The Pale Blue Dot? Chapter Eighteen: Eluer’s Identity, Orbital trajectories, OCP and PMP

Written By Thomas Perez. February 10, 2021 at 7:53PM. Copyright 2021.

Euler’s Identity

Leonhard Euler

Before I begin with this chapter, I would like to give a brief background on Euler’s accomplishments and innovations. Leonhard Euler was born on April 15, 1707, in Basel, Switzerland, and died September 18, 1783, in St. Petersburg Russia. Euler was a Swiss mathematician, physicist, astronomer, geographer, logician, and engineer. He was influential in many discoveries with reference to the various branches of mathematics, such as infinitesimal calculus and graph theory. He also pioneered many contributions to several branches in relation to mathematics; such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. (1). He is also known for his work in mechanics, fluid dynamics, optics, astronomy and of music theory. (2). Euler worked in all areas of mathematics, such as geometry, trigonometry, algebra, he popularized the Greek letter pi in mathematics, the constant e, the modern notation for trigonometric functions, complex numbers, power series, exponential function, and logarithms (now also known as Euler’s number) in analytic proofs, continuum physics, lunar theory and other areas of physics. Euler directly proved the power series expansions for e and the inverse tangent function. (3). 

(1). Dunham 1997. p. 17

(2). Saint Petersburg (1739). “Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae”

(3). Wanner, Gerhard; Hairer, Ernst (2005). Analysis by its history (1st ed.). Springer. p. 63.

Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He invented the calculus of variations including its best-known result, the Euler-Lagrange equation. Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler’s work in this area led to the development of the prime number theorem. (4).

(4). Dunham 1999. Ch. 3, Ch. 4

Euler proved Newton’s identities, Fermat’s little theorem, and Fermat’s theorem on sums of two squares. Euler contributed to theory perfect numbers, Lagrange’s four-square theorem, totient function φ(n), ultimately resulting in what became known as Euler’s theorem. He proved that the relationship shown between even perfect numbers and Mersenne primes earlier proved by Euclid was one-to-one, a result otherwise known as the Euclid-Euler theorem. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss. (5). By 1772 Euler had proved that 231 − 1 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867. (6). He also facilitated the use of differential equations, like the Euler-Mascheroni constant, and improved the numerical approximation of integrals (Euler approximations).

Other acolytes included solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Euler numbers, the constants e π, continued fractions and integrals. He integrated Leibniz’s differential calculu. with Newton’s Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. All of Euler’s collected works would contain 95 volumes. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes.

(5). Dunham 1999. Ch. 1, Ch. 4

(6). Caldwell, Chris. The largest known prime by year

In Physics, Astronomy and Religion

The Euler-Bernoulli beam equation was developed by Euler. It became the corner stone for engineering. Besides successfully applying his analytic tools to problems in classical mechanics, Euler applied these techniques to celestial problems. His work in astronomy was recognized by multiple Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the Sun. His calculations contributed to the development of accurate longitude tables. (7).

(7). Home, R.W. (1988). “Leonhard Euler’s ‘Anti-Newtonian’ Theory of Light”. Annals of Science45 (5): 521–33. 

Euler made important contributions in optics. He disagreed with Newton’s corpuscular theory of light in the Opticks, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light. (8).

(8). Ibid. 

“In optics, the corpuscular theory of light, arguably set forward by Descartes in 1637, states that light is made up of small discrete particles called “corpuscles” (little particles) which travel in a straight line with a finite velocity and possess impetus. This was based on an alternate description of atomism of the time period.”

“Isaac Newton was a pioneer of this theory; he notably elaborated upon it in 1672. This early conception of the particle theory of light was an early forerunner to the modern understanding of the photon. This theory cannot explain refraction, diffraction and interference, which require an understanding of the wave theory of light of Christiaan Huygens.”

Much of what is known of Euler’s religious beliefs can be deduced from his Letter to a German Princess and an earlier work, Rettung der Göttlichen Offenbahrung gegen die Einwürfe der Freygeister (Defense of the Divine Revelation against the Objections of the Freethinkers). These works show that Euler was a devout Christian who believed the Bible to be inspired; the Rettung was primarily an argument for the divine inspiration of Scripture.

Eluer’s Identity

“Euler’s Identity stems naturally from interactions of complex numbers which are numbers composed of two pieces: a real number and an imaginary number; an example is 4+3i. Complex numbers appear in a multitude of applications such as wave mechanics (a study within quantum mechanics) and design of circuits that use alternating current (a common practice in electrical engineering). Additionally, complex numbers (and their cousins, the hyper complex numbers) have a property that makes them especially useful for studying computer graphics, robotics, navigation, flight dynamics, and orbital mechanics: multiplying them together causes them to rotate. This property will help us understand the reasoning behind Euler’s Identity.” (9).

(9). https://www.livescience.com/51399-eulers-identity.html

In 1988, readers of the Mathematical Intelligencer voted it “the Most Beautiful Mathematical Formula Ever.” In total, Euler was responsible for three of the top five formulae in that poll. (10). Standard University mathematics professor Keith Devlin has said, “like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.” (11). And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler’s formula and its applications in Fourier analysis, describes Euler’s identity as being “of exquisite beauty.” (12). A statement attributed to Pierre-Simon Laplace expresses Euler’s influence on mathematics: “Read Euler, read Euler, he is the master of us all.” (13). It is called “The most remarkable formula in mathematics” by Richard P. Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, ei and π

(10). Wells, David (1990). “Are these the most beautiful?” Mathematical Intelligencer12 (3): 37–41. Wells, David (1988). “Which is the most beautiful?” Mathematical Intelligencer10 (4): 30–31.

(11). Nahin, 2006, p. 1

(12). Nahin, 2006, p. xxxii.

(13). Dunham 1999, p. xiii “Lisez Euler, lisez Euler, c’est notre maître à tous.”

Practical Earth Shape Applications

In a paper entitled the Continuation From a Flat to a Round Earth Model in the Coplanar Orbit Transfer Problem, the authors Max Cerf, Thomas Haberkorn and Emmanuel Trélat discuss “The problem of minimization of the fuel consumption for the coplanar orbit transfer problem.” Under the section; The Round Earth Model and the Optimal Control Problem.”

As we have learned in chapter seven, again, we see the words; transference, convergence and transfiguration being used. Other keywords used are; “Orbit transfer problem; Optimal Control Problem (OCP); Pontryagin Maximum Principle (PMP); shooting method; continuation,” In the paper, they transfer, converge and transfigure the Earth mathematically for the saving of fuel consumption by using Euler’s equation results. A consumption that is flown in “high-thrust orbit transfer that we furthermore restrict to be coplanar. …”The resulting numerical continuation process thus provides a new way to solve the problem of minimization of fuel consumption for the coplanar orbit transfer problem.” (14). The problem involves solving shooting calculations, as in initial takeoffs, orbits and trajectories; all this on a coplanar path. 

(14). Max Cerf, Thomas Haberkorn, Emmanuel Trélat: Page 1.

Coplanar: Another keyword found in the paper that we should all focus upon. What does the term coplanar mean? The definition of the word means “Lines, and line segments, that lie on the same plane (and consequently space) are considered coplanar lines.” (15). https://www.storyofmathematics.com/coplanar lines Moreover, “Coplanar means that the lines are on the same flat surface. Non-coplanar means the lines are on different flat surfaces on different planes.” (16).

(16). https://study.com/academy/lesson/coplanar-lines-in-geometry-definition-lesson-quiz.html#:~:text=Coplanar%20means%20that%20the%20lines,flat%20surfaces%20on%20different%20planes.

“There exist various numerical methods to solve such a problem, and we usually separate them in two classes: direct and indirect methods. Direct methods (e.g. surveyed in [4]) consist in discretizing the optimal control problem in order to rewrite it as a parametric optimization problem. Then a nonlinear large scale optimization solver is applied (emphasis T. Perez). The advantage of this approach is that it is straightforward and is usually quite robust. The main drawback is that, because of the discretization step, those methods are computationally demanding and that they are not very accurate in general when compared to the indirect approach. Indirect methods are based on the Pontryagin Maximum Principle (PMP) that is a set of necessary conditions for a candidate trajectory and control strategy to be optimal. The idea is to use those necessary conditions to reduce the search of a solution to the search of the zero of the so-called shooting function (indirect methods are also called shooting methods in this context).” (17).

(17). Ibid. Max Cerf, Thomas Haberkorn, Emmanuel Trélat: Page 2.

Before we continue, it is imperative to understand the definition and/or meaning of a specific word or set of words; beginning with the word “discretizing.” The definition of discretizing means “Gerund or present participle: To represent or approximate (a quantity or series) using a discrete quantity orquantities.” (18). Discretization: “The action of making discrete; and especially mathematically discrete.” (19).

(18). Google Dictionary

(19). https://www.merriam webster.com/dictionary/discretization

Pontryagin Maximum Principle – PMP: “Is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls.” (20). Hence discretizing and PMP go hand and hand together. The PMP was “Formulated in 1956 by the Russian mathematician Lev Pontryagin and his students, (21)(22) and its initial application was to the maximization of the terminal speed of a rocket. (23).The result was derived using ideas from the classical calculus of variations. (24). After a slight perturbation of the optimal control, one considers the first-order term of a Taylor expansion with respect to the perturbation; sending the perturbation to zero leads to a variational inequality from which the maximum principle follows.” (25).

(20). Ross, Isaac (2015). A primer on Pontryagin’s principle in optimal control. San Francisco: Collegiate Publishers.

(21). Boltyanski, V.; Martini, H.; Soltan, V. (1998). “The Maximum Principle – How It Came to Be?” Geometric Methods and Optimization Problems. New York: Springer. pp. 204–227.

(22). Gamkrelidze, R. V. (1999). “Discovery of the Maximum Principle”. Journal of Dynamical and Control Systems5 (4): 437-451. Reprinted in Bolibruch, A, A,; et al., eds. (2006). Mathematical Events of the Twentieth Century. Berlin: Springer. pp. 85–99. 

(23). For first published works, see references in Fuller, A. T. (1963). “Bibliography of Pontryagin’s Maximum Principle”. J. Electronics & Control15 (5): 513–517. 

(24). McShane, E. J. (1989). “The Calculus of Variations from the Beginning Through Optimal Control Theory”. SIAM J. Control Optim27 (5): 916–939.

(25). Yong, J.; Zhou, X. Y. (1999). “Maximum Principle and Stochastic Hamiltonian Systems.” Stochastic Controls: Hamiltonian System and HJB Equations. New York: Springer. pp. 101–156. 

They use the principle of the PMP by discretizing the flat Earth model into a curved Earth model in order to solve the optimal control problem and the orbital transfer problem. They freely admit this as seen in the following quotation. “This paper is organized as follows. First, we state the optimal control problem we want to solve, along with the necessary conditions given by the PMP. Then we introduce the simplified flat Earth model and modify it so as to introduce curvature and make it diffeomorphic to the round Earth model. The next section presents the continuation procedure and explains how to pass from the simplified model to the targeted optimal control problem. A refined analysis is then carried out to provide a robust and efficient algorithm to solve the simplified flat Earth model, which consists in simplifying and specializing the application of the shooting method, due to the particular structure of the problem. Finally, we give a numerical example in which we solve an orbit transfer from an unstable (on a collision course) Sun Synchronous Orbit (SSO) to a nearly circular final orbit. Since our approach involves diffeomorphic changes of coordinates, we explain in the Appendix the impact of a change of coordinates onto the set of adjoint vectors of the PMP.” (26).

(26). Ibid. Max Cerf, Thomas Haberkorn, Emmanuel Trélat: Page 3.

In order to solve the problem of optimal control a Cartesian coordinate system must be used. “In mathematics, the Cartesian coordinate system, or rectangular coordinate system, is used to determine each point uniquely in a plane through two numbers, usually called the x-coordinate and the y-coordinate of the point. To define the coordinates, two perpendicular directed lines (the x-axis or abscissa, and the y-axis or ordinate), are specified, as well as the unit length, which is marked off on the two axes (see Figure 1). Cartesian coordinate systems are also used in space (where three coordinates are used) and in higher dimensions.” (27).

Typical Cartesian coordinate system. Four points are marked in 2,3 in green. 3,1 in red. 1,5-2,5 in blue and 0.0, the origin, in yellow.

Typical Cartesian coordinates on a circle (flat or spherical) with its radius.

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost function. The optimal control can be derived using Pontryagins’s maximum principle (a necessary condition also known as Pontryagin’s minimum principle or simply Pontryagin’s Principle), or by solving the Hamilton-Jacobi-Bellman equation (a sufficient condition).” (28).

(27). https://www.newworldencyclopedia.org/entry/Cartesian_coordinate_system

(28). Ross, I. M. (2009). A Primer on Pontryagin’s Principle in Optimal Control. Collegiate Publishers. 

However, using Cartesian coordinates only causes problems for those who insist that the Earth is a globe. So the “change” must take place in order to fit their prepositional math and spherical Earth model. The convergence and transfer must be obtained if the optimal control problem is to be solved. The problem can only be solved if they assume that the Earth is spherical in shape. And in order to do this they created the PMP as mentioned above.

“The model that we use for the coplanar orbit transfer problem is the following. Assume (emphasis T. Perez) that the Earth is spherical with center O and consider an inertial geocentric frame (O,~i,~j,~k). Since we consider the coplanar orbit transfer problem, we assume that the whole trajectory lies in the plane O + R~i + R~j.” “Instead of Cartesian coordinates, we next use polar coordinates whose definition is recalled.” (29). The definition is the mathematical expression they use. They recall the expression and set it.

(29). Ibid. Max Cerf, Thomas Haberkorn, Emmanuel Trélat: Page 3.

A flat 3 dimensional Cartesian cylinder converted to a 3 dimensional sphere.

A converted spherical Earth with what they see as its polers.

After said conversion and set in, they “Define the flight path angle by once again using the Cartesian system.” “Then, the control system is written in cylindrical coordinates.” “The optimal control problem under consideration then consists in steering the control system from an initial configuration to some final configuration that is either of the form.” 

“The conditions mean that the satellite has to enter a specified orbit at a given point of it. The conditions (7) mean that the satellite must be steered to a final elliptic orbit of energy Kf < 0 and eccentricity ef , without fixing the entry point on that orbit.” (30).

(30). Ibid. Page 4.

“As mentioned formerly, it is difficult to obtain convergence of this method, due to a difficulty of initialization and to the discontinuities of the control. However, we observe that, when assuming that the Earth is flat and the gravity is constant, the corresponding optimal control problem can be easily solved, in a very explicit way. We next introduce this very simplified model and explain our idea of passing continuously to the round Earth model.” (31). Under the section called “Simplified Flat Earth Model” the paper demonstrates how they convert the simplified flat Earth model gradually and continuously into the globe model.

“The motion of a vehicle in a flat Earth model with constant gravity is governed by the control system.” After said simplification, they must change their coordinates from a flat plane to a curve. Even though, “It is admitted that it happens that (OCP)-flat can be explicitly and nearly analytically solved by applying the PMP. This is the object of Section 3 – ‘Analysis of the Optimal Control Problem With the Simplified Flat Earth Model’ – “This resolution leads to an very efficient algorithm based on a shooting method whose initialization is obvious. Based on that observation, it is tempting to try to use this efficient resolution in order to guess a good initialization for the shooting method applied to (OCP). To this aim, the idea is to use a continuation process by introducing parameters such that, when one makes these parameters evolve continuously, one passes from the flat Earth model to the initial round Earth model. Since the coordinates of the flat Earth model are Cartesian, and the coordinates of the round Earth model are polar, this will of course require, at the end of the process, a change of coordinates.”

“Moreover, we would like this modified model to be equivalent, up to some change of coordinates, to the round Earth model. This modified flat Earth model is derived in the next subsection, by defining a change of coordinates that is flattening circular orbits into horizontal trajectories, and then computing the control system from this change of coordinates.” (32).

(31). Ibid. Page 7.

(32). Ibid. Page 8.

The following two pictures are taken from Page 9.

It should be understood that such a control system is achieved by, according to the paper, “Corresponding diffeomorphism, such that F(x, h, vx, vh) = (r, ϕ, v, γ). For the control, the transformation from cylindrical to Cartesian” as seen in the picture above its an application of change. See pages 10 through 13 in paper for mathematical expressions toward this continual gradual conversion from a flat Earth to a spherical Earth; and from a gravitational constant on a flat Earth to a gravitational variable, also from a flat Earth model. Hence they “Make a first continuation on the parameter λ1, keeping λ2 = 0, passing from λ1 = 0 (flat Earth model with constant gravity) to λ1 = 1 (flat Earth model with variable gravity), and then a second continuation, keeping λ1 = 1, passing from λ2 = 0 to λ2 = 1 (modified flat Earth model, equivalent to the initial round Earth model).” (33).

(33). Ibid. Page 10.

In Section 3 of the paper called “Analysis of the Optimal Control Problem With the Simplified Flat Earth Model,” it deals with Application of the Pontryagin Maximum Principle, analysis of extremal equations, and the refined analysis of the strategy Tmax − 0, and algorithmic procedure. While section 4 deals with “Numerical Simulations” and its continuation procedure, its comparison with a direct method, its comparison with other initialization methods, and its restriction to high-thrust orbit transfer. For all continual mathematical convergence from a Cartesian flat Earth to a spherical Earth see actual paper.

The paper concludes; “We have given an algorithmic procedure to solve the problem of minimization of the fuel consumption for the coplanar orbit transfer problem by a shooting method approach, without any a priori knowledge on the optimal solution (and thus on the way to initialize the shooting method). Our method relies on the preliminary remark that, when studying the same problem within a simplified flat Earth model with constant gravity, the optimal control problem can be explicitly solved, and the solution leads to a very efficient algorithm that does not need any careful initial guess. Based on that remark, we have proposed a continuous deformation of this simplified model to the initial model (up to some change of coordinates), introducing continuously corrective terms into the flat Earth model. From the algorithmic point of view, the procedure then consists of solving a series of shooting problems, starting from the simplified flat Earth model which is easy to initialize, and ending up with the sought solution. The whole procedure is time-efficient and provides a way for bypassing the difficulty due to the initialization of the shooting method when it is applied directly to the initial problem.”

“Many questions remain open and from this point of view our work should be considered as preliminary. A first question is to investigate whether this procedure is systematically efficient, for any possible coplanar orbit transfer. Up to now we did not make any exhaustive tests, however it is very probable that one may encounter some difficulties, as in any continuation process, due to the intricate topology of the space of possible continuation paths, this space being not always arc-wise connected. Indeed, the flat Earth model only has one thrust arc while the round Earth model has two or more. Another question is to extend our study to the three-dimensional case, the final objective for an enterprise as Astrium Space Transportation being to have available a reliable and efficient tool to realize any possible orbit transfer without having to spend much time on the initialization of the algorithm.” (34).

One can only wonder whether they are still trying to figure out how to “blast a man or woman” into space, given the “intricate topology of the space of possible continuation paths.” Not to mention the “coplanar orbit transfer.” It also reminds us of chapters 6 and 7, in reference to satellites, the ISS and their constant barrage of “live video” streams and photographs. Yes, these are the things they are telling and showing us. I am merely pointing out the discrepancies of the given facts as told by the experts. You will also recall that “Typology is the mathematical study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.” (Ch 17). Since typology is the application of unchanging properties and spatial relations then why the choice of change to begin with?

Upon this instance many would be quick to say that they do this because the use of a linearized model of a non linear event exists, where the result are minuscule in change to the data but severely complicate the maths. In other words, why do this if its going to complicate the math for no benefit? That alone can tell you that the Earth is spherical. Yes, that much is accurate, but “Lines remains parallel because cylinders are flat.” (Ch 17). Moreover, traditional views are still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and this is how a cylinder is now defined in various modern branches of geometry and topology. The following is a typical cylinder.

Envision this cylinder with a dome that would be considered curved; because domes are curved. Now consider the math applied to compensate for a flat circular cylinder with reference to its convergence at its upward height and its curved r’ on top, the mathematical expressions still remain intact even after said curvature as the paper by Thomas Haberkorn, and Emmanuel Trélat illustrates with their conversion of a flat 3 dimensional Cartesian cylinder converted to a 3 dimensional sphere. But naturally they would want to take it a bit further to include a bottom curved half of the Earth too. Hence justifying their spherical Earth model. And as a result, the math, still stays the same. And since the math stays the same, it is conducive that the model chosen, with reference to Earths shape can be both fixed and set within its parallel proximities to angular reference points and circular objects; like the Sun, Moon and planets (wandering stars) as Euler’s angle expression demonstrates with its reference to fixed mass bodies and moving celestial objects in the firmament. The mathematical expressions and equations are apropos and relate to the Earth models we are discussing. See illustration below.

“The 6DOF (Euler Angles) block implements the Euler angle representation of six-degrees-of-freedom equations of motion, taking into consideration the rotation of a body-fixed coordinate frame (Xb, Yb, Zb) about a flat Earth reference frame (Xe, Ye, Ze).” (35). Hence, the shape of the Earth can indeed be flat covered by a dome.

(34). Ibid. Pages 25-26

(35). https://www.mathworks.com/help/aeroblks/6dofeulerangles.html

Also see…

https://www.mathworks.com/help/aeroblks/simplevariablemass6dofeulerangles.html

Keeping in mind the idea of a dome, another paper, released February 26, 2020 entitled, “On Incompressible Vibrations of the Stratified Atmosphere on the Flat Earth” was written to study atmospheric pressure and vibrations. The paper can be found and downloaded in the link provided below. The researchers behind the study analyzed 38 years of atmospheric pressure data. A study by scientists at Kyoto University and the University of Hawaii at Mānoa has shown that the Earth’s entire atmosphere vibrates in a manner that greatly resembles the vibrations of a ringing bell.” (36). Where do these vibrations come from? What causes them?

(36). https://www.researchgate.net/publication/339471661_On_Incompressible_Vibrations_of_the_Stratified_Atmosphere_on_the_Flat_Eart

“Molecules of carbon dioxide (CO2) can absorb energy from infrared (IR) radiation. This animation shows a molecule of CO2 absorbing an incoming infrared photon (yellow arrows). The energy from the photon causes the CO2 molecule to vibrate.” Hence, “Radiation from the sun is absorbed by the earth as radiant visible light. You feel this effect on a sunny day when you stand in the sunshine vs. the shade. Eventually, the heat from the Earth is re-emitted into the atmosphere as infrared radiation (IR). As an example, infrared radiation is what you can feel and see (slightly) as the red hot burner of an electric stove…” “Certain gases in the atmosphere have the property of absorbing infrared radiation. Oxygen and nitrogen the major gases in the atmosphere do not have this property. The infrared radiation strikes a molecule such as carbon dioxide and causes the bonds to bend and vibrate – this is called the absorption of IR energy.” (37).

Question: Can such a sound and vibration come from a dome enclosed Earth? Can tangible manifestations often left as evidence of trespass, even from so intangible a quarter, be left without further scrutiny? For example, “Where do our carbon dioxide emissions go?” “…Only about 50 percent of the CO2 from human emissions remains in the atmosphere. The remainder is approximately equally split between uptake into the land biosphere and into the ocean.” (38). However, “While hydrogen and helium make up most of the gases in interstellar space, tiny traces of other elements such as carbon, oxygen and iron also exist.” (39). But since the majority of C02 is left trapped, or enclosed if you will, in Earth’s firmament – aka – dome, is it possible that such an oval shape enclosed system is responsible for the given behaviors of C02 molecules in reference to their vibrations? The answer to that question is; yes.

(37). https://scied.ucar.edu/carbon-dioxide-absorbs-and-re-emits-infrared-radiation#:~:text=Molecules%20of%20carbon%20dioxide%20(CO,CO2%20molecule%20to%20vibrate.

(38). http://chemistry.elmhurst.edu/vchembook/globalwarmA5.html https://blogs.ei.columbia.edu/2017/11/13/where-is-all-that-carbon-dioxide-is-going/

(39). https://education.seattlepi.com/gases-found-outer-space-5888.html

“Water vibrations happens every day. “The first energy-rich pulse causes the molecules to vibrate. …A water molecule that is still vibrating will not absorb the energy from the second light pulse. By measuring how much light passes through the water, the number of water molecules still vibrating can be determined.” (40). Moreover, “The production of sound is also the same underwater. When you strike an object, vibrations from the underwater object start to bump surrounding water molecules. The submerged human ear does not hear the sound as easily as above ground. It requires a high frequency or a really loud volume for the human ear to hear it.” (41). 

(40). https://www.sciencedaily.com/releases/2004/02/040206084932.htm#:~:text=The%20first%20energy%2Drich%20pulse%20causes%20the%20molecules%20to%20vibrate.&text=A%20water%20molecule%20that%20is,still%20vibrating%20can%20be%20determined.

(41). https://sciencing.com/water-affect-sound-8510076.html#:~:text=The%20production%20of%20sound%20is,human%20ear%20to%20hear%20it.

Flat Earthers claim that the dome itself is made up of some unseen type of impenetrable glass force. A covering – a “Raqiya” – which means “an expanse” “a visible arch of the sky,” as described in chapter four. If it is an unseen impenetrable glass, then perhaps the vibrations are due to C02 molecules bouncing off the flat Earth firmament/dome. “A glass has a natural resonance, a frequency at which the glass will vibrate easily. Therefore the glass needs to be moved (as pictured on a flat stationary Earth model where the skies/dome is doing the moving – T. Perez) by the sound wave at that frequency. If the force from the sound wave making the glass vibrate is big enough, the size of the vibration will become so large that the glass breaks.” (42).

(42). https://salfordacoustics.co.uk/how-to-breaking-glass-with-sound#:~:text=A%20glass%20has%20a%20natural,large%20that%20the%20glass%20breaks.

However, due to the sheer size of the Earth, if the firmament is a glass, we can rest assured that the unseen impenetrable glass will not “break.” Molecules of C02 are simply not strong enough to shatter a hypothetical impenetrable glass dome, but they can vibrate off of it.  Can an heterogeneous mixture be indicated? “A heterogeneous mixture contains two or more ingredients or phases. The phases might be at least two solids, liquids, or gases, or a solid liquid (suspension), liquid/liquid (emulsion), gas/liquid (aerosol), or gas/solid (smoke). The different phases mix together, but are physically separate.” (43). If we were to consider a heterogeneous causing a separation of atmosphere and water, like when oil and water are combined, that do not mix evenly, but instead form two separate layers, is it not possible or at least feasible, that the components or elements that make up said heterogeneous are indeed made up of a two layer separation of atmosphere and water? The answer to that question is; yes. 

By make up, we all know that water is made up of three atoms; two hydrogen and one oxygen. The universe or “space” itself is made up of a plasma of hydrogen and helium, as well as electromagnetic radiation, magnetic fields, neutrinos, dust, and cosmic rays. But out of this make up, “Helium and hydrogen make up 99.9 percent of known matter in the universe, according to Encyclopedia.com. Even so, there is still about 10 times more hydrogen than helium in the universe”… “Oxygen, the third most common element, is about 1,000 times less abundant than hydrogen.” (44). Electrons represent only ~ 0.0005 of mass.

Hydrogen, when in its pure state, separated from water, is an odorless colorless gas. Hydrogen can also exist in plasma form called hydrogen plasma. The plasma state can be contrasted with the other states: solid, liquid, and gas. “…And when particles from the solar wind are excited by the earth’s magnetic field lines, they can also form a plasma.” (45). Hence heterogeneity may be at play here; separating the atmosphere (the lower parts of the firmament dome with its higher divisional layers of atmospheric conditions) with what is outside the dome, or as it is commonly called the cosmos. To flat Earthers, the dome is the actual universe of expanse or “Raqiya.” where the Sun and Moon exist rotating and in perpetual circular motion around the flat Earth.

(43). https://www-livescience-com.cdn.ampproject.org/v/s/www.livescience.com/amp/58498-why-is-hydrogen-the-most-common-element.html?amp_js_v=a6&amp_gsa=1&usqp=mq331AQHKAFQArABIA%3D%3D#aoh=16128128667653&referrer=https%3A%2F%2Fwww.google.com&amp_tf=From%20%251%24s&ampshare=https%3A%2F%2Fwww.livescience.com%2F58498-why-is-hydrogen-the-most-common-element.html

(44). https://fusenet.eu/what-plasma#:~:text=Hydrogen%20gas%20consisting%20of%20electrons,of%20separate%20protons%20and%20electrons.&text=And%20when%20particles%20from%20the,can%20also%20form%20a%20plasma.

(45). Ibid.

You will recall chapter 13, that the Sun and Moon are local; that is that they are relatively close to us at about 3,000 to 6,000 miles away, and not at 93 million miles (as in the case of the Sun) or 250 thousand miles away (as in the case of the Moon). You will also recall what is written in chapter 6 with reference to the atmosphere.

“The clouds and the sky are said to be at an average height of 6,500 to 20,000 – 22,000 miles high above the Earth. While Earth’s atmosphere is said to be 300 miles thick (from ground to top). With most of it within 10 miles of the surface. There are five main layers to this atmosphere. Working our way up from ground level we have; the Troposphere – 0 – 25 km (0 – 7 miles high), the Stratosphere – 25 – 50 km (7 – 31 miles high), the Mesosphere – 50 – 80 km (31 – 50 miles high), the Thermosphere – 80 – 700 km (50 – 440 miles high), and the Exosphere – 700 – 10,000 km (440 – 6,200 miles high).”

With this understanding in mind, modern scientists are now claiming that the Moon is in our very own atmosphere. For all intent and purposes the Moon is, more than likely, located in the exosphere, which lies between 440 – 6,200 miles just above the thermosphere, and not 238,900 miles away. Hence, an arc of trajectory is not so much dependant only upon an oval shaped dome, or firmament, in so much as they more concerned with the dip during velocity from point A to point B, while attempting to fly by circular moving objects like our Moon, the planets (wandering stars), and various orbital asteroids.

See the following Google pictures for confirmation concerning the claim that the Moon is in our atmosphere, or you can always look it up yourself.