Chapter 17: “The Great Inequality” - solved by the TYCHOS

Back in the 18th century, the spiny question of the observed behavior of Jupiter and Saturn ignited a humongous and long-lasting debate among our world’s most celebrated astronomers and mathematicians (Halley, Flamsteed, Euler, Lagrange, Laplace and Poincaré, to name just a few). What every astronomy historian will know as “the Great Inequality” is a scientific saga of epic proportions. In short, the problem was that the motions of Jupiter and Saturn did not seem to obey either the Newtonian (gravitational) theory or the Keplerian (elliptical) theory. Not a trivial problem, you may say. Surely, Newton and Kepler couldn’t possibly both be wrong, could they?

What had been observed, first by Kepler himself and later by Halley, was that Jupiter appeared to accelerate while Saturn appeared to decelerate. This was truly ominous news for mankind: it meant, according to the Newtonian 'Laws' of gravity, that Jupiter would inevitably end up crashing into the Sun, while Saturn would be driven away into the depths of space!

As we shall see, the TYCHOS can show that these apparent accelerations and decelerations were completely illusory - and that Jupiter and Saturn need not fear any imminent, catastrophic fate. But let's see how the eminent Astronomical Journal described this alarming 'discovery of the Great Inequality' - back in 1895:

Make no mistake, this was no petty matter: the very stability of our solar system was at stake! In fact, the Paris and Berlin Academies set up special prizes to encourage scientists to resolve the pesky and embarrassing matter. Leonhard Euler (the most acclaimed Swiss mathematician of all times) was the first recipient of such a prize, although his calculations showed both Jupiter and Saturn accelerating, contrary to any astronomical observation ever made!

The magnificent Isaac Newton himself had recognized the problem of the apparent “instability” of our Solar System (on the grounds of the observed behavior of Jupiter and Saturn), but he never tackled the troublesome matter while basically saying (freely paraphrasing his words) that “God should take care of this problem in due time and restore the apparent, chaotic nature of our planetary motions”. Kepler also gave up looking into the matter - and admitted that only future generations may eventually unveil the mystery of our Solar System’s apparent instability (suggested by Jupiter and Saturn’s odd behavior). Kepler, for once, was quite right about that...

Now, what you need to know is that, as seen from Earth, Jupiter and Saturn appear to conjuct about every 60 years (or actually a whisker less than 60 years, due to Earth’s 1-mph motion). Since Jupiter employs 12 years to circle around us, while Saturn employs 30 years to do so, the two will regularly “meet up” every 60 years, i.e. 5 x 12 (=60) and 2 x 30 (=60), respectively. These 60-year conjunctions of Jupiter and Saturn move around our celestial sphere in anti-clockwise manner, as illustrated below:

Les conjonctions triples Jupiter-Saturne/ J. Meeus

Here's another book extract describing the most worrisome "Great Inequality" that had all of the astronomers of the time on their toes:

Saturn and its System

Enter Lagrange and Laplace, perhaps the two most acclaimed French mathematicians of all times. The two French science icons engaged in a long struggle to try and resolve the so-called Great Inequality - in order to rescue the sacrosanct Newtonian gravitational laws. Depending on what old text books one may bump into, it was either Lagrange or Laplace who “solved the problem”, basically concluding that (according to their formidably abstruse calculations) the so-called Great Inequality - i.e. the apparently increasing gap between Jupiter’s and Saturn’s celestial longitudes - was only a temporary phenomenon which would eventually reverse course. In other words, the gap would gradually (in the course of about nine hundred years) diminish and cancel out itself. Our Solar System was, after all, a stable one. Phew!

However, it is unclear just how Lagrange and Laplace reached their “mathemagical” conclusions. In academic text books, we may only find some dreadfully complex equations and computational wizardry based on ad hoc assumptions about how “gravitational perturbations” and “tidal friction effects” might cause those puzzling inequalities. As it is, under the Copernican model’s configuration there is no plausible explanation as to why Jupiter’s and Saturn’s celestial longitudes would oscillate back and forth, as observed. In time though - and here’s where it gets funny - Lagrange and Laplace were eventually “proven right”: the apparent, relative accelerations/decelerations of Jupiter and Saturn were then observed, several decades later, as having reversed course:

“In 1773, Lambert used advanced perturbation techniques to produce new tables of Jupiter and Saturn.The result was surprising. From the mid-17th century the Great Anomaly appeared to go backwards: Saturn was accelerating and Jupiter was slowing down! Of course, such behavior was not compatible with a genuinely secular inequality.”

Stability of the Solar System in the 18th century

One of the greatest observational astronomers of the times, William Herschel, had also noticed the “back and forth” oscillations of Jupiter and Saturn: “He [Herschel] describes Saturn’s period as increasing (i.e. Saturn seemed to be slowing down) during the seventeenth century - and Jupiter’s period as diminishing (i.e. Jupiter seemed to be speeding up) and he adds – ‘In the eighteenth century a process precisely the reverse seemed to be going on.’”

So, after all, there was no apocalyptic scenario whatsoever for humanity to fear. Nonetheless, as pointed out by a number of contemporary independent researchers, the “Great Inequality” and its corollary, the very “Stability of our Solar System”, both remain - to this very day - unsolved riddles. For instance, here’s what Antonio Giorgilli (a veteran Italian expert in this peculiar area of astronomical studies and the author of “The Stability of the Solar System: Three Centuries of Mathematics”) warns the reader with:

“Su queste basi cercherò di illustrare che significato si possa dare alla domanda: “il sistema solare è stabile?”[...] Quanto alla risposta, non vorrei deludere nessuno, ma sarà: non lo sappiamo”.

("On this basis I will try to illustrate what meaning can be given to the question: "Is the solar system stable?"[...] As for the answer, I do not want to disappoint anyone, but it will be: we do not know".)

"We do not know"... Well, we obviously cannot attain any firm knowledge of our Solar System’s behavior if we haven’t even envisioned its correct geometric layout, can we? As I will presently illustrate, the TYCHOS model’s geometric layout provides the simplest imaginable explanation for the “Great Inequality”.


Now that we have looked at some historical controversies surrounding the "mysterious variations" of the motions of Jupiter and Saturn, let’s see how the Tychos model can account (without any appeal to “gravitational perturbations” or “tidal friction effects”) for the so-called “Great Inequality”. My below graphic should elucidate - conceptually - just what causes the so-called “Great Inequality”. Again, it is quite simply Earth's slow displacement around its PVP orbit that causes the optical illusion of Jupiter and Saturn alternately accelerating or decelerating. In reality, the two of them only ever move at constant / invariable speeds - just like all their 'family members' in the Solar System:

1: Whenever (in a certain epoch) Jupiter and Saturn are observed, over a 60-year interval, to conjunct in the “upper quadrant” of our celestial sphere, it will seem as if Jupiter is accelerating.

2: Whenever (in a certain epoch) Jupiter and Saturn are observed, over a 60-year interval, to conjunct in the “lower quadrant” of our celestial sphere, it will seem as if Saturn is accelerating.

This is because, as Earth moves slowly (at 1 mph) around its PVP orbit, Jupiter and Saturn will alternately conjunct as they proceed in the opposite or in the same direction as Earth.

Antonio Giorgilli then points out something of paramount interest to the Tychos model’s paradigm. Here’s another quote from his aforementioned paper (“The Stability of the Solar System: Three Centuries of Mathematics”):

“The first long-term simulations have been carried out since the end of the 1980s by some researchers, including A. Milani, M. Carpino, A. Nobili, GJ Sussman, J. Wisdom, J. Laskar. Their conclusions can be summarized as follows: the four major planets (Jupiter, Saturn, Uranus and Neptune) seem to move quite regularly even over a period of a few billion years, which is the estimated age of our Solar System. On the other hand, the internal planets (Mercury, Venus, Earth and Mars) present small random orbital variations, in particular of their eccentricity, which cannot be interpreted as periodic movements: we must admit that there is a chaotic component. Not that the orbits change much, at least not in the short term, but there may be, for example, small variations in the eccentricity of the Earth’s orbit that have very significant effects on the climate: the glaciations appear to be correlated with these variations.”

In other words, this nicely goes to confirm that, as proposed by the Tychos, there are two distinct groups of celestial bodies in our Solar System:

  1. The Binary group (or “the inner orbs”) composed of the Sun, Mars, Mercury and Venus (and of course, the central Earth-Moon system).

  2. The P-type group (or “circumbinary orbs”) composed of Jupiter, Saturn, Uranus, Neptune - and Pluto.

And thus, the Tychos model resolves yet another historical impasse of astronomy: “The Great Inequality". Jupiter and Saturn both travel at stable, constant speeds and will not be crashing into the Sun or migrate to other galaxies. So relax, people: the End of the World is not near!

Clarifying the 12-year and 30-year periods of Jupiter and Saturn

I presently wish to clarify my contention that Jupiter and Saturn may be 'technically' considered to have "integer" periods of respectively 12 years and 30 years (i.e. perfect multiples of the TMSP, our Moon's true synodic period). As every astronomer will know though, the orbital periods of our outer or “Jovian” planets from Jupiter to Pluto are all reckoned to be slightly shorter than a whole, integer number of solar years. Jupiter, for instance, is said to complete one of its orbits in 11.862 years. Saturn is said to complete one of its orbits in 29.4571 years. This means that, in 12 integer years, Jupiter will appear to have precessed by a small amount ('eastwards' in our skies). Likewise, in 30 integer years, Saturn will appear to have precessed ('eastwards' in our skies) by a small amount.

As we activate the "Trace" function in the Tychosium3D simulator, however, we may start envisioning the peculiar geometric motives that allow us to say that - technically speaking - the orbits of Jupiter and Saturn are truly completed in exactly 12 and 30 integer years. My below graphics are screenshots taken from the Tychosium3D simulator.

As we can see, Jupiter will (over an exact 12 year-period) return to a point which is radially equivalent (i.e. at the same distance from Earth) to where it was 12 years earlier. In other words, it will return to the very same point of its characteristic 'teardrop' loop. We may thus consider this (i.e. 12 integer years) as Jupiter's true period - which corresponds to 150 TMSP's of 29.22 days (i.e. our Moon's true synodic period).

Likewise, Saturn will (over an exact 30 year-period) return to a point which is radially equivalent (i.e. at the same distance from Earth) to where it was 30 years earlier. In other words, it will return to the very same point of its characteristic 'teardrop' loop. We may thus consider this (i.e. 30 integer years) as Saturn's true period - which corresponds to 375 TMSP's of 29.22 days (i.e. our Moon's true synodic period).

In conclusion, it would not be unreasonable to affirm that the true orbital periods of Jupiter and Saturn are respectively 12 years and 30 years.

SATURN'S MOTIONS - another Copernican aberration

Did you know that Saturn can return facing the same star in only 252 days? This is an observable fact. Now, make yourself a nice cup of tea - and give it a good thought. As you let this fact sink in, you may then ask yourself this good question: "if Earth revolves around the Sun in about 365 days, how can it possibly reconjunct with Saturn and the same star in only 252 days?"

My below graphic shows two screenshots from the heliocentric Star Atlas simulator. We see that Saturn, in fact, returned facing the same star on:

June 1, 1994 and on February 8, 1995 (a time interval of 252 days) - at the exact same celestial longitude (RA) of 22h56min14s.

Here is how the JS orrery (a heliocentric simulator featuring an overhead / 3D view of our Solar System - much like the Tychosium) depicts these two conjunctions:

And here is how the Tychosium 3D simulator depicts the same two conjunctions:

As you can see, the Tychosium shows Saturn returning to the same line of sight, whereas the JS orrery shows Saturn 'returning facing the same star' on two separate, parallel lines of sight (of course, Copernican advocates will keep arguing that those two lines of sight are "not quite parallel"...).

I will leave it up to the readers to judge for themselves which of the two simulators (the JS orrery or the Tychosium) makes the most sense. May no one ever accuse this author of trying to impose upon the readers his own world view. All I wish is to share it with other thinkers - so as to start a thoughtful and intelligent debate as to which configuration of our Solar System best agrees with all empirical observation assiduously gathered by our world's best astronomers throughout the centuries. One day, of course, I will leave this planet and my earthly life - but hopefully this book will remain available for future generations to assess and consider.


In the next chapter, we shall look at the even more remarkable behaviors of Uranus, Neptune and Pluto - all of which actually return to the exact same point in the sky at the completion of their respective periods of 84, 165 and 248 integer years. As will be demonstrated, the only reason for their slight, apparent precessional advance is the parallax effect caused by Earth's motion around its PVP orbit.