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We figured out from the density of Jupiter that

Â it's probably not made out of rock, in fact

Â it's probably not even made out of ice, even

Â though it's density is higher than that of ice.

Â But it would be compressed on the inside by

Â so much that it's density would be even higher.

Â We suspected in fact that it's made out of gas.

Â What gas?

Â Well we're going to start, by just taking

Â an educated guess, and then work from there.

Â An educated guess might be that Jupiter is made out of the

Â same materials and the same amounts of materials that the sun is.

Â The sun is approximately 75% Hydrogen and 25% Helium.

Â Close enough,but not exactly right.

Â But it's worth exploring what would Jupiter

Â be like if it had this same composition.

Â Can we explain.

Â The size of Jupiter.

Â The density of Jupiter and other properties that we'll

Â talk about later, by assuming a composition like this.

Â The answer is going to turn out to be no.

Â It will be close to this but there will be some very important details that

Â we figure out, but this will be a good start to help us on our way.

Â Okay, so the first thing that we keep talking about is that.

Â Inside of Jupiter there is so much pressure from all the material

Â on top of it that everything is going to be compressed so much,

Â and get higher density so the first question we might ask ourselves,

Â is how can we figure out what the pressures are, inside of Jupiter?

Â Here we''re going to make an assumption, and that

Â assumption is that Jupiter is an hydrostatic [SOUND] equilibrium.

Â Lets look at that for a minute, hydrostatic equilibrium.

Â Hydro means liquid.

Â Static means not moving.

Â We're going to make the assumption that Jupiter, the

Â interior of Jupiter is a liquid that's not moving.

Â An equilibrium in this case means that

Â there's some sort of balance between something.

Â And something else.

Â We'll figure out what that is in a minute.

Â But what we're saying when we say that something's in

Â hydrostatic equilibrium, we're saying that the support for Jupiter, the

Â reason Jupiter doesn't collapse back on itself is not because

Â there is roiling motion inside of there that's keeping everything supported.

Â Or huge temperatures that making, making everything boiled

Â because Jupiter is simply behaving like a stationary fluid.

Â Now, seems crazy I said that, that's it's mostly Hydrogen

Â and Helium, why could we call it a fluid hydrostatic?

Â In fact, the hydrostatic equilibrium was a pretty good assumption.

Â For all of the planets, even the Earth.

Â The interior of the Earth is in something close to hydrostatic equilibrium

Â even though it's actually more like solid than it is like a liquid.

Â And we use hydrostatic equilibrium to discuss things like the

Â Earth's atmosphere which is again a gas and not a liquid.

Â So when we say hydrostatic equilibrium we can mean gases, we can mean liquids.

Â We can even mean solids.

Â And what we really mean is something very specific [SOUND] and that is, that the

Â weight of all the stuff on top of you is balanced by the pressure of you.

Â And by you, I mean a little parcel of, in

Â the case of the earth, a little parcel of air,

Â in the case of Jupiter, a little parcel of Jupiter's

Â interior, in the case of Earth, a parcel of rock.

Â If I look at, let's look at the Earth case, if I look at

Â this little piece of air, in my hands right this minute, well this piece of

Â air in my hands right this minute has a lot of air above it

Â which is pushing down on it and the only reason it doesn't collapse on itself.

Â Is because as it pushes down on it my [INAUDIBLE] also

Â has pressure pushing out and those two things balance, they're in equilibrium.

Â We can use that idea to figure out what the pressure

Â is as the function of altitude in something like the Earth's atmosphere.

Â Let's do it this way.

Â What we'd like to figure out is the pressure.

Â As a function of height, let's say, above the earth.

Â Here's the, here's the surface of the earth.

Â Here's height going up.

Â And we'd like to know what the pressure is as you go

Â up above the surface of the earth, p of z, we'll call it.

Â Well, we know the pressure at the surface of the earth, because we can measure it.

Â You get out your barometer or something else and see what the pressure is.

Â And we can actually figure out pretty easily what the pressure

Â is just a little bit above the surface of the Earth.

Â Let's say that we go up by one meter above

Â the surface of the Earth, not very much, we actually know

Â that the pressure one meter above the surface, surface of the

Â Earth is about the same as the pressure at the surface.

Â But let's say, pressure at zero.

Â Is equal to we'll call it p not pressure

Â at one meter, [SOUND] well it's almost the same as

Â pressure down here at zero, but because it's all due

Â to the weight of the air on top pushing down.

Â But there's less weight.

Â Why is there less weight?

Â Because this little parcel of air is no longer sitting

Â on top of us because we're up a little bit higher.

Â So we subtract the pressure due to this little parcel of air.

Â What is the pressure of that?

Â Well, it's the density of this material, whatever that is, we have the pressure

Â at zero minus the pressure that would have been caused by this little bit

Â of this one metre high slab so that's equal to the density of the gas row,

Â the gravitational pole of the gas and now our one metre size.

Â Let's make sure this makes sense, pressure as

Â you remember as a force per unit area force

Â per meter squared The gravitational force of the

Â earth times mass, will be equal to a force.

Â Which you have instead of a mass, you have a density which is mass divided by volume.

Â We'll multiplying that by the height here to get the mass per unit area.

Â So we have force per unit area is minus row g times one meter.

Â What's the pressure at two meters?

Â Well, the pressure at two meters, [SOUND] is

Â pressure at one meter, [SOUND] minus row g.

Â Times one meter.

Â We can do this forever.

Â I think you see the pattern.

Â And I'm going to write it down in a suggestive way.

Â P(zdeltaz) = P (z)- pg x Delta Z.

Â Now I'm not just talking about one meter steps, I'm

Â talking about any kind of small step that we could do.

Â And I'm going to rewrite this in an even more suggestive way.

Â And I'm going to then take you back to

Â high school calculus [SOUND] and remind you that

Â this thing that I just wrote down, is

Â the definition that you've seen before of the derivative.

Â This is dP.

Â Dz, the derivative of p with respect to z is equal to minus row g.

Â This is a differential equation, but its

Â about the simplest differential equation in the world.

Â And in our simplest possible case, we can write the solution to this

Â differential equation as p of z is equal to p not minus row gz.

Â Lets make sure that this differential equation works, if I take dp dz.

Â This is a constant that goes away, the PDZ is minus

Â row g, that's exactly what our differential equation was over here.

Â It's okay if you haven't seen differential equations in 30 years or perhaps

Â have never seen them all this is saying is that the change in pressure.

Â As a function of height is proportional to row times g.

Â And all this is saying is that the pressure is

Â a linear function of z in a very special case.

Â The very special case is that row and g are not functions of z.

Â They do not change with height.

Â In fact, this is only true if, if row, if the density of the

Â material does not change with compression and

Â this is true for an incompressible fluid.

Â Now, we haven't been talking about incompressible fluid.

Â We've been talking about how the high pressures inside

Â of Jupiter cause the gas to get higher density and

Â that's certainly the case too, true, but there is a

Â good example of an incompressible fluid or nearly incompressible fluid.

Â Where this equation works very well, and that's water.

Â Water, as you know, if you take a piston full of water

Â and you try to compress that water, it's really hard to do.

Â 7:40

Water is, at typical pressures, nearly incompressible.

Â So, what does that mean?

Â That means that if you have water on top of you.

Â That the pressure that you feel from that

Â water increases constantly as a function of Z,

Â the distance that you have, the amount of water that you have on top of you.

Â If you've ever done scuba diving you actually know this already.

Â You know that at the surface before you

Â go underwater, there is one atmosphere of pressure.

Â You know that something like ten meters down.

Â There are two atmospheres.

Â You know that something like 20 meters down there are three atmospheres.

Â And it keeps on going.

Â Every ten meters you get another atmosphere on top of you.

Â That's purely because water is nearly incompressible.

Â And it follows this equation right here.

Â Even if they were made out of water, there, there's so much

Â pressure that water would break down and, and become higher density too.

Â And the gases that they're actually made out of are certainly quite compressible.

Â So we have to think a little bit harder

Â about how we're going to use this equation to figure out.

Â Jupiter, so there are, of course, two complications here.

Â One is that this is the force of gravity and the force

Â of gravity changes as you go up in, say, the earth's atmosphere.

Â The gravity gets less the higher you are.

Â We are going to completely ignore that one right now because

Â it's just an extra mathematical complication that we at least understand.

Â And we can figure out how to deal with but we're not going to.

Â The other interesting thing of course is the density changes as a function

Â of height, really what the density changes as is a function of pressure.

Â The higher the pressure the higher the density.

Â So really row is a function of P and now if you know your differential equations

Â you realize this is no longer a simple

Â differential equation because this is a function of P.

Â And there is no general to this equation, unless we know

Â what this function is right here, row as a function of p.

Â Row as a function of p, row of course

Â is the density, is generically, called an equation of state.

Â And figuring out the equation of state.

Â Figuring out what the density of material is.

Â As a function of pressure is a critically important experimental and theoretical

Â task, for trying to understand the interiors of things like giant planets.

Â All of us know a really simple equation of state

Â though and without even realizing that that's what it was.

Â You probably learned in high school [SOUND], my favorite equation of state

Â the ideal gas law, but if you learned it like I learned

Â it PV equals n RT, this is the pressure, this is the

Â volume of the gas in a balloon or in a piston or something.

Â N is the number of moles, of material which is just

Â a measure of the amount of that gas inside of that thing.

Â R is the ideal gas constant, universal gas constant.

Â And T is the temperature, temperature we haven't talked about before.

Â Now you don't see pressure in here anywhere but you can take this and say.

Â P equals n over v RT and n over v

Â the amount of material per volume, not surprisingly that's actually

Â a density pressure equals density and to get the units

Â right you have to divide by mu the mean molecular mass.

Â So it's something like hydrogen has a very low molecular mass.

Â This is just H2, it's really just two protons.

Â Something like oxygen is much higher but we don't

Â need worry about that right now, we're just going to

Â write it as this and know that we now

Â have an equation for density as a function of pressure.

Â We have an equation of state.

Â Now you may say, this is great!

Â We can now go figure everything out about the interior of Jupiter and you

Â would be wrong, because the ideal gas law is only good for ideal gases.

Â Ideal gases are things that are something like the pressure

Â of the earth's atmosphere is a pretty nice ideal gas,

Â as you compress gasses more and more, their equations of

Â state gets further and further away from the Ideal Gas Law.

Â But it will be good enough for us to at least get a feel for how this is going.

Â I'm going to now solve this equation for density.

Â And say density is pressure times mu divided by rt.

Â And I am going to substitute up in here and get the dpdz is minus g

Â times p mu over rt.

Â This is great because although this is a little

Â bit more complicated than the differential equation we had before.

Â It's a really simple differential equation.

Â If you remember again, anything about.

Â About your derivitives or differential equations you might remember the DDX of

Â EDX equals EVX it's one of the nicest things to differentiate in the world and

Â if we had DDX of A Is equal to a times e to the ax and if you look at this.

Â We have dp, dz is proportional to p and so we can simply write the p of z is

Â equal to some constant which will be the pressure at, at the surface of the earth.

Â Times E to the minus GU over RT times Z.

Â And even if you don't think much about what this

Â stuff is here, you find that the pressure, as you

Â go up, decreases exponentially with altitude, so if I plotted

Â pressure versus altitude, I'm going to do pressure here, altitude this way.

Â It starts out at P not.

Â And it decreases exponentially as it goes up.

Â This is still not quite good enough because we made one more assumption which

Â is that t is a constant and in fact t is not a constant.

Â Temp, the temperature changes greatly, as you know,

Â as you go up in the Earth's atmosphere you

Â cool down and so we actually needed to back here have t as a function of z two.

Â