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That equation of hydrostatic equilibrium is the one that tells us

Â what the pressure is as a function of altitude, which we use z, which is really

Â just a function of how much mass and force is pushing down on that material.

Â The other thing that we needed was the equation of state.

Â The equation of state was critical because it relates.

Â The pressure, and the density.

Â It says,

Â if I have this pressure, this is the density that the material will achieve.

Â And the equation of state is complicated.

Â There are experimental methods to figuring them out.

Â We did one theoretical one where we called,

Â did this thing called the fair me see, where we had, we found that

Â pressure was proportional to the density to the 5/3 for very compressed objects.

Â But that's not the real equation of state of materials.

Â You really do have to do more complicated calculations and

Â the calibrate those with experimental data.

Â We also had the phase diagram.

Â 2:00

Really that's of course the density of Jupiter but

Â it actual will matter that we're actually doing the radius and total mass.

Â How do we do that?

Â Well we start by saying all right here is the upper radius,

Â we know the radius of Jupiter right here.

Â And we're going to start with the top layer of material.

Â We're going to, we know what the temperature is at the very top, so

Â we know the phase of the material at the very top.

Â It's an H2.

Â And we know the energy being dissipated at the very top, that's what we can measure.

Â We measure the photons coming up, we call this the thing that is the photosphere.

Â So we start with this very top.

Â And we slowly add layers.

Â We add the next layer down, we can figure out the pressure of the next layer down is

Â because it's just because of the mass of the material pulling down on this.

Â We know the gravitational field here because we know the total mass of Jupiter

Â 2:48

on the inside pulling it down.

Â We can calculate the pressure, we can calculate the temperature.

Â If we assume what sort of heat flow that we have in there, we can make sure that

Â the phase diagram still tells us what the state of matter is and

Â we calculate a new pressure.

Â We do the same thing one more time, add another little la-,

Â layer of mass in through here.

Â We calculate a new pressure.

Â We calculate a new temperature.

Â We check the phase diagram.

Â We keep on doing the same thing through here.

Â At some point we'll find the phase diagram and it shows that the.

Â Hydrogen has turned into metallic hydrogen.

Â Metallic hydrogen has a different equation of state than molecular hydrogen does, so

Â we'll have to change to that equation of state.

Â We keep on going, and when we get to the center of Jupiter right here,

Â we are either, one of two things is going to happen.

Â Either it perfectly works, we're finished.

Â All the mass, if we add up all the mass that we added in through here.

Â We get the mass of Jupiter or this is what happens in real life if you take

Â a Hydrogen, Helium composition and you start adding these layers in through here,

Â you find that you get to the center of Jupiter, you've added all these layers in

Â through here and you've made a ball that's the same size as Jupiter.

Â But you have mass leftover on the inside.

Â What it does that tell you?

Â That tells you, you cannot make Hydrogen-Helium with the 75% Hydrogen,

Â 25% Helium that we were assuming.

Â You cannot make something the radius of Jupiter and the mass of Jupiter.

Â If you want to make something the mass of Jupiter,.

Â You had to have started at a bigger radius and added the layers

Â together until you get to the center and you perfectly run out of material.

Â Now you can imagine starting too big and you add all the layers together and

Â suddenly you run out of material here and you would have to have a vacuum

Â in the middle of Jupiter and that makes no sense so

Â you have to you wou-, you would iterate around, you'd first do this calculation.

Â Have mass leftover, you would add a little more,

Â you would do this one say oops that's too much and you would come in the middle and

Â say I now know for something the mass of Jupiter what the radius has to be

Â if you're going to make Hydrogen-Helium planet with these ratios.

Â 4:51

And for fun, you can do the same thing for any mass that you come up with, here is

Â mass of Jupiter one in this case and you find that if you take the mass of Jupiter.

Â You need to have a Hydrogen-Helium ball that's this big,

Â which is a little bit bigger then the actual size of Jupiter itself.

Â If you use a little bit less lat mass like, say,

Â the mass of Saturn, you come up with a ball that's this big.

Â It gets smaller.

Â If you use a lot less you come up with something that's this big.

Â Interestingly if you add more mass than Jupiter, two, three,

Â four times the mass of Jupiter here.

Â You get to a maximum and then you start to add more mass and the size goes down.

Â 5:30

This is strange behaviour usually you think of if I have a ball of material and

Â I add more to it it gets bigger, in this case it's the exact opposite and the exact

Â opposite is because of that quantum mechanical effect that equation of state.

Â Which is that the additional pressure is so high as you add more mass in through

Â here that the density increase more than compensates for

Â the extra material that you have adding on.

Â And so your objects actually shrink, some think the mass of Jupiter,

Â made out of hydrogen and helium,

Â is bigger than something 20 times the mass of Jupiter made out of Hydrogen-Helium.

Â This is a cool plot seeing how, how hy, Hydrogen-Helium planet would go

Â as function of mass and it shows once again that Jupiter doesn't fit.

Â What is going on with Jupiter?

Â We'll talk also, when we talk about Saturn we'll.

Â Also, that Saturn doesn't fit either.

Â What is going on with Saturn?

Â One or

Â more of the assumptions that we made in trying to construct Jupiter was wrong.

Â There was really only one way to make Jupiter smaller for the mass that it is.

Â And, that's by adding heavier material to it than just Hydrogen and Helium.

Â Well, that's not a big surprise.

Â We know that the universe is more than just hydrogen and helium.

Â But even if we added these small amounts of additional materials that are in, say,

Â the sun, you would get very little difference.

Â You might drop it down by a little bit.

Â Jupiter has to have more material than just Hydrogen and Helium, and

Â it has to have more of that material than the sun does.

Â And there are two ways to have that material [INAUDIBLE] there are probably

Â more then two ways but there are two end members of ways you could have

Â that material you could imagine that Jupiter just had Hydrogen and

Â Helium and everything nicely mixed throughout.

Â Or, you can imagine, that, Jupiter has Hydrogen and Helium and

Â then a core of solid material that's a different composition.

Â We'll talk a lot about these two different ideas but I'm going to show you first that

Â this calculation, this a line that you see right here goes like this,

Â comes up like that, goes through Jupiter and keeps going.

Â That a line is for the composition of a planet that's mostly

Â Hydrogen-Helium except the inside has a core

Â that is 15 times the mass of the Earth, that's how I'll write Earth mass.

Â That's a little circle with a plus in it is the Earth symbol.

Â 15 Earth masses.

Â Of material on the inside.

Â If I add this 15 Earth-mass core, I can make Jupiter.

Â Do I have to have a truth 15 Earth-mass core?

Â No. I can distribute that

Â material throughout there like this, and

Â we will use additional observations to try to determine whether or not this 15

Â Earth-masses that we need, extra, is down here like this or spread throughout.

Â Now, to be fair, one of the big problems that we have

Â in constructing our model of Jupiter is that equation of state.

Â The equation of state, particularly at the highest pressures, is pretty uncertain.

Â It's uncertain by a factor of a couple, and it's enough uncertain that

Â you could probably get away with using a slightly different equation of state.

Â You could probably get away with just Hydrogen and

Â Helium making it through Jupiter.

Â Most people think that the equation of state requires that there is this

Â additional material inside Jupiter, but it's not 100% certain.

Â What is certain, and again, we'll talk about when we get to Saturn,

Â this is the symbol for Saturn,.

Â When we get to Saturn is that even with a modified equation of state,

Â you can't make Saturn out of just Hydrogen and Helium.

Â There must be additional ma, material inside of Saturn.

Â if there's di, additional material inside of Saturn, and we think there probably is

Â additional material inside of Jupiter, we're going to go on the assumption that

Â it's, that it's probably really there, in one of these two forms.

Â And see if we can figure out what might be going on, on the inside.

Â