Math 5421 Spring 2025
Introduction to Climate Models

Assignment 4
due January 30

Goldilocks Zone

With the discovery of many planetary systems in the galaxy, a natural question arose: Could any of them harbor life that would be recognizable by Earthlings? As a start, one could require that the planet should be located at a distance from the star that would be compatible with having liquid water on its surface. In other words, not too hot and not too cold, a requirement immediately bringing to mind the three bowls of porridge in the fairy tale of Goldilocks and the Three Bears.

A first step from a scientific perspective would be to compute the equilibrium temperature of a black body at a certain distance from the star. Given that the planet would probably have an atmosphere that would allow for a warmer surface temperature, a black body temperature not too far from 0°C would probably allow for liquid water on the surface of a planet located there.

Exercise 1

Assume that the star has radius \(r_s\) meters and surface energy flux \(F\) watts per square meter. Let \(T(r)\) be the black body temperature in Kelvin at a distance \(r\) meters from the sun. Derive the formula \[ T(r) = \left( \frac{F r_s^2}{4\sigma r^2} \right)^{1/4}, \] where \(\sigma\) is the Stefan-Bolzmann constant.

Exercise 2

For our solar system, show that the formula in Exercise 1 reduces to \[ T(r) = \frac{3400}{\sqrt r} - 273, \] where \(r\) is measured in gigameters (\(10^9\)) meters. Assume that the energy flux at the Sun's surface is \(F = 6.29\times 10^7\) watts per square meters and that the Sun's radius is \(6.96\times 10^8\) meters (\( 0.696\) gigameters). Use \(\sigma = 5.67\times 10^{-8}\;\text{Wm}^{-2}\text{K}^{-4}\) for the Stefan-Bolzmann constant, and use -273°C for absolute zero.

Exercise 3

Produce a graph showing the relation between the distance from the Sun in gigameters and the black body equilibrium temperature in Celsius at that distance. Indicate the positions of Venus, Earth, and Mars on the graph. Discuss whether you think it is possible that there was ever liquid water on the surface of Venus or Mars.

Instructions for all Three Exercises

Submit your answers either as a Google Docs document or a pdf. If the later, load your pdf onto Google Drive and share editing permission with me.