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Welcome to the fourth session.
In this session, we will now try to understand
better the pressure temperature profile of exo planet atmospheres.
In particular, we will derive some equations that will
allow us to derive the vertical structure of these atmospheres.
First of all, there are two equations, two basic equations that
we'll need to describe how pressure and density vary with altitude.
These two equations are the two following ones.
On the left side, you will see the equation of hydrostatic equilibrium.
And on the right side, you will have the equation of state of matter.
So first, the equation of hydrostatic equilibrium describes how gravity
and pressure interact to compensate each other actually
to reach an equilibrium where the net force on any element of matter within
the atmosphere has a net force of 0.
So you have the gravity which falls directed
vertically downwards of course and then you will
have a pressure gradient that will oppose gravity, and the sum of the 2 should be 0.
And that's exactly what this equation
of hydrostatic equilibrium is trying to express.
So on the left-hand side of the equation
you have the pressure gradient dP over dz while on the right hand side, you
have minus the density times the gravitational acceleration,
which gives you actually the force of gravity.
You can derive this equation in a relatively simple way.
For example, you imagine that you would multiply the two
sides of the equation with the surface, the initial surface within
the atmosphere on the left hand side multiplying by a
surface dp multiplied by a surface will give you a force.
And that force directed upward while on the right
hand side, if you have dz time s time o and g, this will
just give you the total force of gravity exerted by a volume
element of the atmosphere and this is the force directed
downward of course, and so, there is a minus sign here.
So this is the fundamental equation that
describes how pressure will change with altitude.
On the right of the slide, you will see what we call the equation of state.
In this case, it will be simply the ideal gas law.
So in an exo planet atmosphere we can just assume that we have an ideal gas.
And in this case, as you know, we have a
relatively simple relation between pressure,
density, temperature and chemical composition.
That's the equation that you see there.
So basically pressure is equal to a density times kt divided by mu,
where mu is the mean molecular weight of the matter in your atmosphere.
So for example, for H2 molecular hydrogen gas, you would
have 2, the mean molecular weight would be equal to 2 proton masses.
Usually, we count that in proton masses.
So you have two proton masses per particle.
Now we can combine these two equations together to try
to, to explicitly obtain the pressure as a function of altitude.
That's what we do now.
So in this slide as you can see, we combined the two equations
and again, we are in the similar situation as we did the transfer before.
We have a pressure gradient that is actually proportional to pressure itself.
Again when you have this kind of differential equation, the
solution is virtually straightforward and this is simply an exponential solution.
So you will have the pressure of the function of the altitude is
at an altitude z is equal to the pressure at reference altitude, times the
exponential of minus z over h, where we define here h as being
kT over mu g, and we call it the scale height of the atmosphere.
This is a fundamental parameter of any
atmosphere that we will, that we will encounter
several times in the, future sessions, and
this is a fundamental property of these atmospheres.
So this is the, the height over which
the pressure will change, change by a factor e.
In deriving the solution to the equation, we
made a few assumptions here, mainly that the
temperature mu and g were held constant, but
this is a relatively reasonable approximation in most cases.
So now that we have these vertical atmospheric structure
in terms of pressure of the function of altitude.
We can try, oh, this is for defining h.
We can try to look at the temperature gradient.
This is a bit more difficult because the temperature gradient will depend
on how energy is transported through through the atmosphere.
So, there are two main regimes one is when
energy is transported mainly by radiation, by the radiation field.
And the other regime is when radiation is transported mainly by convection.
Convection means large scale motions of matter of bubbles bubbles of hot
gas rising and transporting heat towards the upper layers of the atmosphere.
And we will see that we will encounter these two regimes
in exo planet atmospheres, so we have to treat them both.
We are not going to derive all the equations here.
I'm just providing here the final results in terms of
the temperature gradients, that you would expect in both cases.
On the left, you have the temperature
gradient when energy is transported by radiation
and basically it contains two terms of
interest here that we can briefly comment on.
The one is Frad and is the total radiative flux.
So the higher this flux, the higher the temperature gradient will be.
And also there is a kappa r which is the mean opacity here.
So the temperature gradient will also depend on opacity.
Now, if energy is transported mainly by convection,
you are in a completely different situation, and the
temperature gradient is given by minus G over Cp,
where Cp is the specific heat capacity constant pressure.
And we'll now try to understand better when each of these cases applies
and what it means in terms of energy transport and atmospheric structure.
So what we can say in a very general way is that radiative equilibrium sets in
when the opacities are low, so that the
radiation field can efficiently transport energy throughout the atmosphere.
On the other hand we will have convective equilibrium when
we have, we are in deeper layers where the opacities are high
and because in this case, radiation cannot efficiently transport energy.
Radiation is struck with matter.
And so convection is the, is the most optimal way of transporting energy.
The convection temperature gradient by the way, is called the idiopathic labstrate.
This is a term that often comes when for example we talk about earth atmosphere.
And also, I would like to note something about the radiative temperature gradient.
It may be inverted.
So it's not always decreasing,
temperature's not always decreasing outward.
Depending on the situation for example if you
have stellar irradiation, heating the atmosphere from above.
In this case, you may have an inverted temperature
gradient and so you may have an inversion layer basically.
And so this will change your energy transport of course.
Let's now have a look at a specific example, earth.
You may be familiar with the atmospheric structure of Earth.
There are a few layers so that we can identify with very different properties.
The, the deepest layer in the Earth atmosphere is called the troposphere
is the next, the first ten kilometers of atmosphere from the surface.
in, in the troposphere, convection is the dominant mode
of energy transport and therefore we have convective equilibrium in
the troposphere and so we have this idea about the
collapse rate which tells you how temperature changes with altitude.
Then then we reach a level which, which is called the
tropopause, which is the interface between
troposphere and stratosphere, the layer that
is just above and this distinction
actually between troposphere and stratosphere comes
from the fact that we are, we changed regime of energy transport.
In the stratosphere, we are in a radiative equilibrium, the density
is much lower, and radiation is the preferred mode of energy transport.
And here actually, what we observe is a temperature inversion.
Why is that?
That is because sunlight is absorbed in the stratosphere, in particular by ozone.
And ozone will heat so this sunlight will heat the,
the stratosphere from above and create this inverted temperature gradient.
So that is a nice example of, of
an application of the equations we we just saw.
Finally we can say a few things about general pressure temperature
profiles and their relations to emerging spectra in exoplanet atmospheres.
Of course, the spectra of exoplanets will depend heavily on
the pressure temperature profile because as we have seen in
previous sessions, the temperature gradient plays a major role in
defining the shape of your spectrum, of your thermal spectrum.
And so basically it all depends on where the photosphere is in the atmosphere.
The photosphere is defined as the location were the thermophotos,
leave the atmosphere and are free to escape to space.
That's the photons we're going to detect.
So depending at which altitude this photosphere is located, you may be in a
region with temperature gradient decreasing outwards or or the opposite.
So depending on that, you would get very different shape of your spectrum.
So now that we have reviewed all these
basics or these basic aspects of atmospheric physics.
We are ready to move to more observational considerations
which will be the topic of our next session.
Thank you.
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