Giraffes, Elephants and other scalings

It's not the first time I am blogging about the as I find amazing fact that with some simple scaling arguments you can estimate quite a number of things without knowing them a priori. You could even argue that this is a core competence of the theoretical physicist: If you consider your self as being one you should be able to guesstimate any number and at least get the order of magnitude right . I have been told of ob interviews for business consultant jobs where the candidates were asked how many bricks the Empire State Building was build from and it's the physicists who usually are quite good at this.

Today, on the arxiv, Don Page gives some more examples of such calculations which I find quite entertaining (even if Lubos argues that they are too anthropocentric and apparently does not understand the concept of order of magnitude calculation where one sets not only h=G=c=1 but 2=pi=1 (ever tried this in mathematica and done further calculations?) as well): Page aims to compute the size of the tallest land animals from first principles and get it basically right.

The basic argument goes like this: First you assume chemistry (i.e. the science of molecules) is essential for the existence and dynamics of the animals. Then the mass of the electron and the fine structure constant give you a Rydberg which is the typical energy scale for atoms (and via the Bohr radius and the mass of a proton gives you estimates for the density both of planets and animal bodies). Molecular excitation energies are down by a factor of proton over electron mass. This implies the typical temperature: It should not be so high that all molecules fly apart but still be warm enough that not all molecular dynamics freeze out.

From this and the assumption that at this temperature atmospheric gases should not at a large scale have thermal energies higher than the gravitational binding energies to the planet gives you an estimate on the size of a the planet and the gravity there. The final step is to either make sure that the animals do not break whenever they fall or to make sure the animals do not overheat when they move or that gravity can be overcome to make sure all parts of the body can be reached by blood (this is where the Giraffes come in).

Of course these arguments assume that some facts about animals are not too different from what we find here (and some assumptions do not hold if all happens within a liquid, the pressure argument and the argument about falling which is why whales can be much bigger than land animals), but still I find it very interesting that one can "prove" why we are not much smaller or larger.

There is a very entertaining paper which makes similar arguments just the other way round (the title is misleading, its really about physics rather than biology): It argues why things common in B movies (people/animals much too large or too small) would not work in real life: King Kong for example would immediately break all his bones if he made one step. On the other hand, if we were a bit smaller, we could fall from any height as the terminal velocity would be much smaller. But simultaneously the surface tension of water would pose severe problems with drinking.

I would recommend this paper especially to the author of an article in this week's "Die Zeit" about nano scale machines that reports amongst other things about a nano-car with four wheels made of bucky balls. Understanding how things change when you try to scale things down show how the whole concept of wheels and rolling does not make sense at very small scales: First of all Brownian movement poses real threats, and then roughness of surfaces at the atomic scale would make any ride very bumpy (the author mentions these two things). But what I think is much more important is that gravity is completely negligible as your nano car would either float in the air or be glued by electrostatic forces (which for example cause most of the experimental headaches to people building submillimeter Cavendish pendulums to check the 1/r law or its modifications due to large extra dimensions) to the surface both perspectives not compatible with wheels and a rolling.

So there are good reasons why we are between one and two meters tall and why our engines and factories are not much smaller.