If Tycho thought the moon's orbital speed was related to its phase, he was, quite simply, wrong. This wouldn't be the only time he was wrong, either. We've learned a lot about the motion of solar system bodies in the four centuries since Tycho's and Kepler's time.
What, exactly, are you complaining about? The moon doesn't actually behave in a way suggested by Tycho? OK. So what?
Again, do you have any reliable data that suggests the moon moves in a way contrary to what the modern heliocentric model predicts? If you do, let's see it. It appears that all you have are quibbles about obsolete ideas. They're obsolete for a reason: they don't work, while others do work.
In astronomy, the variation of the Moon is one of the principal perturbations in the motion of the Moon.
The variation was discovered by Tycho Brahe, who noticed that, starting from a lunar eclipse in December 1590, at the times of syzygy (new or full moon), the apparent velocity of motion of the Moon (along its orbit as seen against the background of stars) was faster than expected. On the other hand, at the times of first and last quarter, its velocity was correspondingly slower than expected. (Those expectations were based on the lunar tables widely used up to Tycho's time. They took some account of the two largest irregularities in the Moon's motion, i.e. those now known as the equation of the center and the evection, see also Lunar theory - History.)
The main visible effect (in longitude) of the variation of the Moon is that during the course of every month, at the octants of the Moon's phase that follow the syzygies (i.e. halfway between the new or the full moon and the next-following quarter), the Moon is about two thirds of a degree farther ahead than would be expected on the basis of its mean motion (as modified by the equation of the centre and by the evection). But at the octants that precede the syzygies, it is about two thirds of a degree behind. At the syzygies and quarters themselves, the main effect is on the Moon's velocity rather than its position.
In 1687 Newton published, in the 'Principia', his first steps in the gravitational analysis of the motion of three mutually-attracting bodies. This included a proof that the Variation is one of the results of the perturbation of the motion of the Moon caused by the action of the Sun, and that one of the effects is to distort the Moon's orbit in a practically elliptical manner (ignoring at this point the eccentricity of the Moon's orbit), with the centre of the ellipse occupied by the Earth, and the major axis perpendicular to a line drawn between the Earth and Sun.
The Variation has a period of half a synodic month and causes the Moon's ecliptic longitude to vary by nearly two-thirds of a degree, more exactly by +2370"sin(2D) where D is the mean elongation of the Moon from the Sun.
The variational distortion of the Moon's orbit is a different effect from the eccentric elliptical motion of a body in an unperturbed orbit. The Variation effect would still occur if the undisturbed motion of the Moon had an eccentricity of zero (i.e. circular). The eccentric Keplerian ellipse is another and separate approximation for the Moon's orbit, different from the approximation represented by the (central) variational ellipse. The Moon's line of apses, i.e. the long axis of the Moon's orbit when approximated as an eccentric ellipse, rotates once in about nine years, so that it can be oriented at any angle whatever relative to the direction of the Sun at any season. (The angular difference between these two directions used to be referred to, in much older literature, as the "annual argument of the Moon's apogee".) Twice in every period of just over a year, the direction of the Sun coincides with the direction of the long axis of the eccentric elliptical approximation of the Moon's orbit (as projected on to the ecliptic).
Thus the (central) elliptical distortion of the Moon's orbit caused by the variation should not be confused with an undisturbed eccentric elliptical motion of an orbiting body. The variational effects due to the Sun would still occur even if the hypothetical undisturbed motion of the Moon had an eccentricity of zero (i.e. even if the orbit would be circular in the absence of the Sun).
Newton expressed an approximate recognition that the real orbit of the Moon is not exactly an eccentric Keplerian ellipse, nor exactly a central ellipse due to the variation, but "an oval of another kind". Newton did not give an explicit expression for the form of this "oval of another kind"; to an approximation, it combines the two effects of the central-elliptical variational orbit and the Keplerian eccentric ellipse. Their combination also continually changes its shape as the annual argument changes, and also as the evection shows itself in libratory changes in the eccentricity, and in the direction, of the long axis of the eccentric ellipse.
OK. That was a cogent and easy to follow description (and vastly better than a video trying to express the same thing).
So what's the "problem" you keep asking about? If Tycho's observations show the moon was moving differently than the tables existing at the time predicted, then either his observations (or conclusions) were wrong, or the tables (which were probably based on observations) were wrong, or the moon's motion suddenly changed by a noticeable amount (which seems exceedingly unlikely). My first guess would be the tables being not sufficiently precise. Where's the problem?
Is the moon's motion behaving as predicted by competent modern techniques, which are based on Newtonian gravity (and lots more than just the moon, earth, and sun being considered) with a dash of relativity when needed for really precise predictions? Yes or no? The reliability and precision of lunar occultation predictions suggests that yes, it is. If you say no, where's your data supporting this conclusion? In other words, where's the problem?
The problem is
an absence (in reality) of THE TILT OF THE EARTH factor (which would contribute
A : to even faster apparent motion of the moon at
equinoxes when the moon is in opposition (syzygy), (
if the moon is lagging less behind the sun at that time), and
B to even slower apparent motion of the moon at
equinoxes when the moon is in a position 90 degr. wrt the syzygy line (at quadrants) OR which would contribute
A to even slower apparent motion of the moon at
solstices when the moon is in opposition (syzygy), (
if the moon is lagging more behind the sun at that time), and
B to even faster apparent motion of the moon at
solstices when the moon is in a position 90 degr. wrt the syzygy line (at quadrants)!!! - A REMINDER :
What would make the difference if we tried to apply Alpha's explanation (see reply #1 above) for the differences in the length of the apparent solar days (due to the amount of the tilt of the earth wrt the sun) in relation to the differences in the length of the apparent moon days (due to the amount of the tilt of the earth wrt the moon)
TILT OF THE EARTH FACTOR :
When the moon is close to the tropics then the apparent moon's day has to be slower (due to the amount of the tilt of the earth wrt to the moon) because the moon goes faster to the east (lags more behind the sun). Going faster to the east contributes to the slower apparent motion of the moon.
When the moon is in the vicinity of the equator, her apparent day has to be faster for the same reason (with the opposite effect, this time).
TYCHO BRAHE'S VARIATION FACTOR :
2. The lunar longitude has a twice-monthly Variation, by which the Moon moves faster than expected at new and full moon, and slower than expected at the quarters.
3. There is also an annual effect, by which the lunar motion slows down a little in January and speeds up a little in July: the annual equation.
1. TILT OF THE EARTH FACTOR :
SOLSTICES = FAST motion
EQUINOXES = SLOW motion
OR
SOLSTICES = SLOW motion
EQUINOXES = FAST motion
2. TYCHO BRAHE's VARIATION FACTOR :
FULL MOON / NEW MOON = FAST motion (moon lags to the lesser degree behind the sun)
or vice versaAT THE QUARTERS = SLOW motion (moon lags more behind the sun)
or vice versaWhen we combine these two factors we get this result (next picture shows the situation in which MOON'S VELOCITY SYNERGY should (if HC theory were true) occur at solstices presuming that at
syzygies (not at
equinoxes) the moon lags to a greater degree behind the sun) :
Do you see the problem?
At equinoxes = moderate situations (moderate/equilibrate length of moon's apparent days throughout all 4 moon's phases)
At solstices = extreme situations (very fast moon's apparent motion at quarters, very slow moon's apparent motion during full moon/new moon phases)
Hypothetical situation in which MOON'S VELOCITY SYNERGY should (if HC theory were true) occur at
equinoxes presuming that at
quarters (not at
syzygies) the moon lags to a greater degree behind the sun) should be presented differently, but MOON'S VELOCITY SYNERGY wouldn't vanish, it would only occur at equinoxes, instead of at solstices.
One similar HC conundrum (so called DEADLY SYNERGY) has already been pointed out (some 4 years ago), here :
https://www.theflatearthsociety.org/forum/index.php?topic=62300.msg1639873#msg1639873 @Jack, it doesn't matter what causes Tycho's variations, what matters is that Tycho's noticed real phenomena :
The main visible effect (in longitude) of the variation of the Moon is that during the course of every month, at the octants of the Moon's phase that follow the syzygies (i.e. halfway between the new or the full moon and the next-following quarter), the Moon is about two thirds of a degree farther ahead than would be expected on the basis of its mean motion (as modified by the equation of the centre and by the evection). But at the octants that precede the syzygies, it is about two thirds of a degree behind. At the syzygies and quarters themselves,
the main effect is on the Moon's velocity rather than its position.
ON TOP OF THAT :
In a letter to
Dr. Bentley. Feb. 25th, 1692,
Newton says ;— “ That gravitation should be innate and inherent in matter, so that one body can act upon another at a distance — is to me SO GREAT AN ABSURDITY, that I believe no man who has, in philosophical matters, a competent faculty of thinking, can ever fall into it .” Yet many have fallen into this “great absurdity.”
Such men therefore—according to Newton — have not "a competent faculty of thinking” in philosophical matters. I am happy to be in agreement with
Sir Isaac on this important point.
Sir Robert Ball says: — “The law of gravitation ... underlies the whole of Astronomy.” (Story of the Heavens, p. 122). It does not speak very well for the Astronomy, if it is founded on an acknowledged “great absurdity.”
“The theory that motions are produced through material attraction is absurd. Attributing such a power to mere matter, which is passive by nature, is a supreme illusion. It is a lovely and easy theory to satisfy any man's mind, but when the practical test comes, it falls all to pieces and becomes one of the most
ridiculous theories to common sense and judgment.” -
Professor Bernstein, “Letters to the British Association”
IN ADDITION :
"Astronomers, and scientific men generally, strenuously oppose any comparison between their theories and the Bible, knowing that they cannot be reconciled. Of what use is it for them to say that their magnificent ideas of innumerable suns and worlds show forth the glory of God, if they cause men to have less respect for the Bible? Revelation and nature cannot disagree : if they seem to do so, man is to blame for it. Sir Horace Walpole became an infidel, because he could not reconcile Christianity with the plurality of worlds, and Modern discoveries in Astronomy and Geology with a divine revelation and the infidel Thomas Paine, and a host of other persons have based their strongest arguments upon the assumption that the Copernican theory is true, which system has been a strong fort with the infidels for many generations. Do the heavens set forth the glory of Newton, or do they declare the glory of God? In the Bible we are led to believe that the sun, moon, and stars are subservient to the earth ; that in consequence of events having taken place on the earth, these heavenly bodies were darkened; that God took five times as long to make the earth as He did the heavenly bodies. Who has a right to say that God, in giving to man an account of His creation, as contained in the First Chapter of Genesis, misrepresented the order and nature of the facts to suit man's capacity? As if man could not have understood them as easily from the Word of God as he does from the mouths of the Astronomers?"