Math 5421 Spring 2025
Introduction to Climate Models

Assignment 4 Solutions

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.

Solution

Let \(F_r\) be the power flux at a distance \(r\) from the star. Then \[ F_r = \frac{F\cdot 4\pi r_s^2}{4\pi r^2} = \frac{F r_s^2}{r^2}. \] This flux is spread over the surface of the planet, which has 4 times the area of the disk intercepting the power. Letting \(F_S\) be the average planetary surface flux, we have \[ F_S = \frac{F_r}{4} = \frac{F r_s^2}{4r^2}. \] The Stefan-Bolzmann equation tells us that \[ F_S = \sigma T(r)^4. \] Solving for \(T(r)\), we have \[ T(r) = \left( \frac{F_S}{\sigma} \right)^{1/4} = \left( \frac{F r_s^2}{4\sigma r^2} \right)^{1/4}. \]

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), and where \(T(r)\) is measured in Celsius. 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 \(-273^\circ\text{C}\) for absolute zero.

Solution

PLugging in the numbers, we have \begin{align} T(r) &= \left( \frac{F r_s^2}{4\sigma r^2} \right)^{1/4} - 273 \\ &= \left( \frac{6.29\times 10^7\times 0.696^2} {4\times 5.67\times 10^{-8} r^2} \right)^{1/4} - 273 \\ &= \frac{3390}{\sqrt{r}} - 273. \end{align}

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.

Solution

Venus: Early in its history, Venus may have had liquid water on its surface, since the black body equilibrium temperature is above freezing but below boiling. However, as the atmosphere thickened, it became impossible for liquid water to remain on the surface.

Mars: At a black body equilibrium temperature well below freezing the possibility of liquid water seems slim. However, with a thick atmosphere, conditions in some places on the surface might have allowed for liquid water.