## Curves of planetary motion in geocentric perspective: Epitrochoids

Usually in a basic class in astronomy we get taught that our planets revolve in elliptical. paths around our sun. This is not quite what you observe when using a telescope yourself, since your point of observation is fixed on the earth which itself revolves around the sun. So the planet's path actually become quite complicated curves with forward and backward motions: They are ellipses around positions on another ellipse. Historically this geocentric view formed the basis of the Ptolemaic system. Mathematically the whole family of curves of circles around moving points on the circumference of another circle is referred to as "epitrochoids". In the applet shown below a planet of our solar system can be chosen, and its trajectory at the sky as seen from the earth is sketched. For example if you choose "sun" as the graph generating celestial body, a perfect ellipse is drawn. Due to the very small excentricity of the earths orbit around the sun, its ellipse is hardly distinguishable from a circle. This figure is exactly the same in the well known heliocentric perspective, only the role of sun and earth are switched. Every other planet will move in ellipses around this line of the sun in the geocentric perspective. The number in the upper left corner indicates the number of earth-years that have passed. Due to the use of polar coordinates instead of Kepler coordinates, there is a slight inaccuracy of the planets velocity for higher eccentricities. To demonstrate this and to make the visualization more complete I included Pluto and two comets, Halley and Hale-Bopp in the app. The resulting curves can be represented as trajectories, line diagrams etc. selected in the second choice element. More explanations below.

START - starts the curve, a second or third curve can be added to the image
CLEAR - clears the screen and rescales the new picture (coordinate system)
"slow <----> fast" regulates the velocity of the drawing

 Semi-major axis earth: AU Semi-major axis planet: AU Orbital period earth: years Orbital period planet: years Eccentricity earth: Eccentricity planet: Relative phase: π One point every: days

fast

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### The planet's paths and their symmetries

Due to the fixed ratio of revolution times between the earth and the other planets quasi-periodic figures arise.
Venus' geocentric path displays a fivefold symmetry, due to the approximate commensurability of 13:8 of the two revolution times. That means during 8 (earth) years Venus revolves approximately 13 times around the sun. Assume we start our observation when Venus and earth are next to each other. Since Venus revolves faster than the earth it will have to pass the earth four more times during the next 8 years (= 8 earth revolutions). Each time earth and Venus "meet" a loop in the geocentric venus path arises due to the overtaking. Since the speed of both of them is almost regular (the excentricity is quite small and enters only quadratically in the ellipse equation) the pattern will display a five fold symmetry.
Note however that none of the mentioned symmetries here is exact, they are due to approximations of the ratios of revolution times which in reality are by no means whole numbers.
Mercury displays a threefold symmetry, due to the approximate commensurability of 1:4 of the two revolution times. A closer view reveals a ratio of 7:29 resulting in a 22-fold symmetry (the degree of rotational symmetry is always given by the difference of the two fractional components of the ratio of the two revolutions).
Mars', trajectory reveals at first sight a onefold symmetry (revolution ratio w.r.t. earth 1:2). A closer view reveals a seven fold symmetry, due to the closer approximation of the ratio by 8:15 and a still closer view results in a 37-fold symmetry, due still closer approximation of 42:79.
Jupiter's trajectory forms an eleven-fold symmetry (revolution ratio Jupiter - Earth 1:12). A closer view reveals a 76-fold symmetry, due to the better description of the ratio by 7:83.

### The planets geocentric path and the sun

To see clearly, how the Epicycloids of the Jupiter path or of any of the outer planets or comets are caused by the earths own movement choose "geocentric path + sun", here the sun's orbit as seen from earth is added to the graph. You will notice that each loop of Jupiter's path corresponds to one circle of the sun that is one earth year.
Observing the path of Venus and the (geocentric) revolution of the sun you will remark how Venus always overtakes the sun and then circles around it to fall behind. The opposite picture yields for Mars. His loops always lag behind the sun whenever he is overtaken.

### Graphical derivation of the Epitrochoids: Connecting lines

Another way to visualize the relationship of the earths orbit with its surrounding neighbours is, to draw a connecting line at fixed intervals between the earth's and the corresponding planet's position on its orbit in the heliocentric perspective. This is meant by the option "connection earth_planet" in the second Choice parameter of the panel. The lines are figuratively the earthly observer's view. The corresponding images are line bundles which form envelopes of epicycloids. For example the connecting-line-image of the Mars orbit is very similar to the well known coffee cup caustics. To arrive at meaningful graphs here, the parameter "draw one point every ... days" has to be chosen carefully. Usually low numbers are appropriate. But for longer runs for example of the planet Venus the parameter 117 days yields astonishing results: Due to the ratio of the periodicities moving pentagrams will be displayed.
Note that the cusps of the envelopes do not occur at the passages of "backward motion" of the planet but at the opposition. To elucidate this behaviour of the two graphical representations a third option has been introduced "geocentric path + connection earth_planet" where both are plotted at the same time. If you add the graph of the fourth option "heliocentric path" you will notice, that each connection line starts exactly on the heliocentric path curve of the earth and ends at the heliocentric path curve of the planet, as it should.
Finally the geocentric path is generated graphically by translating each connection line to the center of the panel, the observer on the earth.

### Heliocentric perspective

To get the well known heliocentric paths of our planet choose "heliocentric path". To get a view of the whole solar system choose "heliocentric paths of all planets".

### Change the point of view

Another application of the applet is to choose a planet, say Venus, and put its values of the revolution times, and excentricity into the fields of the earths revolution times and excentricity. If you press START now, you will see just a point, since you observe only yourself on Venus from your own point of view on Venus. If you choose now a different planet, say Mars, (keeping the altered values of the Earth) you will get the path of the planet Mars as observed from a person in an observatory of Venus. This way you can change your point of view by astronomical scales. Of course you can feed data of exoplanets as well.

### Kepler's music of the spheres

If you regard the ratio of the revolution times of two orbits as ratios between oscillations of a vibrating string (following the ideas of Johannes Kepler ) then
Jupiter - Earth form an interval close to 3 octaves + a fifth,
Mars - Earth form a major seventh,
Earth - Venus form a minor sixth and
Earth - Mercury form an interval of roughly two octaves.
For ways to represent these musical intervals graphically see: curves of Lissajous. This may offer another quite interesting view on our planets motions.

Contact:  higobreitenbach@arcor.de
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