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Lecture 2.04: Hydrostatic equilibrium
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Del curso dictado por Caltech
The Science of the Solar System
285 calificaciones
De la lección
Unit 2: The insides of giant planets (week 1)
Conoce a los instructores
Mike BrownProfessor
Planetary Astronomy
We figured out from the density of Jupiter that
it's probably not made out of rock, in fact
it's probably not even made out of ice, even
though it's density is higher than that of ice.
But it would be compressed on the inside by
so much that it's density would be even higher.
We suspected in fact that it's made out of gas.
What gas?
Well we're going to start, by just taking
an educated guess, and then work from there.
An educated guess might be that Jupiter is made out of the
same materials and the same amounts of materials that the sun is.
The sun is approximately 75% Hydrogen and 25% Helium.
Close enough,but not exactly right.
But it's worth exploring what would Jupiter
be like if it had this same composition.
Can we explain.
The size of Jupiter.
The density of Jupiter and other properties that we'll
talk about later, by assuming a composition like this.
The answer is going to turn out to be no.
It will be close to this but there will be some very important details that
we figure out, but this will be a good start to help us on our way.
Okay, so the first thing that we keep talking about is that.
Inside of Jupiter there is so much pressure from all the material
on top of it that everything is going to be compressed so much,
and get higher density so the first question we might ask ourselves,
is how can we figure out what the pressures are, inside of Jupiter?
Here we''re going to make an assumption, and that
assumption is that Jupiter is an hydrostatic [SOUND] equilibrium.
Lets look at that for a minute, hydrostatic equilibrium.
Hydro means liquid.
Static means not moving.
We're going to make the assumption that Jupiter, the
interior of Jupiter is a liquid that's not moving.
An equilibrium in this case means that
there's some sort of balance between something.
And something else.
We'll figure out what that is in a minute.
But what we're saying when we say that something's in
hydrostatic equilibrium, we're saying that the support for Jupiter, the
reason Jupiter doesn't collapse back on itself is not because
there is roiling motion inside of there that's keeping everything supported.
Or huge temperatures that making, making everything boiled
because Jupiter is simply behaving like a stationary fluid.
Now, seems crazy I said that, that's it's mostly Hydrogen
and Helium, why could we call it a fluid hydrostatic?
In fact, the hydrostatic equilibrium was a pretty good assumption.
For all of the planets, even the Earth.
The interior of the Earth is in something close to hydrostatic equilibrium
even though it's actually more like solid than it is like a liquid.
And we use hydrostatic equilibrium to discuss things like the
Earth's atmosphere which is again a gas and not a liquid.
So when we say hydrostatic equilibrium we can mean gases, we can mean liquids.
We can even mean solids.
And what we really mean is something very specific [SOUND] and that is, that the
weight of all the stuff on top of you is balanced by the pressure of you.
And by you, I mean a little parcel of, in
the case of the earth, a little parcel of air,
in the case of Jupiter, a little parcel of Jupiter's
interior, in the case of Earth, a parcel of rock.
If I look at, let's look at the Earth case, if I look at
this little piece of air, in my hands right this minute, well this piece of
air in my hands right this minute has a lot of air above it
which is pushing down on it and the only reason it doesn't collapse on itself.
Is because as it pushes down on it my [INAUDIBLE] also
has pressure pushing out and those two things balance, they're in equilibrium.
We can use that idea to figure out what the pressure
is as the function of altitude in something like the Earth's atmosphere.
Let's do it this way.
What we'd like to figure out is the pressure.
As a function of height, let's say, above the earth.
Here's the, here's the surface of the earth.
Here's height going up.
And we'd like to know what the pressure is as you go
up above the surface of the earth, p of z, we'll call it.
Well, we know the pressure at the surface of the earth, because we can measure it.
You get out your barometer or something else and see what the pressure is.
And we can actually figure out pretty easily what the pressure
is just a little bit above the surface of the Earth.
Let's say that we go up by one meter above
the surface of the Earth, not very much, we actually know
that the pressure one meter above the surface, surface of the
Earth is about the same as the pressure at the surface.
But let's say, pressure at zero.
Is equal to we'll call it p not pressure
at one meter, [SOUND] well it's almost the same as
pressure down here at zero, but because it's all due
to the weight of the air on top pushing down.
But there's less weight.
Why is there less weight?
Because this little parcel of air is no longer sitting
on top of us because we're up a little bit higher.
So we subtract the pressure due to this little parcel of air.
What is the pressure of that?
Well, it's the density of this material, whatever that is, we have the pressure
at zero minus the pressure that would have been caused by this little bit
of this one metre high slab so that's equal to the density of the gas row,
the gravitational pole of the gas and now our one metre size.
Let's make sure this makes sense, pressure as
you remember as a force per unit area force
per meter squared The gravitational force of the
earth times mass, will be equal to a force.
Which you have instead of a mass, you have a density which is mass divided by volume.
We'll multiplying that by the height here to get the mass per unit area.
So we have force per unit area is minus row g times one meter.
What's the pressure at two meters?
Well, the pressure at two meters, [SOUND] is
pressure at one meter, [SOUND] minus row g.
Times one meter.
We can do this forever.
I think you see the pattern.
And I'm going to write it down in a suggestive way.
P(zdeltaz) = P (z)- pg x Delta Z.
Now I'm not just talking about one meter steps, I'm
talking about any kind of small step that we could do.
And I'm going to rewrite this in an even more suggestive way.
And I'm going to then take you back to
high school calculus [SOUND] and remind you that
this thing that I just wrote down, is
the definition that you've seen before of the derivative.
This is dP.
Dz, the derivative of p with respect to z is equal to minus row g.
This is a differential equation, but its
about the simplest differential equation in the world.
And in our simplest possible case, we can write the solution to this
differential equation as p of z is equal to p not minus row gz.
Lets make sure that this differential equation works, if I take dp dz.
This is a constant that goes away, the PDZ is minus
row g, that's exactly what our differential equation was over here.
It's okay if you haven't seen differential equations in 30 years or perhaps
have never seen them all this is saying is that the change in pressure.
As a function of height is proportional to row times g.
And all this is saying is that the pressure is
a linear function of z in a very special case.
The very special case is that row and g are not functions of z.
They do not change with height.
In fact, this is only true if, if row, if the density of the
material does not change with compression and
this is true for an incompressible fluid.
Now, we haven't been talking about incompressible fluid.
We've been talking about how the high pressures inside
of Jupiter cause the gas to get higher density and
that's certainly the case too, true, but there is a
good example of an incompressible fluid or nearly incompressible fluid.
Where this equation works very well, and that's water.
Water, as you know, if you take a piston full of water
and you try to compress that water, it's really hard to do.
Water is, at typical pressures, nearly incompressible.
So, what does that mean?
That means that if you have water on top of you.
That the pressure that you feel from that
water increases constantly as a function of Z,
the distance that you have, the amount of water that you have on top of you.
If you've ever done scuba diving you actually know this already.
You know that at the surface before you
go underwater, there is one atmosphere of pressure.
You know that something like ten meters down.
There are two atmospheres.
You know that something like 20 meters down there are three atmospheres.
And it keeps on going.
Every ten meters you get another atmosphere on top of you.
That's purely because water is nearly incompressible.
And it follows this equation right here.
Even if they were made out of water, there, there's so much
pressure that water would break down and, and become higher density too.
And the gases that they're actually made out of are certainly quite compressible.
So we have to think a little bit harder
about how we're going to use this equation to figure out.
Jupiter, so there are, of course, two complications here.
One is that this is the force of gravity and the force
of gravity changes as you go up in, say, the earth's atmosphere.
The gravity gets less the higher you are.
We are going to completely ignore that one right now because
it's just an extra mathematical complication that we at least understand.
And we can figure out how to deal with but we're not going to.
The other interesting thing of course is the density changes as a function
of height, really what the density changes as is a function of pressure.
The higher the pressure the higher the density.
So really row is a function of P and now if you know your differential equations
you realize this is no longer a simple
differential equation because this is a function of P.
And there is no general to this equation, unless we know
what this function is right here, row as a function of p.
Row as a function of p, row of course
is the density, is generically, called an equation of state.
And figuring out the equation of state.
Figuring out what the density of material is.
As a function of pressure is a critically important experimental and theoretical
task, for trying to understand the interiors of things like giant planets.
All of us know a really simple equation of state
though and without even realizing that that's what it was.
You probably learned in high school [SOUND], my favorite equation of state
the ideal gas law, but if you learned it like I learned
it PV equals n RT, this is the pressure, this is the
volume of the gas in a balloon or in a piston or something.
N is the number of moles, of material which is just
a measure of the amount of that gas inside of that thing.
R is the ideal gas constant, universal gas constant.
And T is the temperature, temperature we haven't talked about before.
Now you don't see pressure in here anywhere but you can take this and say.
P equals n over v RT and n over v
the amount of material per volume, not surprisingly that's actually
a density pressure equals density and to get the units
right you have to divide by mu the mean molecular mass.
So it's something like hydrogen has a very low molecular mass.
This is just H2, it's really just two protons.
Something like oxygen is much higher but we don't
need worry about that right now, we're just going to
write it as this and know that we now
have an equation for density as a function of pressure.
We have an equation of state.
Now you may say, this is great!
We can now go figure everything out about the interior of Jupiter and you
would be wrong, because the ideal gas law is only good for ideal gases.
Ideal gases are things that are something like the pressure
of the earth's atmosphere is a pretty nice ideal gas,
as you compress gasses more and more, their equations of
state gets further and further away from the Ideal Gas Law.
But it will be good enough for us to at least get a feel for how this is going.
I'm going to now solve this equation for density.
And say density is pressure times mu divided by rt.
And I am going to substitute up in here and get the dpdz is minus g
times p mu over rt.
This is great because although this is a little
bit more complicated than the differential equation we had before.
It's a really simple differential equation.
If you remember again, anything about.
About your derivitives or differential equations you might remember the DDX of
EDX equals EVX it's one of the nicest things to differentiate in the world and
if we had DDX of A Is equal to a times e to the ax and if you look at this.
We have dp, dz is proportional to p and so we can simply write the p of z is
equal to some constant which will be the pressure at, at the surface of the earth.
Times E to the minus GU over RT times Z.
And even if you don't think much about what this
stuff is here, you find that the pressure, as you
go up, decreases exponentially with altitude, so if I plotted
pressure versus altitude, I'm going to do pressure here, altitude this way.
It starts out at P not.
And it decreases exponentially as it goes up.
This is still not quite good enough because we made one more assumption which
is that t is a constant and in fact t is not a constant.
Temp, the temperature changes greatly, as you know,
as you go up in the Earth's atmosphere you
cool down and so we actually needed to back here have t as a function of z two.
And then it gets really super-complicated.
But you get the general idea that you can use this equation of
hydrostatic equilibrium, this one right here,
even though it gets complicated and complicated.
As long as you have the equation of state
of the material in question, and you know the temperature
dependence inside, you can use these things to figure
out what the pressure is as a function of altitude.
If you know the pressure as a function of
altitude you know the density as a function of altitude.
And what you really want to do in this case is, try to
figure out, can we make the density of Jupiter match the measurement.
You take the density that you derive at the begi-, at the center, add up all
the densities as you get to the top, take the average density over the whole planet.
And say.
Did I get the right number?
And we'll try that process, but first we have to do two things.
We have to think a little bit harder about an equation of state for a
Hydrogen and a Helium interior, and we
have to deal seriously with this temperature variation.
If we don't know the temperature, we have no
idea what the interior of the planet is really like.
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