7. Hell, Heaven and Earth

Part 1 - Hell: Source of the Earth's Internal Heat

Introduction to the 'Hell, Heaven, and Earth' Series

The Thick Atmosphere Theory resolves several major scientific paradoxes:

1. How pterosaurs were able to fly,
2. How dinosaurs grew so large,
3. Why dinosaurs evolved a unique body shape, and
4. How Earth maintained a remarkably uniform temperature throughout the Mesozoic era.

Yet, for critical thinkers, this solution demands an explanation: Where did this thick atmosphere come from, and where did it go?

To answer these questions, we must turn to astronomy — or more specifically, planetary science. For decades, planetary scientists have struggled to explain why Earth's atmosphere differs so drastically from those of Venus and Mars. Various theories have been proposed, including differences in planetary size, magnetic field strength, and atmospheric loss due to solar wind. However, these explanations often fail to provide a coherent account of the fundamental differences between these planets.

A far simpler and more compelling explanation lies in the role of Earth's geological activity and life. Given Earth’s orbital position between Venus and Mars — two planets with thick carbon dioxide atmospheres — as well as its size and geological activity, we should expect Earth to have an extremely dense CO₂ atmosphere. And in the past, it did. But unlike its neighbors, Earth took a different evolutionary path because life evolved and flourished here.

The key factor that many planetary models fail to emphasize adequately is that life fundamentally transformed Earth’s atmosphere. This transformation occurred through two primary mechanisms:

1. Carbonate Rock Formation – Organisms such as corals and shell-forming marine life extracted carbon from the ocean, locking it away as limestone and dolomite over millions of years.
2. Photosynthesis – Plants and cyanobacteria absorbed atmospheric CO₂, incorporated the carbon into organic material, and released oxygen as a byproduct, leading to the nitrogen and oxygen atmosphere we have today.

This biological process explains why Earth’s atmosphere is so vastly different from those of its neighboring planets — without requiring complex or speculative mechanisms. The striking simplicity of this explanation, along with overwhelming supporting evidence, sets it apart from the convoluted and unsatisfying models that came before it.

Sometimes we have to go through Hell before we get to Heaven

This chapter, Hell: Source of Earth’s Internal Heat, is the first part of a three-part series — Hell, Heaven, and Earth — which explores how Earth once had a much thicker atmosphere and how it came to have its uniquely balanced atmosphere today. The series begins by investigating the source of Earth’s internal heat, the driving force that first placed an atmosphere on the planet’s surface.

Mankind has long recognized that the deeper one digs into the Earth, the warmer the ground becomes. Volcanic eruptions also provide direct evidence of the Earth's hot interior. Additionally, seismic waves passing through the Earth's center offer convincing indirect evidence that much of the core is molten. While there is scientific consensus that the Earth's interior is extremely hot, geologists and other scientists disagree on the source of this thermal energy.

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

Determining the source of the Earth's internal heat is crucial for understanding how terrestrial planets generate their atmospheres. This, in turn, provides insight into why it is more logical for Earth to have had a thicker atmosphere in the past. Before addressing this question, however, we must first explore the mathematical principles that will aid in identifying the true source of the Earth's internal heat.

Exponential Decay

With regard to the proposed hypotheses concerning Earth's internal heat, a better understanding of exponential decay will help clarify the weaknesses of both the leftover primordial heat and radioactive heating theories. The general equation for exponential decay is given as:

\[ N = N_o \, e^{\text{-kt}} \]

In this equation, N represents a quantity that decreases over time, such as absolute temperature or the number of radioactive isotopes. The variable t represents time, while k is the decay constant.

If we apply this equation to the scenario of an initially hot object cooling to room temperature, the equation takes the form:

\[ T = T_R + (T_i - T_R) \, e^{\text{-kt}} \]

where T represents the absolute temperature of the object at any given time, T_i is its initial temperature, and T_R is the room temperature, with all values measured in Kelvin.

While this equation may appear complex, it can be significantly simplified by understanding the e^{- k t} component. When t = 0, then e^{- k t} = 1, and so T = T_i ; this is marked as point 1 on the graph. At the other extreme, when t approaches infinity, then e^{- k t} = 0, and so T = T_R ; this is marked as point 2 on the graph.

Now that we have identified the endpoints (1 and 2), the equation — and the corresponding curve — defines the temperature for all values of time between zero and near infinity.

To better understand what this graph represents, let’s consider the cooling of a hot cup of coffee. Imagine that after pouring the coffee, we forget to drink it, leaving it on the desk to cool to room temperature. From the graph, we can see that the temperature drops quickly at first, but over time, the rate of decline slows as it approaches the room’s temperature. Eventually, the coffee reaches thermal equilibrium with the surrounding air.

But how long is "eventually"? In other words, how long does it actually take for the coffee to cool to room temperature? According to the equation, an exponential decay event requires an infinite amount of time to fully complete. A mathematician would graph this equation to show that the decay curve never quite reaches the asymptote. However, from personal experience, we know that if we neglect our coffee, it will cool to room temperature in about an hour.

The disagreement over whether the decay process is finite or infinite arises from a subtle yet important distinction between real events and conceptual models of real events. While conceptual models are useful for understanding reality, they are inherently simplifications that capture only the main ideas. In the process of simplification, some information is inevitably lost. Most of the time, this missing information is inconsequential, but in certain cases, it becomes critical to answering our questions.

When examining real decay processes, the decay equation accurately describes the overall statistical behavior of a large sample. However, at a finer level, real decay processes exhibit sporadic variations that only approximate the decay equation as a statistical average. Thus, while it is mathematically true that the decay curve in the conceptual model never reaches the asymptote, this is not true for real-world decay events.

The primary reason for this difference lies in the inherent variability of reality — often referred to as randomness, indeterminacy, or statistical error. For example, when we say that a room’s temperature is 20 degrees Celsius, what we really mean is that it fluctuates around 20 degrees rather than holding a perfectly steady value. Maintaining an absolutely constant temperature in a real environment is simply impossible.

The same principle applies to the cooling of a beverage. While the temperature generally follows the theoretical exponential decay curve, in reality, it fluctuates slightly above and below the idealized path. These fluctuations become especially important when the object's temperature nears that of the surrounding room. At this point, the two "fuzzy" temperature values begin to overlap rather than continuing to asymptotically approach each other, as the mathematical model suggests. Once this overlap occurs, it becomes impossible to distinguish between the two, and for all practical purposes, the decay process is complete.

To summarize: real exponential decay events do not continue indefinitely but instead reach completion after a finite amount of time. While there may be some debate over precisely when a given decay process should be considered finished, there is no debate over whether real decay events persist forever — they do not. Unlike the mathematical equation, real-world exponential decay always concludes within a finite timeframe.

The Debate over the Age of the Earth

During the 17^th and 18^th centuries, scholars and scientists explored various methods to determine the Earth's age. However, these early estimates were heavily influenced by the human perspective of time, leading to exceptionally low figures. As new methods were developed, tested, and debated, the estimated age of the Earth steadily increased. Finally, in the 20^th century, radioactive isotope dating emerged as the most reliable means of determining absolute geological time. This method conclusively demonstrated that the oldest rocks on Earth are billions of years old and that most meteorites date back 4.54 billion years. Based on this evidence, geologists now universally accept that the Earth is 4.54 billion years old.

While determining the Earth's age was a remarkable scientific achievement, some historical debates left behind unintended consequences. Before certain ideas were proven incorrect, they became entrenched as geological dogma, continuing to mislead many geologists today. In the early stages of these discussions, leftover primordial heat was the only explanation geologists considered for the Earth's internal heat. However, once it became clear that the Earth is 4.54 billion years old, this idea should have been discarded.

The Problem with all Hypotheses Involving Primordial Heat

Lord Kelvin received numerous honors during his early career and was so highly respected in his prime that many, if not most, of his ideas went unchallenged by his contemporaries. However, by his later years, so many of his errors had become evident that some of his peers reportedly referred to him as "a silly old idiot." With the perspective of time, he has earned the dubious distinction of having introduced more errors into science than perhaps any other scientist in history.

Around 1897, Lord Kelvin attempted to determine the Earth's age by calculating its cooling rate. He hypothesized that the Earth's interior heat was merely residual thermal energy from its initial molten state. Based on the thermal conductivity of rock, he calculated the Earth's age to be approximately 40 million years. However, this estimate proved disastrously incorrect, primarily because his fundamental assumption about primordial heat was flawed.

Even though the age of the Earth is no longer considered controversial among scientists, the idea that it began in a molten state and that some of its primordial thermal energy still remains has become entrenched in geological dogma. While the notion that the Earth started in a molten state may or may not be true, it is ultimately irrelevant — calculations show that any primordial heat would have dissipated long ago. As discussed in the previous section, real exponential decay events have a finite duration and do not continue indefinitely; naturally, the Earth's cooling would have followed this same principle.

Although Kelvin was mistaken in some of his assumptions, his work did provide a rough estimate of how long it would take for the Earth to cool from a molten state. He based his calculations on the premise that heat transfer from the Earth's deep interior to the surface was solely due to the thermal conductivity of rock. Geologists now understand that convection also plays a significant role, and this process actually accelerates the Earth's cooling. Nonetheless, more recent calculations confirm that Kelvin’s estimate — approximately 40 million years for a molten Earth to cool — is reasonably accurate.

Given that the Earth is now known to be 4.54 billion years old, simple arithmetic shows that its age is about a hundred times greater than the time required for it to cool. This means that if the Earth did indeed start in a molten state, and no additional heat sources were present, it would have fully cooled about 4.5 billion years ago.

The Problems with the Radioactivity Heating Hypothesis

Before the discovery of radioactivity, geologists attributed the Earth's internal heat solely to leftover primordial heat. However, once radioactive decay methods established the Earth's age at 4.54 billion years — far too long for primordial heat to persist — many geologists realized they had no clear answer for the source of the Earth's internal heat. The radioactive heating hypothesis gained popularity more out of necessity than due to strong supporting evidence.

Simply stating that radioactive decay generates heat and concluding that buried radioactive material must be the Earth's heat source is not scientific. A hypothesis requires supporting data, yet such data has not always been readily available. A major issue is that experts in radioactivity often have a strong bias toward their field. Even today, many technical reports favoring radioactive heating rely on circular reasoning, ignoring the fundamental fact that the amount of radioactive material deep within the Earth remains unknown.

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 make informed guesses about what lies deep within the Earth. Some things are well understood: temperature and pressure increase with depth, and materials become progressively denser toward the center. Seismic data from earthquakes also confirm that part of the core is liquid. Beyond these points, a few other properties can be inferred with reasonable certainty. However, when geologists estimate the percentages of iron, nickel, and cobalt in the Earth's interior, these numbers are little more than educated guesses. There is no direct evidence — no "ground truth" — to confirm these figures. Likewise, there is no sampling method capable of determining how much radioactive material exists deep within the Earth.

Without any means of determining what radioactive material, if any, exists deep within the Earth, the radioactive heating hypothesis remains more speculative than scientific. Furthermore, while proponents of this hypothesis are unlikely to ever uncover direct evidence to support their case, substantial evidence can be presented showing that the idea is fundamentally flawed. To assess whether radioactivity should be considered a primordial or constant heat source, let’s revisit some basics of radioactive decay.

When plotting the activity of a radioactive isotope like plutonium-239 on a time scale of tens of thousands of years, we see the expected decay curve. However, on a time scale of billions of years — appropriate for Earth's age — the decay appears as a vertical line. In fact, any radioactive isotope with a half-life shorter than about 100,000 years will graph as a vertical line on such a time scale, meaning these isotopes would have produced primordial heat. Radioactive isotopes can be viewed as either short-lived isotopes that produced primordial heat or long-lived isotopes that contribute nearly constant heat.

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

The power output of each sample will closely match the activity level of each sample. Therefore, 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-lived isotopes on a graph marked out in billions of years, we get nothing but a vertical line.

What these vertical line graphs don't reveal is how isotopes with shorter half-lives release their energy much faster than those with longer half-lives. For example, sample B, with a half-life of one hundred years, will take ten times longer to release its energy than sample A, with a half-life of ten years. Since both samples start with the same amount of energy, the power output of sample B will be ten times less than that of sample A as they transition through the same segments of their respective exponential decay lives. Extending this reasoning to the final sample, sample J, which has a ten-billion-year half-life, will have a billion times less power output than sample A.

Plotting the power output of all our isotopes on a graph will produce an 'L' shaped curve. The short-life isotopes produce a burst of power at the start, while 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 that 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 be thousands or even millions of times more abundant than another? Well, it is still extremely meaningful because, even if a single isotope is a million times more abundant than all 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, making the radioactive heating hypothesis potentially feasible, one of the isotopes with a half-life of several billion years must be hundreds of millions, if not billions, of times more abundant than all the other radioactive isotopes.

This alone is a steep requirement, but when we examine isotopes with long half-lives, additional problems arise. Heavy radioactive isotopes cannot be more concentrated than the highest natural ore concentrations observed in the Earth's crust. Geological evidence shows that when these isotopes become too concentrated, they undergo rapid burn-off through fission decay.

Back-of-the-envelope calculations are useful in science and engineering for quickly determining whether an idea is flawed or if it might have merit. Approximately 9,143 tons of uranium ore were mined to produce the 64 kg of uranium-235 used in the Hiroshima bomb. When detonated, this bomb released an estimated 63 × 10¹² joules of energy, causing mass devastation. However, if this energy were instead released gradually over Earth's 4.54-billion-year history, the power output would be less than a single milliwatt — far too small to detect.

We can take different approaches to perform several more of these back-of-the-envelope calculations, and they all yield similar results: the power output from radioactive decay is about a million times too small to be a significant factor in heating Earth’s interior. Because nuclear energy intimidates most people, there is a strong tendency to rely on expert consensus — often from geologists, whose expertise typically lies in fields other than advanced mathematics. However, for those confident in physics and math, simple calculations make it clear that radioactivity cannot be the primary source of Earth's internal heat.

As a final check on the logic, we can examine the composition of asteroids as indicators of what might exist within Earth's interior. Astronomers have found that 75% of asteroids are C-type (carbonaceous), 15% are S-type (silicate), and the remaining 10% are mostly M-type (metallic), composed primarily of nickel and iron. Notably, astronomers do not generally find asteroids rich in uranium or potassium. Therefore, we can conclude that radioactivity is not the primary source of Earth's internal heat.

Tidal Heating

Every intelligent person knows that the Earth rotates on its axis, yet how many of us realize that this massive spinning sphere possesses a tremendous amount of kinetic energy? In the Solution chapter, a brief explanation was given regarding the gradual slowing of Earth's rotation. But if the Earth is slowing down, its kinetic energy is decreasing — so 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 possesses translational kinetic energy. When we apply the brakes to slow the car, that kinetic energy is converted into thermal energy, heating the brakes. The translational kinetic energy of a moving object is given by the equation:

\[ KE = \frac{1}{2}\, m \, v^2 \]

where m is the object's mass and v is its velocity.

In addition to translational motion, objects can also possess kinetic energy simply by rotating, even if they remain in place. Just as it takes energy to set a wheel in motion, a rotating wheel’s kinetic energy is eventually converted to thermal energy when it comes to a stop. The equation for rotational kinetic energy is similar to that of translational kinetic energy:

\[ KE = \frac{1}{2}\, I \, ω^2 \]

where KE is the kinetic energy, I is the rotational moment of inertia, and ω is the angular velocity.

As the Earth moves through space, it both revolves around the Sun once a year — translational energy — and rotates on its axis once a day — rotational energy. The Earth's translational and rotational kinetic energies have remained nearly constant over the millions and billions of years of its existence. However, while nearly constant, the rotational speed of the Earth is nevertheless gradually decreasing due to tidal forces acting upon it.

As the Moon and Sun exert gravitational torque on the Earth, they cause a slowdown in its rotation, converting some of its rotational kinetic energy into thermal energy within the Earth.

We need to determine the Earth's values for the variables in the rotational kinetic energy equation, ½ I ω², so that we can calculate the change in kinetic energy. For a solid, homogeneous sphere, the moment of inertia can be calculated using the equation

\[ I = \frac{2}{5}\, m \, R^2 \]

where m is the mass of the object and R is its radius. If we assume the Earth is homogeneous, its moment of inertia would be approximately 9.7×10³⁷ kg·m².

However, because the Earth is denser at its center rather than being a uniform sphere, its actual rotational moment of inertia is lower. By approximating the density profile and applying calculus, a more accurate estimate of the Earth's moment of inertia is 8.0×10³⁷ kg·m². This value aligns well with measurements obtained from NASA’s data on the Earth's motion and rotation.

Next, we need the Earth's current angular speed, ω, and its angular speed at an earlier time. The current angular speed of the Earth is determined using its present rotational period, T, which is 24.0 hours. To find the Earth's initial angular speed, we go back 150 million years to when the Earth's rotational period was 23 hours and 11 minutes.

\[ \omega = \frac{2\,\pi}{T} \]

By first converting both rotational periods into seconds and then applying the angular speed formula, we obtain the Earth's earlier angular speed as 7.5284×10⁻⁵ rad/s and the current angular speed as 7.2722×10⁻⁵ rad/s.

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

\[ \Delta KE = KE_f - KE_i \] \[ \Delta KE = \frac{1}{2}\, I \, (ω_f^2 - ω_i^2) \]

From this, we know that the Earth lost 1.52 × 10²⁸ J of energy over the past 150 million years. The average power lost during this period can be calculated using:

\[ P = \frac{\Delta KE}{\Delta t} \]

where Δt is 150 million years. This calculation shows that currently the Earth is losing about 3.2 TW of energy per second due to the slowing of its rotation.

As tidal forces slow the Earth's rotation, part of this power contributes to moving the Moon into a slightly higher orbit. Calculations show that 7.5% of the Earth's rotational energy is transferred to the Moon’s orbit. After accounting for this, the remaining 3.0 × 10¹² W of tidal power is available for warming the Earth's interior.

Think of the Earth as a giant rubber ball that, aside from its size, shares similarities with a small rubber ball that can be held in a person’s hand. If dropped on a hard surface, the ball will bounce back up to nearly the same height. A good rubber ball bounces high, yet each bounce is still slightly lower than the previous one. This happens because, while rubber is elastic, it is not perfectly elastic — some energy is lost each time the ball hits the surface and flexes. From a physics perspective, this "lost energy" is not truly lost but is instead converted from mechanical energy into thermal energy, causing the ball to warm up slightly.

Flexing an object just once may not generate enough heat for us to notice, but continuous flexing can lead to a detectable rise in temperature. This is evident in car tires. When looking at a stationary car, we can see that each tire is slightly flattened where it touches the ground. As the car moves, the rotating tires constantly change shape — flexing — so that whichever part of the tire is in contact with the ground is always slightly flattened. Because of this continuous flexing, tires become noticeably warm while driving on the highway. Similarly, as the Earth rotates each day, it stretches due to the gravitational pull of the Moon and the Sun. This ongoing flexing generates heat, warming the Earth's interior.

Non Uniformity of the Earth's Internal Heat

A unique feature of the tidal force hypothesis is its prediction that more heat would be generated at lower and middle latitudes than at higher latitudes. Tidal forces cause flexing of the Earth in the lower and middle latitudes but not at the poles. Therefore, the hypothesis would be validated if we find that less heat escapes from the Earth's interior in the polar regions than elsewhere.

If not for the Moon, the Sun would be the primary source of tidal forces acting on the Earth. However, because the Moon is much closer, the tidal force it produces is about twice as strong as that of the Sun. When the Sun, Moon, and Earth are aligned, tidal forces are at their strongest, a phenomenon known as a spring tide. Conversely, when the Sun, Earth, and Moon form a right angle, the tidal forces from the Sun and Moon partially cancel each other, resulting in weaker tides called neap tides. The cycle from the strongest to the weakest tides and back to the strongest takes approximately 15 days, as there are high tides on both sides of the Earth and the Moon’s synodic orbital period is 29.5 days.

The greatest tidal distortion of the Earth occurs at the central pulling points, which are on opposite sides of the Earth and aligned with the center of the Earth and the Moon. Because these central pulling points are never more than 28.7 degrees north or south of the equator, the rising and falling of the Earth's surface primarily occur in the lower and middle latitudes. At much higher latitudes — the Polar Regions — there is little to no flexing in response to tidal forces. As a result, tidal forces generate less thermal energy in the Polar Regions compared to the rest of the world.

The reduction of thermal energy emerging from deep beneath the polar regions is unrelated to the fact that these regions have much colder surface temperatures than the rest of the Earth. The presence of ice at the Earth's polar regions is due to the significantly lower intensity of sunlight reaching these areas. In contrast, the amount of heat generated beneath the crust depends on the degree of inelastic flexing of rock caused by the gravitational pull of the Sun and Moon, which varies at different locations on Earth. Across the Earth's surface, solar heating is far greater than the thermal energy rising from within the planet.

Rock is such a poor conductor of heat that most of the heat escaping from the Earth's interior does not rise directly through the rock but instead emerges through volcanic activity. Nearly all ongoing volcanic activity occurs either at the numerous hotspots around the globe or along the chains of volcanoes that form the ocean ridges. Since the tidal heating hypothesis predicts less heat generation beneath the polar regions, we would expect fewer hotspots and fewer ocean ridges in these areas compared to the rest of the world. When we examine a global map, this is precisely what we observe.

Astronomy Evidence in Support of Tidal Heating

Perhaps the most compelling evidence in support of tidal heating comes when we broaden our perspective beyond Earth to consider the sources of internal heating for other large bodies in our solar system. Depending on which planet or moon is under consideration, planetary scientists attribute internal heat to tidal heating, radioactive decay, primordial heat, or some other mechanism — often a combination of these. However, nature tends to be much more straightforward: in most cases, a phenomenon can be attributed to a single dominant mechanism, while other factors are negligible. This patchwork assignment of different explanations to fit the unique characteristics of each planet or moon highlights the fact that scientists are struggling to develop a coherent understanding of our solar system. Instead of this hodgepodge of possible explanations, it is far more likely that the process responsible for heating the interiors of planets and moons is the same throughout the solar system.

After Earth, Jupiter’s Galilean moons, Io and Europa, are the most geologically active bodies in the solar system. Io hosts over 80 active volcanoes, while Europa’s volcanic activity warms a vast subsurface ocean. Among the terrestrial planets, Venus may still have active volcanoes, while Mars and the Moon show signs of past volcanism. This sequence — Earth, Io, Europa, Venus, and Mars — aligns with the gravitational gradients that generate tidal forces. Additionally, tidal forces strongly correlate with planetary density. The following chapters will further explore the relationship between tidal heating, gravitational gradients, and the density of planets and moons.

In summary, the tidal force hypothesis is the only mechanism that:

1. Produces the amount of thermal power observed.
2. Distributes that thermal power around Earth in the manner observed.
3. Correctly correlates tidal forces with geological activity across the planets and moons of our solar system.

In conclusion, the gravitational tidal forces are the source of all hell.