The range of the bird is the intrinsic range of the fuel, 4000 km, times a

factor
*εf*_{fuel}/
(*c*_{d}*f*_{A})^{1/2}.
If our bird has engine efficiency *ε* = 1/3 and drag-to-

lift ratio (*c*_{d}*f*_{A})^{1/2} ≅ 1/20, and if nearly half of the bird is fuel (a fully-laden

747 is 46% fuel), we find that all birds and planes, of whatever size, have

the same range: about three times the fuel’s distance – roughly 13 000 km.

This figure is again close to the true answer: the nonstop flight record

for a 747 (set on March 23–24, 1989) was a distance of 16 560 km.

And the claim that the range is independent of bird size is supported

by the observation that birds of all sizes, from great geese down to dainty

swallows and arctic tern migrate intercontinental distances. The longest

recorded non-stop flight by a bird was a distance of 11 000 km, by a bar-

tailed godwit.

How far did Steve Fossett go in the specially-designed Scaled Composites

Model 311 Virgin Atlantic GlobalFlyer? 41 467 km. [33ptcg] An

unusual plane: 83% of its take-off weight was fuel; the flight made careful

use of the jet-stream to boost its distance. Fragile, the plane had several

failures along the way.

One interesting point brought out by this cartoon: if we ask “what’s

the optimum air-density to fly in?”, we find that the *thrust* required (C.20)

at the optimum speed is independent of the density. So our cartoon plane

would be equally happy to fly at any height; there isn’t an optimum density;

the plane could achieve the same miles-per-gallon in any density; but

the optimum *speed* does depend on the density (*v*^{2} ~ 1/*ρ*, equation (C.16)).

So all else being equal, our cartoon plane would have the shortest journey

time if it flew in the lowest-density air possible. Now real engines’ efficien-

cies aren’t independent of speed and air density. As a plane gets lighter by

burning fuel, our cartoon says its optimal speed at a given density would

reduce (*v*^{2} ~ *mg*/(*ρ*(*c*_{d}*A*_{p}*A*_{s})^{1/2})). So a plane travelling in air of constant

density should slow down a little as it gets lighter. But a plane can both

keep going at a *constant speed* and continue flying at its *optimal* speed if

it increases its altitude so as to reduce the air density. Next time you’re

on a long-distance flight, you could check whether the pilot increases the

cruising height from, say, 31 000 feet to 39 000 feet by the end of the flight.

We’ve already argued that the efficiency of flight, in terms of energy per

ton-km, is just a simple dimensionless number times *g*. Changing the

fuel isn’t going to change this fundamental argument. Hydrogen-powered

planes are worth discussing if we’re hoping to reduce climate-changing

emissions. They might also have better range. But don’t expect them to be

radically more energy-efficient.