620: Wings

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Wings
Please do not try any of this and die or get arrested.
Title text: Please do not try any of this and die or get arrested.

[edit] Explanation

Cueball explains to Megan that on Saturn's moon Titan, the combination of lower gravity and a denser atmosphere make the act of flying simpler. Wings on earth that are only capable of generating 9% of the necessary lift would allow one to fly on Titan.

Cueball has already put together the apparatus necessary to test this: by using a pulley system to offset 91% of his weight, he can recreate the environment for flight on Titan. His wings are a success, and he uses them to fly around in a delighted fashion. Megan points out a fatal flaw: his wings use hot glue, which will soften and melt at high temperatures and Black Hat is waiting nearby with a powerful heat source. Naturally, Cueball falls.

Icarus was a character in Greek mythology who is known his own self-powered flight, which ended when the wax holding his wings together melted and he fell to his death. This supposedly occurred because he ignored instructions not to fly too close to the Sun, a tragic example of hubris — extreme arrogance. Here, Black Hat is bringing an artificial "sun" to "Icarus" to recreate the tragedy.

Note that the calculated figure of 9% is only correct if the temperature on Titan has been raised to be the same as Earth — which, for human-powered flight, would probably be necessary anyway. At Titan's normal temperature, you would only have to generate about 3% of your Earth body weight in lift, as the atmosphere is much denser.

In the title text, Randall asks that xkcd readers do not attempt to reproduce this. The reason for this is explained here 254: Comic Fragment.

[edit] Transcript

Cueball: Titan's gravity is 14% of Earth's, and its atmosphere 50% denser.
Cueball: So if you can generate 9% of your body weight in lift, you can fly on Titan.
Cueball: With wings, a stage harness, a cable, and 91% of my weight in bricks, I want to test this.
[There is a heap of materials on the ground. Cueball is holding a stage harness.]
[Large diagram of a bridge. A rope leads through pulleys tied to the bridge. One end goes to Cueball, one end to a pile of bricks.]
[Cueball is standing with wings attached to his arms.]
[Cueball flaps the wings, and appears to be floating.]
[Cueball glides.]
Cueball: It works!
Megan: Except you have two problems.
Cueball: What?
Megan: You used hot glue on your wing joints and you have friends into Greek mythology.
Cueball: Huh?
[Black Hat is standing on the bridge, with a large lamp labeled "heat lamp" attached to a battery.]
[The wing segments fall off Cueball and he tumbles downward.]

[edit] Trivia

  • Incidentally, 1566 Icarus got an asteroid named for him. It regularly flies very close to the sun.
  • NASA did use a comparable device in the sixties and seventies of the last century, propelled by some rocket engines and called the flying bedstead, aimed to simulate the lunar gravity on Earth conditions.
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Discussion

Cueball's physics has a mistake on this one (or at least assumes we've managed to heat the atmosphere of Titan to Earth's temperature). The temperature of Titan is roughly 1/3 the temperature of Earth on an absolute scale. Starting with the Ideal Gas Law, PV = NkT (k is Boltzmann's constant, N is # of molecules, P is pressure, V is volume, T is temperature), its easy to define the density of a gas, ρ as:

ρ = m/V = (m P)/(N k T) = P (m/N) / (k T)

Titan's atmosphere is 98.4% molecular nitrogen (N2) and on Earth only 78.1% molecular nitrogen (by volume), but for simplicity we'll assume 100% for both. The weight of one molecule of Nitrogen is (m/N) ~ 2 × 14 × 1.67x10-27 (kg/molecule) (there are 28 nucleons per molecule with a mass of about 1.67x10^-27 kg.

The pressure on Titan is PTitan=146.7 kPa, and TTitan = 93.7 K, while on Earth PEarth=101.3 kPa and TEarth = 287 K.

Plugging in numbers, we get ρTitan = 5.3 kg/m3 and ρEarth = 1.2 kg/m3 (note the measured surface density of air on Earth is 1.2 kg/m3 at Earth's mean temperature even without the simplifying assumption of 100% N2).

Hence Titan's atmosphere is 4.4 = (5.3/1.2) times denser than Earth's (or 340% denser); not 50% denser as stated in the comic.

You will get the 50% denser if you assume the same planetary temperature on Titan as on Earth. Titan at 287 K would have a density of ρTitan at 287K ~ 1.73 kg/m3 which is about 50% greater than Earth's.

For the second calculation (panel 2), note lift is proportional to the density of air. If your action on Earth creates a lift of L0 and you weigh W0, on Titan you'd have a lift of 4.4 L0 (Cueball calculated 1.5 L0) due to the greater air density. Your weight would only be 0.14 W0, due to Titan's lower surface gravity. If lift balances weight, you would be able to fly on Titan, that is if 4.4 L0 = 0.14 W0. That means to fly on Titan you need a lift on Earth of L0 = 0.03 W0, that is 3% of your Earth weight. Substituting Cueball's Titan density you would get the critical value from the comic: L0 = 0.14 W0/(1.5) = 9% W0.

PS: I largely adapted this my writeup on xkcd forums from 2009 when the comic was made. Jimbob (talk) 05:44, 8 June 2013 (UTC)
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