The Solution to the Big Dinosaur Paradox
Summarizing what is known up to this point:
These are simple facts supported by undisputed evidence. Through the years, many researchers have recognized the paradox presented by the exceptional size of dinosaurs, so hypothesizing that the Earth must have somehow changed between then and now is not a new idea. The problem has been the lack of success in developing a feasible hypothesis as to how the Earth could have changed.
Ground Rules for Finding Science Solutions
Given that this paradox has gone unresolved for at least a couple centuries, it would be best to start with an open mind in listing all reasonable hypotheses rather than make the mistake of possibly excluding the correct answer without giving it a serious review. Also as we review these hypotheses we should not expect the answer to be obvious or else it would have been discovered a long time ago. But while having an open mind 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:
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.
Going with our first rule that it should be a simplistic 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 science 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. Often it is assumed that students understand the significance of a numerical answer. But sadly, many students make no connection between the numerical answer and the original question. Therefore, at the end of almost every mathematical investigation science educators should ask the key question: “Comparing this numerical result to another known numerical value, what does it mean?”
As an example of how numerical data facilitates making good decisions, let us take the example of determining if classes should be canceled because of winter weather conditions. To only be told that snow is forecasted is usually not enough information since the snowfall may easily range between about a half a centimeter to a half a meter or more. The projected amount of snowfall, that 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.
Returning to the present problem, in this investigation we will find that many of these hypotheses might produce a small change in the physical environment, but what is needed is a solution that can produce a large enough change to account for our scaling factor. So our first step is to determine the scaling factor so that we can make these logical comparisons.
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.
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.
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 greatest impact on a particular species. Ideally, we would like to be confident that these changes in the size of a species corresponded only to changes in the global physical environment.
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 dominate 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 dominate 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. Today birds, with their feathers for improved aerodynamic wings and thermal insulation, their unique and highly efficient respiratory system, and their elevated high-energy metabolism, now clearly rule the sky over the dragonflies. Today the larger a dragonfly grows, the slower it flies, and consequently the more likely it will become a tasty meal for a bird.
The horsetail, a plant, followed a similar path to that of the dragonfly. Like the dragonfly, the horsetail also appeared during the Carboniferous and during this time it grew exceptionally large, up to 30 meters. 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 so much of its life in the water that 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. Yet despite these problems, the crocodile does appear to be a species that has survived through time as the large dominate species of its environment. While the aquatic lifestyle hurts the crocodile’s ability to be a precise indicator of the changing scaling factor, additional research investigating the size of the crocodile through geological time may be helpful in confirming, or disproving, the ideas presented here.
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 different classifications of animals and yet they are both vertebrates. The point is that these groups share more common traits than differences. Most important to this discussion is that they are constructed from the same biological building materials.
Over the years paleontologists have presented hypotheses suggesting that both the dinosaurs and the pterosaurs were substantially lighter than what their size would indicate. Yet if not for the scaling confusion concerning these Mesozoic animals, it is unlikely that any of these types of arguments would have ever been taken seriously.
The bone material, common to all of these vertebrates, the hydroxyapatite crystals, has a set chemical bonding arrangement:
Chemically there would no difference in the bonding arrangement of each group's supporting bone material. The importance of this is that that the chemical bonding arrangement of the elements determines a materials 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.
While it is much harder to find fossils of muscle tissue, the arguments concerning muscle tissue are similar to the argument for bone: it is extremely unlikely that any of the major biological building materials of these different groups were substantially lighter, stronger, or in any other way either different or superior to modern animals.
To bring closure to the issue, the process of evolution does not ever de-evolve. If ever a species were to evolve a superior biological building material, the genetic code for that superior trait is passed forwarded to future generations. If the dinosaurs were to evolve a way for bones or muscles to be lighter or stronger, that genetic code for those superior traits would have been brought forward to the present time.
Clearly the size of the African elephant is almost entirely limited by physical constraints rather than biological constraints. Two indicators of this are that 1) elephants do not jump and 2) African elephants have less than 25% relative muscle strength. Because of scaling properties, volume being a function of L^3 while area is a function of L^2, the large size of elephants pushes the limit of how much stress their leg bones can tolerate without breaking. The force of impact after a fall is typically several times greater than the weight of the animal and so it is too dangerous for an elephant to fall any distance at all. The African elephant is typically between six and seven tons, and so because of scaling properties the relative strength of African elephants is less than the smaller Asian elephant. Unlike Asian elephants, African elephants have such low relative strength that they do not have the strength needed to do anything beyond just moving themselves and so they are not normally useful as work animals. But by pushing the physical limit of how large a terrestrial animal can be elephants are able to intimidate lions and other carnivores from attacking them.
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 elephant. The similarity can be seen in that for both groups 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 intimidate carnivores from attacking them. Comparing the size of the largest terrestrial vertebrates of the present and the past is one way of producing a scaling factor.
However it is still debatable over just how big were the biggest sauropods. Because of their large size, few of these individuals were every buried quickly enough to preserved everything in place. So it is rare to find the fossilized bones of such large creatures to be completely intact. Another problem is that the African elephant is actually not the largest ‘present-day’ terrestrial animal. Woolly Mammoths, a slightly larger relative of the African elephant, died out only four thousand years ago, and so Woolly Mammoths should be included as a present-day terrestrial animal.
Because of this fuzziness over what might be the largest of the dinosaur and what might be the largest elephant, we can do no better than produce a range of scaling factors. A comparison between the largest terrestrial creatures of each time, the sauropods compared to the elephants, would suggest a range of scaling factors between 2.5 and 3.5.
A more precise scaling factor is possible through a comparison of the tallest animals. Because the blood pressure applied to the heart is directly related to the height of column of blood above it, a comparison of the cardiovascular heights appears to be most reliable method for determining our scaling factor. The density of blood that produces the pressure and the strength of the heart and cardiovascular system needed to deal with these pressures would all appear to be constants.
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 neck is at its maximum height as determined by the stress placed on the cardiovascular system. The vertical 8.0 m heart-to-head difference of the Brachiosaurus compared to the vertical 2.5 m heart-to-head difference of the giraffe gives a scaling-factor of 3.2.
From the earlier discussion of scaling properties it was shown that both stress and pressure are equal to the same three variables: g D L where g is the acceleration due to gravity, D 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.
Where the M subscript represents Mesozoic and the P represents the present day. Setting the Mesozoic stress equal to the present-day stress gives:
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:
our new scaling-factor of 3.2 is equal to the inverse of (L-P / L-M). Finally inserting the present value for the acceleration due to gravity of 9.81 m/s^2 gives the Mesozoic acceleration due to gravity as 3.1 m/s^2. 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 at or 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. These stationary objects still feel an attraction to the center of the Earth. This attraction is the force due to gravity or the weight of the object.
The acceleration due to gravity g is the constant that is often used to calculate the weight of an object when the object's mass (inertia) is known:
The acceleration due to gravity g is defined as 9.807 m/s^2. The acceleration due to gravity is often referred to as the acceleration due to gravity constant even though 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:
where F is the force, G is the universal gravity constant equal to 6.67 E-11 N m^2/kg^2, M-1 and M-2 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:
This equation is the same as the first except that now M-E is the mass of the Earth, m is the mass of an object on the Earth's surface, and R 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:
Investigating the Possibilities
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 M-E, or the radius of the Earth R-E 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, let us take a moment to clarify 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 reaching our solution. 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 E8 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. The distance of the universe is possible infinite, if not nearly so. 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. But instead what we observe is uniformity between the known physics properties on Earth to what is observed throughout the universe. So no, the universal gravity constant G has not changed over time.
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.
Now 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 creator the evidence of that creator is eventually washed away by erosion. But on the Moon 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 and so it would be the same for the Earth. From this we can conclude that the amount of asteroid matter that has been added to the Earth for the last three billion years has been insignificant in comparison to the overall mass of the Earth. 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/ r^3, when the Moon was half as near as what 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 interesting in, the last 150 million years.
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 3.1 m/s^2, 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. This would produce an effective g that is acting on these objects. The author will define an effective g as the constant that gives us the effective weight of an object as a result of other forces in addition to gravity acting on an object.
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 a buoyancy force - decrease the effective weight of an object on the Earth's surface. At our present time these two forces are small forces such that they are usually neglected. Currently the maximum centrifugal force is only 0.34% of the weight and the present buoyancy force is only 0.13% of the weight (for a typical object with a density of 1000 kg/m^3).
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
where F-c 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
F-c 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 to the higher latitudes would feel substantially heavier than when they are near the equator.
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 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.
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 be 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. The Earth’s rotation causes the alignment of the stretching to lead the line drawn between the Earth and the Moon by 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. Scientists have measured the Moon to be moving away from the Earth at a rate of 3.8 cm per year thus confirming these statements.
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. Within the Earth-Moon system, as the Earth’s rotation slows down it’s lost of angular momentum is being transferred to the Moon. Eventually, billions of years from now, the Earth will stop rotating, locking 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 logical 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 and light gas 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/m^3 as compared to the density of water at 1000.0 kg/m^3. For any object submerged in a fluid, a high density fluid is more capable of providing effective buoyancy than a low density fluid.
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. It is the absence of the weight limitation that allows some of these aquatic species to display gigantism. It is the buoyancy of water that allows the whales, the largest animals of today, to grow so large. But 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:
where D-F is the density of the fluid, D-s 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 3.2 and an overall vertebrate density of 970 kg/m^3, the Earth's atmospheric density during the late Jurassic period can be calculated to be 670 kg/m^3. 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 be 2/3’s of 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. The thick atmosphere solution violates no property of science. It is the correct solution.
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Gravity and Size