0:17

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

Â coming all in one direction toward the Earth.

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

Â on its own temperature with light that's going in all different directions

Â around the surface of the Earth.

Â And what we're doing to solve for

Â the temperature of the Earth is resolving for the condition where the energy

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

Â That's the eventual steady state.

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

Â and then water stars to build up in the sink and it gets deeper and deeper.

Â And as it gets deeper in the sink there's more pressure of water pushing it down

Â the drain, and so the rate at which the water leaves the sink

Â is a function of how deep the water is in the sink.

Â And so, the water level in the sink, if it starts all the way down will rise

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

Â Or if you start out with too much water,

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

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

Â and you started out with no water, but

Â there'd be no more coming in than out initially, and so

Â the water would build up and it would tend to relax to that condition of balance.

Â Or if you walk up to it with a big bucket and

Â dump a big bucket all at once in there, you'd start out with too much water and

Â it would relax downward toward that steady state value.

Â So what we were doing in this calculation is going right for

Â that steady state value.

Â 1:57

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

Â And so, we'll look at this in pieces, the energy going out you now know is given by

Â the Stefan-Boltzmann formula, epsilon, sigma, temperature to the fourth,

Â where the temperature is the temperature of the Earth, and we're assuming for

Â now that it's all the same temperature, cuz it's a very, very simple model.

Â 2:24

Epsilon for the Earth is pretty close to one.

Â Most solids and liquids,

Â most condensed matter has a pretty good black body properties.

Â And so, it has epsilon pretty close to one.

Â The exception to that we'll get to, is greenhouse gases.

Â A very important exception.

Â But for now, we can kinda just call that one and not worry about it.

Â And this is Stefanâ€“Boltzmann constant, which you can just look up in a book.

Â 2:48

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

Â But if we wanna do the planet overall,

Â we have to multiply by the area of the surface of that sphere.

Â So this area,

Â to get rid of the watts per square meter in the area of a sphere is 4 pi R squared.

Â 3:10

Now, on the other side of the equation, we have the energy coming in from the sun.

Â And this is given by a solar constant,

Â which is a number in watts per square meter which is determined

Â by how bright the sun is, but also how far we are away from the sun.

Â So it's how bright the sunshine is if you look at it straight on

Â at the distance from the sun to the Earth.

Â It's about 1,350 watts per square meter of area.

Â So if you had a solar cell one square meter in size

Â you could about run a normal sort of, hair dryer on that.

Â That's kind of the energy flux coming through that.

Â But not all of that energy actually is absorbed as heat by the planet.

Â Some of it gets reflected back out to space and never gets absorbed.

Â And that fraction is reflected in the albedo,

Â which is given the Greek letter Alpha.

Â 4:06

So 1 minus albedo is the fraction that gets absorbed.

Â The value of the albedo for the earth is about 30%,

Â because of the clouds mostly reflecting light back out to space.

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

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

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

Â But this is a bit trickier now, because the area of the Earth isn't all

Â facing directly at the Sun, perpendicular to the way the Sun is coming.

Â So we could do a complicated integral where we add up all

Â the square meters of the Earth and figure out which ones are sort of oblique, and so

Â they're not getting as intense sunlight.

Â But there's a tricky, easier way to do it.

Â And that is to realize that the amount of light that is intercepted by this planet

Â is given by the size of the shadow of the Earth.

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

Â it's oriented in the right way.

Â 5:17

[SOUND] And so, this area now is not the area of the sphere that we had before,

Â but it's the area of a circle, which is only pi R squared.

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

Â constant times the fraction that's absorbed times

Â the area of the shadow, and here's the infrared emission,

Â the black body radiation, and the area of the sphere.

Â But we can now divide by pi R squared, in fact we can even divide by

Â 4 pi R squared to get this equation which is in the most convenient form,

Â it's now in units of watts per square meter of the surface of the sphere,

Â so per square meter of the earth's surface.

Â So what we have now is an equation that only has one unknown in it, and

Â that's the temperature of the planet.

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

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

Â 6:26

[SOUND] So this is the table of how that calculation goes for the Earth and

Â our sister planets, Venus and Mars.

Â The solar constant is much higher for Venus,

Â because Venus is closer to the Earth than the Earth is and it's lower for Mars.

Â That's because the intensity goes down as you get further from the source.

Â The reflectivity of Venus is very, very high.

Â That's because Venus is covered with clouds all the time.

Â 6:54

Venus is the second brightest thing in the night sky.

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

Â It's not a star.

Â It's just reflecting the light that's coming in from the sun,

Â because of these clouds.

Â And then, Mars doesn't really have clouds or much of any ice,

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

Â 7:20

So here, is where we calculate the temperature of these planets,

Â given their solar constant and albedo values.

Â And it's interesting, Venus, because it's so

Â reflective, would be even colder than the earth.

Â Even though it's closer to the sun,

Â it's just wasting all that energy by reflecting it out to space.

Â And then, the Earth is warmer than Venus, and the Mars is cool again,

Â because it's so far from the sun.

Â 7:50

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

Â temperatures that the planet is actually have.

Â Venus is much, much hotter than we just predicted.

Â Earth is hotter than we predicted, and so is Mars.

Â They are all, the climate model as we've done it so far is always too cold.

Â [MUSIC]

Â