Chapter 14: Curing Newton's headache - the Moon
14.1 The Moon’s bewildering motions
To Sir Isaac Newton, the Moon’s motions were notoriously problematic, causing him much misery and sleepless nights. An astronomy essay published by S. M. Alladin and G. M. Ballabh in August 2005 contains the following amusing anecdote regarding Sir Isaac’s exasperation with the seemingly intractable motions of the Moon:
“The motion of the Moon is very complicated. Sir Isaac Newton is supposed to have told his friend Halley that lunar theory made his head ache and kept him awake so often that he would think of it no more." "Dynamics of the Sun-Earth-Moon system" - by S.M. Alladin and G.M. Ballabh(2005) (opens in a new tab)
The reason they caused so much torment to Sir Isaac can be gleaned from Figure 14.1.
Fig. 14.1 The Moon’s motions traced out by the Tychosium 3D simulator
It is quite ironic that the greatest astronomical controversies revolve around our own Moon’s motions. After all, our Moon is the nearmost and most extensively studied celestial body. One would presume the scientific community had fully settled the matter after all this time. How can the Moon’s motions still be such a hotly debated question?
“Lunar theory attempts to account for the motions of the Moon. There are many small variations (or perturbations) in the Moon’s motion, and many attempts have been made to account for them.” "Lunar Theory" - Wikipedia (opens in a new tab)
Attempts. Just attempts! The Wikipedia entry on ‘lunar theory’ goes on to say that “after centuries of being problematic, lunar motion is now modeled to a very high degree of accuracy”. Well, that is simply untrue since today’s scientists are still hard at work trying to wrap their heads around the Moon’s inexplicable and seemingly anomalous orbital motions, as this abstract from a 2011 scientific paper concludes:
“Thus, the issue of finding a satisfactory explanation for the anomalous behavior of the Moon’s eccentricity remains open.” "On the anomalous secular increase of the eccentricity of the orbit of the Moon" (opens in a new tab) by Lorenzo Iorio (2011)
Back in the day, the Moon’s baffling motions caused much pain in Newton’s brain as they stubbornly refused to comply with his theory of universal gravity. Let us take a brief look at what the Moon controversy was all about, as documented in the astronomy literature:
Fig. 14.2 Extract from “The controversy surrounding the secular acceleration of the Moon’s mean motion”.
To this day, no consensus has been reached regarding these apparent variations of the Earth’s and the Moon’s orbital and/or rotational speeds. The astronomy literature offers only frail theories and unending flame wars about ‘non-gravitational effects’ which are supposed to account for the observed phenomena. A host of whimsical effects have been proposed over time, such as ‘tidal forces’, ‘core-mantle coupling’, assorted ‘turbulences’ and ‘planetary perturbations’. Astronomy historian John Phillips Britton remarked in a 1992 essay that the Moon’s acceleration...
...“was proving an embarrassment to theoretical astronomers, since no gravitational explanation for this phenomenon could be found.” "Models and Precision: The Quality of Ptolemy’s Observations and Parameters - App. 1, Secular accelerations of the Sun and Moon” (opens in a new tab) by John Phillips Britton (1992)
Eventually, astronomers turned to geologists for assistance and a provisional ‘lunatic’ consensus was crafted: ‘tidal friction forces’ were said to slow down Earth’s rotation and at the same time speed up the Moon’s motion! However, in the introduction to his 1972 paper “Non-gravitational Forces in the Earth-Moon System”, Robert Russell Newton (renowned for his extensive work on the apparent changes of the Earth’s rotation rate) curtly states:
“There are no satisfactory explanations of the accelerations. Existing theories of tidal friction are quite inadequate.”
Further on in his paper, R.R. Newton concludes:
“We are seriously lacking in mechanisms to explain the non-gravitational forces. The only mechanism of tidal friction (the ‘shallow seas’ model) that has been evaluated quantitatively provides only one-fourth of the necessary amount of friction, and it does not provide for much change with time within a period as short as historic times.”
In fact, the Moon’s motions were―and still are―in serious conflict with Newton’s gravitational laws. It is a matter of historical record that his theories were contradicted by the Moon’s “inexplicable, renegade behavior”, and that this plain fact ignited a humongous controversy in the scientific community, which, incredibly enough, remains unresolved to this day. The reason why I am stressing this point is that you shouldn’t let anyone tell you the old Moon controversy has already been settled. That would be a barefaced lie, flying in the face of what has repeatedly been admitted in the academic literature, past and present, as I am partially documenting here.
Yet, what astronomy students are taught today is that the Moon’s utterly bewildering motions were eventually ‘figured out’ by some of the most revered scientists of our times (e.g., Euler, Horrocks, Lagrange, Laplace, Clairaut, Dunthorne, Mayer, Einstein, to name but a few), all of whom contributed with a plethora of ‘terms’, ‘perturbations’ and ‘non-gravitational effects’ intended to account for the observed anomalies. Eventually, a disjointed hodge-podge of assorted theories was concocted in order to rescue Newton’s sacrosanct gravitational laws. The grievous affair is described in the Wikipedia entry on ‘lunar theory’:
“The analysts of the mid-18th century expressed the perturbations of the Moon’s position in longitude using about 25-30 trigonometrical terms. However, work in the nineteenth and twentieth century led to very different formulations of the theory so these terms are no longer current. The number of terms needed to express the Moon’s position with the accuracy sought at the beginning of the twentieth century was over 1400; and the number of terms needed to emulate the accuracy of modern numerical integrations based on laser-ranging observations is in the tens of thousands: there is no limit to the increase in number of terms needed as requirements of accuracy increase.” "Lunar Theory" - Wikipedia (opens in a new tab)
As you can see, there is apparently no limit to the number of terms required to explain the Moon’s motions within the Copernican framework. The number of terms grows year after year, with no end in sight, and astronomy professors and students are assuredly not encouraged to question the validity of the same, unless they are prepared to be labeled ‘heretics’ by their respective institutions. To say the least, this Moon business is not the most commendable page in the history of science. In his book “Pierre-Simon Laplace, 1749-1827: A Life in Exact Science”, Charles Coulston Gillispie writes:
Fig.14.3 Extract from "Pierre-Simon Laplace- A Life in Exact Science" (opens in a new tab) by Charles Coulston Gillispie (1997)
Likewise, the Edinburgh Review or Critical Journal highlighted the fact that the Moon’s observed motions, with its ‘anomalies’ and ‘inequalities’, contradict Newton’s gravitational theories:
Fig.14.4 Extract from The Edinburgh review or Critical Journal (1808) (opens in a new tab)
The controversies over the Moon’s motions ranged from its observed periodic (short-term) motions all the way to its secular (long-term) motions over the centuries. The latter triggered a gigantic and still unsettled debate since studies of the ancient solar/lunar eclipses suggested that the Moon, as viewed from Earth, was continually ‘accelerating’ over time, despite the fact that its orbital speed was, paradoxically enough, said to be slowing down! Other theories proposed that it was actually Earth’s rotation that was decelerating. In short, and to put it bluntly: a sorry mess.
“Astronomers who studied the timing of eclipses over many centuries found that the Moon seemed to be accelerating in its orbit, but what was actually happening was that the Earth’s rotation was slowing down. The effect was first noticed by Edmund Halley in 1695, and first measured by Richard Dunthorne in 1748, though neither one really understood what they were seeing.” ["Is the Moon moving away from Earth? When was this discovered?"(https://www.tychos.info/citation/134A_Moon-Moving-Away.htm (opens in a new tab)) by Britt Scharringhausen (2019)]
Perhaps the most cringeworthy attempt to salvage Newton’s ‘inviolable laws’ was that of Paul Dirac, hailed as “one of the most significant physicists of the 20th century”. F. R. Stephenson published a paper in the Journal of the British Astronomical Association, saying that:
“The most plausible cause of a non-tidal acceleration is a possible time rate of change of G, as was first proposed by Dirac. Such a change would affect the planets as well as the Moon, producing accelerations (or decelerations) in the exact ratio of the mean motions.”
How so? A “time rate of change of G”―the all-important ‘gravitational constant’? It’s like saying, “Hey, gentlemen, let’s just tweak that ‘constant’ and make it a ‘non-constant’ et voilà: Newton wins again!” It is quite comical to read about the countless ad hoc ‘remedies’ whipped up by the watchdogs of heliocentrism in their feverish travails to patch up the cracks in their crumbling edifice. The time has truly come to clear up this sorry state of affairs. I shall start with these supposed secular ‘accelerations’ and/or ‘decelerations’ of the Moon and demonstrate how the TYCHOS can account for them in the simplest manner imaginable. The supplied graphics should help visualize what has caused so much confusion and controversy over our Moon’s observed motions.
14.2 Is the Moon accelerating? Or decelerating?
Isaac Newton credited his own mentor and sponsor, Edmond Halley, with having first discovered the ‘secular acceleration of the Moon’. Interestingly, as shown in Figure 14.5, Halley was spurred by “a perceived need to prove that the universe was not eternal”...
Fig.14.5 Extract from “Ancient Astronomical Observations and the Study of the Moon’s Motion”,
by John M. Steele.
The diagram in Figure 14.6 illustrates how and why the Moon will indeed appear to accelerate over the centuries, although this is only an illusion caused by the Earth-Moon system circling around the PVP orbit, as proposed by the TYCHOS model:
Fig. 14.6 The apparent acceleration of the Moon - and apparent deceleration of the Earth's rate of rotation - are illusory.
Once you know that Earth is slowly moving around its PVP orbit, the apparent accelerations and decelerations of the Moon and/or of Earth’s rotation are seen for what they are: an optical illusion. Our solar system is in reality an extremely stable and reliable ‘clockwork’, with all its components moving in perfectly circular orbits and rotating around their axes at perfectly constant speeds. Ever since the advent of heliocentrism, astronomers and physicists have been busy filling our universe with perturbations and aberrations that simply are not there. The simplicity, harmony and utter regularity revealed by the TYCHOS are not only intuitively gratifying but also provide a cure for the academic cognitive parallax of observing one thing and believing another. So, is the universe eternal? Perhaps not, but it’s probably not going to drift apart or grind to a halt anytime soon.
I shall now further demonstrate how the illusion arises that our Moon (or Earth’s rotation) is subject to ‘accelerations’ and ‘decelerations’. We are told by mainstream astronomers that the Earth’s rate of rotation (axial spin) is gradually slowing down. We are also told that the Moon’s orbital motion is slowly speeding up in relation to Earth, and yet, at the same time, slowing down in relation to the stars. It sounds utterly bewildering but it is nevertheless easily explained by the gradually changing perspectives inherent in the Earth-Moon system’s slow, clockwise revolution around the PVP orbit.
Imagine a man in London always looking in the direction of Earth’s 1.6 km/h motion around the PVP orbit (for the sake of argument, let us assume the man is able to sense this direction at all times). As a Copernican disciple, the man is unaware of Earth’s PVP orbit. For 6336 years, our immortal man carefully monitors the Moon’s celestial positions as it moves against the background stars:
Fig. 14.7 A man in London (L) is always facing in the direction of Earth’s orbit. The Moon’s constant point of return (M) is (almost) facing the same stars, or its so-called ‘sidereal period’.
As shown in Figure 14.7, by the end of that 6336-year period our man in London will find himself at a 90° angle from where he started (year 0) in relation to the universe. Now, 90° is of course ¼ of 360°, and ¼ of Earth’s equatorial circumference (40075 km) amounts to 10018.75 km. This means that our man in London has been, so to speak, ‘slipping out of synch’ with the Moon each year by about 1.6 kilometers of Earth’s circumference or, more precisely, by 1.58124 km annually (10018.75 km / 6336 = 1.58124 km).
10018.75km / 6336 = 1.58124km (per year)
Well, this is most interesting, because 1.58124 km equals 0.0039457% of 40075 km. As we saw in Chapter 12, the Earth moves annually by 14036 km, corresponding to 0.0039457% of the PVP’s orbital circumference of 355 724 597 km. This motion makes us ‘meet up’ with the Sun at a slightly earlier point of its own orbit: 37087 km ‘earlier’, or 0.0039457% of the solar orbit’s circumference of 939 943 910 km. And remember, 1 solar year equals 0.0039457% of 1 TYCHOS Great Year (25344 years).
At the end of these 6336 years of patient observation, our man in London will probably conclude that Earth’s rotation has decelerated by 6 hours of RA (25% of 24 hours) in relation to the Moon. Or he might wonder whether it is the Moon’s orbital motion that has accelerated in relation to Earth. However, to his growing puzzlement, the latter hypothesis clashes with the fact that the Moon has, on the other hand, appeared to decelerate in relation to the starry background.
It is easy to see that our man in London will remain stumped at his own observations as long as he believes the Earth scurries around the Sun and is unaware of the PVP orbit. To be sure, under the heliocentric paradigm, the observed secular motions of our Moon are not only bewildering: they are utterly inexplicable from any rational, optical, geometrical or physical perspective.
To cut a long story short, the apparent accelerations and/or decelerations of the Earth and the Moon are completely illusory, as the above diagrams have hopefully clarified. The two bodies move at constant speeds in circular (albeit somewhat eccentric) orbits, much like all the other bodies in our solar system. Another misconception currently promoted by Copernican astronomers, namely that the Moon is receding from Earth at about 4 cm per year, will be clarified in section 14.5.
14.3 The Moon’s evection explained by the TYCHOS
We shall now examine what astronomers define as the largest observed inequality or anomaly of the lunar motion: the so-called ‘lunar evection’ (or longitudinal oscillation).
“In astronomy, evection (Latin for “carrying away”) is the largest inequality produced by the action of the Sun in the monthly revolution of the Moon around the Earth. The evection, formerly called the Moon’s second anomaly, was approximately known in ancient times, and its discovery is attributed to Ptolemy. Evection causes the Moon’s ecliptic longitude to vary by approximately ± 1.274° (or ± 4586.45″ seconds of arc), with a period of about 31.8 days. The evection in longitude is given by the expression +4586.45'' sin(2D-L), where D is the mean angular distance of the Moon from the Sun (its elongation), and L is the mean angular distance of the moon from its perigee (mean anomaly). It arises from an approximately six-monthly periodic variation of the eccentricity of the Moon’s orbit and a libration of similar period in the position of the Moon’s perigee, caused by the action of the Sun.” "Wayward moon receding from Earth" (opens in a new tab) by Bruce Mc Clure (2017)
This ‘evection’ causes the Moon’s ecliptic longitude to vary by approximately ±1.274° over a period of about 31.8 days. However, to compare this variation with one annual 360° solar revolution we need to know how much the Moon oscillates during just one of its 27.3-day sidereal orbits around the Earth:
This means that the Moon (in the course of ONE revolution around the celestial sphere) will appear to oscillate back and forth by about:
27.3 days / 31.8 = 0.85848
Ergo, the east-west oscillation of the Moon will add up to (in degrees):
0.85848 x 1.274° ≈ 1.0937°
Total 27.3-day east-west oscillation = 1.0937° x 2 = 2.1874°
Viewed from Earth, the Moon subtends ~0.54° on average. We see that 2.1874 / 0.54 ≈ 4.05. The diameter of our Moon is 3474 km. Hence, the total east-west displacement of the Moon, or what might be termed the ‘kilometric amplitude of the evection’, will add up to:
3474 km x 4.05 ≈ 14069.7 km
This is nearly identical to 1 EAM (Earth’s annual 14036-km motion around the PVP orbit)
Fig. 14.8 The Moon would always return to the same position if it were equidistant from Earth at all times. However, the Moon is observed to oscillate in relation to its mean position by about two lunar diameters either eastwards or westwards, which is known as the Moon’s evection.
That is yet another of the Moon’s pesky ‘inequalities’ put to rest by the EAM, this time with regard to longitudinal oscillation. In Chapter 13, we saw that the radial oscillation of the Moon’s perigee amounted to about 14044 km and that the radial oscillation between its perigee and apogee amounted to 42108 km (3 x 14036 km). The explanatory power of the TYCHOS model is truly astonishing!
We may thus confidently assert that the major, apparent "inequalities, irregularities and anomalies" of the Moon's spatial motions are, once more, all directly related to (and are a natural consequence of) the Earth-Moon system's annual 14036-km motion around its PVP orbit.
14.4 Computing the apparent velocity variation of the Moon
In section 14.3, we showed that the Moon’s evection (longitudinal oscillation) is near-identical to the EAM (14036 km). Let us now verify whether this oscillation―ascribed by Kepler to periodic variations of the Moon’s orbital speed and to its orbit’s alleged ellipticity―is related to the Earth’s orbital speed of ~1.6 km/h. Assuming a constant orbital speed of 3656 km/h, the Moon would employ 230.34 minutes to cover a distance of 14036 km:
14036 km / 3656 km/h = 3.839 hours (or 230.34 minutes)
Our civil calendar year consists of 365 days (or 525600 min). This is the timespan against which we gauge the annual lunar oscillations. We see that 230.34 min amounts to 0.04382% of 525600 min. We have thus obtained the percentage value of the Moon’s apparent orbital velocity responsible for the so-called ‘lunar evection’.
0.04382% of 3656 km/h (the Moon’s constant orbital velocity) = 1.602 km/h
This is in excellent agreement with the Earth’s orbital speed of 1.601169 km/h, as proposed by the TYCHOS model. The graphic in Figure 14.9 will help understand why Kepler and his fellow heliocentrists fell for the illusion of velocity variations in our Moon’s orbital motions around the Earth:
Fig. 14.9 The "0.8 factor".
We see that the variation amplitudes will be 1.6 at the June and December solstices, 0.8 at mid-season and 0 at the March and September equinoxes. Note that, since these are amplitude variations, even the negative values should take the + sign when computing the average amplitude of the Moon’s oscillations. Hence, the mean variation coefficient of the Moon’s apparent orbital speed—and indeed of all our surrounding planets—will be:
0 + 0.8 + 1.6 + 0.8 + 0 + 0.8 + 1.6 + 0.8 = 6.4 / 8 = 0.8
This mean variation coefficient (henceforth, MVC) of 0.8 will obviously affect our perception of the motions of all of the bodies of the system in relation to the stars, creating the appearance of alternate acceleration and deceleration. In Chapter 24 we will see how Dayton Miller’s interferometer experiments lend support to the MVC. In Chapter 25 we shall see how even the minuscule parallactic behaviour of our nearmost stars can be accounted for by Earth’s slow progression around its PVP orbit, thus resolving the ‘mystery’ of the coexistence of positive and negative stellar parallaxes.
For now, what you’ll need to envision and keep in mind is that Earth travels at snail pace (1.6 km/h) around the PVP orbit, like a huge merry-go-round in slow motion, giving the ‘short-term impression’ of moving in a straight line (it only curves by about 1.42° per century). It is this formerly unknown motion of the Earth-Moon system which gave Sir Isaac Newton so many headaches and sleepless nights.
14.5 Is the Moon waving good-bye to Mother Earth?
Modern astronomers will tell you that the Moon recedes from Earth each year by a little less than 4 centimeters. We shall now see how the TYCHOS model can account for, and thus dismantle, the rather alarming notion that the Moon is slowly ‘waving good-bye’ to Mother Earth.
“The Moon is gradually receding from the Earth, at a rate of about 4 cm per year. This is caused by a transfer of Earth's rotational momentum to the Moon's orbital momentum as tidal friction slows the Earth's rotation.” "What will be the fate of our Moon?" - Physlink.com (opens in a new tab)
“Although the moon’s distance from earth varies each month because of its eccentric orbit, the moon’s mean distance from Earth is nonetheless increasing at the rate of about 3.8 centimeters (1.5 inches) per year. That’s about the rate that fingernails grow.” "Wayward moon receding from Earth" - EarthSky.org (opens in a new tab)
According to the TYCHOS, the Moon is not receding from Earth and is not going to vanish in space. As we shall now demonstrate, the Moon’s annual ‘4-cm recession’ is yet another illusory effect arising from astronomers’ unawareness of the Earth’s PVP orbit. Their computations related to the Moon’s apsidal oscillations will thus always fail to account for the ‘secular revolution’ of the Earth-Moon system.
We know that the Moon cyclically approaches and recedes from the Earth. As we saw in Chapter 13, the Moon’s orbit oscillates back and forth by 42108 km, a value we shall call the Moon’s maximal apsidal oscillation (MMAO). So, could the secular drift of the MMAO along the TGY (25344 solar years) be responsible for what astronomers believe to be an annual ‘4-cm recession’ of the Moon? Let us find out.
If we consider that the Earth-Moon system completes a full 360° revolution in 25344 years, we can envision how the MMAO―the spatial orientation of which remains ‘fixed’ to the Moon’s orbit―will slowly revolve once in relation to the Earth in 25344 years. This makes it possible to calculate by how much the MMAO would appear to ‘drift’ annually, as viewed from the Earth and in relation to the stars:
Actual amount of the MMAO’s annual precession: 42108 km / 25344 years = 1.66 km
Now, astronomers will be using their grossly inflated star distances as a benchmark to gauge the fluctuating Moon-Earth distances since their instruments are calibrated according to heliocentric parameters. However, as will be thoroughly expounded in Chapter 23, the TYCHOS model stipulates that the stars are ~42633 times closer to us than currently believed:
Amount of apparent annual lunar recession corrected by the TYCHOS reduction factor:
1.66 km / 42633 = 0.0000389km = 3.89 cm
In conclusion, the Moon will not be parting with us anytime soon. What astromomers think is a slight annual ‘4-cm recession’ is nothing but the slow 25344-year secular precession of the MMAO, given by the tranquil 1-mph motion of the Earth-Moon system around the PVP orbit. In our current epoch, the oscillation of the MMAO is evidently in its ‘receding phase’. Over time though, there will be an ‘approaching phase’ which will reverse the apparent recession and bring things back to normal, in the good tradition of the wondrously stable and reliable ‘Swiss clockwork’ that is our solar system.