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That equation of hydrostatic equilibrium is the one that tells us

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

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

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

The equation of state was critical because it relates.

The pressure, and the density.

It says,

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

And the equation of state is complicated.

There are experimental methods to figuring them out.

We did one theoretical one where we called,

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

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

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

You really do have to do more complicated calculations and

the calibrate those with experimental data.

We also had the phase diagram.

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Really that's of course the density of Jupiter but

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

How do we do that?

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

we know the radius of Jupiter right here.

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

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

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

It's an H2.

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

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

So we start with this very top.

And we slowly add layers.

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

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

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

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on the inside pulling it down.

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

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

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

we calculate a new pressure.

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

layer of mass in through here.

We calculate a new pressure.

We calculate a new temperature.

We check the phase diagram.

We keep on doing the same thing through here.

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

Hydrogen has turned into metallic hydrogen.

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

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

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

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

Either it perfectly works, we're finished.

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

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

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

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

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

But you have mass leftover on the inside.

What it does that tell you?

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

25% Helium that we were assuming.

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

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

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

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

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

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

in the middle of Jupiter and that makes no sense so

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

Have mass leftover, you would add a little more,

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

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

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

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

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

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

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

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

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

It gets smaller.

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

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

four times the mass of Jupiter here.

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

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

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

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

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

here that the density increase more than compensates for

the extra material that you have adding on.

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

made out of hydrogen and helium,

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

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

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

What is going on with Jupiter?

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

Also, that Saturn doesn't fit either.

What is going on with Saturn?

One or

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

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

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

Well, that's not a big surprise.

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

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

the sun, you would get very little difference.

You might drop it down by a little bit.

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

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

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

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

that material you could imagine that Jupiter just had Hydrogen and

Helium and everything nicely mixed throughout.

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

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

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

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

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

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

Hydrogen-Helium except the inside has a core

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

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

15 Earth masses.

Of material on the inside.

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

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

No. I can distribute that

material throughout there like this, and

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

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

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

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

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

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

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

You could probably get away with just Hydrogen and

Helium making it through Jupiter.

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

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

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

this is the symbol for Saturn,.

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

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

There must be additional ma, material inside of Saturn.

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

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

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

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