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Calculating the Animal Power Available for Flight

Birds and other flying animals are no different from planes in that they must yield to the laws of physics in fulfilling the aerodynamic criteria of flight. The fact that birds and other flying animals use their wings for both lift and forward propulsion while planes separate the two functions is not a significant difference or at least not a serious distinguishing factor in determining the flying ability of these objects. Yet there is an important difference in regards to available power: an aerodynamic engineer can assign almost whatever power deemed necessary for a plane yet birds and other animals are stuck with just the limited power created by their metabolism. It is this limitation of power that limits the weight of flying animals.

For animals the relative power is directly related to the level of the muscle cells’ metabolism. The metabolism level of a cell is for the most part directly related to how fast food and waste can be delivered and removed from the cell. Thus the power output of an animal is mostly determined by the collective efficiency of a team of physiology systems and organs delivering supplies to the cells. This logistics team includes the respiratory system providing oxygen, the circulatory system including the heart, the stomach, small intestine, liver and numerous other organs.

The fuel these organs provide to the cells is glucose and oxygen. Within an animal’s cell the mitochondria is the powerhouse that turns this fuel into energy in the form of ATP and waste in the form of carbon dioxide and water. ATP stands for adenosine triphosphate. It is a molecule that stores an incredible amount of energy, either 7.3 kcal/ mol or 10.9 kcal/ mol depending on how the bonds are broken. It is often referred to as the energy currency of the cell since it acts like a tiny powerful battery that the cell can transport to wherever energy is needed to perform an energy consuming activity.

C6H12O6 + 6O2 --> 6CO2 + 6H2O + energy (ATP)

The efficiency of the respiratory and circulatory systems and all the other operations involved in bringing fuel to the cells it a function of the Square-Cube law such that the smaller animals run at a higher relative metabolism. The Square-Cube Law comes into play in that as we look at larger animals the cross sectional areas of the fluid passages do not increase at the same pace as the volume of cells that they service. For the respiratory system the cross sectional area of the bronchial tubes are increases as a function of the square of the multiplier while the size of the lungs that they service is increasing as a function of the multiplier cubed. Once the oxygen is transferred to the blood the circulatory system is faced with a similar proportionality problem. The cross sectional area of the arteries and veins increase as a function of the multiplier square while the volume of cells being serviced is increasing as a function of the volume cube. Consequently a mouse uses 23 times more oxygen per body weight than the much larger elephant.


The high rev metabolism of small animals means that the smaller animals race through life much faster than the larger animals. The higher metabolism rate of the smaller animals can be roughly determined by comparing heart beats per minute. For comparison purposes the heart rate of a mouse is 670 beats per minute while the heart rate of an elephant is 28 beats per minute. The mouse heart rate is 18 times faster than the elephant compares well with its oxygen intake per mass that is 23 times greater. Thus a mouse is speeding through life at a rate about twenty times faster than the elephant. Since every mammal is allotted roughly a billion heartbeats for its lifespan the mouse has a lifespan of about three or four years while an elephant can live to be about 70 years old.

Roughly speaking the smaller animals with the highest metabolism will have the shortest lifespan while large animals with low metabolism will have the longest lifespan. So while the small warm-blooded animals such as a mouse or hummingbird will live only three or four years the large cold-blooded giant turtles can live to be over 150 years old. Greater lifespan correlates with lower metabolism and lower metabolism means lower power output per unit mass.

Because the power output of an animal is a function of the square-cube relationship an empirical equation for calculating animal power output as a function of weight can be stated as:

P3 = K W2.

Where P is the power, K is a constant, and W is the weight of the animal.


To determine the K constant we can use the use the industry standard of one horse power being equal to 746 W yet we will soon see that this industry standard will need some adjustment. This standard of one hp equaling 746 W came into being during the industrial revolution when objective of the tractor industry was to convince farmers to replace their horses with tractors. From this perspective it not surprising that the power rating that the tractor industry assigning one horsepower as equaling 746 W is not a fair assessment of the power output of a horse. Perhaps 746 W average power output is about right if a horse was worked for 24 hours, but to be fair a 5000 N horse can produce about 1200 W of power for several hours before taking a rest. Furthermore, unlike a machine, animals can vary their power output according to whether the need is for a sprint or a marathon.

Solving our equation for K gives us 69 W3 / N2.

So then

P3 = (69 W3/N2) W2

In this equation be careful to distinguish between the W that stands for watts a unit of power and the underlined W that stands for the weight of the animal. This equation enables us to determine the approximate power output of various size mammals.

German Shepard - a warm blooded mammal

There are actually two primary variables involved in determining the power output of a vertebrate: 1) the size or mass of the animal and 2) its overall metabolism strategy base on its use of enzymes. Generally warm blooded or endothermic vertebrates keep their body at a constant elevated temperature, a narrow temperature range where the set of enzymes operate at their maximum efficiency. While the body temperature of cold blooded or ectothermic vertebrates is more at the mercy of the prevailing surrounding environment meaning that the body temperature of ectothermic animals can vary substantially. When the surrounding temperatures are cold ectothermic animals do not have the ability to generate power so as to move. This external temperature dependency makes it difficult to assign a power output value to cold blooded animals.

Classifying vertebrates as either warm-blooded / endothermic or cold-blooded / ectothermic is not all that precise since many animals do not clearly fall into one category or the other. Many otherwise cold-blooded ectothermic species are at least partial warm-blooded. Furthermore lumping mammals and birds together as being warm-blooded is to ignore numerous differences that boost the metabolism of birds. It may be appropriate to distinguish birds as being hot blooded.

Beyond evolving wings and feathers for flight birds evolved a superior respiratory system that boosts their metabolism so as to give them the power they need for flight. Birds have a multi chamber air sac system that circulates the air so that the oxygen concentration in the lungs is at the same level as the atmosphere.

The superior respiratory system of birds delivers more oxygen to their cells causing their ‘fires’ to burn hotter. The result is not only more fuel for their cells but a higher body temperature that further speeds up the rate of chemical reactions and therefore increase their level of metabolism. The body temperatures of various species of birds fall between 40 and 42 degrees Celsius while mammals normally have lower body temperatures that lie between 34 and 40 degrees Celsius. The feathers surrounding their bodies are just as important as their wing feathers since these body feathers save energy by holding in the heat.

Flying seagulls at the harbor

Just how much more powerful a bird’s metabolism is than a mammal’s is an area that needs more research. But one indicator is the difference in heart rate of birds compared to mammals. For birds and mammals having the same mass the bird will have a heart rate that is about 50% faster than the same size mammal. If this difference in heart rate directly correlates with the cellular respiration within the animal then it means that when comparing same size birds to mammals the birds will be about 50% more powerful than their comparable size mammal. Thus to account for birds being more powerful than mammals the initial results determining the power output of a mammal based on its mass needs to multiplied by 1.5 to be more accurate as to the true power output of birds.

It is because of these advantages that the largest flying birds grow to be more than ten times heavier the largest bats. Bats, as the only flying mammals, barely meet the minimal power requirements to be able to fly and it is because of this that most bats are hardly any larger than a mouse. With limited relative power available the Square-Cube Law favors the smallest animals yet an animal as small as a mouse is about as small as a vertebra can be. Consequently when we step down to the next lower power classification of the cold blooded or ectothermic vertebrates we should not wonder why we can not find any present-day flying reptiles.

The Paradox of Flying Pterosaurs

Fossil evidence shows that millions of years ago there were birds and flying reptiles that were much larger than the flying birds of today. Even without knowing anything about the importance of size, flight, or the physiology of these animals this anomaly should drive our scientific curiosity to investigate further. Yet instead of working to solve this puzzle, the paleontology community has put their efforts into convincing the public that there is nothing odd about such large animals being able to fly. Paleontologists claim that the pterosaurs were able to fly because they were extremely light. This claim begs the question of how did these paleontologists weigh the pterosaurs.

Pterosaurs have been extinct for over 65 million years so no one as actually weighed a pterosaur. All that we have are the fossilized bones. With these bones we know the size and shape of these animals. We would know the approximate mass of these pterosaurs if we can find a similar size and shaped living animal for that we know the mass. We could also calculate an estimate of a pterosaur’s mass if we found a smaller yet similar shaped living animal of known mass. When there are two closely related species, one extinct and the other living, these types of estimates are usually accurate to within plus or minus 20%. But when there is no close match there is a much wider numerical spread for possible error.

Quetzalcoatlus compared to a car

Currently the 12 m wingspan Quetzalcoatlus is the largest known pterosaur, but it is not alone as some freak of nature. There are at least half dozen pterosaurs with wingspans over five meters. For comparison, a few of the largest flying birds of today are the 2.9 m wingspan 10.4 kg California Condor, the 3.5 m wingspan 11 kg Wandering Albatross, and the 2.4 m wingspan 15 kg Mute Swan.

When flying the Quetzalcoatlus has the same profile as the Mute Swan. When we scale the data for the Mute Swan to the Quetzalcoatlus’ 12 m wingspan the estimated mass for the Quetzalcoatlus is 1900 kg.

M2 = M1 (L2 / L1) 3

M2 = 15 kg (12.0 m/ 2.4 m)3

M2 = 1900 kg

Mute Swan in flight

While standing the Quetzalcoatlus stands as tall and has a similar profile to a giraffe. Male giraffes have a mass 1400 kg while the mass of the shorter female is only 700 kg.

The profile of a bat does not match up well with a pterosaur but nevertheless some readers may be interested in seeing this comparison since both animals fly using a stretch skin membrane as their wing. The Golden-crowned flying fox has a 1.5 m wingspan and 1.2 kg mass. Using this data and the same equation as what was used for the Mute Swan, the Quetzalcoatlus’ mass estimate is calculated to be 600 kg. We can continue on with numerous other comparisons but in the end they are all going to land between a half ton and two tons.

As scientists we are obligated to ponder the ‘magic’ of the Mesozoic era that allowed reptiles to not only fly but to further break the rules of aerodynamics to become the largest flying animals that ever existed.


The largest pterosaurs, the Quetzalcoatlus, had a wingspan greater than the recreational plane Cessna 172. The size of a Quetzalcoatlus could also be compared to an extremely large horse, yet it is questionable if the cold-blooded Quetzalcoatlus could produce the power of a horse. The Cessna flies because it is equipped with a 160 horsepower engine. If an object of this size requires 160 hp to fly yet does not have even one horse power, then any rational aerodynamic engineer would conclude that the Quetzalcoatlus is not even close to having the power required to fly in today’s atmospheric environment.

Once a scientific paradox is acknowledged, we can begin an objective investigation of the evidence in the hopes of finding the solution. In these baffling situations we need to be careful and systematic in sorting out the facts that we are certain to be true from the beliefs that we merely assumed to be true.

  1. The pterosaurs did fly. The wings cannot be fully evolved without actually serving the purpose of flying.

  2. Size does matter. The larger the animal the less relative strength and power it has and heavier flying objects require substantially more power than lighter flying objects. These realities explain why the small humming birds have such an abundance of power that they can hover while the largest birds can only soar. The largest birds seek out rising air thermals to allow them to fly which indicates that these birds are at their upper allowable weight limit for flight.

  3. The laws of physics are invariable over the billions of years of the Earth’s existence. Thus identical animals flying in either the past or present environments would generate the same amount of lift if those environments were essentially the same.

  4. The pterosaurs were not substantially lighter or stronger than similar flying animals of today. To suggest that there were super pterosaurs is to ignore the Theory of Evolution. Species evolve through the selection of the superior traits; given this forward process to better traits it is not possible for the physiology of modern animals to be substantially inferior to Mesozoic animals.

  5. The Earth’s gravitational field could not have substantially changed so as to reduce the weight of the pterosaurs. This hypothesis is discussed in the next chapter.

  6. It is possible for the density of the atmosphere to change. This may be important since air density is one of the variables in the flight equations. A lot can happen in millions of years and there is evidence that some changes in the Earth’s atmosphere did occur. This will be addressed in the later chapters.

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