We're coming to a very exciting part now because we have laid the foundation with black-body radiation. We're going to build the first super-simple model of the planet's climate using what we have learned about black-body radiation. The model we're going to build first is super, super, super simple. But even in simplicity, it will allow us to understand a lot of things. For instance, it will allow us to understand the difference in climate between the Earth and other planets, the current changes in climate on the Earth, the changes in climate in the medium and the distant past, and what is most important, the importance of greenhouse gases for the change in temperature of a planet, not just of the Earth, but of any planet. This model only gives high level prediction, say for the Earth as a whole. It can be used to localize a wide geographical area. But it's still extremely useful to understand what drives the changes in average surface temperature of a planet. By the way, this is exactly the type of reasoning I said at the beginning of the course that in the 19th century, Fourier, Arrhenius, and Tyndall began to create the first models predicting climate change. They were exactly using or converging onto the type of model that we're going to describe now. Let's begin with a motivating picture here. There is a picture with, we said about the eccentricity. The eccentricity of the Earth has been fantastically exaggerated here. But this is how normally things are depicted for artistic drama. Here we see the first planets, Mercury, Venus, and the Earth. After looking at this picture, which planet do you guess is warmer? Mercury or Venus? Now, let's get some more precise information. Mercury is the closest planet to the Sun, and Venus is twice as distant. Surely Mercury is much hotter than Venus. Or is it? Mars is a long, long way from Sun, is 240 kilometers from the Sun. Remember that the energy goes down as the square of the distance. Therefore, 250 kilometers is less than twice as far than the Earth, but you have to raise to the power to see how much energy reaches Mars. Surely Mars is much, much colder than the Earth, or is it? Well, actually, the average surface temperature of Venus is between 1/2 and two times higher than the surface temperature of Mercury. The surface temperature of Mars is pretty close to within 50 Kelvin. Kelvin is the same as magnitude as Celsius, not as Fahrenheit, never use Fahrenheit, as the surface temperature on Earth. How can this happen? What explains these surprising differences? In this session, we are going to learn that for a given level of irradiant, so for a fixed level of energy coming in, two main variables affect the surface temperature of a planet. The albedo, which is the reflectivity. That is easy. We can just put it to one side. Then what is called the effective number of layers of the atmosphere. We have to understand what this is. The key to understanding the effective number of layers is black body radiations. Since it is so important, let's remind ourselves of the key concepts behind black-body radiations. We have seen that every object, I don't want tor say every body because when I say everybody sounds like every person, every object at a non-zero temperature emits electromagnetic radiations. When a body receives energy from the outside, a bit of it is reflected so the albedo, and we just subtract that part of the energy, we just disregard it. As for the energy which is absorbed, As it absorbs more and more energy, it warms up and therefore emits black-body radiation. This is the key. The frequency of emission of radiation which is emitted has nothing to do with the frequency of the incoming radiation. It only depends on the temperature of a black body. As we shall see, understanding this last bit that I repeat again, the frequency of the emitted radiation has nothing to do with the frequency of the incoming radiation. This is key to understanding the greenhouse effect. We're going to do some maybe calculations now. We want to estimate first how much energy from the sun reaches the earth. Let's begin from the beginning. The sun emits approximately 4 times 10^26 watts of energy per second in the form of photons. Photons are the tiniest units of light. Now imagine that you draw an imaginary sphere around the sun. You can make the sphere as large as you want. No photon gets lost because it's a sphere which is all around the sun. Now make this sphere larger and larger until it just touches the earth. Now the earth rotates 150 million kilometer. Otherwise, it would be quite hot, 150 million kilometers from the sun. This imaginary sphere, we can calculate the surface of the sphere. We all know that the surface of a sphere is 4 Pi r-squared. For r, we have 150 million kilometers, and we get that huge number there, 2.8 times 10^23 meter squared. Now the total energy emitted by the sun can be divided by the surface area of the sphere, and I can have the energy per square meter on this imaginary sphere that just touches the Earth. That is the number there, 1,360 watts per meter squared. This is an important quantity which is called the solar constant for the Earth, and it's often denoted by the symbol S. This is energy per square meter. The next question is, how much energy actually falls on the Earth? Now we do another little trick. We're going to place a screen immediately next to the Earth or if you want, cutting through the equator of the Earth perpendicular to the direction of arrival of a sun. There it is. We have for radius of the Earth, which is called R, and we have the incoming radiation. In a way, we are looking at the shadow cast by the earth on a screen through the equator. Well, it's very easy to calculate the circle with the diameter of the Earth. We know the diameter of the Earth is 6,400 kilometers, so we're using Pi r-squared again to obtain that the area of this circle here is 1.3 10^14 meter square. Therefore, the total energy falling on the earth every second is 180,000 terawatts. It is given by the S, which is the solar constant, times the area of a circle. 150,0000 terawatts is a huge number. By comparisons, humans currently use 16 terawatts. That is why solar energy in principle is so important, and as such a promise in energetic terms. As we know, not all of the energy is absorbed by parties reflected. We know we have seen that the albedo is 0.3, so we can just multiply the number we have obtained times 0.7, 0.7 is 1 minus the albedo, to obtain the effective energy in as 120,000 terawatts. This means that the effective energy per unit area is given by the effective energy in divided by the area, so I have the expression of the numerator divided by expression of the denominator. Notice that not only the Pis, but also the radius of the Earth cancels out, and I'm left with an expression which is 238, let's say recurring figure that we shall find over and over again, watts per square meter. Watts is power, so is energy per second per square meter. It is the net effective amount of energy absorbed per square meter, and it does not depend on the size of the Earth. Of course, that is only an average value, it depends on the latitude, on the hemisphere. For instance, the Southern Hemisphere has got a lot more water. The water reflects poorly, therefore, it has a lower albedo. It totally depends on the time of day, on the season, but we're just looking at the planet Earth as a whole. As we have seen, if the incoming energy is constant, at equilibrium, the amount of energy coming out must be equal to the amount of energy coming in. This is true for the Earth and is true for anything else, and this is the condition that ensures that the body is a thermal equilibrium. In the previous session, and this is now the key, we have seen that the energy per unit area emitted by a black body was given, you remember, the energy out, the Stefan-Boltzmann constant, times temperature to the fourth. Now it's very very easy. We simply equate the effective energy in with the energy out. The effective energy in is S, the solar constant, 1 minus the albedo divided by 4. The four it is what remains of the 4Pi r squared because the Pi and the r square canceled out, equal to Sigma, the Stefan-Boltzmann constant, T to the fourth. What is the temperature? Well, that is the temperature of the surface of the Earth. It takes nothing to solve that equation for T. I simply bring Sigma to the numerator, I take the fourth square root and it gives me a number. This number I obtained as a first cut prediction of the temperature of the Earth, 255 Kelvin, which is minus 18 Centigrades. Not very good, a tiny bit chilly, but not bad for such a simple model. But this raises the question, why do we make a mistake? Why do we get such a low value for the surface temperature of the Earth which is about 15 Centigrade? I'm off by about 33 Centigrades. Giving the simplicity and crudeness of the model, this is not bad, but we can do much better. Now, we are getting to the heart of the matter. In order to get a better agreement with the actual temperature of the surface of the Earth, we have to introduce the greenhouse gases. At the moment, our Earth was naked. It had no atmosphere. We're going to do this by introducing a one-layer model, a two-layer model,... and n-layer model. I'm going to apply the same model, not just to the Earth, but also to the other planets. Therefore, in one swoop, I will explain not just the temperature of the Earth, but also the temperature of Mercury, the temperature of Venus, the temperature of Mars, and everything. Understanding this is at the very heart of the climate change problem, so we have to understand it well.