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

Chapter 21: A Man's Yearly Path, the Analemma - and the number 137

A Man's Yearly Path

In the TYCHOS, Earth proceeds at 1.6km/h(or ≈1mph), covering an annual distance of ca. 14036km (a distance only 1280 km longer than Earth's own diameter of 12756 km). 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 30° or so. It follows that, due to these combined rotational and translational motions, the path traced by a man (standing still in one spot for a full year) will be a trochoid (opens in a new tab) 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 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 (and 'heights') of these stellar trochoidal paths will vary depending on the celestial latitude of any given star - and on the observer's earthly location. However, in the case of stars located along the Earth's equatorial ecliptic, these will NOT exhibit any visible trochoidal curve; instead, they will just appear to proceed along a straight line - while periodically retrograding (i.e. reversing direction) whenever Jim, our astronomer, is temporarily 'carried backwards' in relation to the Earth's forward motion (see above graphic).

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

Source: "Heliocentric parallax" - by Michael Richmond (2014) (opens in a new tab)

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 exhibit such 'looping' paths, went on to formulate an abstruse theory that he named "Stellar Aberration" - or "the Aberration of Light". But more about that in Chapter 22.

The below image is from a 'mainstream' astronomy website. It shows that even our nearmost star, Proxima Centauri, is observed to proceed along a similar trochoidal path as Vega's (albeit a 'flatter' one, due to Proxima's different celestial latitude / viewing angle):

Source: "Hunting for Planets Around Proxima Centauri" - by David Dickinson (2016) (opens in a new tab)

Note the absurd, official explanation: “The looping action is the result of the Earth’s motion around the Sun”. Pray tell, how could this be the case? Surely, if our Solar System were moving at 800000 km/h (as officially claimed), thus covering some 7 billion kilometers annually, this looping pattern should be far more elongated - by several orders of magnitude; to wit, the spaces between the loops should be enormously larger and the loops themselves (which would only represent the 300 Mkm diameter of the Earth's supposed orbit around the Sun) would hardly be noticeable: 300 Mkm is a mere 0.0043% of 7 billion kilometers!

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's yearly motion 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 an 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. For the remaining part of the year, our clocks will 'drunkenly' be slipping in-and-out of sync with the Sun by as many as +16.5 minutes or -14 minutes, depending on the season / time of year.

What exactly causes this wondrous analemma phenomenon? Of course, the vertical component (December-June) of the analemma is due to the Sun’s shifting elevation between winter and summer (23.5°X 2 = 47°) caused by Earth's 23.5°-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 current theory has it, it is caused by Earth's elliptical orbit and its variable velocity around the same*. This, we are told, would account for the Sun’s zenith to oscillate in our skies by more than half-an-hour. What sort of magical forces would cause Earth to speed up and slow down? And why would its orbit be elliptical? Here on Earth, there are no similar phenomena to be observed in nature. Yet, this has somehow been accepted as scientific fact - in absence of any experimental corroboration.

*NOTE: Current theory is falsified by the observable fact that the Sun appears to "accelerate" around June & July, when the Sun-Earth distance reaches its maxima (i.e. when a "deceleration" would be expected) - in direct contradiction with Kepler's 'laws'.

Indeed, the most 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 (opens in a new tab) 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 full annual east-west oscillation of the Sun around its ‘mean zenith’ amounts to 31.44 minutes. As you will recall, we already met this peculiar figure in Chapter 19 where I mentioned how one might ‘mathematically expect’ a TYCHOS solar year to last for 365.22057 days, i.e. circa 31.5 minutes less than the Gregorian solar year of 365.2425 days. However, such a calculation doesn’t take into account either the trochoidal path of the terrestrial observer nor the alternating Sun-Earth orbital directions and fluctuating Sun-Earth distances, nor the 23.5° tilt of our planet’s polar axis.

Now, to find the average rate of oscillation of the Sun over the four quadrants of our celestial sphere (i.e., the four seasons), we must divide this figure by 4:

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 sidereal year there are 525969.17 minutes. We see that our 7.86 value is 0.00149438% of 525969.17 - i.e. very nearly 0.00149326% !

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

  • 1 All astronomical observations must necessarily take into account the trochoidal rotation 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 All apparent stellar motions and parallaxes are affected by what we may call "a man's yearly path".

My below graphic is my best attempt at a 'visual summary' of the all-important looping motion of any earthly observer:

The TYCHOS and the "Magic" 137 number

In Chapter 12 we saw that the solar year is shorter than the sidereal year. Our earthly estimates of the average daily distance covered by the Sun are of course based on the shorter solar year. However, a hypothetical observer on the Sun – let’s call him “Prof. Sunstein” - will gauge his own ‘mean daily motion’ against the full sidereal year of 1440 min (which has the Sun returning in practically the same ‘absolute’ celestial position), rather than the solar year of 1436.024 min – a 0.2672% difference. In fact, here on Earth we see the Sun moving daily by 3.976 min of RA (on average) which amounts to about 0.2672% of 1440 min. Since the Sun revolves once around the Earth (completing its annual 939 943 910-km journey) it will subtract one day–or "a 0.2762% slice"–from our earthly calculations. Hence, our Prof. Sunstein (who won't be subjected to this illusion) will more correctly estimate the Sun to move by 0.2672% of its orbital circumference every day.

0.2672% of 939 943 910 km ≈ 2 596 125 km

This 2 596 125 km value turns out to be quite interesting, because it is approximately 1/137th of 355 724 597 km, i.e. the circumference of the Earth’s PVP orbit:

355 724 597km / 2 596 125 km ≈ 137.02

In other words, we may say that the distance covered by the Earth in one Great Year (as it completes one full PVP orbit in 25344 solar years) is about 137 times larger than the daily distance covered by the Sun. Or, to put it another way: for each daily rotation of the Earth, the Sun covers a distance corresponding to 1/137th of the Earth's orbital circumference.

Well, it so happens that this peculiar 1/137 ratio is one of the most 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) (opens in a new tab)

In fact, the number 137 has preoccupied our world's most eminent physicists for decades, 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" - a video featuring Professor Eaves) (opens in a new tab)

Without delving too deeply into nuclear physics, a domain beyond the scope of this book, suffice to remind the reader that electrons have long been thought to revolve around an atom's nucleus (in 1912, Niels Bohr proposed his famous model of the atom, where the electrons orbit around the atomic nucleus "much like planets orbit the Sun"). Today, it is believed that the 'orbital velocity' of electrons is 1/137th the speed of light. Theoretical physicists refer to this 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) (opens in a new tab)

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, the Sun may be conceptualized as the "electron" that revolves around - and in the opposed direction of - the spinning "nucleus" (which we may envision as 'the central magnetic field' constituted by the Earth's PVP orbit). As we saw above, for every diurnal rotation of the Earth, the Sun moves by a distance corresponding to 1/137th of the PVP orbit's circumference. Could this be entirely accidental? Or could it perhaps be a meaningful clue towards a better understanding of the 137 riddle that has our world's theoretical physicists vainly scratching their heads? There certainly couldn't be any more fascinating prospect than discovering that the microcosm and the macrocosm both obey to the same 'universal ratios'.

We then have this quote by John K. Webb of the University of New South Wales (Australia), a most industrious 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?" (opens in a new tab)

In another speech, John K. Webb further muses about the perplexing issue of the observed, spatially-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..." Once more, the TYCHOS model offers a plain explanation for this 'very strange' phenomenon: since the Earth slowly proceeds through space at 1.6 km/h (along a virtually straight line rather than around an annual circle), the stars "to our left" will seem to move in the opposite direction of the stars "to our right". This is also why stars exhibit both 'positive' and 'negative' parallax - as will be thoroughly explicated 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, 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 of Aplha amounted to about "0.0000057" (of the 137 value).

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

Well, in the TYCHOS model - as you may recall from Chapter 11 - the Earth moves each day along its PVP orbit by 38.428 km

So could this minuscule 'Alpha variation' that John K. Webb detected possibly be related to the Earth's diurnal motion? If not, we shall have to chalk this up to yet another extraordinary coincidence, the odds of which you may choose to characterize as "astronomical" or "atomical".

In conclusion, as assessed within the TYCHOS paradigm, the 1/137 ratio would not only seem to be 'reflected' by the Sun's daily motion vis-à-vis the "nucleus" represented by the Earth's PVP orbit, but its tiny observed variation can also be shown to be ascribable to the daily motion of the Earth itself! One can only marvel at the explanatory power of the TYCHOS model which, as we progressively test its tenets against empirical observations, would even appear to extend to arcane quandaries of physics such as the mysterious "1/137 fine-structure constant".

A recommended viewing (were it only to realize how much effort is being deployed by modern-day physicists to try and make sense out of the "1/137 dimensionless ratio"): "Why Is 1/137 One of the Greatest Unsolved Problems In Physics?" - by PBS SPace Time (opens in a new tab)

At this juncture, it would appear that we have a solid groundwork with which we can start to dismantle the heliocentric theory - once and for all. However, we will first need to demonstrate, in methodical fashion, that our world's most celebrated scientists were all "at sea" as to the true geometric configuration of our Solar System. A few of them 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 convoluted theories put forth by these two icons of science were based on illusory observations, fallacious interpretations and - quite literally - on thin air.