Design-a-Planet Sample Question

 

This is a sample of the kind of reasoning that will be used in the Design-a-Planet activity. To make life simpler for those of you who don't have the Adobe PDF reader installed, I am doing this sample in plain HTML. That means I can't easily use Greek characters or complicated formatting, but hopefully the formulae will be legible enough. sigma = Stefan Boltzman Constant (5.67 x 10-8 (W/m2)/degK4). pi = 3.14159...

 

Problem: A planet has radius r. It has zero albedo (i.e. absorbs sunlight perfectly). It has no atmosphere. At a distance of one a.u., the energy flux of the planet's star is 1000 W/m2. At what distance should I put the planet to achieve a temperature of 273K?

Answer: Energy in = Energy out. Energy in is solar absorption. Energy out is infrared radiation.

Let R be the distance from the star, in a.u. The energy flux incident on the planet is 1000/R2, by the inverse square law. The energy absorbed by the planet is pi r2 1000/R2, since the planet casts a shadow with area pi r2. The energy lost by radiation is 4 pi r2 sigma T4, since the surface area of the planet is 4 pi r2, and the Stefan Boltzman law states that each square meter radiates energy at a rate sigma T4. Equating input to output,

pi r2 1000/R2 = 4 pi r2 sigma T4

which is the same as 250/R2 = sigma T4

From this, you can solve for R as a function of T, and find what distance is needed to get the desired temperature.

Make sure you know how the answer to this would be modified by a nonzero albedo (reflectivity) or infrared trapping by a greenhouse gas.