that material has it's gravitational potential which is trying pull itself together, but there are things resisting being pulled together. One of them that we talk a lot about is the temperature, is the heat. So, even within a disk you, you try to push yourself together, but the higher the temperature is, the faster the motions of the gas are and the more they resist being pushed together. The one other thing that resists being pushed together is that everything is going around the star here and there's that rotational kinetic energy that this patch of material has that also is keeping it from trying to collapse together. It's trying to sheer itself apart. This stuff is moving a little bit faster. This stuff is moving a little bit slower. So every time you try to collapse down, the sheer wants to pull you apart. So, you can balance these 3 things, the, the gravitational potential, the thermal energy, and the rotational kinetic energy. And we'll call that 1, 2 and 3, and if 1 is greater than 2 plus 3, there's more gravity. There's more energy. The energy available in the gravitational potential can overcome the thermal energy and the rotational kinetic energy, and boom, you get this collapse. We can write down a relatively simple formula for what is required for this gravitational potential to dominate. And this formula looks something like this. Q is called the Toomre Q. Do I pronounce it right? I have no idea if I pronounce that right. Toomre Q parameter and it involves the sound speed, sound speed, sound's complicated but sound speed really just means the velocity that the individual gas particles have. The sound speed is related to the temperature in a very simple way. You can say that the kinetic energy of a, of a gas particle is one half m, v squared and that that is something like the thermal energy. The thermal energy is, k, t where k is the Boltzmann constant. And what is the Boltzmann constant? It's just a way to convert temperature to energy. If I give you a temperature and I want to know how much energy that temperature corresponds to, I multiply it by the Boltzmann constant. Which means that if I tell you a gas temperature and I want to tell you what's typical velocity of that gas, well that typical velocity of the gas is something like, I'm going to ignore factors of two, but it's something like squared of kt over m. So this sound speed, which is something like the typical gas velocity of something like square root of kt over m. The most important point is that it's dependent on that temperature. At least the square root of that temperature. Because that's the thermal energy. This is the thermal energy. It depends on it over here. So what do we get? We get the thermal energy over here. This is the rotational frequency of the disk, how fast it's rotating. So notice that the temperature is up here on the top. The rotational frequency is up on the top. Increase the temperature, queue gets big. Make it rotate faster. How do you make it rotate faster? Well, you have to make a bigger star to make it rotate faster. But do any of those two things and this Q parameter gets bigger, and how do you make it smaller? There's only way to make it smaller. Increase Pi? Mmh, no you can't do that. Increase G? That's the gravitational constant, can't do that. Increase this, this if you remember is the surface density of the disk. Make the disk more massive. It makes sense. A more massive disk is going to have more material in here, trying to make itself collapse. A faster rotating disk will sheer itself apart. A hotter disk will not allow itself to collapse. So, this is, this is an intuitively reasonable balance of parameters. How do you make this disk have a Q less than 1? Well the easiest way to do it is to make the mass of the disk the total mass of the disk, something like 10% of the mass of the star. If that happens, and with some other assumptions that we don't need to worry about, if that happens, the, the, there are patches on the disk that become unstable, and collapse, instantaneously. You can figure out how big those patches are, then you can figure out, therefore, how big of a collapse happens, and, when you do that, you find out that the mass that collapse is something like a Jupiter mass. Pretty amazing. We have a entirely separate way to make a Jupiter mass object. Now, interestingly, this would, that's just the initial fragment. There are a lot of fragments that might potentially happen. So we have a way of maybe making a lot of Jupiter mass objects, which is a little bit uncomfortable. One thing it doesn't do, of course, is make a core in the middle of that Jupiter mass object. It just takes everything from the disk and collapses it instantaneously. it's, it's similar to the way the, the sun would have formed. It would be a, a representative sample of everything out there. So, in this case, in this disc instability model, how do we get a core of material? How do we get all of that extra material inside of Jupiter? Now, I'll remind you, we don't know for sure that Jupiter actually has a core. All of our best models say it does, but there are still uncertainties enough that maybe there's not an actual core in the bottom. But we do know that Jupiter has extra material. It's enhanced in things that are not hydrogen and helium compared to the sun. It has extra planetary materials in it. One reason could be the core, but the other could be that there's material spread throughout Jupiter or maybe a little bit concentrated on the inside, but could be spread throughout. In the disk instability model, you could imagine that that process happens because when the disk goes unstable and makes a initial Jupiter sized objects, there are still all of these planetesimals around, these solid bodies around. And now there's Jupiter sitting here in the middle of this sea of all these planetesimals, and it's big, and it has the ability to attract a lot of those planetesimals in, they go in, they dissolve in the atmosphere as they, as they burn up on the inside and they distribute their materials throughout. Perhaps they even make it in to the center, and could make something even resembling a core. Well it's possible that this process happens, and certainly, it's possible that this process happens elsewhere in the universe. My bet is that the core instability, or something similar to the core instability is what's really going on, and that this process was not the one that made Jupiter. Part of the reason for my assumption that that's true, comes from looking at the other giant planets in our solar system. That is Jupiter, and Uranus and Neptune. And we'll do that in the next lecture.
