Planet Surface Temperature

by Mee Wong-u-railertkun, John Pharo, David Vartanyan

Abstract

We investigate the relationship between the energy emitted from a star and the surface temperature of a planet. We assume that a star and a planet are perfect spheres and they are perfect blackbodies, i.e. power absorbed equals to power emitted. Moreover, they radiate isotropically. Later, we compare between the theoretical surface temperature of the Earth with the real data.



Introduction

Stefan-Boltzmann law relates flux with temperature as following,

sigma is called the Stefan-Boltzmann constant. With this equation, we relate flux with luminosity; flux depends on the distance from a source while luminosity is not. (Note that this definition is for astronomy only and is not consistent with the definition in E&M)

Solution

Assume T*, L*, and R* be the temperature, luminosity and radius of the star respectively.

First, we have to find the luminosity from flux at the surface of the star.

Now, assume a planet be at a distance "a" away. Since the flux depends on distance from a star, we find the flux of a star at the planet.

Since radius of a planet is much less than the distance from a star to a planet, we can see the surface of the planet that get radiation from a star as a disc with radius of a planet. So, the power absorbed is the multiplication of flux at that distance and the surface area.

Since we assume that a planet is a perfect blackbody, the power absorbed is equal to the power emitted. Now, we have to find the flux of a planet by divide the power emitted with the total surface area of a planet.

With the Stefan-Boltzmann law, we can find the surface temperature of the planet from the flux.

Discussion

Now, we substitute in real data for the sun to find the surface temperature of the Earth.

If we substitute in these numbers, we get the surface temperature of the Earth to be,

The theoretical answer is 278 Kelvin or 5 degree Celcius. It is colder than the real temperature on Earth may be because the Earth's atmosphere acts as a greenhouse and warms up the Earth.

Acknowledgement

We thank WolframAlpha for the data of temperature and radius of the sun and a distance between the sun and Earth. All the equations are made by the online LaTex editor.