7. Hell, Heaven and Earth
Part 1 - Hell: Source of the Earth's Internal Heat

The Thick Atmosphere Solution resolves several major science paradoxes: 1) how the dinosaurs grew so large, 2) how it was possible for the pterosaurs to fly, 3) a logical explanation for the dinosaurs' unique shape of having much larger rear legs and a large muscular tail, and 4) how the Earth had such a uniform temperature from the equator to the poles throughout the Mesozoic era. Yet to be a complete scientific theory there needs to also be a logical explanation and supporting evidence showing how this thick atmosphere came into being and what occurred such that the Earth's present atmosphere is much thinner than what it was before.

This chapter is the first of the three chapters, titled Hell, Heaven, and Earth, presenting the evidence showing how the Earth's atmosphere and oceans came into being and how the Earth evolved into its present state. The series starts by investigating the source of the Earth's interior heat; this heat is the driving force that first placed an atmosphere on the Earth's surface.

Mankind has long recognized that as one digs deeper into the Earth the ground becomes warmer. Volcanic eruptions also give evidence that the Earth has a hot interior. Finally, the seismic waves passing through the Earth's center give convincing indirect evidence that much of the core is melted. But while there is scientific agreement that the Earth's interior is extremely warm, geologists and other scientists do not agree as to what is the source of this interior thermal energy

In discussing this topic it is customary for geologists to list the following possibilities

1. Leftover primordial heat
2. Radioactivity heating
3. Tidal heating

Resolving the question of what is the source of the Earth's internal heat is necessary to understand how terrestrial planets generate their atmosphere. This in turn clarifies why it is more logical for the Earth to have a thicker atmosphere. To get started there first needs to be a discussion of the mathematics that will be helpful in determining the source of the Earth's internal heat.

In regards to the proposed hypotheses concerning the Earth's interior heat, an improved understanding of exponential decay will clarify the weaknesses of the leftover primordial and the radioactive heating ideas. The general equation for exponential decay is given as:

In this equation N would represent the absolute temperature of something, the number of radioactive isotopes, or some other quantity. The t is the time and k is the decay constant. If we write this equation specifically for the event of an initially hot object cooling off to room temperature, the starting equation is:

where T represents the absolute temperature and T_R is the room temperature with both of these values given in Kelvin. Solving this equation for the temperature as a function of time gives us:

In this equation T_i is the initial hot temperature while T is the temperature of the object at any time after that.

While this equation may look a little complicated it can be simplified considerably by understanding the e^{- k t} component. When t = 0, then e^{- k t} = 1, and so T = T_i: marked as point 1 on the graph. At the other extreme when t = infinity, then e^{- k t} = 0, and so T = T_R: marked as point 2 on the graph. Now that we know the end points 1 and 2 the equation gives, and the curve shows, the temperature for all the time between zero and near infinity.

To get a feel for what this graph is telling us, let us examine the cooling of a hot cup of coffee. Imagine that after pouring our cup of coffee we forget to drink it and so it sits on our desk as it cools to room temperature. We can see from the graph that the temperature of the beverage falls quickly at the beginning but after a while the decline slows as it approaches the room’s temperature. Eventually the temperature of the beverage matches the temperature of the room.

But just how long is eventually? That is, how long does it actually take for our beverage to cool to room temperature? According to the equation, an infinite amount of time is required to complete an exponential decay event and a mathematician would graph this equation showing that the decay curve never quite reaches the asymptote. Yet almost everyone knows from personal experience that if we neglect to drink our hot beverage that it will cool to room temperature in about an hour.

The reason for this disagreement as to whether the decay process is finite or infinite comes from the subtle yet important difference between real events and conceptual models of real events. While conceptual models of real events are usually helpful in understanding real events they are nevertheless not the same as the real event. The model is a simplification that gives only the main idea(s) of the real event. Yet in this process of simplification some information is lost. Most of the time the lost information is inconsequential, yet occasionally the missing information is critical to answering our questions.

In regards to the real decay processes, when we are looking at the overall statistical result of a huge sampling then the equation accurately describes the main features of the real event. But when we look closer the real decay process shows sporadic moment that only follows the decay equation as a statistical average. Thus, while the mathematicians claim that the decay curve never touches the asymptote is true for the conceptual model it is not true for real decay events.

The primary reason for the difference between the mathematical model and reality is a result of the inherent fuzziness of reality. This fuzziness is usually known as random, indeterminate, or statistical error. So for example when we say that the room temperature is twenty degrees Celsius, what we really mean is that the room temperature is around twenty degrees Celsius rather than it being precisely twenty degrees Celsius. This is because it is simply not possible to hold the value of a real object to be absolutely steady.

The same thing goes for the temperature of the beverage as it cools. While the object is cooling the actual temperature of the object is very close to yet still not precisely following the theoretical exponential decay curve. Because the actual temperature is fluctuating the actual temperature is often found slightly above or below the ideal curve. This sporadic behavior of the decaying temperature and the room reference temperature becomes important once the two temperatures are nearly the same. At this point the two fuzzy lines begin overlapping each other rather than continuously approach each other as our simple mathematical model would suggest. Once the two temperature plots start overlapping each other, it become impossible to tell one from the other and so at this point, for all practical purposes, the exponential decay process is completed.

To summarize the key concept of this section: real exponential decay events do not go on indefinitely but rather they are complete after a finite amount of time. Granted, depending on the nature of the decay event there can be some debate on when we should call it over, but there can be no debate on whether real decay events go on forever or not. Unlike the equation, real exponential decay events are completed after a set amount of time.

During the 17^th and 18^th centuries, scholars and scientists were considering different methods that they could use to determine the age of the Earth. Since these early ideas were biased by the human perspective of time, the first dates that were given were exceptionally low. But as different methods were tested and the evidence presented and debated the accepted age of the Earth continued to increase. Finally in the 20^th century radioactive isotope dating methods proved to be the most reliable means for determining absolute geological time. Radioactive dating showed conclusively that the oldest rocks on the Earth are billions of years old and that most meteorites are 4.6 billion years old. Based on this evidence geologists no longer debate the fact that the Earth is 4.6 billion years old.

While determining the age of the Earth is a wonderful science achievement, there has been some negative fallout from these historical debates. Before some of these ideas were proven incorrect they became embedded as geology dogma and now these incorrect ideas are still confusing many geologists. At the beginning of these debates, leftover primordial heat was the only explanation geologists could think of as being the source of the Earth's internal heat. This is an idea that should have been discarded once it was known that the Earth is 4.6 billion years old.

Lord Kelvin was awarded many honors during his young active years and he was so highly respected during his prime that many if not most of his ideas went unchallenged by the other scientists of his day. But by his waning years, so many of his mistakes had become apparent that some of his contemporaries began referring to him as 'a silly old idiot'. The test of time has given him the final dubious distinction of being the scientist that introduced more errors into science than possibly anyone else in history.

Around 1897 Lord Kelvin felt that he could determine the age of the Earth by calculating its cooling rate. He postulated that the source of the Earth’s interior heat was left over thermal energy as the Earth cooled from an initial molten state. Based on the thermal conductivity of rock material he calculated the age of the Earth to be roughly 40 million years old. This answer proved to be horribly wrong primarily because Lord Kelvin’s initial assumption regarding primordial heat was wrong.

Even though the age of the Earth is no longer considered controversial among scientists, the idea that the Earth started in a molten state and that some of that primordial thermal energy could still exist is now a part of geology dogma. While the idea that the Earth started as a molten state may or may not be true, calculations show that it is impossible for there to be any left over primordial heat still within the Earth. As it was just discussed in the earlier section, real exponential decay events have a finite life and so they do not go on forever, and of course the cooling of the Earth would have been a real event.

While Kelvin was wrong in a couple of his assumptions, there is something that can be salvaged from his work in that he did produce a rough estimate of how long it would take for the Earth to cool from a molten state. Kelvin made his calculations based on the idea that the transfer of heat from the deep interior to the surface was entirely due to the conductivity of the rock material. Geologists now know that convection also plays a role and that this convection process actually speeds up the cooling of the Earth. In addition, there have been more recent calculations regarding the cooling of the Earth confirming the fact that Lord Kelvin’s estimate that a molten Earth would cool in 40 million years is ‘in the ball park’.

Because we now know that the Earth is 4.6 billion years old it is simple math to show that the age of the Earth is about a hundred times older than the amount of time needed to cool. This tells us that if the Earth started off in a hot molten state, and there were no other source of heat available, then the Earth would have completely cooled more than 4.5 billion years ago.

Until radioactivity was discovered geologists had only one explanation of what was the source of the Earth’s internal heat: leftover primordial heat. But ever since radioactivity decay methods placed the age of the Earth at 4.6 billion years, much too long of time for primordial heat, many geologists realized that they no longer had an answer for what is the source of the Earth’s internal heat. The popularity of the radioactivity heating hypothesis came more out of the need to fill this blank rather than through any merits of the hypothesis.

It is hardly scientific to simply state that radioactive decay produces some heat and then conclude the radioactive material buried within the Earth must be the source of the Earth’s interior heat. We need data to determine if a hypothesis is reasonable. However appropriate data has not always been readily available. One problem is most experts of radioactivity tend to be strongly biased in favor of their interest in radioactivity. Even now the majority of technical reports favoring radioactive heating are nothing more than long-winded circular arguments. These reports ignore the all encompassing fact that there is no way of conclusively determining the amount of radioactive material that lies deep within the Earth.

Radioisotopes	Half-life (years)
Cobalt 60	5.26
Cesium 137	30.2
Radium 226	1600
Carbon 14	5730
Plutonium 239	2.41 E4
Uranium 235	7.04 E8
Potassium 40	1.25 E9
Uranium 238	4.46 E9
Thorium 232	1.41 E10

The best we can do is guess as to what exists deep within the Earth. We can be rather certain the temperature and pressure increases with depth. We know material becomes progressively denser as we approach the center. We know from world-wide seismic readings of earthquakes that part of the core is liquid. In addition, there are a few other things that we can figure out with a reasonable amount of certainty. But when geologists give percentages for the amount of iron, nickel, and cobalt that might exist within the Earth, these are nothing more than educated guesses. There is no ground truth that can confirm these figures and there is no sampling method that can tell us how much radioactive material is deep within the Earth.

Without a means of directly falsifying the radioactivity heating hypothesis, it is debatable if the radioactivity heating hypothesis should even qualify as a scientific hypothesis. But even without the ground truth there is still plenty of indirect evidence that weighs in against the radioactivity heating hypothesis. Let us start with a refresher on radioactivity since this will help in determining whether radioactivity should be considered a primordial heat source or a constant heat source.

When we graph the activity of a radioactive isotope such as plutonium 239 on a time scale with increments of ten thousand years we see the usual radioactivity decay curve. But if we graph the plutonium 239 on a time scale with increments of a billions of years, a time scale appropriate for the age of the Earth, our plutonium 239 activity plots as a vertical line. In fact, all of the radioactive isotopes with a half-life less than about hundred thousand years are going to graph as a vertical line when the time increments are billions of years, and so all of these isotopes would have been producing primordial heat. All radioactive isotopes can be thought of as either short-life isotopes that produced primordial heat or as extremely long-life isotopes that are producing nearly constant heat.

We can get a feel for how much thermal power these long-life isotopes can produce by doing a thought experiment. Imagine we have ten different radioactive materials and each sample is capable of releasing a hundred joules of energy. The first radioactive sample contains an isotope with a half-life of ten years. The next sample has a half-life of a hundred years, the following one a thousand years, and so forth all the way up to the last one having a half-life of ten billion years. Now, we start all of these samples decaying at the same time and then plot the thermal power output over billions of years.

The power output of each sample will closely match the activity level of each sample. So, the plot of an isotope’s thermal power output will look almost identical to its exponential decay activity plot. Similar to when we plotted plutonium 239, when we plot the power outputs of the short-life isotopes on the graph marked out in billions of years we get nothing but a vertical line.

What is not revealed by these vertical line graphs is how the isotopes with the shortest half-lives give up their energy much quicker than the longer half-life isotopes. That is, sample B with half-life of a hundred years will take ten times longer to give up its energy than the sample A with a half-life of ten years. Since both samples have the same amount of starting energy the power output of sample B will be ten times less than the power output of sample A as they transition through the same segments of their respective exponential decay lives. If we carry this line of thought out to comparing the end samples, the final sample J that has a ten billion year half-life will have a billion times less power output than sample A.

Plotting the power output of all of our isotopes on a graph will produce an 'L' shaped curve. The short-life isotopes produce a burst of power at the start and then the long-life isotopes will produce millions to billions of times less power for the remainder of the time. The reason the power output of the long-life isotopes is so low is they are spreading out the release of their energy over a period of billions of years.

But how can this thought experiment, using the same amount of each isotope, be meaningful when one radioactive isotope can thousands or even millions of times more abundant than another? Well, it is extremely meaningful because if a single isotope is only a million times more abundance than all of the other isotopes, the plot of the total power output would still look the same as the plot from the thought experiment. To change the shape of this graph so the radioactivity heating hypothesis might be a feasible hypothesis, one of the isotopes with a half-life of a several billion years must be hundreds of millions if not billions of times more abundant than all the other radioactive isotopes.

This, by itself, is a rather steep requirement, but when we look at the actual isotopes with long half-lives there are addition serious problems. Heavy radioactive isotopes can not be more concentrated than the highest natural ore concentrations that are already observed in the Earth's crust. Geological evidence shows that when these isotopes are too highly concentrated they initiate a rapid burn through fission decay. So the amount of thorium 232, uranium 235, or uranium 238 can not be significantly greater than the present quantities and so these heavy long life isotopes are eliminated.

Once the three heavy long-life isotopes are removed from the list of possibilities all that is left is potassium 40 with a half-life of 1.25 billion years. Potassium is already 2.1% of the elements found in the Earth's crust while potassium 40 is only 0.01% naturally abundant among the potassium isotopes. Even if we imagined that the entire interior of the Earth is filled with potassium the thermal output of this radioactive material would still be millions if not billions of times less than what is required.

As a final check on the logic we can look at the composition of the asteroids as indicators of what might exist within the interior of the Earth. Astronomers have found that 75% of the asteroids are type C-carbonaceous asteroids, 15% are S-silicate asteroids, and the remaining 10% are mostly M-metal asteroids consisting of mainly nickel and iron. Astronomers generally do not find uranium or potassium asteroids. So we can conclude that radioactivity is not the source of the Earth's internal heat.

Every intelligent person knows that the Earth rotates on its axis, yet how many of us are aware that the spinning of this massive sphere represents a tremendous amount of kinetic energy. In the Solution chapter a short explanation was given regarding the slowing down of the Earth’s rotation. But if the Earth is slowing down then its kinetic energy is decreasing. We should be asking ourselves, “Where is this energy going?”

Any object in motion has kinetic energy. For example a car moving down the road has translational kinetic energy. When we apply our breaks to slow down the car that kinetic energy is converted to thermal energy that warms the car’s breaks. A moving object's translational kinetic energy is calculated as ½ m v² where m is the mass of the object and v is the speed of the object.

In addition to objects having kinetic energy as they change their location, objects can also have kinetic energy by just rotating regardless if the object is changing its location or not. Similar to the translation energy example with the car, it takes energy to get a wheel rotating, and later the rotating wheel’s kinetic energy becomes thermal energy when the wheel is brought to a stop. The equation of the rotational kinetic energy looks similar to the equation for translational kinetic energy

As the Earth moves through space it is both revolving around the Sun once a year - translation energy - and it is rotating on its axis once a day - rotational energy. The Earth’s translational and rotational kinetic energies have been nearly constant over the millions and billions of years of the Earth’s existence. But while nearly constant, the rotational speed of the Earth is nevertheless decreasing due to the tidal forces being applied to it. Thermal energy is being generated within the Earth as the Earth is being slowed down as a result of the Moon and Sun applying a gravitational torque to the Earth.

We need to determine the Earth’s values for the variables of the rotational kinetic energy equation ½ I w² so that we can then calculate the change in kinetic energy. For a solid homogenous sphere we can calculate the moment of inertia with the equation I = 2/5 m R² where m is the mass of the object and R is its radius. If we were to assume that the Earth is homogenous the moment of inertial for the Earth would be calculated as 9.7 E37 kg-m².

However the Earth is actually denser at its center rather than being a homogenous sphere and so this reduces its rotational moment of inertia. By approximating the density profile and applying calculus a much better approximation of the Earth's moment of inertia is calculated as 8.0 E37 kg-m². This value agrees with the results generated from NASA's data recording of the Earth's movement and rotation.

Next we need the current angular speed of the Earth w and the angular speed of the Earth at an earlier time. The current angular rotation of the Earth w is determined from the current period of the Earth's rotation T being 24.0 hours. We can obtain our initial angular speed by going back 150 million years to when the Earth's rotational period was 23 hours and 11 minutes.

By first converting our two rotational periods into seconds and then inserting into the above equation we obtain the Earth’s earlier angular speed as 7.5284 E-5 rad/s and the current angular speed as 7.2722 E-5 rad/s.

Now we are ready to calculate how much rotational kinetic energy the Earth lost over the past 150 million years.

From this we know that the Earth lost 1.52 E28 J of energy during the last 150 million years. The average power output over the past 150 million years can now be calculated as

where delta t is our time of 150 million years. This gives the thermal power that is being generated within the Earth as a result of the slowing down of the Earth's rotation as being 3.2 TW.

As the tidal forces slow down the rotation of the Earth part of the power is going towards placing of the Moon in this slightly higher orbit. Calculations show that 7.5% of the Earth's rotational power is going towards placing the Moon in a higher orbit. After accounting for the power being transferred to the moon we are left with 3.0 E12 W of tidal power going towards warming the interior of the Earth.

A feature that is unique about the tidal force hypothesis is the prediction that more heat would be generated at the lower and middle latitudes than that of the highest latitudes. The tidal forces produce flexing of the Earth at the lower and middle latitudes but not at the poles. So we will have validation of the tidal heating hypothesis if we find that there is less heat leaving the Earth’s interior at the Polar Regions than elsewhere.

If not for the Moon, the Sun would be the primary source of the tidal forces acting on the Earth. But because the Moon is much closer to the Earth the tidal force produce by the moon is about twice as strong as that of the Sun. When the Sun, Moon, and Earth are all in line the tidal forces are the strongest and when this happens this is called a spring tide. When the Sun, Earth, and Moon are at a right angle the Sun and Moon’s tidal forces tend to cancel each other to produce weaker tides. These weak tides are called neap tides. The time period required to go from strongest to weakest then back to the strongest tides is about 15 days since there are high tides on each side of the Earth and the synodic orbital period of the Moon is 29.5 days.

The greatest tidal distortion of the Earth occurs at the central pulling points that are on opposite sides of the Earth and in line with the center of the Earth and the Moon. Also the large area surrounding the central pointing points is being distorted but to a lesser degree. Because these central pulling points are never more than either 28.7 degrees north or south of the equator the rising and falling of the Earth’s surface occurs primarily through the lower and middle latitudes. The Polar Regions are the only locations on the Earth that are not flexing each day in response to the tidal forces. Thus the tidal forces are producing less thermal energy at the Polar Regions than what is being generated throughout the rest of the world.

The central pulling points of the Moon are usually near the equator and never more than either 28.7 degrees north or south of the equator. As the Earth bulges in response to the pull of the Moon, the greatest distortion of the Earth occurs at these central pulling points. Since these central pulling points are always on opposites sides of the Earth and they are never far from the equator, the rising and falling of the Earth’s surface occurs primarily through the lower and middle latitudes. The locations on the Earth that receive the least amount of flexing are the north and south poles. Thus the tidal forces produce less thermal energy at these higher latitudes than what is being generated throughout the rest of the world.

The reduction of thermal energy emerging from deep beneath the Polar Regions has no connection with the fact that the Polar Regions have a much colder surface temperature than the rest of the Earth. The reason there is ice at the Earth’s Polar Regions is due to the much lower intensity of the sunlight that reaches these locations. Whereas the amount of heat generated under the crust is dependent on the amount of inelastic flexing of the rock that occurs due to the pull of the Sun and Moon changing the gravitation field at different locations on the Earth. Everywhere on the Earth surface the thermal heating from the Sun is far greater than the thermal heating rising up from within the Earth.

Heat naturally flows from hot to cold locations, so within the Earth heat should travel from the warmer locations of the lower and middle latitudes in the direction of the cooler higher latitudes. In addition since the Earth’s outer core is liquid convection currents should produce a more nearly uniform hot temperature near the Earth’s center. So we should not expect the Polar Regions to be completely absent of heat but rather we should expect that the heat escaping from below the Polar Regions to just be less than the heat escaping elsewhere.

Rock is such a poor conductor of heat that most of the heat escaping form the Earth’s interior does not come up through the rock but rather it emerges as part of volcanic activity. Nearly all of this ongoing volcanic activity is either taking place at the many hot spots around the globe or the chains of volcanoes that make up the ocean ridges. Since the tidal heating hypothesis would produce less heat under the Polar Regions we would expect fewer hotspots and fewer ocean ridges at the Polar Regions than what exist elsewhere. When we look at global map this is what we observe.

Perhaps the most compelling evidence in support of tidal heating comes when we broaden our perspective beyond Earth to consider the sources of interior heating for other large bodies that make up our solar system. It is simply more reasonable that the process for heating the interior of a planet or moon should be the same throughout the solar system. Most planetary scientists have abandon primordial heating and radioactivity heating in favor of tidal heating for explaining the internal heating of planets and moons.

After Earth, the Galileo moons nearest to Jupiter, Io and Europa, are geologically the next most active bodies in the solar system. More than 80 active volcanoes have been identified on the surface of Io. Far less visible is Europa’s volcanic activity that warms a vast liquid water ocean beneath its icy outer crust. Going back to the terrestrial planets, Venus may still have active volcanoes and then Mars and the Moon show evidence of volcanic activity in their past. This order of geological activity: Earth, Io, Europa, Venus, and Mars correlates with the order for the changing gravitational gradients experience by each of these bodies that produce tidal forces on these bodies. The next chapter will further explain this relationship between tidal heating and changing gravitational gradient.

The tidal force hypothesis is the only mechanism that can 1) produce the amount of thermal power that is observed, 2) it is the only mechanism that would distributes that thermal power around the Earth in the manner that is observed, and 3) correctly correlates between tidal forces and geological activity for the planets and moons of our solar system. In conclusion, the gravitational tidal forces are the source of all hell.

Exponential Decay

• Exponential Decay - Eric W. Weisstein
  
• Radioactivity Decay Graph - LAWRENCE BERKELEY NATIONAL LABORATORY
  

Determining the Age of the Earth

• PALEOBIOLOGY: FOSSILS AND TIME - M.J. Farabee
  
  

Radioactivity

• Knowledge about Radioactivity - Josh Baxter
  
• ABC's of Nuclear Science - LAWRENCE BERKELEY NATIONAL LABORATORY
  
• Uranium - JEFFERSON LAB
  
• Uranium - INSTITUTE FOR ENERGY AND ENVIRONMENTAL RESEARCH
  
• Potassium 40 - AMERICAN ELEMENTS
  
• Natural Abundance of Potassium - Mark Winter
  
• Potassium - JEFFERSON LAB
  

The Interior of the Earth

• The Interior of the Earth - Eugene C. Robertson
  
• Seismic Waves and Earth's Interior - Charles J. Ammon
  
• Geothermal Gradient - E_NOTES
  

Tides

• Orbit of the Moon - ThinkQuest
  
• Moon Tides
  
• Tides and Water Levels - NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION
  
• Tides - Matt Oltersdorf