So far the things we've learned about Mars have come from basic observations, or maybe slightly sophisticated observations. Basic observations, I mean like, looking at Mars and watching it rotate, and seeing that it has features on it that you can map, or features that rotate so you can see how long its day is, or the obliquity of the planet. Even things like that detection of water vapor was a, was a sort of basic observation. You saw the Earth's water vapor absorption lines, and then you could see slightly shifted, the ones from Mars. Even the measurement of the temperature of Mars though I would say a fantastic and sophisticated at the time measurement, really was just a direct measurement. Put an aperture on Mars, let the thermal emission from Mars reach your telescope, reach your detector, see how much heat comes from it. All of these are, purely observational approaches to understand Mars. One of the other really important, purely observational, things that was discovered and that we saw were the polar caps. And the polar caps were known by this point to get bigger and smaller with the season. There were differences in the North and the South. And, an obvious question that you might ask is, what are those polar caps made out of? You saw in a previous lecture when we talked about the temperature of the surface of Mars that, that water could be frozen on the surface. But we also saw in a previous lecture that another thing that was in the atmosphere of Mars was carbon dioxide. Carbon dioxide can also freeze on the surface in the form of dry ice, frozen carbon dioxide. And there was a, a debate in this point in time, this is the 60's still, a debate about what those ice caps were made out of. So what do you do? Well, one, one approach to resolving a debate like this and, and often a very good one in planetary science, is to go look, is to go observe those polar caps and see if you can tell what they're made of. We looked already at the process of spectroscopy of atmospheric gases, which have the signatures of those gases. You can do the same thing with ices. Different ices reflect sunlight in different ways, different spectral ways. And we'll talk a lot about that, as we talk more about the surface composition of Mars. But at the time, in the 1960s, the instruments to do this sort of spectroscopy were just not good enough. There were some indications that water, ice, frozen water, had been detected on the polar caps. There were some indications that this was not true. And so, as I said before, the debate was raging. What do you when the observations are inconclusive and you'd still like to know the answer? One very powerful technique that's often used in science is the technique of modeling. What does it mean to make a model of a physical system? It means to take equations, equations that you understand, that describe the physics and the chemistry of what's happening. And try to understand how the system that you're thinking about would evolve under the behavior of those equations. Here is the surface of Mars. And what I would like to know is, at the temperature that this point is on the surface of Mars, will water ice or CO2 ice condense out of the the atmosphere? Or if there is water ice or CO2 ice on the surface, will it evaporate and go into the atmosphere? If the temperature is cold it will continue to condense, if the temperature is hot, whatever pile is there will continue to evaporate. And that will describe, that should describe, the coming and going of, of seasonal polar caps. As the temperature goes down in the winter, from, the gas from the atmosphere will freeze out, frost will form on the surface. As the spring time heats up, that frost will evaporate and go back into the atmosphere. This entire model is going to be based on, on energy conservation. So let's think about where all the energy comes from, and what the temperatures are that, that are relevant. The main energy source that we have coming in here is sunlight. We'll have the energy flux of that sunlight. We'll call it S nought. One of the ways in which we lose energy in this problem is by radiation, thermal radiation, coming off of the surface, based on the temperature of that surface. Now you remember that the amount of energy coming from a unit area of a surface is given by sigma t to the fourth. Sigma is the Stefan-Boltzmann constant. T to the fourth t, of course, is in, in Kelvin. And in this case, we're going to add a correction factor in front of that because just like we were talking about in a previous lecture, some of this energy goes up into the atmosphere and is absorbed by the atmosphere and comes back down again. And so not all of the energy gets away. We're going to call that correction factor epsilon. Epsilon is an emissivity. And we're just going to make up a plausible number for that. In fact, if we had nothing else going on we would just simply balance the amount of sunlight coming in being absorbed by the surface with the thermal radiation going out. There are a couple of other things to consider. One is that some of the sunlight that comes in is purely reflected off of the surface. It doesn't absorb into the surface. It doesn't heat the surface up. The amount that's reflected is, is determined by the albedo. Albedo is a number between zero and one. Zero means that it is completely dark, it absorbs all of the sunlight. One means that it reflects all the sunlight, real surfaces are somewhere in the middle. Mars has an average albedo of something like 15%. So, we'll use that for our number. So, that means 15% is reflected. If 15% of it is reflected, that means that 85% is absorbed. So we could simply balance s nought times 1 minus the albedo, by something like epsilon sigma t to the fourth. And we would get something like the surface temperature of Mars. And, in fact, this is a pretty easy calculation to do. So maybe in the previous lecture, when I pretended like we had no idea what the surface temperature of Mars was going to be, I was being a little bit disingenuous, but it is true. Because we didn't know enough about the atmosphere, we didn't know about this epsilon correction factor. This is, essentially, the greenhouse effect. If this number is very small, that means very little radiation goes out. And you have a very strong greenhouse effect. You have to have a large temperature for the radiation to get away. If you have a body with no atmosphere, like the moon, for example. This, this value would get very close to one. And taking this model, by the way, from this paper by Leighton and Murray, published in Science in 1966. I really like this paper. Because, in my mind, this is one of the first papers that does this idea of taking a model, taking the basic physics behind something, and doing what we planetary scientists do all the time now, putting it into a computer. I think they were very proud of the fact that they were putting it in this computer. They put it into a 7094 computer. They specifically say that in their paper. 7094, that refers to the IBM 7094, of course, and here's what they looked like. Well this is actually two of them, so this is a exaggerated size. But this is a, this is a NASA group working on the 7094s to help with the Mercury astronaut program, the very first astronauts who went into orbit around the Earth. And this is the sort of thing that these planetary scientists decided to use to try to understand what the polar caps were made out of Mars. I find that just sort of an amazing story. Now, you might ask why do you need a computer to solve this? And the answer is because they didn't just do this. This is a simple equation that we could write down and come up with a guess for what the surface temperature is close to. But you, what you really want to do is realize that of, that of course any patch on the surface, let's just think about this patch right here. Any patch on the surface sees the sun rise and the sun set and the amount of sunlight changes with both the day and with the year. And so they changed the amount of sunlight by cosine of Z, Z is the angle from here, this is the zenith angle. So if you're straight on you get all the sunlight. If your angle is 90 degrees all the way over here, well cosine of 90 is zero, so you get no sunlight. And in between you modulate the amount of sunlight that you're getting, so that's one change. The other important change they do is energy is not just being transferred right at the very surface of Mars. In fact, the heat from the sun is being transmitted down through the soil of Mars. In the daytime, that soil will heat up a little bit, down through a few layers. In the nighttime, that soil will cool down, down through a few layers. Over the course of the year, that soil will heat up over a larger area and cool down over a larger area in something that we call the, the thermal wave that gets transmitted down through the soil. So we have to calculate that part, too, otherwise you're not going to get your polar caps right. And the final thing they had to do is talk about, think about this evaporation and condensation of the CO2 or the H2O to volatiles, they would call them. How does that work? Well you know that to evaporate a solid you have to give it energy. You have to give it energy equal to the latent heat. And we don't think about this part as much, but if you condense something out of the atmosphere onto the surface, you release energy. You release that same energy of the latent heat. Okay, so now we have to add that extra step in there too. And in fact, we're going to end up with something that's not just one equation but several equations, because this is energy. This is also energy, but the energy might be modified. Plus or minus latent heat, plus or minus internal heating or cooling. Okay, now, so we need to figure out the equations for these last two parts, the latent heat and the internal heating. Latent heat, we can simply look up what the latent heat of CO2 or H2O is. So we'll know that if it evaporates or if it condenses, that's how much heat it takes, or how much heat is released. But how do we know how much condenses? Well, here's where we need to review the concept of vapor pressure. If I have an ice chamber, doesn't matter what the ice is, we'll call it water ice for now, and the water ice is held at a certain temperature. Above that water ice, in the chamber, there will be water vapor. If I increase the temperature of the water ice, the partial pressure of that water vapor will increase. If I increase the temperature again, the partial pressure will increase again. There's a balance between ice evaporating and gas condensing when the partial pressure of the gas above the ice is equal to the vapor pressure of the ice at that temperature. Vapor pressure is a function of temperature. It's not very surprising. At a very low temperature, there's very little evaporation of this ice, and so the vapor pressure, the partial pressure above the ice is very small. At a very low temperature, an ice cube is almost like a rock. As the temperature gets higher, though, there's more evaporation and the partial pressure above here gets higher. Imagine now what would happen if I had instead of ice, if I just had this chamber and I had a plate right here, and I had a partial pressure. If this plate is hot, nothing happens. The molecules don't stick to it, they hit and go away. What if I cool this plate, though? Eventually, I'll cool this plate to the point where at the temperature of this plate, the vapor pressure of the ice is equal to the partial pressure. Remember, when the vapor pressure is equal to the partial pressure, things want to start to stick. Okay, let's try a slightly different experiment. Let's make a chamber, and we'll have the ice, we'll make some very cold ice in here. And we'll have a pretty high partial pressure in here. In fact, a higher partial pressure than the very low vapor pressure of this very cold material right here. What happens? Well, not very much is evaporating because the vapor pressure is low. And yet a lot of it is condensing, every time a molecule hits it wants to condense. And so this continues to condense and continues to condense. Every time it condenses, it releases heat. When it releases heat, it heats this object here. Eventually, what's going to happen, ice is going to continue to grow on this until a balance is achieved again. When is the balance going to be achieved again? When the temperature at this point reaches the point where the vapor pressure at that temperature is equal to the partial pressure that's left inside that chamber. Now as long as the chamber was big and there was sort of an infinite supply of this stuff, then the partial pressure maybe isn't changing. But if there wasn't very much of this to begin with it, you can imagine all of it plates out on to the surface here. This might, or might not be, what's happening on Mars. But the important point is that we can calculate for Mars what this temperature is, below which things will start to condense. The temperatures on Mars so that CO2, for example, will start to condense is something like 145 degree Kelvin. If the temperature of the surface is above 145 degree Kelvin, the vapor pressure of any CO2 ice, which is above 145 degree Kelvin, would be so high that it would all be quickly driven into the atmosphere. If the vap, if the temperature of the surface is below 145 degree Kelvin, vapor wants to condense onto the surface. It wants to condense onto the surface until that balance is achieved again. That balance is achieved because latent heat is released and that temperature is heated up again. So that surface stays at 145 degree Kelvin. So in our calculation, what we're going to do is calculate what the surface temperature should be. And then if the surface temperature is below 145 degree Kelvin, we're going to, instead of letting it be below 145 degree Kelvin, we are going to condense CO2 on it until that temperature is back to 145 degree Kelvin. If there's ice on the surface, the temperature's going to remain 145 degree Kelvin and if I, if I calculate that the temperature should be higher, what I will do is instead evaporate CO2 and let it go away. The one other key point is about this internal heating that we talked about before, and the way that we're going to deal with that is just consider layers underneath the surface of Mars. And we'll have a temperature at every layer, and we'll calculate how much heat flows between the different layers. How do we do that? Heat between a layer is very simple to calculate. If we have the temperature of layer one and we have a temperature at layer two, and I would like to know how much heat is going between layer one and layer two, it is simply proportional to the difference in temperature. If the, if the two temperatures are the same, no heat is flowing. If there's a big temperature difference, there's a big heat flow. There's a thermoconductivity term that you need in there, and in this case, we also have to divide by the distance. We'll call that delta Z. You can imagine that if these layers are very thick and, and very far apart, the temperature flow is very slow. If they're close together and they have a big temperature difference, the temperature, the heat flow is very fast. So, we'll add this term in there. We'll consider this case and, and we'll have this. And if we put all those together and crank it through our monster mainframe IBM computer for days and days and days, we'll eventually get an answer. First, I just want to show you what the temperature does over the course of one day on Mars. It starts very low, 150 degree Kelvin, and spikes all the way up to here, goes way back down again. And it does slightly different things depending on the precise value of k, the thermal conductivity, this was the thermal conductivity. If a little more is, heat is transmitted into the surface, you get a little less extreme variation on the top. These points right here you might recognize. These are the points with air bars from Sinton and Strong, those original measurements, this model does a very nice job of falling through these measurements. Notice though, the temperature never gets below 145 here. At these locations, these were equatorial locations, you never get polar caps. Not a big shock. You can do this calculation for many different locations on the surface of Mars, and for the entire Martian year, as Mars goes around the sun. And then you can do things like ask yourself, what is the coldest it ever gets at this particular spot? And you can see plots that look something like this. These are the latitudes from the South Pole all the way to the North Pole, and over the course of one Martian year. So you can see, the coldest that things get are really quite cold. This is the line of 145 degrees CO2 condensation. And you can see it at something like this curve, at minus 50 degrees, never gets below here. 50 degrees does, so you should have CO2 polar caps that extend down this far in the dead of the winter. 70 degrees, very cold, 90 frigid. And from all of that, you finally get to my favorite plot in the entire paper, which is, the extent of the polar cap with season. And, they compare it to the measurements. These are the data in days from the southern summer solstice. This is the southern polar cap, at southern summer solstice, zero. The measured polar cap, there is still some residual stuff. It's still going down. And in the 100 or more days before summer solstice, the cap is quite large, 35 degrees and going down. There are two lines on here. One is for water ice, one is for CO2 ice. Water ice is this line here. There's so little water in the Martian atmosphere that it doesn't have the ability to grow and shrink very large water polar caps. CO2 though, there's a lot of CO2 in the atmosphere, and the temperatures allow that cap to grow from something like 40 degrees before the summer solstice to, before finally disappearing at zero. Now, which of these is a better fit? I think it's pretty clear that the CO2 does a better job of fitting what the data are showing than the water does. Is it perfect? No, it's not perfect. And one of the, the explanations that Layton and Murray suggest is that, yes, you still see the polar cap after summer solstice, but this model assumed a perfectly smooth sphere. And you could imagine having frost still in shadowed regions in craters or behind hills or something, and that's probably what this extra stuff in through here. Overall, though, Layton and Murray were one of the very first to use a model like this. A simple model coupled with this new-fangled computer thing to make an important statement about what is happening on Mars. That important statement is: based on everything we know, watching the polar caps, it sure looks like these polar caps are due to CO2. The coming and going of the polar caps is due to CO2. Does that means there's not water ice? No, it doesn't mean there's not water ice. In fact, as we will discuss in detail, these water ice caps do exist, and in fact there are huge, permanent water ice caps. But the growing and the shrinking caps, that were the thing that really caught people's eye at this point are due to CO2, and not due to water. Okay, It's 1966. Mars is cold. Mars doesn't have much water vapor in the atmosphere. There's no hint of water vapor coming and going at the poles. Doesn't really seem like a very hospitable place. What to do next? Send a spacecraft there. Check it out.