# Surface Temperature of Planets

by Eric S. Mukherjee, Nathan Baskin, and I forget who else (sorry).

Abstract
In this problem we consider the how the temperature of the sun affects the temperature of the earth.  This is possible to estimate by assuming both the sun and the earth to behave like perfect blackbodies.

Introduction
Assuming the Earth has constant surface temperature and that it behaves like a blackbody, we can estimate the surface temperature using the energy emitted by the sun.  We also assume the sun to be a perfect blackbody.  Under these assumptions we can find the surface temperature of the Earth by knowing the temperature of the sun, the radius of the sun, the mass of the sun, the mass of the earth*, and the radius of the earth.

Methods
We start with the equation for flux at the surface of a blackbody (σ is the Stefan-Boltzmann constant):

$F=\sigma T^{4} \: \frac{erg}{cm^{2}\, s}$

From this we derive the luminosity of the sun by multiplying through by the surface area:

$L_{\odot }=\sigma T^{4}\left ( 4\pi R_{\odot }^{2} \right )\: \frac{erg}{s}$

Then the flux of the sun at the surface of the earth is (where a is the astronomical unit):

$\frac{L_{\odot }}{4\pi a^{2}}=\sigma T^{4}\left ( \frac{R_{\odot }}{a} \right )^{2}$

If we consider the area of the earth through which the flux passes, it is the circle of area π R2.  Multiplying through by this quantity we get the power input to the earth from the sun.  We then realise that this is necessarily equal to the power output of the earth due to energy conservation which, at the surface of the earth, is equal to σT4π R2.   Thus we have an equation of the form:

$\frac{\left ( \frac{R_{\odot }}{a} \right )^{2}T_{\odot }^{4}}{4}=T_{\oplus }^{4}$

Using this equation with R= 695,500 km and T= 5778 K, we get T⊕ = 279 K.

Conclusions
This temperature that we calculate is around 5.5°C which sounds reasonable for an earth without accounting for atmospheric greenhouse effects and allowing for the temperature at the poles.  The true average temperature of the earth is around 16°C but that is measured with the warming effect of the atmosphere.  The sun is not a perfect blackbody which also contributes to the difference between our calculation and the true value.

Acknowledgements
I'd like to thank the entire Ay 20 class and teachers for collective brainpower due to the fact that I can't remember who exactly worked on this problem and I'm sure we drew from the knowledge of many people in the room.  I'd also like to thank the superior computational power of Wolfram Alpha for bringing to my attention that there exponents matter when calculating ratios and that the temperature of the earth is most definitely not 1270 K.

*Note: It has been brought to my attention by Professor Johnson that the masses of the earth and sun do not actually factor into this calculation at all unless we need them to derive some of our other known constants.

## 3 Responses to Surface Temperature of Planets

1. Very nice writeup! However, you state in the Intro that the mass of the earth and Sun are important. How so?

Also, what would happen if you account for the Earth's albedo?

http://en.wikipedia.org/wiki/Albedo

2. Ah, the mass of the earth and sun are actually not important. Only if we don't already know the astronomical unit or either of the two radii. Sorry, I saw mass on the worksheet and didn't think before writing.

For the albedo, presumably a factor of (1-A) where A is the albedo when calculating the power input would correct for it since the reflected power wouldn't be absorbed to contribute to the Earth's output?

3. Ah, it is nice to know that at least some elements of science haven't changed in four decades! I was thrilled to see that Stefan-Boltzmann has remained constant instead of varying. Be still my beating heart!