Chapter 2

The previous chapter’s explanation of Galileo’s Square-Cube Law showed that size matters. Yet when we go back in time to the Mesozoic era, the dinosaur world becomes a magical place where the laws of science no longer seem to apply. In all likelihood, if dinosaur fossils had never been discovered the science community would have long ago 1) incorporated the teaching of Galileo’s Square-Cube Law in elementary science education and 2) concluded that largest terrestrial animals of today are at the largest possible size.

There are four problem areas illustrating why the largest dinosaurs and pterosaurs present a paradox to science:

1. Inadequate bone strength to support the largest dinosaurs
   
2. Inadequate muscle strength to lift and move the largest dinosaurs
   
3. Unacceptable high blood pressure and stress on the heart of the tallest dinosaurs
   
4. Aerodynamics principles showing that the pterosaurs should not have flown

Before we can discuss the arguments regarding bone strength and muscle strength, we need an estimate of the mass of the largest dinosaurs, the sauropods.

The sauropods were the largest terrestrial animals to ever exist. The Apatosaurus, that has the usual dinosaur shape of larger rear legs than forward legs; the Diplodocus, similar to the Apatosaurus yet has a much longer neck and tail; and the Brachiosaurus, known as the ‘arm lizard’ because it has the unusual feature of its forward legs being longer than its rear legs. In addition to these better known sauropods, numerous other large yet similar sauropods with various names since paleontologists tend to compete for bragging rights over who has found the largest land animal. For making these comparisons we will use the Brachiosaurus.

To get a ballpark value of an animal’s mass, determine a scaling factor by looking for similar shaped animals and then comparing similar features such as shoulder height. Cube the scaling factor to account for volume and hence weight of the animals, and then multiply this by the weight of the known animal. If several appropriate animals of about the same density and shape are used then the average of the results should give a good approximation.

In this equation M ₁ and M ₂ are the mass each animal and L ₁ and L ₂ are the lengths of similar comparable parts. For example if we are comparing a deer to an elk, or a fox to a wolf, the lengths could either be shoulder heights or the distances between the shoulder blades and the hips. By knowing the lengths and the mass of one of the animals the mass of the unknown animal can be determine to within about 20 % of the actual value.

While this method works well between numerous animals of similar shapes it loses its accuracy when applied to dinosaurs since dinosaurs are not shaped like modern animals.

If an accurate scale model is available, then another simple method is possible for determining the weight of the animal in question. Simply submerge the model in water and measure the volume or mass of the water displaced. Then if, for example, the model is one fortieth of the actual size, multiply 40³ or 64000 by the volume of water displaced, then multiply this result by the suspected density of the animal. The vast majority of terrestrial vertebrates are slightly less dense than water so a fast approximation is achieved by simply multiplying the scaling factor cubed by the mass of the water displaced. For a little more precision, multiply the displaced water volume by 0.97 to account for the animal being slightly less dense than water.

Where M is the unknown mass, V is the volume of the water displaced, D is the animal’s overall density which should be about 0.97 for terrestrial animals, and S.F. is the scaling factor.

As long as the model correctly matches the shape of the dinosaur the second method should be the most precise. The author’s estimations by using these methods show that the mass of a Brachiosaurus lies between 50 and 200 tons.

Relative bone strength can be defined as the strength of the bone divided by the weight being supported by the leg bones. Likewise the relative muscle strength can be defined as the strength of the animal divided by its weight.

The relative bone strength and the relative muscle strength are grouped together because they are similar scaling problems. For both, strength is function of the cross-sectional area. If we look at the longest length of a bone or muscle and then imagined cutting this length in half, the newly exposed area is the cross-sectional area. The strength of either a bone or a muscle is directly proportional to this cross-sectional area, so both bone and muscle strength are two dimensional attributes. Yet body mass is a function of volume, a three dimensional attribute. In accordance to the Square-Cube, as we look at increasing larger animals the mass of each animal increases at a faster rate than the cross-sectional areas of either the bone or the muscle. Thus, larger animals have less relative muscle strength and less relative bone strength than that of smaller animals.

In regards to relative bone strength, the larger animals are at a much greater risk of breaking their bones than the smaller animals. The likelihood that a broken bone will cut an animal’s life short is a strong possibility for the larger animals. This possibility of broken bones affecting the animal’s survival thus becomes a limitation on the size of the largest animals.

For example, race horse can easily shatter a leg just by running. These breaks usually occur at various places within the lower front leg. Yet it is possible for other parts of both the forward legs and the rear legs to be injured as well. This indicates that the breaks are not a result of a specific inherent weak spot within the leg. But rather it is just the simple physics of the heavy weight of the horse producing an impact that exceeds the material strength of the bone within the leg.

Another indication that these 500 kg animals are pushing the size limitation are the problems that arise when attempts are made to heal one of these magnificent animals after they have shattered a leg. Horses often sleep standing up as a successful evolutionary survival technique so that they can quickly flee from predators. But if day and night a horse is able to stand on only three good legs and it does this for an extended time, these overloaded good legs may develop a condition known as laminitis. Soon it becomes just as painful to stand on these legs as on the original broken leg. It is because of these and other associated complications that it is often more humane to put the horse down rather than have it suffer through its final days.

While it is easy to show that the largest animals have the lowest relative bone strength and the lowest relative muscle strength, it is more challenging to determine precisely the largest possible terrestrial vertebrate. One problem is that as we look at ever larger animals they change their behavior so that they can stay within the limitations.

While race horses are large they are far from being the largest terrestrial animals. Weighing in at about a ton, the Clydesdales horses have twice the mass of the typical racehorse. The typical male African elephant has a mass of five to seven tons.

Here the lower muscle strength of the larger animals is actually beneficial to the larger animals in reducing the possibility of them carrying out potentially dangerous activities that may cause a broken bone. By running more slowly or in the case of elephants by running more slowly and not jumping, the largest terrestrial animals usually manage to avoid the higher impact forces that may break a leg.

Bones break when the stress applied to bone exceeds the bone material’s breaking point. To get an idea of the greater risk that larger animals have of breaking their bones, we can compare the stress on the leg bones of animals as they do nothing more than support their own weight.

The table below lists a representative selection of mammals ranging from the smallest to the largest. From the measured data of the front and rear leg bone circumferences the amount of cross sectional bone area is determined. The stress being applied to the bone while the animal is standing can then be calculated by dividing the weight of the animal by the bone cross sectional area. The final column on the right is the stress on the animal’s legs as it is standing.

The stress on bones can be many times greater when an animal is landing after a fall. The stress on the bone that comes at the end of a fall generally depends on how large the animal is and how far it falls. A mouse can easily survive a fall from a tall building. At the other extreme, an elephant can be contained with a one meter (about three feet) deep dry moat, because an elephant falling from this height would most likely result in one or more fractures. There is truth to the expression “the bigger they are the harder they fall”.

The low relative muscle strength is similar to the relative bone strength in that the largest animals have the lowest relative strength. Absolute strength can be defined as how much weight an animal can lift regardless of the animal’s own weight, and clearly the larger animals have greater absolute strength than the smaller animals. But when we look at relative strength, the lifting ability of an animal relative to its own weight; it is the smallest animals that have the greatest relative strength. For example, an ant can lift an object fifty times its own weight, a strong person can lift another person, while an Asian elephant can lift only one fourth of its own weight. The larger five to seven ton African elephant is not a working animal because its relative strength is even less.

For anyone who has watched large farm animals such as cattle or horses pick themselves up off the ground it is clear that these animals are exerting all the strength that they have. The same is true of other large animals such as elephants and giraffes that need all their strength to perform this task that is not challenging for the smaller animals. As a consequence of these difficulties, it is not surprising that many of these larger animals evolve the behavior of sleeping while standing up.

Yet numerous dinosaurs were much larger than these animals. Their greater size would mean that their relative strength would be substantially less than that of the large animals of today. It is not realistic to imagine that the large dinosaurs never fell or otherwise found themselves lying on the ground through their entire lives. If a Jurassic Park was actually created, any sauropod or other exceptionally large dinosaur lying on the ground would have been as helpless as a whale stranded on a beach.

To explain how the sauropods could grow so large we were first told that these animals spent most of their time in the lakes so that the buoyancy of the water helped support their massive weight. Later this idea was abandoned and replaced with the idea that the bones of these animals were somehow structurally superior to modern animals and that the massive bodies of these sauropods were disproportionally light. Along this theme it has become a common practice of paleontologists to grossly underestimate the weight of the larger dinosaurs. While it may not be considered polite to question the opinion of these paleontologists, the advancement of science absolutely depends on beliefs being questioned and validated by evidence. The extent of the fudging of these dinosaur weight estimates is revealed to anyone that takes the time to submerge a model of a dinosaur in water and make the relatively simple calculations.

Many researchers have questioned how it would be possible for a Brachiosaurus to supply blood to its head. Several unlikely hypotheses have been suggested. Some paleontologists have suggested that Brachiosaurus had a massive heart to produce the needed pressure to lift the blood. Another proposal is that the Brachiosaurus evolved a series of several evenly spaced hearts in the neck as a pumping system that would get the job done. More recently a popular idea is that the Brachiosaurus never lifted its head up but instead just moved it back and forth horizontally.

The assortment of hypotheses comes from the problems associated with pumping blood to a greater height. In a column of a fluid the pressure increases during the descent from the top of the fluid to a lower level according to the relationship P = g D h, where P is the pressure, g is the acceleration due to gravity, D is the density, and h is the distance below the surface. Because of this, a pump and the tubing at the bottom of a column of fluid must be strong to withstand fluid pressure near the bottom of the column.

To better understand how blood pressure can vary with height, consider the way a person’s blood pressure is taken. The cuff is wrapped around the bicep of the arm while a person is sitting because at this point the blood is at the same height as the heart. The blood pressure taken this way is a close approximation to the pressure of the blood as it leaves the heart. If during the measurements a person were to raise their arm the reading would be much less, or if the measurement is taken at the ankle rather than at the arm the blood pressure would be much greater. This is because blood pressure is a function of height.

It is easy for the heart to pump blood to parts of the body that are at the same elevation as the heart. For these horizontal circuits, the heart only has to overcome the viscous drag of pushing the blood through the arteries. For these circuits there is almost no loss of blood pressure until the blood moves through the capillaries. For this reason the blood pressure taken at the bicep is a close approximation to the blood pressure as it leaves the heart.

When the heart pumps blood to the lower parts of the body the work is even easier since gravity is helping the blood flow downward. However, once the blood passes through the capillaries in the feet it has to travel back up to the heart. This is accomplished in part by being pushed along by the weight of the blood in the arteries. Also valves in the veins take the pressure off the lower parts of the veins during the time between the beats of the heart. In addition, the valves in these lower veins allow the leg muscles to work like the heart in squeezing the blood up to the next level whenever the leg muscles contract. The reason we feel discomfort while standing for long periods or sitting during a long plane flight is because our leg muscles are immobile and that causes the blood to accumulate in our lower veins.

The heart has to work the hardest when it is elevating the blood up to the head. This is because with every beat the heart must lift all the blood within the vertical column that is in the arteries going up to the head. We can use the equation P = g D h to calculate how high the blood pressure P must be as it leaves the heart so that it can reach a height of brain h. For an upright adult the top of the head is about 45 cm above the heart and thus the minimum pressure the heart needs to reach this height is 35 mm Hg. Once it reaches this height there needs to be still more pressure to push the blood through the capillaries. To accomplish the complete task of lifting the blood and pushing it through the capillaries a normal person requires a blood pressure of about 120 / 80 mm Hg. The reason the heart is located closer to our head than it is to our feet is because of the challenges of pumping blood up a vertical distance.

A couple examples will give additional insight of how height affects the cardiovascular system.

The adventurous person that has attempted inverting themselves so as to stand on their head knows that this is a mildly painful position. In this unusual position blood pools in the head causing the face to turn red. Yet we need not wonder why bats and other small mammals do not care about which side is up because their bodies are too small to experience much of a pressure difference between the highest and lowest parts of their bodies. It is only the larger, taller terrestrial animals that must deal with the challenges of a large blood pressure gradient due to elevation.

Besides standing on our heads, a much more common experience people have is the dizzy feeling we sometimes get when we stand up too quickly. While resting horizontally our heart is not working nearly as hard as when we are standing or exercising. When we stand up the heart must suddenly work much harder to pump blood up to the brain. When we stand up quickly the blood momentary fails to reach the brain and the cells in the brain momentary starved for oxygen causing us to feel faint.

At approximately six feet or 2.0 meters human beings stand tall among most terrestrial vertebras, yet at 18 feet or 5.5 meters the giraffe is the much taller modern-day champion of height. Our occasional feeling of light headedness when standing up is hardly comparable to the 15 ft or 5.0 m elevation change a giraffe goes through in obtaining a drink of water. If not for valves in the veins and arteries of its neck, the extreme pressure would cause the blood vessels to break when the giraffe lowers its head, and conversely the giraffe would pass out from lack of blood when it later lifts its head.

Another potential problem is the extreme pressure that exists in the giraffe’s lower legs while it is standing. Anyone who has a job where they are standing most of the day is aware of how uncomfortable it can be as the blood pools in the lower legs, and yet a giraffe is three times taller and so the pressure it three times greater. Furthermore, if their legs were similar to other animals then even a small cut on the leg would bleed profusely and potentially be life threatening.

To prevent blood from pooling in its lower legs, the legs are surrounded with a tough thick skin that counteracts the blood pressure to prevent the blood from pooling. Inside the skin there is a thick inner fibrous tissue and the leg’s blood vessels are far from the surface so as to avoid the potentially lethal problem of bleeding from a cut.

Yet the giraffe’s greatest cardiovascular problem is having a strong enough heart to lift blood up to its brain. To produce the necessary blood pressure the giraffe’s heart is a huge muscle with walls up to three inches (eight cm) thick and weighing 25 pounds (11 kg). But even more impressive is that the giraffe’s resting heart rate is 65 beats per minute. This is about twice what is expected for an animal of its weight. The giraffe’s massive ‘revved up’ heart produces the 300 / 180 mm Hg blood pressure needed for the blood to reach the giraffe’s head. Giraffes have a relatively short lifespan of only 20 years and are prone to heart attacks as a consequence of their cardiovascular adaptations.

Yet if the giraffe is an amazing animal in overcoming all of these cardiovascular problems to achieve its height, what should we think of the Brachiosaurus that stood at a height of 13 meters? While the giraffe’s head is 2.5 to 3.0 m above its heart, the brachiosaurs’ head was 8.0 to 9.5 m above its heart. As the variety of unlikely proposals show, paleontologist are baffled by this problem.

The sauropod blood pressure paradox has been debated for several years and now it is showing up in physics textbooks. Increasingly, paleontologists are coming to the belief that the Brachiosaurus could not have held its head up, and likewise the other sauropods could not have reared up on their hind legs to reach the higher foliage.

Yet remounting all the brachiosaur exhibits so as to lower the head is not the solution. This ad hoc solution does not explain why the Brachiosaurus has a posture for reaching up high. The Brachiosaurus, the ‘arm lizard’, and its cousins, are the only dinosaurs with longer forward legs than rear legs. The logical explanation for the longer forward legs is that the addition of longer legs and its long neck serve the purpose of extending the Brachiosaurus’ reach up to the highest foliage. Thus we have the paradox of having an animal that is built for its head and mouth reaching the maximum height and yet at this great height its heart lacks the ability to pump blood up to its head.

The paradox of how the giant pterosaurs flew is the subject of the next chapter.

The Problems with Big Dinosaurs

• The Science Paradox of Big Dinosaurs and Pterosaurs - Octave Levenspiel
  
• Mysteries of the Dinosaur Epoch - Chris Yukna
  
• Body Design of Sauropods - George Johnson
  
• Logical Inconsistencies Regarding Dinosaurs - C Johnson
  

Brachiosaurus and other Sauropods

• Brachiosaurus - DINODICTIONARY.COM
  
• Brachiosaurus - Bob Strauss
  
• Diplodocus - WIKIPEDIA
  
• Sauropods - Scott Hartman
  
• Brachiosaurus
  
• The Largest Animals to ever Walk the Earth - Gavinry Mill
  
• Brachiosaurus - Brian Roesch
  

Determining the Weight of Dinosaurs from Models

• Dinosaur Weight
  
• How to Weigh a Dinosaur - COPERNICUS
  
• Mass of a Whale - Glenn Elert
  

Relative Strength

• Relative Strength - Ron Lakes
  
• Strongest Creature - EXTREME SCIENCE
  
• Relative Strength in Regards to Weightlifting Training - Mark Twight
  

Horses Breaking Their Legs

• Physics-Strain/Stress?
  
• Fatal Breakdowns - Sarah McCarthy
  
• Why racehorses are cracking up - Glenn Robertson Smith
  
• Broken Leg is Bad News for Horse - Daniel Engber
  
• Clydesdale - WIKIPEDIA
  

The Blood Pressure Problem

• Brachiosaurus - WALKING WITH DINOSAURS
  
• Blood Pressure - WIKIPEDIA
  

Blood Pressure and the Human Cardiovascular System

• Circulation of Blood - Larry M. Frolich
  
• The Human Cardiovascular System - Kenneth R. Koehler
  

Giraffes

• Giraffes' Necks & Fluid Mechanics - Tim Pedley
  
• Why Giraffes Don't Get Dizzy - LIVE SCIENCE
  
• Giraffes - SAN DIEGO ZOO
  
• Giraffe - HONOLULU ZOO
  
• Giraffe Information - HUNTING SOCIETY
  

Proposed Solutions to the Brachiosaur Blood Pressure Problem

• Hearts, Neck Posture and Metabolic Intensity of Sauropod Dinosaurs - Roger S. Seymour and Harvey B. Lillywhite
  
• Sauropod Necks - Yann Oliver