Here is what the motions of Jupiter and Saturn look like over the
course of a year against the same stellar background -

http://apod.nasa.gov/apod/image/0112...loop_tezel.jpg

Here is what the same images look like as time lapse footage -

http://apod.nasa.gov/apod/image/0112/JuSa2000_tezel.gif

There is no giant mathematical contortions behind those images as each
planet takes a certain length of time to return to the same position
among the constellations - almost 20 years for the nearer Jupiter and
close to 30 years for the further Saturn,all the while the Earth is
orbiting and periodically overtaking these planets hence the
retrogrades and the relative distance each planet is from the Earth
and the central Sun.

Although Galileo makes a minor error in using epicycles in relation to
retrogrades (they exist only in the orbital circular/elliptical
arguments and refinements),his approach to retrogrades and their
resolution is spot on and nothing like the empirical junk which ruined
the arguments -

Salviati: "In the Ptolemaic hypotheses there are the diseases, and the
Copernican their cure. . . . With Ptolemy it is necessary to assign to
the celestial bodies contrary movements, and make everything move from
east to west and at the same time from west to east, whereas with
Copernicus all celestial revolutions are in one direction, from west
to east. And what are we to say of the apparent movement of a planet,
so uneven that it not only goes fast at one time and slow at another,
but sometimes stops entirely and even goes backward a long way after
doing so? To save these appearances, Ptolemy introduces vast
epicycles, adapting them one by one to each planet, with certain rules
about incongruous motions -- all of which can be done away with by one
very simple motion of the Earth.

"Sagredo: I should like to arrive at a better understanding of how
these stoppings, retrograde motions, and advances, which have always
seemed to me highly improbable, come about in the Copernican system.

Salviati: Sagredo, you will see them come about in such a way that the
theory of this alone ought to be enough to gain assent for the rest of
the doctrine from anyone who is neither stubborn nor unteachable. I
tell you, then, that no change occurs in the movement of Saturn in
thirty years, in that of Jupiter in twelve, that of Mars in two, Venus
in nine months, or in that of Mercury in about eighty days. The annual
movement of the Earth alone, between Mars and Venus, causes all the
apparent irregularities of the five stars named. . . .

[Here Salviati explains Jupiter's motion, then follows with:]

Now what is said here of Jupiter is to be understood of Saturn and
Mars also. In Saturn these retrogressions are somewhat more frequent
than in Jupiter, because its motion is slower than Jupiter's, so that
the Earth overtakes it in a shorter time. In Mars they are rarer, its
motion being faster than that of Jupiter, so that the Earth spends
more time in catching up with it. Next, as to Venus and Mercury, whose
circles are included within that of the Earth, stoppings and
retrograde motions appear in them also, due not to any motion that
really exists in them, but to the annual motion of the Earth. This is
acutely demonstrated by Copernicus . . .

You see, gentlemen, with what ease and simplicity the annual motion --
if made by the Earth -- lends itself to supplying reasons for the
apparent anomalies which are observed in the movements of the five
planets. . . . It removes them all and reduces these movements to
equable and regular motions; and it was Nicholas Copernicus who first
clarified for us the reasons for this marvelous effect." 1632,
Dialogue Concerning the Two Chief World Systems

It is time to really push the great astronomical arguments for
planetary dynamics using the power of contemporary imaging thereby
getting rid of the unintelligent ideas about planetary dynamics and
solar system structure .