Chapter 21: A Man's Yearly Path - and the Analemma

In the TYCHOS, Earth proceeds at 1.6km/h(or ≈1mph), covering an annual distance of ca. 14036km. You will note that this distance is just a little longer (by 1280km) than Earth's own diameter of 12756km. The Earth makes a 360° rotation around its axis every 23h56min (a sidereal day), but every 24 hours (a solar day) it will rotate by 361°. Hence, from one month to another, it will rotate by another 30° or so. It follows that the path traced by a man (standing still for a full year) will be a trochoid or, more precisely, a so-called 'prolate trochoid':

A trochoid is simply a curve traced by a point fixed to a circle as it rolls along a straight line. So if we imagine a stationary astronomer in London (let's call him "Jim") looking through his telescope for a full year, our Jim will be carried around a trochoidal path (Jim is patiently monitoring the annual motions of the star Vega over a full year). Hence, here's what Jim's yearly path will look like - as viewed from above our North Pole:

Of course, unless Jim is aware of his own trochoidal motion (i.e. his 'ever-looping frame of reference"), he will be baffled and stupefied at star Vega's seemingly 'inexplicable' behavior: as Jim records (over a full year) the successive positions of star Vega on a fixed photographic plate (which, of course, will gyrate in the same manner as himself), the annual motion of star Vega will appear to trace a peculiar geometric curve known as a 'prolate trochoid':

Note that the shapes (or 'heights') of these stellar trochoidal paths will vary depending on the celestial latitude of any given star. However, in the case of a star located along the Earth's equatorial ecliptic, that star will NOT exhibit any visible trochoidal curve; instead, it will just proceed along a straight line - while periodically retrograding (i.e. reversing direction) whenever Jim, our astronomer, is briefly 'carried backwards'(in relation to the Earth's motion) - as shown in my above graphic.

Here is a modern diagram of the observed motions of the star Vega (over a 3-year period):

Source: "Heliocentric parallax" - by Michael Richmond, for Rochester Inst. of Technology (2014)

Our Jim then decides to monitor another star, then another - and then another. He finally realizes that ALL the stars in our sky exhibit trochoidal motions and/or short retrograde periods! Jim, who just isn't ready to abandon his long-nurtured convictions, may then come up with all sorts of ad hoc theories to 'explain the unexplainable'. This was, in fact, precisely the case with Astronomer Royal James Bradley who famously monitored the star Draconis for extended periods of time and, having later noticed that ALL the stars actually exhibit similar annual trochoidal paths, went on to formulate an abstruse theory that he named "Stellar Aberration" - or "the Aberration of Light". But more about that later.

The below image is from an astronomy website. It shows that even our nearmost star, Proxima Centauri, is observed to proceed along a similar (albeit 'flatter') trochoidal path as Vega:

Source: Hunting for Planets Around Proxima Centauri - by David Dickinson / 2016

Note the bizarre, official explanation: “The looping action is the result of the Earth’s motion around the Sun”. (???). Pray tell... HOW exactly would THAT play out?

We shall now proceed to show how this annual trochoidal motion (of any earthly observer) can account for other still unexplained, or dubiously interpreted celestial phenomena. Perhaps the most curious of them all is the observed annual motion of the Sun – as it traces an elongated “8”-shaped pattern in our skies. This geometric pattern traced by the Sun is known as the “Analemma”.

THE ANALEMMA - a qualitative analysis

Any patient photographer can empirically verify the existence of the Analemma by setting up a tripod and snapping pictures of the Sun at noon (say, every ten days or so) for a full year. What will be obtained is a vertically-elongated “8”-shaped curve (wider at the lower end) well-known to astronomers. In the past, the Analemma used to be printed on the pretty globes adorning people's living rooms. For some reason though, this is no longer the case.

Everyone has heard of the proverbial broken clock “which will nonetheless show the correct time twice a day”. However, not everyone knows that our earthly clocks are, strictly speaking, almost never on time. In fact, our clocks only agree with the Sun’s midday zenith 4 times a year. The remaining part of the year, our clocks will drunkenly be slipping in-and-out of sync with the Sun by as many as +16 minutes or -14 minutes, depending on the season / time of year!

What exactly causes this curious analemma phenomenon? Of course, the vertical component (December-June) of the analemma is due to the Sun’s shifting elevation between winter and summer (23.4°X 2 = 46.8°) caused by Earth's 23.4°-degree axial tilt - so no mystery there.

On the other hand, the lateral component of the analemma (i.e. the alternating East/West drift of the Sun) has not, to this day, been adequately explained in any satisfactory manner. As Keplerian theory has it, it is caused by Earth "accelerating or decelerating around its elliptical orbit”*. This, we are told, would account for the Sun’s zenith to oscillate in our skies by more than 30 minutes. What sort of magical forces would cause Earth to speed up and slow down is just unfathomable; here on Earth, there is no similar phenomenon to be observed in nature. Yet, this has somehow been accepted as scientific fact - in absence of any experimental corroboration.

*NOTE: This Keplerian theory is directly contradicted by the fact that the Sun's motion (or 'the Earth's motion' - under the heliocentric theory) appears to accelerate in June & July, i.e. when its velocity, according to Kepler, would reach its MINIMA - and as the Sun-Earth distance reaches its MAXIMA.

Another curious aspect of the Analemma is its highly asymmetric 8-shape (thinner at the top and thicker at the bottom). Now, what could possibly cause this uneven 'distribution' of the Sun's annual East-West oscillation? Various theories have been advanced - yet none have definitively settled the question. However, as we consider a "Man's Yearly Path" (see above), we see that the trochoidal motion of an earthly observer (over a full year) has a lateral displacement ratio of 3:1. And in fact, the asymmetry of the Analemma also exhibits a similar 3:1 ratio; I trust that my below comparative diagram should instantly clarify this matter. (Also, the reader may now recall the 3:1 ratio of the Moon's trochoidal apsidal precession - as illustrated in Chapter 13).

Note how the four occasions when our earthly clocks 'agree with the Sun" (i.e. June 16, Dec 24, Aug 29 and Apr 15) neatly 'coincide' with the observed Analemma - as shown in the two graphics compared above. At this point, it should be intuitively evident to the reader that the Analemma is - at least qualitatively speaking - closely related to what I like to call "a Man's Yearly Path". Let's now take a closer look at the maths involved - but don't worry: I will, as ever, keep my computations as simple as possible.

THE ANALEMMA - a quantitative analysis

So let’s see: if our clocks, in the course of the year, can be “ahead” by about 16.5min and “behind” by about 14min, the total East-West offset of the Sun in relation to the true zenith would thus amount to 30.5 minutes. Now, how then can we possibly accurately measure time and calibrate our clocks (which, of course, tick at constant speed) with the solar motion if our celestial timekeeper (the Sun) keeps “accelerating and decelerating”? Well, we simply can't.

The so-called Equation of Time is a clever man-made convention devised to deal (to the best of our capacities) with this pesky lateral oscillation of the Sun. In fairness, the Equation of Time has provided an ingenious solution to this problem. Yet, the fact remains: our clocks, as useful as they are for our daily purposes, are cosmically-speaking almost always 'offset' in relation to the Sun.

Note that the total observed annual “lateral drift” of the Sun adds up to 30.5 min (16.5min + 14 min) of RA. However, this is without accounting for the fact that an extra 3.93 min. is added by convention - via the leap-year gimmick - every four years or so. To be precise, 3.76 min. are added (over longer periods of time - since some leap years are skipped). Therefore, 0.94min (¼ of 3.76 min) should be added to the annual count of the Sun’s lateral drift, giving us a total of:

30.5 min. + 0.94 min. = 31.44 min

In other words, the true mean annual (East-West) oscillation of the Sun around its zenith amounts to 31.44 minutes. To find the average rate of oscillation of the Sun over the four quadrants of our celestial sphere (i.e. over our four seasons), we may simply divide this number by four:

31.44min / 4 = 7.86min

Note that it matters not whether this mean figure of 7.86 should take the "-"(minus) sign or the "+" (plus) sign, since the Sun's motion can be either co-directional or counter-directional vis-à-vis Earth's motion. Let's now verify whether these ca. 7.86 minutes may be related to Earth’s orbital speed of 1.601169 km/h, which amounts to a mere 0.00149326% of the Sun’s orbital speed of 107226km/h.

In one solar year there are 525948 minutes. We see that our 7.86 value is 0.00149444% of 525948 minutes or - lo and behold - almost exactly 0.00149326%!

In conclusion, we may conceptually envision the Analemma as Earth's "speedometer" - since its mean rate of East-West oscillation reflects our planet's 1.6km/h orbital speed. In addition to this important realization, we may stipulate what follows:

  • 1 All astronomical observations must necessarily take into account the annual trochoidal motion of our earthly reference frame.

  • 2 This trochoidal motion is the root cause (along with other minor factors) for our need of the Equation of Time.

  • 3 Our earthly trochoidal motion needs to be considered in all matters pertaining to stellar motions and parallaxes.

In order to summarize this chapter - and hopefully further clarify some of its contents - I submit the following graphic:


As we just saw, the Sun is empirically observed to oscillate (from east to west) by as many as 31.44 minutes in relation to an ideal / fixed / virtual meridian (or solar apex) - and consequently, in relation to our earthly clocks. We may thus say that the ‘mean solar apex oscillation’ amounts to 31.44min / 2 = 15.72min.

Now, the mean duration of ONE DAY (as generally-considered) here on Earth is 23h and 56min. However, this is without considering that the Sun is observed to oscillate, in the course of a year, by 31.44min - or by 15.72min (of 'terrestrial time') on either side of the ‘mean solar apex’. We may therefore say that the true, average diurnal distance covered by the Sun is accomplished in the 'solar time' of 23h56min + 15.72min = 23h71.72min, or 24h11.72min - or roughly 24h12min (i.e. 24.2 hours).

In 24.2 hours, the Sun will move along its orbit by 107226.1km/h X 24.2h = 2 594 872 km. This value of “2 594 872km” turns out to be most interesting, because it is almost precisely 1/137th of the circumference of the Earth’s PVP orbit of 355 724 597km (as proposed by the TYCHOS model): 355 724 597km / 2 594 872 km = 137.08.

In other words, the distance covered by the Earth in one Great Year (as it completes one PVP orbit in 25344 years) is just about 137 times the daily distance covered by the Sun. Well, it so happens that this peculiar 1/137 ratio is one of the most famous and hotly-debated ‘mysteries' in physics!

Why the number 137 is one of the greatest mysteries in physics. Does the Universe around us have a fundamental structure that can be glimpsed through special numbers? The brilliant physicist Richard Feynman (1918-1988) famously thought so, saying there is a number that all theoretical physicists of worth should “worry about”. He called it “one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man”. That magic number, called the fine structure constant, is a fundamental constant, with a value which nearly equals 1/137. Or 1/137.03599913, to be precise. It is denoted by the Greek letter alpha – α.(...) Appearing at the intersection of such key areas of physics as relativity, electromagnetism and quantum mechanics is what gives 1/137 its allure." "Why the number 137 is one of the greatest mysteries in physics" - by Paul Ratner (2018)

The number 137 preoccupied other great physicists as well, including the Nobel Prize-winning Wolfgang Pauli (1900-1958) who was obsessed with it his whole life: “When I die my first question to the Devil will be: What is the meaning of the fine structure constant?” Pauli joked. Physicist Laurence Eaves, a professor at the University of Nottingham, thinks the number 137 would be the one you’d signal to the aliens to indicate that we have some measure of mastery over our planet and understand quantum mechanics. The aliens would know the number as well, especially if they developed advanced sciences. "Fine Structure Constant - Sixty Symbols" (a short Youtube video featuring Professor Eaves)

Without going too deeply into nuclear physics, which is well beyond the scope of this book, suffice to remind the reader that electrons are believed to revolve at high speeds around an atom's nucleus (in 1912, Niels Bohr proposed his now-famous model of the atom, where the electrons orbit around the atomic nucleus "much like planets orbit the Sun"). Today, theoretical physicists refer to the perplexing, more recently-discovered 1/137 ratio as the "fine-structure constant α" (or the "coupling constant") of the electromagnetic force that binds atoms together.

"Perhaps the most intriguing of the dimensionless constants is the fine-structure constant α. It was first determined in 1916, when quantum theory was combined with relativity to account for details or ‘fine structure’ in the atomic spectrum of hydrogen. In the theory, α is the speed of the electron orbiting the hydrogen nucleus divided by c. It has the value 0.0072973525698, or almost exactly 1/137. Today, within quantum electrodynamics (the theory of how light and matter interact), α defines the strength of the electromagnetic force on an electron. This gives it a huge role. Along with gravity and the strong and weak nuclear forces, electromagnetism defines how the Universe works. But no one has yet explained the value 1/137, a number with no obvious antecedents or meaningful links." "LIGHT DAWNS" - by Sidney Perkowitz (2015)

The "magic" 137 number is also described as a constant related to an electron's magnetic moment, or the "torque" that it experiences in a magnetic field. In the TYCHOS, of course, the Sun could be considered as the "electron" that revolves at high speed around the spinning "nucleus"(i.e. the Earth). As we saw above, for every diurnal rotation of the Earth, the Sun moves by a distance corresponding to 1/137th of the circumference of the Earth's inner PVP orbit. Could this Sun-Earth 1/137-relationship (exhibited in the TYCHOS model) be purely coincidental? Or may it perhaps help elucidate and demystify this major riddle that has our world's theoretical physicists obsessively scratching their heads?

Dulcis in fundo, we then have this quote by John K. Webb (of the University of New South Wales, Australia), arguably the world's foremost researcher of the mysterious 137 number, a.k.a. the 'fine-structure constant Alpha': "There's something stange going on... a spatial variation... because when we look in one direction of the Universe we see Alpha being a little bit smaller - and when we look in exactly the opposite direction it's a little bit bigger." Source: "Is Our Entire Universe Held Together By One Mysterious Number?"

In another speech, John K. Webb further muses about the 'strange issue' of the observed, opposed variations of the constant Alpha. He explains that the two sets of data he studies are collected by two of the world's largest observatories (the Keck Observatory in Hawaii and the VLT in Chile), located practically on the opposite sides of our planet: "Using the Keck telescope, it seems as if Alpha decreases, while using the VLT, it seems as if Alpha increases. Very strange..." The TYCHOS model offers, of course, a plain and simple explanation for this 'strange phenomenon': since the Earth is slowly proceeding at 1mph along its PVP orbit (along a virtually straight line), the stars "to our left" will seem to move in the opposite direction of the stars "to our right". This is also why the stars exhibit both 'positive' and 'negative' parallax - as will be thoroughly expounded and illustrated in Chapter 25. But the best is yet to come - with regards to professor Webb's most rigorous and exacting research:

As we can read in the Wikipedia's "Fine-Structure Constant" entry, John K. Webb's first - and groundbreaking - findings (published in 1999) reported a minuscule variation of the Alpha constant ("In 1999, a team led by John K. Webb of the University of New South Wales claimed the first detection of a variation in α.") Now, this variation amounted to about "0.0000057" (of the 137 value).

Well, 0.0000057 is 0.0000041% of 137 - and 0.0000041% of 939 943 910 km (i.e. the Sun's orbital circumference) is 38.537km.

By Jove! As you may recall from Chapter 11, in the TYCHOS model, the Earth moves each day along its PVP orbit by 38.428km!

Hmm... Could this minuscule 'Alpha variation' that John K. Webb detected be related to the Earth's diurnal motion? If not, you will have to chalk this up, once more, to some phenomenal coincidence - the odds of which should make any probability statistician holler in disbelief.

In conclusion, as assessed within the TYCHOS paradigm, the "1/137 fine-structure constant Alpha" would not only seem to be 'reflected' by the daily motion of the Sun (in relation to the "nucleus" / or "magnetic field of opposed polarity" represented by the Earth's orbital path), but its tiny observed variation would also appear to be 'reflected' by the daily motion of the Earth itself!

At this juncture, I submit that we have a solid groundwork with which we can start to dismantle the heliocentric theory, once and for all. However, we first need to demonstrate - in methodical fashion - that some of our most celebrated scientists were all "at sea" as to the true geometric configuration of our Solar System. Yet, they are still hailed today for having "definitively proven that Earth revolves around the Sun" - in absence of any experimental evidence in support of this contention. Two names come to mind: James Bradley and Albert Einstein. In the next chapter, we shall see how the strange and convoluted theories put forth by these icons of science were based on illusory observations, fallacious interpretations and - quite literally - on thin air.