Dinosaur dimensions

This is part 2 of three posts on the Physics of Prehistoric Animals. The first is on trilobite optics and the last is on finite element modelling of bite strength.

The main thing we all know about dinosaurs is that they were big. The largest sauropods were 40 m long and weighed 100 tonnes. A blue whale has about twice that mass, but it is supported by water – the sauropods were the largest land animals.

We can use scaling laws to try to understand how huge dinosaurs functioned. First, image a small animal (say a mouse) getting larger. If its length doubles, then its volume and therefore its mass will increase approximately as its length cubed: it will weigh eight times as much. However, the strength of its muscles and bones depend on their cross-sectional area, which only increases with the square of length. Our giant mouse will be twice as long and eight times as heavy, but its legs will be only four times as strong, and it will be unable to move. This is why heavy animals (elephant, hippo, rhino) need to have disproportionately thick legs to support their massive bodies. The concept of scaling laws like this was put forward by Galileo in 1638.

There’s a UCL connection here as well. One of the early popular descriptions of dimensional analysis was by a UCL Professor of Genetics and later of Biometry, J B S Haldane. He did a wide range of research, mainly around theoretical and mathematical biology, which was probably for the best as much of his practical work involved self-experimentation which apparently resulted in various injuries and illnesses. He was also an enthusiastic populariser of science, through articles written for the Daily Worker. He described dimensional analysis in an essay called “On being the Right Size” in 1926, before he came to UCL. In it, he explains why “you can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away … A rat is killed, a man is broken, a horse splashes.” His explanation is that the acceleration force due to gravity increases with the animal’s mass (proportional to length cubed), but the resistive force from the air, which slows it down, increases with its surface area (proportional to length squared). The velocity ends up being proportional to the square root of mass/area, or to the square root of length. A horse which is 10,000 times heavier than a mouse would be twenty times taller, longer and fatter, and so land with five times the speed of the mouse, and therefore would splash. In 1956, Haldane left UCL and moved to India, either in protest at the UK Government’s actions during the Suez Crisis or “to avoid wearing socks”, depending on your source. He died, in 1964, of cancer, and wrote a poem about it which begins “I wish I had the voice of Homer// to sing of rectal carcinoma” and includes some of the very few medical physics-inspired lines of poetry:

They pumped in BaS0₄.
Till I could really stand no more,
And, when sufficient had been pressed in,
They photographed my large intestine.

Back to dimensional analysis: it can also be applied to temperature control. Heat is generated by metabolism throughout the volume of the body (proportional to length cubed), but is only lost through the body’s surface. Animals in cold environments should therefore minimise their surface area-to-volume ratio by either minimising their surface area (Allen’s rule) or maximising their volume (Bergmann’s Rule). Arctic foxes, for example, are plumper than desert foxes and have small ears whereas desert foxes have huge ears to maximise their surface area for efficient cooling. In this case, the physicist who, when asked to predict milk production, wrote a report beginning “consider a spherical cow…” could have been onto something.

Dinosaurs, being big, had a large mass within which to generate heat, but a relatively small surface area through which to lose it. We tend to think of animals as being either cold-blooded (like the lizard who sits on a rock to warm up in the sun), or warm-blooded (a hummingbird eats its bodyweight in food each day to maintain a body temperature of 42°C). It is likely that the largest dinosaurs were in some intermediate category – they were large enough that their body temperature was substantially warmer than their environment, but they couldn’t actively control it. This suggests that they might have even more active than we might expect, but they wouldn’t need to eat as much as an equivalently-sized warm blooded animal, enabling them to reach such large sizes. Some large sharks have a similar metabolism, partly warm and partly cold blooded. This argument wouldn’t apply to smaller dinosaurs, who would have a relatively larger surface area and therefore lose heat. They therefore evolved feathers and survived the extinction of the dinosaurs by becoming warm-blooded birds.

Some of the most well known large dinosaurs (Triceratops, Stegasaurus, Spinosaurus) had anatomical features which increase their surface area (a neck frill, plates along the spine, or a sail). We don’t know what purpose these features had, but it’s certainly possible that they played a role in temperature regulation. These large, flat features, have a large surface area compared to their volume, so they would gain and lose heat efficiently, in the same way that an elephant’s ears can help to keep it cool.

An elephant charging me in Kruger National Park. Note the big, floppy ears, ideal for heat exchange and threat display, and the slightly blurry photograph.

Think now of energy rather than temperature. The energy generated by an animal has to escape through its surface, or else the animal will get hotter and hotter. The energy used at rest, called the basal metabolic rate, must, therefore scale with the animal’s surface area, which is the square of its length. Because mass is length cubed, this is the same as saying that the basal metabolic rate scales with mass to the ⅔ power, or equivalently, it is proportional to M^⅔. However, if this exponent is measured, it turns out to be a bit larger, and is closer to M^¾. Either way, this is remarkably constant. Apart from steps from single-celled animals to cold-blooded animals, and from cold-blooded animals to warm-blooded animals, this relation holds from cells, weighing a tiny fraction of a gram, to whales, over an extraordinary 20 orders of magnitude.

This is possibly the most extraordinary graph in biology. Think about it. Animals from bacteria to mammoths could barely be more different, but they all follow the same constant scaling laws.

The argument as to why the exponent is ¾ instead of ⅔ is complex and has only recently been worked out. Part of the justification says that the energy is generated on the surfaces of objects within cells, and the total surface area involved is so large and tangled up, the surface actually scales with the volume. Imagine stuffing a bedsheet into a washing machine. Even though the sheet is pretty much a 2D area, the amount we can get into the washing machine depends on the washing machine’s volume rather than the surface area of its drum.

There’s a neat little side-argument here: metabolism is supplied by the volume of blood in a fixed time, so if metabolism scales with M^¾, then so must the volume of blood supplied in a fixed time. The volume of blood leaving the heart in a fixed time (proportional to M^¾) is equal to the volume of the heart (which is proportional to the volume, or mass, of the body) multiplied by the heart rate, so the heart rate must be proportional to M^-¼. A similar argument can be made to show that an organism’s lifespan is proportional to M^¼. If we multiply the lifespan and heartrate together, we find that the total number of heartbeats is independent of the animal’s size, and is in fact constant. Hence, remarkably, a straightforward argument from physical principles can show that all animals have approximately the same number of heartbeats in a lifetime, althought it is, of course more complicated than this. This number is about 2 billion, which puts a human lifespan at about 60 years, which is pretty close, particularly given that we have a longer lifespan than we should due to medicine and sanitation etc.

Finally, let us come back to dinosaurs. A recent article proposed that a non-linear curve fitted the graph above slightly better than the simple linear power law we’ve been talking about so far. The proposed curve gets steeper as the animals get larger. At a mass of about 100 tonnes, the curve reaches a gradient of one, where it becomes much harder to provide energy to the body of an animal bigger than this mass. This may give a soft limit to the maximum size that an animal can achieve and still function efficiently. This happens to be about the same size as that of the largest dinosaur, so at last, we can predict, using scaling laws how big the largest dinosaurs could be.