0:17

So we have a sunlight that's coming from the sun and

coming all in one direction toward the Earth.

And then, we have Earth that is shining its own light based

on its 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 resolving for the condition where the energy

coming in to the planet balances the energy that's leaving.

That's the eventual steady state.

And an analogy to that would be if we had a kitchen sink and you turn on the faucet,

and then water stars 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 all the way down will rise

until it comes close to the level at which the water budget balances and stay there.

Or if you start out with too much water,

it'll sink until it goes to that steady state value and stay there.

[SOUND] So the water, if this is the rate of water flow in from the faucet,

and you started out with no water, but

there'd 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 out 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.

1:57

So we're looking for the condition where the energy in is equal to the energy out.

And so, we'll look at this in pieces, the energy going out you now know 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, cuz it's a very, very simple model.

2:24

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 kinda just call that one and not worry about it.

And this is Stefan–Boltzmann constant, which you can just look up in a book.

2:48

So this tells us the energy flux in watts per square meter.

But if we wanna 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 a sphere is 4 pi R squared.

3:10

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 we are away from the sun.

So 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 1,350 watts per square meter of area.

So 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.

4:06

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 to space.

And then, so this is in watts per square meter, but it's in per square meter of

this energy that's coming from the sun, sort of looking straight on, and

we have to multiply it by an area to get just the total energy coming in.

But this is a bit trickier 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

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.

And that shadow is all directly straight on to the sun, and so

it's oriented in the right way.

5:17

[SOUND] And so, this area now 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.

[SOUND] So if we equate those two sides, we have the solar

constant times the 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 in it, and

that's the temperature of the planet.

And so, we can rearrange that and calculate what the temperature should be

for any given combination of the solar constant and the albedo.

6:26

[SOUND] So this is the 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.

6:54

Venus is the second brightest thing in the night sky.

It's not because Venus is so hot that it's shining red light, or white 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,

and so it's mostly sort of darkish rock, and so it has a fairly low albedo.

7:20

So 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 the Mars is cool again,

because it's so far from the sun.

7:50

But, the real interesting thing comes when we compare with the real

temperatures that the planet is 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 climate model as we've done it so far is always too cold.

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