4. The Solution to the Dinosaur Paradox

Summarizing what is known up to this point:

  1. Size matters. It is over 370 years since Galileo first explained how size affects the properties of objects. It is a major failure of science education that so few people are aware of Galileo’s Square Cube Law showing why size matters.

  2. Galileo's Square-Cube Law is especially important for biology. The Square-Cube Law becomes increasingly more important as we encounter objects of greater complexity. Advanced life forms such as highly evolved animals are at the top level of this complexity and so for these species size most definitely matters. This means that the highly evolved animals are the most limited in the range of size that they can have.

  3. On a global scale, numerous terrestrial Mesozoic era species far exceeded the maximum size obtainable by modern day species. We can either see this as a general comparison of the typical dinosaur or pterosaur to that of present terrestrial animals or by comparing the largest or tallest examples of Mesozoic species to the present; for example we can compare Brachiosaurus to that of both the present-day elephant and giraffe.

Ground Rules for Finding Scientific Solutions

The solution to the paradoxes of the large dinosaurs and the large flying pterosaurs must not be an obvious solution otherwise someone would have figured this out a long time ago. Therefore we should keep our mind open to all remotely reasonable hypotheses rather than make the mistake of excluding the correct answer without giving it a serious review. But while being open-minded in considering all possibilities, once a hypothesis looks promising it must then meet some extremely steep requirements before it can be accepted as the winning solution.

The correct solution to any scientific problem should fulfill the following general guidelines:

  1. It should be a simple solution. This is often stated as Occam's razor, "All things being equal, the simplest solution is most likely the correct solution". Whether it is F = m a, E = m c2 or the key point of evolution - that only the individuals who survive to reproduction maturity can pass on genetic code - almost all correct ideas in science are simple at their core. Some of my science friends complain that I make everything seem so simple, so for my fellow scientists, “Non sunt entia multiplicanda praeter necessitatem”.

  2. Preferably it should not require the discovery or invention of a new fundamental relationship of science. While occasionally it does happen that the solution is the discovery of the new relationship of science such as the E = m c2 just listed, this option is usually only available to the fundamental sciences and it should only be considered as a last resort. Furthermore, any hypothesis claiming a new law or property of science requires substantial additional evidence for its justification. A more common problem in science is that researchers in one science discipline will sometimes suggest hypotheses that violate known science properties or evidence of another science discipline. More needs to be done to improve cooperation between different science disciplines while working on these interdisciplinary problems.

  3. Most importantly, the solution needs to be supported by evidence. This is by far the most important point. Physical evidence is what separates meaningless opinions from what is actually good science. Before Galileo's time it was often believed that logic by itself was enough, and even today some people still make this mistake. But while logic is helpful in the formation of hypotheses it should never be considered foolproof in regards to understanding nature. Instead, once a scientific hypothesis is formed it needs to be investigated and tested. This is always possible because for a hypothesis to be a real scientific hypothesis there has to be something unique about it that allows the possibility of it being proved false. To state this in positive terms, when a good hypothesis is investigated, supporting evidence will be discovered rather than a fatal flaw.

Before beginning our search for the solution, we must resolve our minds to the fact that the only way for the Mesozoic species to be substantially larger than the present species is for there to have been a global change in their physical environment. While it is true that many earlier investigators have failed in discovering the solution to this paradox, our belief that we live in a rational reality must not waver. With confidence in science, we will proceed through a step-by-step process, investigating all reasonable hypotheses until the solution is found.

Scaling Factor

In seeking a simple solution, there should be just one phenomenon that affected all of the species on a global scale that would account for the larger size of Mesozoic animals. Furthermore, it is necessary to do more than just acknowledge that the Mesozoic animals were larger. In order to judge the value of different hypotheses, we need a number, in this case a scaling factor expressing the magnitude of the size of previous species compared to the present.

The role of mathematics in science is not always clearly understood so let us take a moment to clarify the necessity of numbers, or data, in helping us apply our reasoning skills to scientific questions. The primary reason why scientists collect data and work with numbers is for the purpose of making logical comparisons.

However the significance of using numerical values to process logic is often lost in science education classes. Usually a scientific problem involving calculations will end with a numerical answer. Many science educators will stop at this point based on the assumption that the students, the future scientists and engineers, understand the significance of the numerical answers; which is not often true. So to check understanding, science educators need to finish these investigations by asking “Comparing this numerical result to another known numerical value, what does it mean?”

snow storm Winter Weather

To show how numerical data facilitates making good decisions, let us take the example of determining if classes should be canceled because of winter weather conditions. Since snowfall may easily range between about a half a centimeter to a half a meter or more, being told that snow is forecast without numerical information is not enough information to make a wise decision. The projected amount of snowfall, a number with units, is needed so that it can be compared to a cutoff value established by officials as to what is reasonable for students to still be able to attend class. Throughout science it is often absolutely necessary to place information into numerical form so that these types of comparisons can be made to facilitate drawing the correct conclusions.

In trying to solve the paradox of how the dinosaurs grew so large, we need a scaling factor indicating how much larger the dinosaurs were than the present terrestrial animals. Otherwise we have no way of knowing if a given proposed hypothesis could be adequate for producing the required change in size. So our first task is to determine this scaling factor.

This scaling factor represents the size of Mesozoic species in comparison to the size of present-day species. But it would be unlikely that a catastrophic event occurred in the Earth's history that suddenly changed its physical environment and as we will see, the evidence does not support such a premise. It is much more likely that a gradual physical change took place that caused the size of these terrestrial species to change size over time. Thus, eventually, in the process of our research we should be able to generate a graph of the scaling factor as a function of geological time.

Graph Showing Scaling Factor

This is a project that can be addressed later. For now, for the sake of keeping the discussion simple in solving the immediate problem, let us focus on the peak of the dinosaur's gigantism that occurred about 150 million years ago. Our immediate goal is to determine the scaling factor that corresponds to approximately the middle of the Mesozoic era, the age of the dinosaurs.

There are several methods available for the purpose of determining the scaling factor between the species of the Mesozoic and present species. But as is usually the case in science, some of these methods will be much more precise than others. Because of this, it is unwise to just average the results of all of the methods. A better strategy is to first put some effort into determining which methods are more likely to be the most precise.

dragonfly Dragonfly

One method for producing a scaling factor would be to compare the present-day size of long-lived species such as the dragonfly, horsetail, and the crocodile, to their Mesozoic era size. While this may at first appear to be the most reliable method, it is not.

Our objective is to produce a scaling factor that represents the global change in the physical environment that had an effect on all terrestrial species. However the environment of a species that determines the size, form, and the behavior of a species, consists of both a physical and a biological environment, and often times it can be difficult determining which of these two environments is having the greater impact on a particular species. Ideally, we would like to be confident that these changes in the size of a species correspond only to changes in the global physical environment.

bird eating insect Bird Eating an Insect

Unfortunately, the biological environment, the interaction between species, tends to muddle our ability to draw clear conclusions on the relationship between the size of species and the changes in the global physical environment. Yet there is a way to remove the confusing biological factors and that is to choose large dominate species whose size would not be limited by their competitive interaction with other species.

However, neither the dragonfly nor the horsetail fulfills the requirement of being dominant terrestrial species throughout their existence.

When the dragonfly first appeared about 350 million years ago, they were gigantic in having 70 cm wingspans and they were the dominant predator of the sky. But by the early Mesozoic era the first pterosaurs appeared and in addition during the last third of the Mesozoic era birds had also appeared.

The respiratory system of insects is not efficient for larger animals. This was not a serious problem for dragonflies and other insects when they did not have competition from flying vertebrates. However once birds evolved they clearly ruled the skies over the dragonflies. Birds have a high temperature metabolism coupled with a highly efficient circulatory respiratory system that is even superior to the mammals. The high metabolism along with excellent aerodynamic form gives birds both greater speed and maneuverability than the flying insects. Dragonflies no longer grow so large because now the larger a dragonfly grows, the slower it flies, and consequently the more likely it will become a tasty meal for a bird.

crocodile Crocodile

The horsetail, a plant, followed a similar path to that of the dragonfly. Like the dragonfly, the horsetail also appeared during the Carboniferous period and during this time it grew exceptionally large, up to 30 meters tall. But since this time other plant species have evolved that have proved to be superior. With only a few exceptions, most variations of horsetails today grow no more than 1.5 m tall.

Unlike the dragonfly and the horsetail, the crocodile has its own problems in regards to being useful as a scaling tool. The crocodile spends much of its life in the water and so it may not be a good gauge for changes affecting terrestrial species in general. It may even be possible that the crocodile has made a behavior change in that in the past it may have hunted as much on the land as it did in the water. While the crocodile does appear to be a species that has survived through time as the large dominate species, its aquatic lifestyle compromises its usefulness as a precise indicator of the changing scaling factor.

Our next option is to attempt comparisons between different species that are still closely enough related that they are constructed of the same biological materials. Mammals and dinosaurs are of course different classifications and yet they are both vertebrates.

The bone material, common to all of these vertebrates, the hydroxyapatite crystals, has a set chemical bonding arrangement:

dinosaur fossil Dinosaur Bones
Ca10 [PO4]6 [OH]2

Chemically there would no difference in the bonding arrangement of each group's supporting bone material. The importance of this is that the chemical bonding arrangement of the elements determines a material's overall properties such as strength and density. Thus it is highly unlikely if not impossible for the bones of dinosaurs and pterosaurs to be substantially lighter or stronger than the bones of reptiles and mammals of today.

What do we know about dinosaur muscle, or any type of muscle for that matter? Unlike the bones, muscle has almost no chance of being fossilized, and yet amazingly, scientists have found a couple of small samples of fossilized dinosaur muscle. However it is not enough to tell us much.

Muscle strength is determined by its cross sectional area and whether it is a type I or type II muscle. For all of the non vegetarian readers, type I muscle is the dark meat of a chicken leg while the type II muscle is the white meat of the breast. Type I muscle has greater endurance while type II has greater strength. The two types of muscles also have a slightly different density.

Without knowing much about what type of muscles the dinosaurs had, the best we can do is assume that the overall strength, endurance, and density of dinosaurs were not that much different from modern day animals. It is reasonable to believe that we are no more than about 20 percent off by making these assumptions.

It is difficult to imagine how either the bone or muscle tissue of dinosaurs could have been significantly different from the same biological material of reptiles, birds, and mammals of today. Nevertheless, in their effort to explain away the large dinosaur paradox, some paleontologists have caused confusion by claiming that dinosaur bones could be hollow and still just as strong. More than just being illogical, the fossil evidence does not support either aspect of their claim.

Clearly the size of the African elephant is limited by the strength of its muscles and bones in dealing with its weight. Unlike the smaller Asian elephant, the relative strength of the large African is so low that it usually can not be used as a work animal. Similar to other large animals it can barely pick itself off the ground. Elephants may be the only terrestrial vertebrates that do not jump; the impact after even a short fall is enough to break their already overstressed legs.

elephant Mother Elephant and Calf

The clade known as sauropods that include the species Apatosaurus, Brachiosaurus, and Diplodocus would have possibly filled a similar environmental niche to that of the present-day African elephant. The similarity can be seen in that with both sauropods and the modern large animals there is a gap in size between the largest terrestrial animal and the remaining animals. That is, it could be that both the sauropods and the elephants are applying the same strategy of pushing the physical size limitation as a means of intimidating carnivores from attacking them. A comparison of the size of the sauropods to that of the elephants, would suggest a scaling factor ranging between 2.3 and 3.5.

However, a comparison of the tallest animals may produce a more precise means for producing a scaling factor. In the effort to reach the greatest height it is more apparent that each tall species cardiovascular system is being pressed to its limits.

Giraffes of the present-day and Brachiosaurus of the late Jurassic period each fill a similar environmental niche during their respected times. Each species evolved long necks for the purpose of extending their reach to foliage that was or is beyond the reach of other animals. For both species, maximizing the height of the mouth, in both cases by the extension of the neck, enhanced the species' survival. The extension of the legs also assisted in achieving the goal of extending the position of the mouth to the maximum height. Brachiosaurus is unique among most sauropods and dinosaurs in general, by having longer forward legs than rear legs. A reasonable hypothesis for the longer forward legs would be to extend the vertical reach of Brachiosaurus, thus enabling this species to reach the highest foliage. The conclusion from this is that for Brachiosaurus the head is at its maximum height as determined by the stress placed on the cardiovascular system.

The vertical 8.8 m heart-to-head difference of the Brachiosaurus compared to the vertical 2.8 m heart-to-head difference of the giraffe gives a scaling-factor of 3.1.

Effective Gravity

From the earlier discussion of scaling properties it was shown that both stress and pressure are equal to the same three variables: g ρ L where g is the acceleration due to gravity, ρ is the density of the biological material, and L is the size or height.

Based on the above discussion there can not be a substantial difference between the Mesozoic vertebrates and the present-day vertebrates concerning biological materials. Therefore both maximum stress variables can be set equal to each other.

sM = gM ρM LM


sP = gP ρP LP

Where the M subscript represents Mesozoic and the P represents the present day. Setting the Mesozoic stress equal to the present-day stress gives:

gM ρM LM = gP ρP LP

In all of these arguments the density of the biological materials may be referring to different things: blood, muscle, bone, or overall body density. Yet in all comparisons, the density of the Mesozoic vertebrates can not be substantially different from the present-day vertebrates so that:

ρM = ρP

gM LM = gP LP

gM = gP (LP / LM)

our new scaling-factor of 3.1 is equal to the inverse of (LP / LM). Finally inserting the present value for the acceleration due to gravity of 9.807 m/s2 gives the Mesozoic acceleration due to gravity as 3.2 m/s2. This value represents the lowest effective acceleration due to gravity that occurred during the Mesozoic era.

This would be a good time to review the acceleration due to gravity constant g. The acceleration due to gravity is the acceleration that an unrestricted object will have towards the center of the Earth if it is released near the surface of the Earth. Most of the objects surrounding us are restricted from falling because they are either resting directly on the Earth’s surface or resting on some other intermediate object. The force of attraction that an object feels towards the center of the Earth is known as the force due to gravity or the weight of the object.

The acceleration due to gravity g is the constant used to calculate the weight of an object when the object's mass is known:

F = g m

The acceleration due to gravity is not an elementary constant of physics but rather it is a derived constant. The g value comes from the general gravitational force equation that is then applied to the earth:

F = G M1 M2 / R2

where F is the force, G is the universal gravity constant equal to 6.67 E-11 N m2/kg2, M1 and M2 are the mass of object one and object two, and R is the distance between the center of the mass of the two objects. This is the more general equation for calculating the attractive force between any two objects. It applies to everything from objects on your desk to planets and stars throughout the universe. When applied to an object on the Earth's surface the equation becomes:

F = G ME M / R2

This equation is the same as the first except that now ME is the mass of the Earth, M is the mass of an object on the Earth's surface, and RE is the radius of the Earth. By comparing this equation to our initial equation F = g m, we see that the acceleration due to gravity g is:

g = G ME / R2

Investigating the Possibilities

stars2 Picture of Space

This gives us three possible variables that if one or more of these variables were to change then it could change the acceleration due to gravity. However, it is difficult to imagine how either the universal gravitational constant G, the mass of the Earth ME, or the radius of the Earth RE could have changed significantly between the Mesozoic era and the present. Both the physical evidence and simple calculations of what is physically possible confirm that none of these values could have significantly changed during the last hundred and fifty million years. But still, in the spirit of keeping an open mind, let us take a moment to investigate why none of these values could have changed by a significant amount over the last 150 million years.

There has actually been a suggestion that the universal gravity constant G could have changed. But as stated earlier, our preference would be to avoid hypothesizing a change in a fundamental property of science as a means of resolving a science paradox. Yet there may still be someone who wants to pursue this hypothesis on the argument that without a time machine how we can be certain that G has not changed. But the problem with this argument is that in effect we do have a ‘time machine’.

Light travels at a constant speed of 3.0 x 108 m/s. This is an exceptionally high speed in comparison to the speeds of most objects on Earth. Yet this is not a fast speed in regards to the time required to transverse the universe. It is unknown whether the size of the universe is finite or infinite, yet even if it is finite it is still so vast that from our limited prospective of distance it may just as well be infinite. Thus it requires thousands, millions, and even billions of years for light to reach us from the various parts of the universe, so that when we look at stars and galaxies, we are looking at both distant objects and looking back in time. A changing universal gravity constant G would create havoc in our attempts to understand the movements of stars and galaxies.

To some scientists, havoc in the distant universe is precisely what we are observing. Among other things, the rotational speeds of distant galaxies do not work out correctly unless there is more mass among the stars than what is observed. They have proposed the solution that Dark Matter may constitute more than four fifths of the universe. Yet other cosmologists have suggested more radical hypotheses to solve the gravitational problem. Perhaps over the course of billions of years the gravitational constant G can slowly change. Such an idea may not be as crazy as it first seems. For example, with special relativity Einstein showed that such ‘constants’ such as mass and time actually do change when the speed of an object approaches the speed of light. In truth, we do not even know what gravity really is and so how can we be certain that the gravitational constant G is forever constant.

Moon Moon

Nevertheless the author is going to rule that the changing gravitational constant hypothesis is not a realistic explanation for how the dinosaurs grew so large. Changes in the gravitational constant G that may, or may not, be possible over vast distances of billions of light years will not work to account for the huge change in the size of terrestrial animals that occurred on the Earth a mere hundred million years ago. Occam's razor directs us to toss out the changing gravitational constant hypothesis; for if we try to use the changing gravitational constant hypothesis to account for the large size of the dinosaurs this in itself creates so many unsolvable problems that it completely muddles our understanding of the laws of reality.

The next possible variable is the total mass of the Earth. About 4.6 billion years ago, in the earliest stages of the birth of the Earth, the mass of the Earth grew rapidly as it, along with the other planets, swept up the debris of the early solar system. However relatively quickly, within a matter of the first several million years, almost every possible collision between objects that could have occurred, would and did occur. So for all practical purposes, the mass of the Earth has been constant for billions of years. The physical evidence supporting this last statement comes from the study of the craters left on the planets and moons, in particular we can learn about the Earth by studying the Moon.

Earth & Moon Earth and Moon

The Earth and Moon revolve around their common center of mass as they occupy the same orbit around the Sun, and as these two objects orbit around the Sun they are occasionally bombarded with asteroids. The small asteroids that approach the Earth usually burn up in the atmosphere and even when a large meteorite reaches the surface to create a large crater the evidence of that crater is eventually washed away by erosion. Consequently there is only a small partial record of these meteorite impacts on the Earth. But on the Moon, where there is no atmosphere or oceans, every meteorite impact leaves its mark for nearly all of time. So we can study the permanent crater record left on the Moon with the understanding that the rate of asteroid impacts on the Moon will correspond to the rate of asteroid impacts on the Earth.

The vast majority of craters on the Moon are over three billion years old. Thus we can conclude that there have been relative few meteorite impacts hitting either the Moon or the Earth over the last three billion years. Likewise, for all practical purposes, we conclude that the mass of Earth has been constant for at least three billion years. Thus the mass of the Earth has not changed by a significant amount between the Mesozoic era and the present.

The final variable in the equation is the radius of the Earth. If the Earth were to compact down by a significant amount then this would reduce the radius while increasing the acceleration due to gravity g. The Earth does have a means of compacting because the Moon and the Sun both pull on the Earth causing the tides. This tidal pull raises and lowers the ground in addition to the raising and lowering of the ocean waters. When it does this there is the possibility of a reshuffling of the rocks within the interior of the Earth. The planets that experience the greatest tidal forces correspond to being the planets that have the greatest density, an indication of this compacting of the Earth. With the Moon so close exerting a strong tidal force on the Earth, the Earth has the highest density of all of the planets within our solar system.

But while there is evidence that compacting has taken place, the majority of this compacting took place during the earliest stage of the Earth's development. Billions of years ago the Moon's tidal force that it applied to the Earth was considerably greater than what it is today. There is both physical evidence of this and it is logical from the fact that it is know that the Moon is presently moving away from us. So billions of years ago the Moon would have been much closer to the Earth creating substantially larger tides. Since the tidal force is a function of 1/ r3, when the Moon was half as far as it is today the tidal force was eight times greater. So the compacting of the Earth would have been much greater billions of years ago, while comparatively little compacting would have taken place during the time that we are interested in, the last 150 million years.

diagram of density layers

Another reason why a changing radius is not the answer is because it is not physically possible to compact the Earth enough to significantly effect the acceleration due to gravity g before coming to the problem of density inversion. Calculations show that if we tried to go backwards in time, inflating the Earth to produce an acceleration due to gravity of 3.2 m/s2, the average density of the Earth would be less than water. This is not a rational possibility. Unless the movement of material is blocked, the layers of material must increase in density as we approach the center of the Earth.

It may appear that we have gone through all possible hypotheses without finding the solution. Yet fortunately, there are still a couple more possibilities available when we realize that there could be other forces involved besides the gravitational force (weight). These forces could be pushing up on an object so as to diminish the strength of the gravitational force. A force pushing up on an object could produce an effective g that is less than the present gravitational constant.

On the Earth's surface there are two other forces besides the gravitational force that act on an object so as to reduce the net force acting downward on the object. The two forces - the pseudo-centrifugal force due to the Earth's rotation and the buoyancy force - decrease the effective weight of an object on the Earth's surface. At our present time the maximum centrifugal force is only 0.34% of the weight and for an object with a density of 1000 kg/m3 the buoyancy force is only 0.13% of the weight. Because these forces are so low their effect is usually insignificant for most calculations as so these forces are usually neglected. But if one of these forces were much stronger in the past then perhaps they could reduce the Earth's effective gravity.

Centrifugal Force Forces Acting on an Object, such as a Person, on a Rotating Earth

The pseudo-centrifugal force is due to the Earth's rotation. Anywhere on the surface of the Earth, with the exception of the north and south poles, the spinning of the Earth makes a weak attempt to throw objects off of its surface. The magnitude of this force is at its greatest at the equator and it is a function of the speed of the Earth's rotation. This centrifugal force is calculated as

Fc = M v2 / R

where Fc is the centrifugal force, M is the mass of an object, v is the object's speed, and R is the distance to the rotational axis. This equation can also be written as

Fc = m r w2

Fc is the centrifugal force, m is the mass of an object, R is the distance to the rotational axis, w is the angular speed of the Earth’s rotation. For computational purposes the mass of the Earth is 5.98 E24 kg, the Earth’s average radius is 6.38 E6 m, and the present angular speed of the Earth’s rotation is 7.272 E-5 rad/s.

The pseudo-centrifugal force actually points away from the rotational axis rather than directly upward against gravity, so that the greatest effect of the centrifugal force is at the equator while there is no effect at the poles. In our present world the centrifugal force is so small that this change is unnoticeable, but if we are to believe that the centrifugal force is responsible for reducing the effective weight of dinosaurs the effect would be dramatic. Dinosaurs migrating from the lower to the higher latitudes would feel substantially heavier when they are farther away from the equator.

Effective weight changes with latitude Vector Addition of Forces Acting on an Object at Different Latitudes

During the Mesozoic era the Earth did spin slightly faster. Physical evidence comes from coral samples and tidal rhythmites showing that 150 million years ago during the Mesozoic era the sidereal year was 378 days long compared to the present 365.25 days while each solar day was only 23 hours 11 minutes long compared to the present 24 hour day. The tidal rhythmites are sedimentary rocks from an ancient shore line. The alternating dark and light silty deposits can be counted to determine the number of days in a year. The physical evidence confirms the logical conclusion that the tidal forces are slowing down the spinning of the Earth.

The point of all of this is that scientists know quite well how fast the Earth was rotating during the Mesozoic era. We are certain that the Earth had a rotational speed w of 7.526 E-5 rad/s, 150 million years ago, the approximate peak of the dinosaur gigantism. Yet this faster rotation would still just produce an insignificant reduction of the effective gravity of the Mesozoic era. The maximum centrifugal force during the Mesozoic was at best only 0.37% of the gravitational force at the equator while treating the Earth as a perfect sphere.

Complicating the issue a little bit is that the centrifugal force causes the Earth to bulge out slightly at the equator so that the Earth's radius at the equator is slightly greater than the Earth's radius at the poles. While this bulging further increases the effect of the centrifugal force, the total effect is still less than one percent of the gravitational force. Therefore the centrifugal force would not have been strong enough to be a significant factor for reducing the effective g of the Mesozoic era.

Diagram of Earth's Tidal Bulge Tidal Lag: the Moon slows down the rotation of the Earth while simultaneously the Moon slowly moves away from the Earth.

Let us take a moment to understand why the Earth's rotation is slowing down.

Most people are familiar with tidal forces in connection to the twice daily high ocean tides that are observed at most beaches around the world. But in addition to the movement of the water, the land or Earth itself bulges up twice a day in response to the pull of the Moon. The reason there are two tidal bulges is because while the Moon is applying a gravitational pull towards itself, a pseudo-centrifugal force is being applied to opposite side. This pseudo-centrifugal force is a result of the Earth and Moon completing their once a month dance around their combined center of mass. It is also necessary to have both the gravitational force and the equal yet opposite pseudo-centrifugal force acting against each other to fulfill Newton's physics requirement that all forces are equal and opposite.

The gravitational pull of the Moon, causes the Earth to stretch almost in line with the Moon. The key phrase here is ‘almost in line’ rather than being directly in line. This stretching of the Earth does not line up directly with the Moon because the Earth is rotating. The Earth's rotation causes the alignment of the stretching to lead the line drawn between the Earth and the Moon by about three degrees. Because the axis of this tidal bulge is not directly in line with the Moon, a torque is being applied to the Earth that is slowing down its rotation.

The tidal lag also causes a forward force to be applied to the Moon. This pulls the Moon into a higher orbit as it circles the Earth, or in other words, the Moon is moving away from the Earth. The Moon is moving away from the Earth at a speed of 3.8 cm per year.

Hot Air Balloons Hot Air Balloons Floating in Front of Pikes Peak Mountain, Colorado

The slowing down of the Earth along with the movement of the Moon away from the Earth is a display of another fundamental physics property: conservation of angular momentum. The Earth's angular momentum is being transferred to the Earth-Moon system as the Earth's rotation slows down and the Moon moves away from the Earth. Eventually, billions of years from now, the Earth will lock one side to always face the Moon. At the same time the Moon will stop its movement away from the Earth.

It was necessary to give the arguments that eliminate the incorrect hypotheses in order to prepare the reader for the solution.

The buoyancy force is best described by Archimedes' principle that states that when an object is partially or fully submerged in a fluid, an upward buoyancy force lifts up on the submerged object that exactly equals the weight of the fluid displaced. The buoyancy force is often associated with objects either submerged in water or floating on the water's surface. But buoyancy also applies to air. It is what gives a lifting force to hot air balloons. The main difference in the buoyancy effect provided by these two fluids is the amount of fluid volume that needs to be displaced to achieve flotation. This is because sea level air has a low density of only 1.29 kg/m3 as compared to the density of water at 1000.0 kg/m3. For any object submerged in a fluid, a high density fluid is more capable of providing effective buoyancy than a low density fluid.

Buoyancy force vector The upper fluid buoyancy force reduces the normal force that is the effective weight felt by the dinosaur.

For terrestrial vertebrates, it is the net force produced by their weight that often limits their size. But this is not true for species that exist in the water. For the latter species it is not their weight but rather other factors, such as the availability of food that might limit the size of these species. Without the weight limitation some of these aquatic species grow to display gigantism. It is the buoyancy of water that allows the whales, the largest animals of today, to grow so large. Without this buoyancy to counteract gravity, the poor whale that finds itself stuck on a beach is soon having its bones broken from its own weight.

To produce an effective buoyancy force on dinosaurs the Earth's atmosphere would have to be thick enough to have a density comparable to the density of water. By summing the forces acting on a typical dinosaur such as a Brachiosaurus the density of the necessary atmosphere is calculated as:

ρF = ρS (1 - 1/S.F.)

Derivation of Fluid Density Equation

Fb + FN = Fg

V ρF g + m gef = m g

V ρF g + V ρS gef = V ρS g

ρF g + ρS gef = ρS g

F - ρS)g = - ρS gef

ρF = ρS - ρS (gef/g)

ρF = ρS (1 - 1/S.F.)

where ρF is the density of the fluid, ρs is the density of the substance submerged in the fluid such as the dinosaur, and S.F is the scaling factor. Inserting into this equation a scaling factor of 3.1 and an overall vertebrate density of 980 kg/m3, the Earth's atmospheric density during the late Jurassic period can be calculated to be 660 kg/m3. This says that to produce the necessary buoyancy so that the dinosaurs could grow to their exceptional size, the density of the Earth's air near the Earth's surface would need to about two thirds (2/3) the density of water.

It may be difficult for some people to imagine how the Earth could have had such a dense atmosphere. But nevertheless, the wonders of our reality often exceed the limitations of many people's imagination. Esker’s Thick Atmosphere Theory violates no property of science. It is the correct solution.


How Dinosaurs "Evaded" the Square-Cube Law: David Esker's Answer

External Links / References

How Do We Know What is True?

The Role of Mathematics in Science

Dominate Species

Biological Materials



Investigating the Possibilities

Centrifugal Force, Non Spherical Earth, and Effective Weight

Moment of Inertia and Conservation of Angular Momentum


Comments, Questions, and Answers

Selected comments and questions are given with the permission of the parties involved.


I read your Dinosaur Theory with fascination. I appreciate your writing style of building concept upon concept, reinforcing, moving forward, reinforcing, moving forward. It makes your arguments very clear and understandable. Only once did you seem to make a claim without an abundance of supporting explanation. That was at the critical juncture of stating:

“To produce an effective buoyancy force on dinosaurs the Earth's atmosphere would have to be thick enough to have a density comparable to the density of water.”

Reading this, I wanted to ask, “What is ‘an effective buoyancy force’? – What determines the minimum buoyancy needed and the maximum allowable? Is ‘effective’ half way between? What does 'effective' actually mean?”

You go on to say:

“By summing the forces acting on a typical dinosaur such as a Brachiosaurus the density of the necessary atmosphere is calculated as: ”

ρF = ρS (1 - 1/S.F.)…”

and then you explain the symbols. However, the equation seems to come out of thin air. In order for your readers to embrace your conclusions fully it would be helpful to understand the derivation of the equation.

Thank you for your fascinating and thought-provoking theory. I am definitely intrigued, and do hope you can help me past this small, critical step to reach solid confidence your conclusion.

April 2017

Hi Jim,

Wow! Thanks you for your excellent questions and comments. For your first question regarding effective buoyancy I give the following response.

When we weigh an object on a scale we usually do not think about the fact that the atmosphere is providing a small upward buoyancy force on the object. Yet because of the atmosphere the reading on the scale is not the precise weight of the object. We neglect this fact because the reading is typically off by no more than about .1 or .2% at the most. To get a more precise weight of an object we should place both the scale and the object in a vacuum.

Yet even though we neglect to account for the buoyancy force provided by the atmosphere, most science minded people are aware that they should not neglect the buoyancy force if they are weighing an object while it is submerged in water. If they weigh an object that is capable of sinking in water then that object could weigh considerable less than its usual weight. The reason this is so is because the water provides an upward buoyancy force on the object that reduces the weight considerably.

So what I mean by effective is that the buoyancy force is great enough that we cannot neglect its effect without introducing considerable error. If the atmosphere's density was a quarter, half, or three fourths as dense as water then certainly we could no longer neglect its effect without introducing considerable error.

Your second question asked why I left out the derivation of the ρF = ρS (1 - 1/S.F.)…” equation.

I left out the derivation just to same myself time and effort, but now that you have made me aware that some readers would like to see this I have now added the derivation to the website page.

P. S. After posting the derivation, Jim informed me that he also worked out the derivation. His derivation is actually better than mine since it is shorter and so more directly to the point. Here is his derivation of the ρF = ρS (1 - 1/S.F) equation:

It seemed to me what should be “effective” is the density of the dinosaur, which is simply biological density minus fluid density. So, I set out to re-derive your equation from that perspective.

First, I decided the objective was to show that the stress on the leg bone cross-sectional area of a fluid-supported dinosaur should be equal to the stress of a modern animal of smaller size, with negligible support, where the scaling factor S is the ratio of heights (I would suggest to the centers of gravity of the two animals). Since area increases as S2 and volume and weight increase as S3 stress, which is Wt./Area, increases as S3/S2, which is to say, stress increases directly with S.

Ap stand for present animal leg bone cross-sectional area, and
S2 x Ap stand for Mesozoic animal leg bone cross-sectional area,
ρb for biological density,
ρf for fluid density,
ρeff for “effective density”, = (ρb - ρf),
Vp for present volume, and
S3 x Vp for Mesozoic volume,
I made the Mesozoic leg bone stress equal to present leg bone stress as follows
(S3 x Vp x ρeff )/ (S2 x Ap) = (Vp x ρb )/Ap

This simplifies to:
S (ρb - ρf) = ρb,
which in turn simplifies to:
ρf = ρb (1-1/S),
confirming your final equation.