The second model you're going to be working on is an iterative model of the Ice-Albedo Feedback and how it affects the temperature of the Earth. So instead of stepping through the time in this model, we're going to be making successive guesses of what the right answers are and the guesses will converge and get closer. You get the same answer every time you guess as you get towards the end, so it's a fundamentally different kind of calculation. So the idea is that if planet is cold, it will have ice and snow, which is very reflective, and so that will reflect incoming energy. Whereas if the planet is warmer there's none of that stuff, and so the sunlight is absorbed more effectively. So if this is the energy balance of the planet, here is the incoming solar [COUGH], and here is the outgoing infrared which is a function of the temperature of the planet. The Albedo here kind of comes off the top of the incoming solar, gets reflected away, rather than absorbed and this is a function of ice, which is a function of temperature. So, the calculation gives you a linear function of temperature to describe the latitude, that ice will form on a planet, so the colder it is, the more the ice can be found closer and closer to the equator. And another equation is given with the Albedo as a function of temperature, and again, you're going to fit a straight line to the data that you're given. And so the idea is to start with a guess for the Albedo, and then using that Albedo, calculate the temperature. And then from this temperature get another guess for the Albedo, and then go back for another temperature, and then back and forth, back and forth, iterating between Albedo and temperature. This is what the snowball code looks like in Python, it has one loop going up and the other loop going down where we're looping over different values of the solar constant. So, for going down we start with a high solar constant, and so, the temperature is warm and the Albedo is probably ice free and that's true for many of these different values of L. But then, at some point you cross a value of L where it starts to freeze a little bit and so then it takes a few iterations, this is the number of iterations here for the temperature and Albedo to stop changing. And then at some point there's a critical solar constant value at which all of a sudden the feedbacks run all the way to the equator, and the temperatures go all the way down, the Earth gets covered with ice like a snowball. So if i change the plot type here and run that again, we can see. We are now looking at the equilibrium temperature after the iterations as a function of the solar constant in this dimension. And the reason why this isn't a simple line is because there is hysteresis in the system, if you start up here from an ice-free state and you cool the sun down in this dimension, the temperature, of course, gets colder but you don't start to get any ice until you reach an insulation that's sort of here, solar sunlight rate insulation. And then at some point you drop down into the snowball make it colder, it's just ice frozen all the way anyways, so just the sun effect here. But then as you warm it back up, the ice sheet is able to perpetuate its own existence by reflecting sunlight back to space. And so to get out of the snowball, it takes considerably more work, than it took to get- than it took sort of coolness to get into it, this is what we call hysteresis, or path dependence.
