Now you have the pieces you need to construct
our very first climate model of a naked planet.
We have sunlight that's coming from the sun
and coming all in one direction toward the Earth.
And then we have Earth that is shining it's own light, based on it's own
temperature, with light that's going in all
different directions around the surface of the Earth.
And what we're doing, to
solve for the temperature of the Earth,
is we're solving for the condition where the energy
coming into the planet balances the energy that's leaving.
That's the eventual steady state.
An analogy to that would be if we had
a kitchen sink, and you turn on the faucet and the
water starts to build up in the sink and it gets
deeper and deeper. And as it gets deeper in the sink,
there's more pressure
of water pushing it down the drain, and so the rate at which the water
leaves the sink is a function of how deep the water is in the sink.
And so, the water level in the sink, if it starts out all the way down, will rise,
until it comes close to the level which the water budget balances and stays there.
Or if you start out with too much water, it'll sink
until it goes to that steady state value and stay there.
If this is the rate of water
flow in from the faucet, and you started out with
no water, there would be no more coming in
than out initially, and so the water would build up.
And it would tend to relax to that condition of balance.
Or, if you walk up to it with a big bucket and dump a big bucket all at
once in there, you'd start with too much water,
and it would relax downward, toward that steady state value.
So, what we were doing in this calculation,
is going right for that steady state value.
So, we're looking for the condition where
the energy in is equal to the energy out.
We'll look at this in pieces. The energy going out is
given by the Stefan-Boltzmann formula, epsilon, sigma, temperature to the fourth.
Where the temperature is the temperature of the Earth, and we're assuming for now
that it's all the same temperature because this is a very, very simple model.
Epsilon for the Earth is pretty close to one.
Most solids and liquids, most condensed matter
has a pretty good black body properties.
And so it has epsilon pretty close to one.
The exception to that we'll get to is greenhouse gases.
A very important exception.
But for now we can kind of just call that one and not worry about it
and this is the Boltzmann constant which
you can just look up in a book.
This tells us the energy flux in watts per square meter, but if we want to
do the planet overall, we have to multiply by the area of the surface of that sphere.
So this area, to get rid of the watts per square meter, in the area of the sphere,
is 4 pi R squared. Now on the other side of the equation, we
have the energy coming in from the sun, and this is given by a solar constant.
Which is a number in watts per square meter which is determined by how
bright the sun is, but also how far away we are from the sun.
It's how bright the sunshine is if you look at it straight on.
at the distance
from the sun to the earth it's about 1350 watts per square meter of area.
If you had a solar cell.
One square meter in size,
You could about run a normal sort of hair dryer
on that, that's kind of the energy flux coming through that.
But not all of that energy actually is absorbed as heat by the planet.
Some of it gets reflected back out to space and never gets absorbed.
And that fraction is reflected in the
albedo, which is given the Greek letter alpha.
So, 1 minus albedo is the fraction that gets absorbed.
The value of the albedo for the
Earth is about 30%, because of the clouds, mostly.
Reflecting light back out into space.
This is in watts per square meter
but it's in per square meter of this energy, that's coming from the sun,
looking straight on, and we have to multiply it
by an area, to get the total energy coming in.
But this is a bit tricker now, because the area of the earth isn't all
facing directly at the sun, perpendicular to the way the sun is coming.
So we could do a complicated integral, where we add up all this,
the square meters of the earth and figure out which ones
are sort of oblique and so they're not getting as intense sunlight.
But there's a tricky, easier way to do it,
and that is to realize that the amount of light that is intercepted by this
planet is given by the size of the shadow of the earth.
That shadow is all directly straight onto
the sun, and so it's oriented in the right way.
This area is not the area of the sphere that we had before.
but it's the area of a circle which is only pi R squared.
So, if we equate those two sides we have
the solar constant, times a fraction that's absorbed,
times the area of the shadow, and here's the infrared emission, the black-body
radiation, and the area of the sphere, but we can now divide by pi R squared.
In fact, we can even divide by 4 pi r squared.
To get this
equation, which is in the most convenient form, it's now in units of watts per
square meter of the surface of the sphere,
so per square meter of the Earth's surface.
So what we have now is an equation that only has one unknown,
and that's the temperature of the planet.
We can rearrange that, and calculate what the
temperature should be, for any given combination of the solar constant,
and the albedo.
This is a table of how that calculation goes
for the Earth and our sister planets, Venus and Mars.
The solar constant is much higher for Venus because Venus is closer
to the Earth than the Earth is and it's lower for Mars.
That's because the intensity goes down as you get further from the source.
The reflectivity of Venus is very, very high.
That's because Venus
is covered with clouds all the time.
Venus is the second brightest thing in the night sky.
Not because Venus is so hot that it's shining its own
light, it's not a star.
It's just reflecting the light that's coming
in from the sun because of these clouds.
And then Mars doesn't really have clouds
or much of any ice, it's mostly
darkish rock and so it has a fairly low albedo.
Here is where we calculate the temperature of
these planets given their solar constant and albedo values.
And it's interesting, Venus, because it's so
reflective, would be even colder than the earth.
Even though it's closer to the Sun.
It's just wasting all that energy by reflecting it out to space.
And then the Earth is warmer than Venus and then
Mars is cool again because it's so far from the Sun.
But the real interesting thing comes when we compare
it with the real temperatures that the planets actually have.
Venus is much much hotter than we just predicted
Earth is hotter than we predicted, and so is Mars.
They are all the same, the climate model as we've done
it so far is always too cold.