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Hydrostatic Equilibrium
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Imagining Other Earths
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Planetary Atmospheres
This lecture begins by introducing the basic physics of gases, liquid and solids. The lecture then describes how the balance between gas pressure and gravity shapes the structure of planetary atmospheres.
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David SpergelCharles Young Professor of Astronomy on the Class of 1897 Foundation and Chair
Department of Astrophysics
Welcome back.
Now, let's take these ideas and
apply them to see what sets the overall structure of the atmosphere of a planet.
But before we do that, we need one more ingredient and that ingredient is Gravity.
It's Gravity that holds the atmosphere.
Down is what keeps it bound to the planet and what determines its structure.
If I'm looking at an atmosphere of a planet, most atmosphere's are relatively
thin, so we can make the approximation, that the strength of gravity.
Pulling down on the atoms and molecules in the atmosphere is nearly constant.
In the case of the Earth, that strength of gravity is going to
depend on the Earth's Mass and the Earth's radius.
And the gravitational acceleration that we experience downwards when we jump.
And the atoms and
molecules feel, as they're being pulled downwards and what determines our weight.
On Earth it's about 9.8 meters per second squared.
That's the gravitational acceleration we feel.
The force that the Earth's exerting on me is my mass,
times that gravitational acceleration.
And that same force is felt not just by my body, but by the atoms and
molecules in the air around me.
If I consider a cylinder of air around me, the mass in
that cylinder is going to equal the density of air, times the volume.
And let me remind you that for a cylinder,
the volume of the cylinder is it's area times its height.
So combining this and this, we see that the force exerted downwards on that
cylinder is the mass of gas in that cylinder times the strength of gravity.
Now recall, that the gravitation, that pressure, is force divided by area.
So the pressure that gravity, is exerting effectively or
pulling down, on that air, is going to be force divided by area.
So the density of air times the strength of gravity,
times the height of that cylinder.
Now this pressure which is trying to compress the air,
is going to be balanced by air pressure.
So we're going to have gravity pushing down with, this way, and
air pressure, and air pressure arises.
Because the gas at the bottom of the atmosphere is at high pressure,
the gas at the top of the atmosphere is at low pressure.
And these two will be in Equilibrium.
That gas pressure could be understood most easily if we assume that
the air around us and the air in our atmosphere is at constant temperature.
That's not exactly true the, as we'll see soon.
The temperature of our atmosphere decreases as we go up.
But in order to make our calculation simple,
we're going to make the approximation that the atmosphere of the Earth and
the atmosphere of Mars are Isothermal.
Isothermal means constant temperature.
If you're Isothermal and we call the pressures equals to density times
temperature, the change in pressure, from the bottom of the atmosphere
to the top will go as the change in density times these constants and
the fact that we're keeping temperature constant means for giving.
Planets say this is just a constant.
This is the temperature of the planet's atmosphere.
This depends on its composition.
This is the mass of the molecules or atoms that make up the atmosphere.
And that's just a constant in nature, in number.
So the way an atmosphere is going to stay in equilibrium.
Is to have high pressure gas on the bottom, low pressure gas on the top, and
have this pressure gradient, due to a variation in density balance gravity.
So, let's make that balancing act happen.
We equate the two terms.
We equate the pressure gradient.
To the force of gravity setting those to equal we get this equation here
that relates the change in density to the height of the atmosphere.
This gives us an equation that we can solve.
Solving this equation requires.
Some calculus. We're not going to use that,
instead I'm just going to tell you the answer.
The solution to this equation is that the density of gas in
the atmosphere drops exponentially with height.
So that as we go up in the atmosphere the density drops.
And there's a characteristic number that we call the Scale height,
which describes how far you have to go up in the atmosphere,
before the density changes significantly.
And the Scale height of the atmosphere is sent by the balance between
pressure and gravity.
Pressure.
It's going to depend on temperature, gravity is going to depend on g,
the gravitational acceleration, and you can see dimensionally
this is equal to a velocity square divided by an acceleration.
So we now have a Simple Model.
That describes the atmosphere of planets,
also describes the basic properties of stars.
This simple model that says if things are in equilibrium, and
the temperature's constant, then the density profile of any atmosphere will
be an exponential profile with a scale height.
Set by Temperature, gravity and composition.
We can apply this to say the Earth and Mars.
The black curve here, is a description of the Earth's atmosphere.
The Earth has a thicker atmosphere at its base,
the total mass in this atmosphere is larger.
So, where we are here, near sea level, the density is quite high, but
as we move up towards Mount Everest, then the height of a typical airplane,
the density starts to drop, and you can see as we go from sea level.
To say 50 kilometers up, the density drops by about a factor of 10,000,
so there's an enormous difference between the density of the, our atmosphere at sea
level, and say, you know, at the altitude flown by,
kind of experimental planes and, above the altitude flown.
50 kilometers is higher than your.
Your typical airplane will fly but you can see the density varies enormously.
Mars starts out with
a significantly lower density, about a 100 times lower at its surface.
But the scale height of the Martian atmosphere is actually a bit larger.
While the temperature is lower in Mars, gravity is weaker on
the Martian surface because Mars is a much smaller planet than the Earth.
When you plug those numbers in for
Mars you'll find the characteristic scale for Mars.
Is, not quite double but roughly that compared to the Earth.
So that our model predicts a profile for
the Martian atmosphere that falls off more slowly.
And in fact if you're high enough of in the Martian atmosphere you'll actually
find, at the same distance from the surface,.
A higher density of atoms and
molecules in the Martian atmosphere than in the Earth's atmosphere.
The power of these equations, they're remarkably general.
I can tell you what, or you can tell me what the mass of your planet is.
What the strength of gravity is at its surface.
That specifies the gravitational acceleration.
Once you know how far the planet is from its host star, if the atmosphere's thin,
the equations we worked out in the last lecture tells you the temperature.
Of the planetary atmosphere.
If you know the atmosphere's composition, that gives you enough information
to determine the scale height, and to get a general solution for the density profile
in the atmosphere, under the approximation that the temperature is constant.
And that's a reasonable approximation.
This is the Earth's atmosphere where it temperatures vary some but
you'll notice the axis here go from,
say 200 Kelvin here to 280 Kelvin here.
So zeroes way over here.
And as we go up from the Earth's surface in the Troposphere the temperature drops.
In the Stratosphere the temperature increases.
When we get up to about 50 kilometers, the temperature isn't so different from what
it is at the base, and we can make make a reasonable approximation to the profile.
Lectures and some average temperature that works over a pretty wide range of radius,
and that will give us our exponential decrease in pressure and density.
And this shows how rapidly the pressure drops, dropping by a factor of
a thousand as we move from the earth's surface to the stratus and
as we move up in the atmosphere, it drops yet another factor of 1,000,
by the time we get up to 90 kilometers above the surface of the Earth.
So you should think about the Earth's atmosphere and
really the atmosphere of any other planet, as being typically relatively thin,
with most of the molecules concentrated with in that single scale height.
The same is true of Mars, here's the Martian atmosphere,
it's temperature drops as you go up, you still have approximated on calculations
of having the characteristic temperature here, but we can see over all
the temperature drops slightly, but the general picture's about the same.
Almost all of the atoms and molecules in the Martian atmosphere is
concentrated in this relatively thin layer here, and
the density falls off exponentially dropping by factors of ten.
As you go up in height by about 15 or, or so kilometers.
So this very rapid drop in pressure and density with height
is inevitable as you try to balance gravity against the pressure gradient.
So to summarize, this is our characteristic density profile,
this is our characteristic height.
For the atmosphere.
And we'll see this basic relationship will work not only for planetary atmospheres,
but for stars we start talking about stellar properties in a couple weeks.
So let me stop here and
then have you apply these ideas and see that you're now actually in the position.
To start to think about in detail, the properties of a planet around other stars.
Thanks.
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