Flying animals or true 'weight lifting'

In response to a question on the Speculative Evolution website I thought it might be useful to write a short post on animal flight, with an eye on other worlds besides Earth. It will turn out that the logic is very similar to that of leg design. That subject, focussing on bone thickness, was discussed in two earlier posts, here and here. There is some math involved, but nothing more complicated than understanding powers and roots.

This post will only deal with the most basic aspect of flight, which is staying aloft. Let's start by considering an animal that is in stable flight. This means that is neither losing nor gaining height, with some kind of propulsion we will ignore (the same reasoning will also hold when the animal stably glides downwards, so lift is a bit less than weight). When the animal says aloft without sinking two forces must be equal in size: gravity pulls the animal down and lift pulls it up. We need to take a closer look at each. First, weight; it is the force induced on a body by gravity:

   weight = gravity constant (g) * body mass (m),

Click to enlarge; copyright Gert van Dijk

The image above shows several views of a general avian of the genus 'Avidisneius'. We will deal with the size of the animal's body; the wing will come later. For that we need to understand that its mass equals the product of its density and its volume. We will not alter density at all but will play with the volume. Volume is determined by its length, height and width. If this sounds as if the animal is shaped like a rectangular brick, that is essentially correct. But a more complex shape than a brick does not alter the principle that its volume depends on the product of its length, width and height; there will just be various constant factors thrown in, that we ignore. All three are measurements of length that we can label as 'L'. Volume thus equals L* L* L, or L to the third power: L^3. Weight was g*m, and we now replace 'm' with L^3:

    weight =  g * density * L^3.

Now we can go on to lift. Textbooks will tell you that it depends on a fairly simply equation:

  lift = rho * area * v^2.

Rho is the density of air.  'Area' is the wing area as seen from above, and 'v^2' is the square of the velocity of the animal with respect to the air. What this tells us is that there are three ways to get more lift. Obviously we cannot change the first, atmospheric density, but the equation tells us that an atmosphere with double the density doubles lift. Halve that density and you get half the lift, which is bad news for animals trying to fly on planets such as Mars. We can and will play with wing area: double the wing area and you obtain double the lift. The real winner here is velocity: doubling velocity gives four times the lift, because the formula contains velocity squared.

Of course, a wing can also be described by length and width and height. We will not use 'L' here but 'W' to specify that we are dealing with the wing. The mass of the wing will be W^3, but its area is proportional to W^2. So the equation for lift becomes:

  lift = rho * W^2 * v^2.

Click to enlarge; copyright Gert van Dijk

The two equations for weight and lift are all we need, for now. Let's take 'Avidisneius' and either double or treble its size as in the image above. To get the new forms, we replace 'L' in the weight equation with '2L' or '3L'. The volume becomes (2L)^3 or (3L)^3, giving us 8*L^3 and 27*L^3. The mass and weight change linearly with volume, so weight will increase by a factor of 8 or 27.

Unfortunately, this 'simple scaling' disturbs the balance between weight and lift. Why? Remember that lift depends on area, and hence on W^2. So if we double or treble W just as we did for L, we get new wing areas of (2W)^2 and (3W)^2, or 4*W^2 and 9*W^2. These wings, even though they are larger, are too small to hold the larger weight up. It's because of that infernal third power for weight versus the square for lift. (As an aside, the usual expression for 'scaling every dimension by the same amount' is 'geometrically similar'.)
              

Click to enlarge; copyright Gert van Dijk

Alas, the wings will have to be made even bigger. The image above shows the results, first for the previous 'simple' scaling and for a 'corrected' scaling attempt. We have seen that doubling body size (L=2) will make weight increase 8 times. What we therefore need is a new wing with 8 times the area. Similarly, if we make the body three times larger (L=3) then the new wing area must be 27 times the original one. We can find out how much we need to change the wing dimension 'W': area was the square of W, so to find the new 'W' we take the square root of 8 and of 27: the numbers are 2.82 and 5.20 respectively. So, if the body dimension (L) is to be 2 times bigger, the wing dimension (W) has to become 2.82 times bigger, and if the body becomes 3 times larger, the wing has to become 5.2 times larger.
 
Is everything solved now? Alas again... The additional increase in wing generates just the right amount of lift to compensate for the larger body. But the wing itself will also become heavier. Remember that volume corresponds to length to the third power? If our new wing dimension W is 2.82 times the original, the new wing mass will be 2.82^3 larger than the original, or 22.43 as much! This is not funny anymore. For the animal that became 3 times larger, the wing dimension W had to become 5.2 instead of 3, meaning the new wing volume is 140 times the original one, even though the body became only 27 times heavier. The 'corrected' scaling definitely falls short...

There is no escape from these cubic effects. Here is another way to look at its devastating effects. Suppose that the mass of the wing was originally about 20% of the mass of the animal. For an original Avidisneius of 500g in total, the wing would have a mass of 100g and all the rest has a mass of 400g. Let's take the animal we made three times larger using the 'corrected' scaling: the new mass of the body will be 27*400g, or 10,800g. The wing mass of 100g becomes 140*100g, or 14,000g. So our original 0.5 kg beastie now weighs 24.8 kg, and a staggering 56% of it is now wing. That's good if you like wing meat, and the animal should be easy to catch: can its heart and lungs even keep up with these massive wings? Mind you, in reality the bones and muscles themselves also need to increase by additional amounts, as their strengths depend on diameters (for examples see the posts mentioned above or my discussion of the giants in Game of Thrones here). 

The lesson is that, if you increase a flying animal's size, the wing dimension must increase more than the body. This actually happens in nature: larger birds have relatively larger wings. But there is no escape from the merciless differences caused by weight depending on cubic effects and lift, while bone and muscle strength depend on cross-sectional areas. At some size, the only weight that the wing can lift is that of the wing itself! But long before that point is reached, the construct will no longer be a viable animal. It is difficult to say where the 'Limit of Flight Plausibility' lies. On Earth now, kori bustards are arguably the largest flying birds. They weigh up to 18 kg, with a 275 cm wingspan. But extinct birds may have weighed 40 kg or even 72 kg. Did pterosaurs really weigh up to 250 kg? There is room for speculation here (which convinces me that Furaha needs some gargantuan beasts in the sky). However, please do not just geometrically enlarge a sparrow and present it as a 250 kg avian: approaching the Limit calls for profound changes in anatomy and flight efficiency.      

Other planets

The lessons for other planets are not very complicated: if you increase gravity by a certain amount, you increase weight by the same amount. If you transplant an Earth-like flying animal to a heavy world, you should make certain that lift increases by the same amount. Enlarging the wings will do that, but, because of the heavier wing, you should trim off a considerable amount of weight wherever you can. You could also make your animal fly faster, but do not think that that solves everything: your animal must be able to fly at low speeds to start and stop. You could evolve the propulsion system in such a way that it provides additional lift.

A high atmospheric density is a luxury, in contrast: if you double density, you can get away with half the wing area, which means that 'W' need only be 0.7 of what it would be on Earth. The soupier the air is, the more you can dream of avians with short stubby wings, resembling flying penguins.
  
But could you have something as large as a 'Game of Thrones' dragon flying around on a planet with an apparently Earth-like gravity and atmosphere? Of course not. Don't be silly. Dragons fly through magic. Birds don't.