The Flowering of Celestial Mechanics

Inner Planets Hypotrochoid Orbits

Inner Planets Hypotrochoid Orbits

Johannes Kepler was the first person to successfully model planetary orbits to a high degree of accuracy and his First Law of Planetary Motion [published in 1609] is deeply embedded in modern celestial mechanics.


Kepler's First Law of Planetary Motion

Johannes Kepler’s first law of elliptical orbits is repeated like a mantra throughout the mainstream literature and is visually reinforced in many illustrations.

Orbits of the bodies in the Solar System

Following Kepler’s laws, each object travels along an ellipse with the Sun at one focus.

By careful analysis of the observation data, Johannes Kepler found the planets’ orbits were not circular, but elliptical.


However, there are significant problems with elliptical orbits.

Firstly, the mainstream gravitational mechanics underlying the [symmetrical] orbital ellipse are impossible when the second focus [of the ellipse] is absent i.e. a ghost.

Miles Mathis - Ellipse

This is physically impossible. Using the given motions, the ellipse is impossible to explain. The logical creation of an ellipse requires forces from both foci, but one of our foci is empty. It is a ghost.

Drawing an orbital ellipse [on a sheet of paper] requires:
a) Two foci [the green and yellow pins].
b) A loop of string representing the centripetal force [gravity] exerted by each foci on the planet.
c) A pencil [representing a planet] to mark the elliptical orbit around the two foci.

Drawing an ellipse

Clearly, drawing the orbital ellipse would be impossible with only the yellow pin [which represents the sun]. Remove the green pin and you can only draw a circle – not an ellipse.

Secondly, and most importantly, planets DO NOT follow elliptical orbits [around the Sun].

Precessing Kepler orbit

The orbit of a planet around the Sun is not really an ellipse but a flower-petal shape because the major axis of each planet’s elliptical orbit also precesses within its orbital plane

Planets actually follow flower-petal [hypotrochoid] orbits around the sun.

Perihelion precession

Orbits of the Inner Planets_Table

Inner Planets Hypotrochoid Orbits

Inner Planets Hypotrochoid Orbits

The mechanics underpinning hypotrochoid orbits is very simple.


Anyone familiar with a Spirograph will probably be familiar with hypotrochoid orbital patterns.

In the illustration [above] the black cog represents a rotating planet [as it orbits around a central sun] and the outer red cog represents a rotating vortex which creates the centripetal force which counter-balances the centrifugal force generated by the orbiting planet [black cog].
Spirograph pattern

Anyone familiar with flowers will also be familiar with hypotrochoid orbital patterns.


The mechanics of hypotrochoids is easily grasped by children.

The mechanics of hypotrochoids aren’t explained by Newton or Einstein.

The mechanics of hypotrochoids orbits are partially explained by Leon Hall.

Trochoids, Roses and Thorns - Beyond the Spirograph

The “Perihelion Precession of Mercury” problem is a wonderful example of how the mainstream resorts to mathematical magic and logical contortions when they are faced with observational evidence that clearly falsifies their cherished beliefs and laws.

Clearly the orbit of Mercury falsifies two strongly held mainstream beliefs [which they call laws]:

1. Kepler’s First Law of Planetary Motion is falsified because the orbit of Mercury is hypotrochoid.

2. Newton’s Law of Universal Gravitation is falsified because it fails to predict the orbit of Mercury.

Under Newtonian physics, a two-body system consisting of a lone object orbiting a spherical mass would trace out an ellipse with the spherical mass at a focus. The point of closest approach, called the periapsis (or, as the central body in our Solar System is the sun, perihelion), is fixed. A number of effects in our solar system cause the perihelia of planets to precess (rotate) around the sun. The principal cause is the presence of other planets which perturb each other’s orbit. Another (much more minor) effect is solar oblateness.

Mercury deviates from the precession predicted from these Newtonian effects.
This anomalous rate of precession of the perihelion of Mercury’s orbit was first recognized in 1859 as a problem in celestial mechanics, by Urbain Le Verrier. His re-analysis of available timed observations of transits of Mercury over the Sun’s disk from 1697 to 1848 showed that the actual rate of the precession disagreed from that predicted from Newton’s theory by 38″ (arc seconds) per tropical century (later re-estimated at 43″)

However, instead of acknowledging the patently obvious falsification of their hallowed “laws” the mainstream proceeded to construct a series of elaborate intellectual edifices they can hide behind.

This was not an easy task primarily because Newtonian mechanics can only fully resolve the mechanics for two gravitational bodies in isolation. When three [or more] bodies are involved then Newtonian mechanics becomes very wobbly and this issue is referred to as the “Three Body Problem”.

In its traditional sense the three-body problem is the problem of taking an initial set of data that specifies the positions, masses and velocities of three bodies for some particular point in time and then determining the motions of the three bodies, in accordance with the laws of classical mechanics (Newton’s laws of motion and of universal gravitation).

Instead of acknowledging that Newtonian Mechanics had some very real-world problems [because there are a multitude of bodies in the Universe] the mainstream response was to develop Perturbation Theory.

Perturbation Theory enabled the mainstream to calculate all the gravitational tugs associated with each body involved in any particular “Three Body Problem” [provided they ignored all other gravitational tugs generated by bodies outside the scope of their particular study]. .

Another example of a classical three-body problem is the movement of a planet with a satellite around a star. In most cases such a system can be factorized, considering the movement of the complex system (planet and satellite) around a star as a single particle; then, considering the movement of the satellite around the planet, neglecting the movement around the star. In this case, the problem is simplified to the two-body problem. However, the effect of the star on the movement of the satellite around the planet can be considered as a perturbation.

Perturbation theory has its roots in early celestial mechanics, where the theory of epicycles was used to make small corrections to the predicted paths of planets. Curiously, it was the need for more and more epicycles that eventually led to the 16th century Copernican revolution in the understanding of planetary orbits. The development of basic perturbation theory for differential equations was fairly complete by the middle of the 19th century.

However, even Perturbation Theory failed to fully resolve the “Perihelion Precession of Mercury” problem for the mainstream and it is this failure that created a glorious opportunity for Albert Einstein to invent a brand new dimension called “spacetime” which could be warped [curvature] around the hallowed laws of Kepler and Newton.

A number of ad hoc and ultimately unsuccessful solutions were proposed, but they tended to introduce more problems. In general relativity, this remaining precession, or change of orientation of the orbital ellipse within its orbital plane, is explained by gravitation being mediated by the curvature of spacetime. Einstein showed that general relativity agrees closely with the observed amount of perihelion shift.
This was a powerful factor motivating the adoption of general relativity.


In physics, spacetime (also space–time, space time or space–time continuum) is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as existing in three dimensions and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions. From a Euclidean space perspective, the universe has three dimensions of space and one of time. By combining space and time into a single manifold, physicists have significantly simplified a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.

Unfortunately, Albert Einstein’s inventive imagination has caused the mainstream to dive headlong into a logical “black hole” which [unlike a rabbit hole] cannot be seen with the naked eye.


It is called “black” because it absorbs all the light that hits the horizon, reflecting nothing, just like a perfect black body in thermodynamics. Quantum field theory in curved spacetime predicts that event horizons emit radiation like a black body with a finite temperature. This temperature is inversely proportional to the mass of the black hole, making it difficult to observe this radiation for black holes of stellar mass or greater.

Gravitational distortions caused by a black hole

This strange [bizarre] tale of mental gymnastics has enabled the mainstream to develop a theory of “gravitational radiation” [as predicted by Albert Einstein in 1916].

In physics, gravitational waves are ripples in the curvature of spacetime which propagate as a wave, travelling outward from the source. Predicted to exist by Albert Einstein in 1916 on the basis of his theory of general relativity, gravitational waves theoretically transport energy as gravitational radiation.

Thus, the final intellectual edifice is deployed [by the mainstream] to explain the “Perihelion Precession of Mercury” by relying upon “gravitational radiation” that is associated with the oblateness of the Sun and its Quadrupole Moment.

The mass quadrupole moment is also important in general relativity because, if it changes in time, it can produce gravitational radiation, similar to the electromagnetic radiation produced by oscillating electric or magnetic dipoles and higher multipoles. However, only quadrupole and higher moments can radiate gravitationally. The mass monopole represents the total mass-energy in a system, which is conserved—thus it gives off no radiation. Similarly, the mass dipole corresponds to the center of mass of a system and its first derivative represents momentum which is also a conserved quantity so the mass dipole also emits no radiation. The mass quadrupole, however, can change in time, and is the lowest-order contribution to gravitational radiation.

The simplest and most important example of a radiating system is a pair of mass points with equal masses orbiting each other on a circular orbit (an approximation to e.g. special case of binary black holes).

This whole horrific history of mainstream science is summarised by Wikipedia in one small table.

Sources of the precession of perihelion for Mercury

In this situation the use of Occam’s Razor is child’s play [and a pleasure].

Childs Play

hypotrochoid art set

Gallery | This entry was posted in Astrophysics, Gravity, Inventions and Deceptions, Science, Solar System, Vortices. Bookmark the permalink.

10 Responses to The Flowering of Celestial Mechanics

  1. oldbrew says:

    Wait about 12 seconds between changes…

    Whether it’s all true, who knows – but a few points of interest at least?

    Reply: More than a few points to take a look at…

  2. Chaeremon says:

    Hello, your comment at the tallbloke blog (on Phi): I can send a menu of diagrams (in .png format) which plot the orbits of Earth or Moon as seen in hypotrochoidial fashion, from which you can choose (and if you like) for your illustrations here. My software draws a black bracelet and passages of stars as inlets (small bars colored in lapislazuli and jade green). Examples: Precession of Equator (3 bracelets, years 673BCE / 1169CE / 3011CE, also for years 4037BCE / 1169CE, 6374CE); an Equinox Sandwich (3 bracelets, Dec / Mar / Jun 2012); a Solstice Sandwich (3 bracelets, Mar / Jun / Sept 2012); also triplets of eclipses on the Saros and Inex cycle (each 3 bracelets), and min/max close/far encounters of Mercury and Moon (2 bracelets), and many more. Inside each bracelet is an unspectacular oval which adapts visually to the min/max perigee/apogee reported resp. perihelion/aphelion reported) for quick comparision, this is also (and more accurate) reflected by each bracelet’s width. The data was taken from numerical integration and not changed except for scale.

    Reply: Amazing… replying be email – thank you!

  3. Roger Clague says:

    Very interesting post.
    I have never been convinced that he orbits of planets are an ellipse. There is only one place where force comes from, the sun. Not two as are needed to create an ellipse.

    It seems right that as you suggest the orbits are, for the inner planets, hypotrochoids. A repeated ellipse producing a smeared out circular orbit. Movement in two lots of superimposed circles, like Ptolomy’s epicycles.

    And the centre of the orbits is the sun. The sun itself is orbiting the Solar System Barycentre, in an hypotrochoid distorted by the outer planets.

    Maybe the smeared out quantum mechanical electron is also a hyprotrochoid.

  4. Alan says:

    If you were making an orrery based on this, I suppose you’d set up a planetary gear system. Then, one of the planetary gears would have an off-centre point to represent the planet. Now it’s just a not-so-trivial matter of setting the gear ratios, to have the right number of hypotrochoid “petals” on the orrery.
    I believe Pluto’s orbit intersects that of Neptune. That could prove difficult.

  5. tallbloke says:

    Great post Tim, don’t know how I missed this one.
    The other interesting thing about this is that Newton is in double trouble here.

    The gravitational potential energy plus the kinetic energy at perihelion doesn’t equal the GPE+KE at aphelion…

  6. tallbloke says:

    Tim: You’ll like this; mainstream confirmation of your vortex theory
    Also, check this site out – meaty stuff.

    Thanks Roger… will do so digging – Tim

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