The other day, we were arguing in the office about Icarus’s flight to the sun. As you all know, the legend goes that Icarus made himself some wings out of wax and feathers. Unfortunately his hubris led him to fly higher and higher until his wings melted due to the heat from the sun.
Our argument was about whether, having abandoned the safety of the atmosphere, would the wax melt or freeze when in sunlight? In order to answer this question, we need to make use of the principle of detailed balance. According to detailed balance, energy exchanges between bodies at equilibrium must cancel out. Therefore if Icarus is receiving a certain flux of energy from the sun, he must also be emitting the same amount of energy. The corresponding equation is:
The only way for Icarus to emit energy is by radiating black body radiation. Since the intensity of black body radiation is related to the temperature of the body by Stefans law, we know that:
where T is Icarus’s temperature, is Stefan’s constant and is Icarus’s total surface area. Similarly, the incoming energy depends on the solar flux and on the surface of Icarus that is exposed to sunlight:
I will assume that the surface of Icarus exposed to the sun is half of the total surface area. Therefore Icarus’s temperature will be:
Now, the solar flux is approximately 1 at the distance from the Earth to the sun, therefore Icarus’s temperature is approximately 306 degrees Kelvin. Or about 40 degrees Celsius. The melting point of wax is 60 degrees, so Icarus would be fine. In your face legends!