One of the great contemporary scientific efforts of today is the study of the climate of planet Earth. Climate fundamentally effects how we live. Life in Pennsylvania is different from life in Los Angeles or life in Miami. And the climate today is not seem to be the same as it was just a few thousand years ago. Where people were buried in boats in Tibet 4,000 years ago and paint pictures of swimming in Egypt 10,000 years ago, there is now only dessert. Geologists have discovered that before that, the earth has been through sporadic ice ages where much larger portions of the norther and southern latitudes have been covered in glaciers. And over the last two hundred years, our population and industry have grown so large that we ourselves are changing earth's climate.
The term "climate" is used to represent the aggregation of all the earth's chaotic weather conditions that are varying in time and space. This is not the same as the weather we experience each day  no one person can see the entire state of the earth's climate. Each of us can observe and experience only a thin, insignificant slice of the climate. Without extensive personal experience, it is hard for us to develop intuition and foresight about changing climatic conditions. Modelling is one approach that can help us develop that intuition. In this chapter, we will explore two simple models of earth's climate. The first model attempts to predict the average temperature of the earth using only geometry and one wellstudied law from thermodynamics. The second model is a generalization that supports a theory claiming the earth once froze solid and stayed that way for millions of years  a snowball earth. Along the way, we'll come up with a new kind of mathematical model and some ideas for studying it.
One approach to modelling climate is to make use of the principle of conservation of energy. The idea is that the climate of Earth depends on the amount of energy in our atmosphere. One way to frame this is to say the temperature we experience here at the earth's surface is the consequence of energy entering and leaving the atmosphere through four sources  the sun, the sky, the oceans, and the earth's crust. Intuition and orderofmagnitude analysis tells us that the majority of our atmosphere's energy enters the atmosphere from the sun than from the crust or the rest of the sky. Just think how quickly the temperature can drop at night when the sun goes down, and how much colder it gets on long winter nights than short summer nights. The earth's core is heated by radioactive decay, and cools by releasing energy into the atmosphere directly through the crust and indirectly through the oceans. Sometimes this release of energy takes the spectacular form of a volcanic eruption. However, the presence of permafrost, glaciers, and ice sheets at the earth's poles is a clear indicator that this is a relatively small contribution over all. The oceans act mostly as large reservoirs for heat, which they absorb, redistribute, and reemit. The cooling of the atmosphere then occurs mainly through emission of energy out into the rest of the sky.
If we hypothesize that the earth's average temperature is roughly at steadystate, we can study the atmosphere's energy using an energybalance model. The conservation of energy is a fundamental principle of the theory of thermodynamics. If, at steadystate, our atmosphere's energy is conserved, then the rate of change of the earth's atmosphere's energy must be equal to the power (energy per time) input (\(I\)) from the sun minus the power emission (\(E\)) by the earth back to the sky. Put as an equation,
\[I  E = 0.\]
To determine the power input and emission, we must apply some knowledge from thermodynamics and geometry. Naively, the toughest part is determining the power outputs. However, there is a very interesting and useful observation from the thermodynamics called the StefanBoltzmann relation. The StefanBoltzmann relation says that the power (P) emitted from an ideal body in the form of light is proportional to fourth power of the body's absolute temperature (\(\theta\)), \(P \propto \theta^4\). So, the hotter a body is, the brighter it shines. If we know the temperatures of the earth and sun, we can determine their power outputs. The rest, then, is geometry.
To work out the geometry, we can draw a simple diagram of the solar system consisting of the earth (represented with astronomical symbol \(\oplus\)) and the sun (represented with symbol \(\odot\)). Let \(r _ \odot\) be the radius of the sun, \(d\) be the average distance from the sun to the earth, and \(r _ \oplus\) be the radius of the earth. Through study, we have determined that \(r _ \oplus \approx 6.4 \times 10^6\) meters (using half the Karman line), \(r _ \odot \approx 7 \times 10^8\) meters, and \(d \approx 1.5 \times 10^{11}\) meters.
The total power output from the sun is the output per unit area times the area of the sun. Approximating the sun as a sphere, we know this area is 4 times the crosssectional area. Thus, the total power output will be given by \[P _ \odot A _ \odot = ( \sigma \theta _ \odot^4)(4 \pi r _ \odot^2) \] where \(\sigma = 5.67 \times 10^{8}\) watts per quartic degrees is the StefanBoltzmann constant. Scientists have studied the sun closely and found it's surface temperature \(\theta _ \odot = 5800^{\circ}\) K.
The fraction of that power that hits the earth will be the fraction of the spherical shell occluded by the earth, divided by the area of the spherical shell with a radius equal to the the earth's average distance from the sun, \(d\). For a planet that is small and far from the sun, the area occluded is closely approximated by the crosssectional area of the earth, so the fraction occluded is \((\pi r _ \oplus^2)/(4 \pi d^2)\). Thus, the earth's power input
\[I = P _ \odot A _ \odot \frac{\pi r _ \oplus^2}{4 \pi d^2}.\]
Even though we don't usually think of the earth as an incandescent object emitting energy, the exact same reasoning we used for the sun's power output applies for the earth. \[P _ \oplus = \sigma \theta _ \oplus^4\] So, the earth should emit energy at a rate \[E = P _ \oplus A _ \oplus = (\sigma \theta _ \oplus^4) (4 \pi r _ \oplus^2).\] Applying conservation of energy, the power absorbed by the earth must balance the power emitted back into space by the earth: \(I = E\). So, by substitution, \[\begin{gather} P _ \odot A _ \odot \left(\frac{\pi r _ \oplus^2}{4 \pi d^2}\right) = P _ \oplus A _ \oplus \\ \left( \sigma \theta _ \odot^4 \right) \left( 4 \pi r _ \odot^2 \right) \left( \frac{\pi r _ \oplus^2}{4 \pi d^2} \right) = \left( \sigma \theta _ \oplus^4 \right) \left(4 \pi r _ \oplus^2 \right) \end{gather}\] Solving for the earth's temperature, we get the simpler formula \[\theta _ \oplus = \theta _ \odot \sqrt[2]{\frac{r _ \odot}{2 d}} = 280^{\circ} K = 44^{\circ} F = 7^{\circ} C\] So, this geometric model predicts that the earth's temperature is about 44 degrees Fahrenheit. We should really caveate this statement, as the prediction is a longterm prediction for the whole earth, rather than the temperature at any one place and time. Our model predicts that the earth's global average annual temperature is about 44 degrees. Using satellite technology and other resources, climatologists estimated the earth's average temperature in 2013 was \(58^{\circ} F = 15^{\circ} C\). So our estimate is definitely in the right ballpark. If fact it seems very good!
It should be noted, however, that we've left out one particularly important component in our calculation  albedo. The albedo of a body is its tendency to reflect incident light at the same wavelengths it arrives, rather than to absorb and reemit that energy as blackbody radiation. This is important because the reflected light typically has higher frequency than the emitted light, and thus has a different impact on temperature. From satellite measurements, we know the earth's albedo \(\alpha \approx 0.3\). When we incorporate albedo into our energy balance equation, \[ \left( \sigma \theta _ \odot^4 \right) \left( 4 \pi r _ \odot^2 \right) \frac{\pi r _ \oplus^2}{4 \pi d^2} (1\alpha) = 4 \pi r _ \oplus^2 \sigma \theta _ \oplus^4\] Solving for the global average annual temperature again, we find \[\theta _ \oplus = \theta _ \odot \sqrt[4]{\frac{ (1\alpha) r _ \odot^2}{4 d^2}} = 256^{\circ} K = 1^{\circ} F\]
It was a happy accident that our first calculation came so close to the earth's actual average temperature. This formula that includes albedo is a more accurate formula, and predicts an earth that is significantly colder than we experience. Part of the explanation for the difference between observed and predicted temperatures is the earth's atmospheric chemistry and dynamics, which reduce the rate of energy emissions back into space.
You may have wondered why we're modelling the global average annual temperature. You may have even asked yourself what use is the "global average annual temperature" anyway? As a person at one place and time, we never experience this average temperature. We experience the local temperature with it's regular variation from latitude, season, time of day, and weather. And you would be right. Global average annual temperature is a useful summary statistic for tracking certain trends, but a better model will capture some of the variation in temperature we know is important.
One important component of climate temperature variation is the variation of temperature with latitude. The differences in climate and weather between polar and tropical latitudes is dramatic, and global weather patterns are largely attributable to the atmospheric mixing driven by these temperature differences. One model of temperature variation with latitude and time is Budyko's model.
The basic setup of Budyko's climate model is based on the earth's energy balance  energy comes in from the sun and is radiated back out, and over decade time scales, will be in equilibrium when input rates match the output rates. However, energy input and output rates are influenced by some important things that we need to take into account. For one thing, the rate of energy input depends on latitude  because of the earth's spherical shape, the north and south poles get less light than the equator. Latitudinal temperature differences are further effected by snow and ice that alter the earth's albedo and reflect light back into space. The north and south poles have lots of ice, while the equator has very little. Weather works to undo the differences in heating between the poles and equator  cold air gets moved south, while warm air gets moved north. We can summarize this energybalance dynamic as \[\frac{\partial \theta}{\partial t} \propto I  E + D\] where \(\theta(y,t)\) is the typical temperature at latitude \(y\) at time \(t\), \(I\) is the rate of energy input from the sun, \(E\) is the rate of energy emission from the earth back into space, and \(D\) is the rate of atmospheric mixing evening out the temperature. We will use \(y\) as the rectilinear latitude (scaled linearly from 0 at the equator to 1 at the north pole), and \(t\) is time (in years).
The rate of energy input \(I\) from the sun can be treated geometrically, as we did with our energybalance model, with additional terms that account latitude dependences on geometry and albedo. The average annual input rate per unit of latitude \[\begin{gather*} I = Q s(y) [1 \alpha(y, y _ c)] \end{gather*}\] where \(Q\) is the per unit area, \(s(y)\) is the effective area at latitude \(y\), and \(\alpha(y, y_c)\) is the effective albedo at latitude \(y\) when the iceline is at latitude \(y_c\). Using integration with observations of the earth's tilt and orbit, the geometry term can be reasonably approximated by the parabola \(s(y) = 1  0.241 (3 y^2  1)\). The albedo is relatively constant except for the jump it takes at poles because of ice, so it may be approximated by the piecewise constant function \(\alpha(y, y _ c) = 0.32 + 0.3 H(y  y _ c)\), where \(H(x)\) is the Heaviside function (which we leave undefined at zero for the moment).
The rate of energy emission \(E\) could be approximated with the StefanBoltzmann relation, but for the range of temperatures we are interested in, it is simpler and just as accurate to use a linearized version of this relation, so \[\begin{gather*} E = A + B \theta. \end{gather*}\] Now, we must include something for the term \(D\) that accounts for movement of heat between latitudes. Since mixing just moves energy around, we still expect the total energy to be conserved, so \[\int _ {1}^{1} D dy = 0.\] The idea Budyko used was that global mixing moves local temperatures towards the global average temperature according to Newton's law of cooling: \[\begin{gather*} D = C (\overline{\theta}  \theta ) \end{gather*}\] where \(C\) is a constant and \(\overline{\theta}\) is the global average temperature.
Putting all of these pieces together, \[\begin{gather} \frac{\partial \theta}{\partial t} \propto Q s(y) (1\alpha(y,y_c))  (A+B\theta) + C \left( \overline{\theta}  \theta \right), \\ R \frac{\partial \overline{\theta}}{\partial t} = Q (1\overline{\alpha}(y_c))  (A+B\overline{\theta}), \\ \overline{\theta}(t) = \frac{1}{2} \int _ {1}^1 \theta(y,t) dy = \int_0^1 \theta(y,t) dy, \\ \overline{\alpha}(y_c) = \int _ {1} ^{1} s(y) \alpha(y,y _ c) dy, \end{gather}\]where \(Q = 340\) watts per square meter, \(A + B \theta = 202 + 1.9 \theta\), \(C = 1.6\), \(\alpha(y, y _ c) = 0.32 + 0.3 H(y  y _ c)\), \(H(x)\) is the Heaviside function, and \(s(y) = 1  0.241 (3 y^2  1)\). This is the dynamic Budyko model for global temperature dynamics. The Budyko model can be rewritten as the single partial differential equation \[R \frac{\partial \theta}{\partial t} = Q s(y) \left(1  \alpha{\left (y,y _ c \right )} \right)  (A + B \theta) + C \left( \int_{0}^{1} \theta{\left (y,t \right )}\, dy  \theta{\left (y,t \right )} \right)\]
Budyko's theory is puzzling at first. It's equations look and feel very different from the models we have previously studied. At first, it seems like a partial differential equation for \(\theta(y,t)\), but there are no partial derivatives with respect to \(y\), only a definite integral and some functional dependences. There is also a discontinuity in our albedo function at the ice line latitude \(y_c\), and it's not entirely clear yet how we should handle this ice line.
Budyko's theory is an example of a Stefan problem  a problem where an unknown internal moving boundary plays an important role in the solution. Stefan problems were originally studied in the context of freezing lakes, where we were interested using temperature observations to estimate ice thickness. Today, generalized versions of Stefan problems called free boundary problems are of wide interest. In our case, the importance of the ice line is not evident a priori, but will become clear momentarily.
To make sense of Budyko's model, let us start by hoping the dynamics will approach a steadystate solution after sufficient time, and attempt to find this steadystate.
At steadystate, \(\partial \theta/\partial t = 0\), so the steadystate temperature \(\theta^*\) solves \[0 = Q s(y) \left(1  \alpha{\left (y,y _ c \right )} \right)  (A + B \theta^*) + C \left( \overline{\theta}^*  \theta^* \right)\] Since the local temperature \(\theta\) appears linearly on the right hand, we can rearrange to show \[\theta^*(y,y_c) := \frac{ Q s(y) \left(1  \alpha{\left (y,y _ c \right )} \right)  A + C \overline{\theta}^* }{B+C} \] where the steadystate global annual average temperature \[\overline{\theta}^ * (y _ c) := \frac{Q}{B} \left( 1  \overline{\alpha}(y _ c)\right)  \frac{A}{B}.\] For the moment, let's treat the iceline latitude as an independent variable, even though its location should depend on the temperature. When we know where the ice line latitude \(y _ c\) is, we can calculate the steadystate average temperature \(\overline{\theta}^ * (y _ c)\).
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Combining our descriptions of albedo \(\alpha(y,y _ c)\) and solar geometry \(s(y)\) with the average temperature \(\overline{\theta}^ * (y _ c)\), we can calculate the steadystate latitudinal temperature profile for different ice lines.
Note that our steadystate latitudinal temperature profiles never overlap  each point between the maximum and minimum solutions falls on exactly one steadystate solution. This is not aways the case, and a special consequence of piecewise continuity and monotonicity of albedo. We also see that the annual average temperature decreases monotonely with latitude.
The catch with our analysis so far is that the ice line's location is not an independent variable. The location of the ice line should be a function of the earth's temperature profile. The ice line will be close to latitudes where the annual average temperature is \(0\) degrees Celsius. It won't be at exactly 0, since seasonal variation in temperature will melt ice during the summer at that latitude. Let's call the critical annual average temperature below which ice persists year round \(\theta _ c\). Evidence suggests the critical annual average temperature \(\theta _ c = 10^{\circ}\) C.
Where is the average temperature equal to the critical temperature? There are two extreme cases  a Dunelike planet without ice (\(y _ c = 1\)) and a Hothlike planet encased in ice (\(y _ c = 0\)). In the Dune case, we see \(\theta^*(y,1) > \theta _ c\) everywhere, suggesting the planet is plenty warm enough to stay icefree. But this will only hold as long as \(\theta^*(1,1) > \theta _ c\). On the other hand, in the Hoth case, we see \(\theta^*(y,0) < \theta _ c\) everywhere, suggesting the planet would be cold enough to stay icecovered everywhere as long as \(\theta^*(0,0) < \theta _ c\). For our estimated parameter values, both the Dune and Hoth scenarios are valid steadystate solutions.
Now, what about intermediate iceline latitudes? Are there latitudes where our model predicts the temperature at the ice line will be the critical temperature \(\theta_c\)? Well, from the plots above, we see the latitudes where the iceline reaches critical temperaure are at the ice line?!?! Our model predicts a big jump in local average temperature whereever the ice line occurs, and for the parameter values we are considering, the local temperature jumps actually straddle the critical temperature!
Without some more information, we can not say precisely where the temperature profile will cross the critical temperature. We need some other condition. Perhaps what we need is to define the annual average temperature at the ice line? A natural (but not necessarily optimal) idea is to define the temperature at the iceline latitude \(y _ c\) as the average of the temperatures at the immediately higher and lower latitudes: \[\begin{gather} \theta^*(y_c,y_c) = 1/2 \; \lim_{\epsilon \rightarrow 0} \left[ \theta^*(y_c + \epsilon, y_c) + \theta^*(y_c  \epsilon, y_c) \right]. \end{gather}\]We call this extra equation a "jump condition" or "Stefan condition". Stefan conditions are a simple case of the more general freeboundary boundary conditions that can appear in the study of metal melting, bubble movement, and blood circulation. At steadystate, we expect the Stefan condition to give the iceline's critical latitude \(y_c^*\) implicitly by \(\theta _ c = \theta^*(y_c^*, y_c^*)\). We can plot the average temperature at the iceline as a function of the iceline latitude to determine solutions.
What we discover is that the iceline's temperature crosses the critical threshold of \(10^{\circ}\) Celsius at two separate points. These two crossing points each correspond to different steadystate solutions \(\theta^ * (y, y _ c^ * )\). In between these two points, our jump condition predicts the temperature will not be cold enough for ice year round. Let's call the larger of these two steadystate solutions Earth (since it seems closest to our current circumstances), and let us call the smaller solution Gethen, in honor of Ursula K. Leguin's speculative fiction world.
If we plot the iceline latitudes of the 4 steadystate solutions (Dune, Earth, Gethen, and Hoth) as a function of the incoming solar power, we get a more complete picture of the situation. For our realistic parameter estimates, all four steadystates exist. The Hoth and Dune steadystates only exist for temperatures where there temperature profiles don't intersect the critical temperature. If there is much less incoming power, then only the Hoth steadystate exists. If there was much more power coming in, then only the Dune steadystate would exist.
So far, our model gives us multiple predictions about the steadystate of earth's climate. But just because it is a steadystate doesn't mean it is realistic. A steadystate also has to be stable, in some sense, in order for it to correspond to something we are likely to observe in real life. Unfortunately, we'll have to develop a new method to assess stability of a system with a discontinuity.
Recall from our elementary differential equations studies that for a onedimensional autonomous ordinary differential equation \(\dot{u} = f(u)\), any steadystate solution \(u^*\) solving \(f(u^*)=0\) is locally stable when \(f'(u^ * ) < 0\). Let's try to apply this same reasoning to analyze the stability of our 4 steadystates using the evolution equation for global annual average temperature. Dynamically, \[\begin{gather} \frac{d \overline{\theta}}{dt} = Q \left(1  \overline{\alpha}(y _ c(\overline{\theta})) \right)  ( A + B \overline{\theta} ). \end{gather}\] Here, we've used the implicit function theorem to treat the iceline location as a function of the average temperature. Differentiating the right hand side with respect to \(\overline{\theta}\), we find the average temperature will be stable provided \[  Q \frac{\partial \overline{\alpha}}{\partial y_c} \frac{\partial y_c}{\partial \overline{\theta}}  B < 0.\] But the inequality is inconclusive because of the partial derivatives we have not yet defined. But, now, let's do something tricky  let's differentiation our steadystate condition \[0 = Q (1  \overline{\alpha}(y _ c) )  ( A + B \overline{\theta} )\] with respect to the power \(Q\). \[\begin{gather} 0 = (1  \overline{\alpha}(y _ c) )  Q \frac{\partial \overline{\alpha}}{\partial y_c} \frac{\partial y_c}{\partial\overline{\theta}} \frac{\partial \overline{\theta}}{\partial Q}  B \frac{\partial \overline{\theta}}{\partial Q}. \end{gather}\] After substitution, our stability condition becomes \[\begin{gather} (1\overline{\alpha}) \left( \frac{\partial \overline{\theta}}{\partial Q} \right)^{1} < 0 \end{gather}\] which we can simplify to \[\begin{gather} \frac{\partial \overline{\theta}}{\partial Q} > 0 \end{gather}\]for interior steadystates. This is a classic result known as the slopestability theorem by (Cahalan and North, 1979) with an elementary interpretation  any equilibrium global average annual temperature with an interior iceline will be a stable equilibrium if the global temperature increases as power increases. Looking at our plot, this tells us that Earth's situation is stable, while Gethen's situation is unstable. These results inturn imply stability results for the extreme scenarios of Hoth and Dune. Hoth will be stable for the range of low power inputs and inputs where Gethen exists, while it will be unstable for larger power inputs. Dune will be unstable for low power inputs and when Earth exists, while Dune will be stable for higher power inputs. So our model says that there are two stable steadystate climates we could have  the "Earthlike" climate we have today, or a snowball earth "Hothlike" climate that some geologists have speculated about.
You shouldn't accept this stability analysis uncritically. It seems to suggest that the stability of our temperature profiles can be determined entire in terms of a single ordinary differential equation. That's probably not so. If our stability condition above fails, the steadystate will be unstable, since an appropriately picked perturbation is sure to grow. However, the inverse does not necessarily hold. There may be perturbations to our steadystate solutions that are not encompassed in our simple analysis and lead to instability. However, the presence of a discontinuity in our steadystate solution breaks the conventional methods we would usually try to use. We will have to leave the questions concerning the local stability of the steadystates in SellersBudyko theory incompletely answered.
Fortunately, our planet is not like Hoth rightnow. But the geological record does indicate we've had iceages and maybe even snowball earths in the past. That means there are probably some slow but powerful effects that are moving our climate back and forth allong this spectrum. Pushed in one direction, our climate will steadily warm up until there are no more ice caps. This change would be reversible. Pushed the other way, climate will cool for a bit. But if it was cooled enough, it would hit a turning point of no return, where the spread of the ice sheet would accelerate and encase the whole planet in glaciers. That change could be irreversible on the time scale of human lives. Eventually, the transition back to our familiar climate might be equally rapid. This process of long periods of slow change alternating with short periods of rapid change is called a relaxation oscillation. Relaxation oscillations can be very dangerous because there way be very little warning or opportunity to prepare before the onset of rapid change.
Budyko's climate theory is provoking, in that, beginning with the familiar theories of temperature, conservation of energy, and the simplystated StefanBoltzmann law, we can derive climate predictions for entire planets that are surprisingly good. It's even more provocative because there is geological evidence supporting the appearance of ice ages, and maybe even snowball earths. But we should view Budyko's model with a fair degree of skepticism as well, given our relative naivety about the complexities of climate science. The model has multiple equilibria because of the particular Stefan condition we've adopted. There are two interior equilibria because the Stefan condition for the iceline creates a concave function with an interior peak. If we examine that condition closely, it begins to appear somewhat arbitrary, and we might wonder if there are other equally reasonable alternatives that will flip that concavity. Over the course of a year, we expect the ice line to have strong seasonal oscillations which would demand a more complicated stability analysis, perhaps based on Floque theory. At a higher level, our definition of global annual average temperature is suspect because it seems to weigh all latitudes equally. And our submodel for heat redistribution by atmospheric mixing seems little more than a phenomenological convenience, about as realistic and complete a representation of Rembrant the person as one of his selfportraits.
Overcoming these criticisms and building really good predictive models of our planet's global climate is an incredibly challenging research problem. There are many details that have to be including in the accounting, like mountain ranges, ocean circulation, ice sheet melting, cloud formation, daily weather patterns, and even salinety freshwater inflows from large rivers. Even our largest, most expensive computers are barely able to manage the demanding task currently, and a great deal of research remains for those interested and up for the challenge.
there is not just one tipping point, but many  melting of glaciers, collapse of ocean circulation, decaying of permifrost, ... We don't know where these tipping points are or how/if we can avoid them. Steffen, 2018
What fraction of the sun's power output does the earth reemit back to the sun?
In our calculations above, we ignored power entering the atmosphere from the light bouncing off the moon. Estimate the average magnitude of power from moonlight relative to direct solar power.
In our energybalance model, we used an approximate formula for the fraction of power output from the sun occluded by the earth. Find the exact formula for the fraction of power output from a point source occluded by a sphere of radius \(r\) a distance \(d\) from the source (distance being measured to the nearest point on the sphere. Show that our approximate formula is the correct asymptotic approximation as \(r/d \rightarrow 0\). How much error does this approximation introduce to our calculations?
If energybalance theory of global climate is correct, it should be applicable not just to the earth but to other bodies in our solar system as well. Lets consider the special case of Mars.
What data would you need to predict the average temperature of Mars?
Where can you find this data on the internet? (Please choose a source or sources that are trusted and have a verifiable citation.)
Using the methods from class, predict the average temperature on Mars.
In 1976, the Viking 1 lander became the first spacecraft to reach mars. As part of its mission, Viking 1 recorder the temperature on Mars (http://wwwk12.atmos.washington.edu/k12/mars/data/vl1/part1.html
). Plot the actual temperature over 10 sols (a Martian day), and discuss the relationship between the predicted and observed values.
(Hard) Extend our climate model to predict daily temperature variation on Mars, and compare your results with Viking 1's data.
If we want to build a ringworld in our solar system that has an average climate of 295 degrees Kelvin and an albedo 30 %, what should the ringworldâ€™s radius be (in astronomical units)? You can assume the outer sides of the ringworld are perfectly insulated and do not lose heat.
to finish and test After a sudden weather change, a pond that was uniformly 35 degrees Fahrenheit is subject to a cold snap of 15 degrees Farenheit for a week. How thick is the ice on the top of the pond at the end of the week?
Estimate the power entering the atmosphere from the earth's crust.