## Would Icarus’s wings _actually_ melt on way to the sun?

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:

$E_{incoming} = E_{outgoing}$,

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:

$E_{outgoing}=S\sigma T^4$.

where T is Icarus’s temperature, $\sigma$ is Stefan’s constant and $S$ 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:

$E_{incoming} = S_{solar} \Phi$.

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:

$T = ( \frac{\Phi}{2 \sigma} )^{1/4}$ .

Now, the solar flux $\Phi$ is approximately 1 $kW m^{-2}$ 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!

I am a researcher in solar energy at the University of Oxford. I am interested in mathematics, programming and trying to understand why things work. I also like the great outdoors and riding my bike.
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### 3 Responses to Would Icarus’s wings _actually_ melt on way to the sun?

1. Assuming the assumptions are correct (albedo = 0 for both emission and absorption? these will be very different frequencies – 6000K vs. ~300K…), Icarus would also have to spin fairly quickly to even out the temperature differences. Otherwise the sun-facing side may melt while the rear is chilled. I seem to recall the daylight temperature on the moon is around 100C.

A tinfoil solar shield & he should be OK I’d guess though.

Though as the legend says, he did try to fly to the sun, and the solar flux is going to ~1/R^2.

• I will amend the legend to include a tinfoil solar shield.
Also I guess that in outer space the flying would be more like ion propulsion?

• From what I recall the light pressure acting outwards on a body causes its orbit to decay, hence the solar system being cleared of dust (which has a relatively high surface area to mass ratio). So Icarus is definitely being taught his lesson, it’s just a matter of when and whether he burns up in the Earth or the Sun.

I do now want to waste time writing a finite element Icarus…