Curves of planetary motion in geocentric perspective: Epitrochoids
Usually in a basic class in astronomy we get taught that our
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 motion,
they are ellipses around positions on another ellipse.
Historically this geocentric view formed the basis of the
Mathematically the whole family of curves of circles around points on the circumference of another
circle is referred to as
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
of the earths orbit around the sun, the 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.
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
"slow <----> fast" regulates the velocity of the drawing
"time of graph ... years" counts the earth's revolutions during the drawing process.
The words "years" and "days" refer to earth years and days.
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.
geocentric path displays a fivefold symmetry, due to the approximate
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.
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).
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.
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.
Pressing CLEAR adapts the virtual size of the applet to the planets path. So if you want an overall
image of the solar system you have to CLEAR and start with the outermost neptune before
graphing the other planets. A reasonable picture is feasable for all outer planets Neptune to Mars or
for the inner planets Mars to Mercury.
As an orientation its always helpful to add the sun's orbit to the graph. This is included in the
second choice "geocentric path + sun"
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
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.
(for example for longer runs of the planet Venus the parameter 117 days
gives good results, due to the ratio of the periodicities.)
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 fourth option "heliocentric path" you will
notice, that each connection line starts exactly on the heliocentric path curve, 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.
Change the point of view
Another interesting application of this 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.
To get the well known heliocentric paths of our planet choose "heliocentric paths". To get a view of
the whole solar system you have to start with the outermost planet neptune.
A reasonable picture is feasable for all outer planets Neptune to Mars or
for the inner planets Mars to Mercury.
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
Jupiter - Earth form an interval close to 3 octaves + a fifth, Mars - Earth a major seventh,
Earth - Venus 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 planets motion.
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Last modified: Feb. 2010