Human-powered flight on Titan



*  XKCD article on flight


For a flying object,

Drag_Force = 1/2 * Atmosphere_Density * Cross_Sectional_Area * Velocity^2

A wing deflects air downward and converts drag force to upforce.

Up_Force = C * Drag_Force

C is a function of the wing tilt. The more efficient the wing, the large the achievable value of C.
Typical commercial aircraft have C ~ 6 and a streamlined glider can have C up to 20.

Flight is possible if the upforce exceeds gravity.

Up_Force > Mass * Gravity

The power expended by the airplane is

Power = Drag_Force * Velocity = 1/2 * Atmosphere_Density * Cross_Sectional_Area * Velocity^3

The faster the airplane goes, the larger the upforce. The "stall velocity" is the
minimum velocity for flight, and it also represents the minimum power required
for flight.  The stall velocity occurs when

Mass * Gravity = 1/2 C Atmosphere_Density * Cross_Sectional_Area * Velocity^2

The power expended by the airplane when flying at the stall speed is

Power ~ (Mass * Gravity)^3/2 * (1/2 C Atmosphere_Density * Cross_Sectional_Area)^(-1/2)

Flight is easiest when gravity is small and the atmospheric density is high.

        Gravity Atmosphere Density
         m/s^2   (kg/m^3)
Venus    8.87    67.
Titan    1.35     5.3
Earth    9.78     1.2
Mars     3.8       .020

Suppose a human-powered aircraft has Cross_Sectional_Area = 1, and the mass of
the human plus aircraft is 100 kg. How much power is required to fly on Mars,
the Earth, Titan, and Venus?  What is the stall velocity on each of these
objects?

On Titan, you can electrolyze water to produce hydrogen, and the hydrogen
can be used to inflate a balloon for buoyancy.
What volume of balloon is required to lift a 100 kg person?

On the Earth, human powered flight is possible by specialized athletes for a duration of ~ 10 minutes.