Welcome back, in this final section we are going to apply what we learned in the previous two sections to figure out what makes planets habitable, to start with our own Solar system and then look at habitability in other systems. We'll then turn to the effects of an atmosphere and see how that changes and complicates and enriches our story somewhat. So, let me remind you of the important result that we derived in the last section. The planet's temperature depends on the temperature of the star. The albedo of the planet, how much light bounces back. The size of the star, and the distance of the star. Notice that the size of the planet does not enter into this equation. The planet's temperature is entirely determined by the star's properties, the planet's reflectivity, and the size of the star and its distance. Let me also remind you again about our kelvin scale works. If the temperature is between 273 and 373 degrees Kelvin, water, at the pressures we have in this room, will be in the form of a liquid. Life thrives on our planet, in this temperature range. So, we can ask where is the habitable zone? Where is the range of distances for a given star where life could thrive? And that's what's plotted here. This is shown first for our own solar system. Our sun is a G type star. It's temperature recalls about 6,000 degrees. For our sun, we can ask, over what range of distance. Let's go back to that equation. We'll plug in the sun's temperature. We'll plug in the sun's size. Assume the same reflectivity, say, as the Earth, and then ask, over what range of distance does this temperature lie between 273 and 373 degrees? That range of distance is shown right here. And you can see Venus is just on the edge and Mars is just outside the habitable zone. So they're just marginally too hot and too cold, and the Earth, right now lies right in the middle of the habitable zone. Makes the Earth a nice place to live. Jupiter, Saturn, Uranus, and Neptune, all are going to be too cold. We do not expect there to be liquid water on these planets. Mercury, certainly too hot. Now what's nice about this equation is we can apply it just not to our Sun, but to other stars. What's plotted on this axis is the star's mass. And we'll talk a lot more about stars properties later on in the course. But as stars become more massive, moving up in this plot, they become hotter. So these A stars are hot. These M stars are cold. For colder stars, you need to be closer in, in order to have the same temperature. Now here we can apply this to some of the planets that have been discovered. This is the Kepler 22 system. A planet that's been found by the Kepler Satellite through [INAUDIBLE], we'll talk a lot more about Kepler, as we move on in the course. This remarkable telescope has discovered large numbers of planets around stars. And Kepler 22 was arguably the first planet that we discovered around a nearby star that lies in the habitable zone. We can take its, the distance from this star, Kepler's host, 22's host star, to the planet, the temperature of the star, the radius of the star. Compute the habitable zone of the star and the very exciting result that the Kepler telescope found was this newly discovered planet lies within the habitable zone. Now, you'll notice in this figure, this is from a NASA diagram, they put Mars within the habitable zone. The exact location of the habitable zone depends upon your assumptions about the albedo of the planet and also your assumption about the planet's atmosphere. And we're going to turn to the effect of the atmosphere next. So, now we want to talk about the greenhouse effect and how that changes the temperature of the planet. The greenhouse effect is in some ways very simple. Remember, when we looked at the, what determined the temperature of a planet, we balanced the energy in with the energy out. This story changes if you have an atmosphere that absorbs the energy emitted from the surface of the planet. So, let's start with this example here where the energy flows in from the star, the energy from the planet strikes the atmosphere, is absorbed entirely by the atmosphere, and then half the energy absorbed by the atmosphere is emitted out to space, and half goes back down to the ground where it's absorbed. Now let's apply our energy balance. Let's first apply it to the atmosphere. The atmosphere's heated by infrared radiation coming off the planet. And cooled by energy it emits to space and reflecting back down. Or emitted back down towards the planet. We can equate the flux on the planet to the flux on the atmosphere. Since this energy goes in here the atmosphere emits in both directions. The flux per unit area from the planet should be twice the flux from the atmosphere if this system is in equilibrium. At the planet's surface, we have two sources of heat at the planet's surface. There's energy flowing in from the star. That's this term. We've talked about that in the previous section. But now what changes is there's energy flowing in from the atmosphere. So, we have to include this term. And we now have a set of two equations that relate the flux coming in from the star, to the flux from the atmosphere and the flux from the planet. What we can do next is solve these two equations, and see how the atmosphere changes the story, and the effect of having a greenhouse that absorbs all the light from the planet, changes the planet's temperature. We'll do that here. Here are our two equations. This is our energy balance equation at the surface where we equated the flux coming in from a star and the flux absorbed from the atmosphere to the flux radiated from the planet. Here's our second equation. Again, requiring an equilibrium between the flux from the planet and the flux emitted from the atmosphere. We can solve these equations. Let's divide this by factor 2. Plug this in here. We now have an equation expressed in terms of the flux of the planet. We'll bring this over to this side. We'll solve for the flux from the planet, in terms of the flux from the star. The big change, compared to what we talked about in the previous section, is this factor 2. Because, the atmosphere radiating back has provided a new source of heat, to the planet's surface. Plugging in, our equations from before, we now have a new equation that relates the temperature of the planet to the temperature of the star, the reflectivity of the surface and the radius and distance in the star. And the change is this factor two to the one quarter. The effect of having a greenhouse is to increase the temperature of the planet by about 20%. Now 20% doesn't sound like that much, but remember that the Earth's temperature expressed in kelvin is about 300 degrees. So at 20% increase in temperature corresponds to a 60 degree celsius increase. So greenhouse heating could potentially be enormous. This is much larger than the kind of Greenhouse Effect that we talk about due to carbon dioxide in our own planet. But you can see that greenhouse heating can raise the temperature by a great deal and change the properties of the atmosphere and the surface. A better approximation to the Earth and to many other planets, might be called a leaky greenhouse, where we balance the energy coming in with the energy coming out. Some of the flux emitted by the planet goes through the atmosphere. We assume the atmosphere absorbs a factor tau climbs the flux of the planet, that's the Greek letter tau that we often use to refer to the fraction of light that's absorbed. So here, you could see a situation where say, carbon dioxide in the atmosphere absorbs some of the radiation from the Earth's surface. But at other wavelengths, the atmosphere's transparent. So, some escapes out to space. Again, we want to balance our energy equation. So, let's look at what's happening here in the atmospheric layer. The amount absorbed is the difference between this term and this term. So Tao times F planet is absorbed into the atmosphere. That balance is the amount that is emitted by the atmosphere, and then at the Earth's surface we have the same equation here. We now want to go through the same exercise. Equate the energy in and out of the surface, equate the energy in and out of the atmosphere. Solve these equations. Let's divide this by two. This gives us f atmosphere equals Tao over 2 F planet. We plug this term in here. That gives us this equation here. We now solve for F planet. That gives us this equation here. We then substitute in for the temperature, from what we talked about in the first part of this lecture. We now have an equation that I think is one of the important equations that we're going to need in this course and if there's one thing that you want take away from this lecture is that by doing energy balance by looking at this equilibrium here we are able to relate the planets temperature to the properties of the star. And the properties, the planet's surface and atmosphere. So now, let's again, go through the basic terms here. The planet's temperature increases, the star's temperature goes up. But now we have two terms that depend on the property of the planet that go in different directions. As the planet becomes more reflective. So, we'll paint the planet white. A increases, that means more light goes back out to space. A planet that's white will be cooler. A planet's that's plain, painted black would be hotter. The planet's temperature also depends on the properties of the atmosphere. If the planet's atmosphere absorbs a lot of the radiation emitted from the surface. So, let's say it absorbs 20 or 30%. That optical depth is higher. That will make the planet's temperature hotter. So the properties of a planet is going to depend, it's temperature is going to depend not just on its distance from the star but also on the planetary properties. So we go back and apply this equation to different systems, we actually don't know where Kepler 22B lies in the habitable zone, because we don't not yet know the properties of this planet. We don't know its obito and we don't know it's optical depth. In the case of Venus, Earth, and Mars these terms have a very important affect. Venus has a very high optical depth. This term being large. In fact, Venusâ optical depth is so large, that the approximation we used in this lecture aren't valid and just underestimates the temperature. The planet's temperature is much larger. You could also see from these equations. How global warming works with the Earth. As we increase the amount of carbon dioxide in the Earth's atmosphere, that increases the optical depth. More of the light emitted from the Earth's surface is absorbed in the atmosphere and then re-emitted back down to Earth. By increasing the optical depth, we increase the temperature of the planet. You can also see all the different terms that come in when we think about habitability of planets around other stars. If you'd like a planet to be habitable and for now let's define this as having a comfortable temperature, one where water could be a liquid, you can see there are a bunch of different terms we can balance. You can have a star that, say, is much hotter than the sun and a planet that's much further away. If the star temperature goes up, the distance go up, the two can balance each other and keep the planet's temperature in a comfortable regime. Alternatively, you can have a planet like Mars. It's a bit further away from the Sun. And if we took Mars and put a lot of carbon dioxide in its atmosphere, this was what it probably was like on early Mars, the optical depth was larger. That makes the temperature higher in Mars' past. Another complication here, and we'll talk more about this later in the course, is the early sun had a different temperature. The early sun was actually a bit cooler. So, all these different factors enter in. Now, for some of you, looking at these kinds of quantitative equations might be a new experience. For those of you I'd like you try to go back, go through this lecture again, work through it the most important result is this final one here. What I want you to take away from this class is the idea that we can take basic properties we observe about stars and planets, and see from that, or learn from that, the property of the planet. And the derivations, I think, are important for you to get an understanding that these things, we don't make them up from thin air. We start with often some very in some ways simple concepts, the idea that systems are in balance, that the energy in balances the energy out and we express these concepts as equations. What we learned is the relationship between a planet's temperature and its environment. So, let me stop there. Look forward to seeing you in the next lecture. [BLANK_AUDIO]