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Heliocentric and geocentric motion of the bright planets

At the left the applet is simulating the Copernican heliocentric model of the solar system,
whereas the simulation at the right is showing the motion from the geocentric point of view,
propagated by Tycho Brahe (Tychonic model).
Both models are equivalent, and both of them decscribe the phases of the Planets correctly.

 Planet Sid. Orbital period Synodic period Multiple of syn. per. Multiple of sid. year Gap Eccentricity Orbital Inclination ° Mercury 87.96934 d 115.8776 d 3 syn. per. 347.633 d 1 trop. year 365.25636 d 17.62 d 17.4° 0.2056 7.004 Venus 224.70069 d 583.9206 d 5 syn. per. 2919.603 d 8 trop. years 2922.051 d 2.45 d 2.4° 0.0068 3.394 Mars 686.960 d 779.9617 d 7 syn. per. 5459.732 d 15 trop. years 5478.845 d 19.11 d 18.8° 0.0935 1.850 Jupiter 4333.2867 d 398.8781 d 11 syn. per. 4387.659 d 12 trop. years 4383.076 d -4.58 d -4.5° 0.0489 1.308 Saturn 10756.1995 d 378.0956 d 28 syn. per. 10586.677 d 29 trop. years 10.592.434 d 5.76 d 5.7° 0.0565 2.488 Earth 365.2564 d 0.0167 0

The Ptolemaic Model

The phases
of the planets as seen by Ptolemy's model are not correct.

 Ptolemy Copernicus

 Ptolemy explained the retrograde motion of the planets by his epicycle theory: the planets are assumed to move around a small circle (epicycle e), the center of which moves along a larger circle (deferent D). Both circles rotate counterclockwise and are roughly parallel to the Earth's plane of orbit (ecliptic). To improve the agreement with observational data Ptolemy introduced the concept of the equant: The equant point (E) and the Earth are equidistant on either side of  the deferent center (X). The center of the epicycle is rotating on the deferent circle uniformly with respect to the equant.
 Mercury: 2008, Jan 1 - Mar 28 elliptic orbit (e=0.2056) retrograde Jan 30 - Feb 19 Mercury according to Ptolemy, assuming a circular orbit.

Diagrams: geocentric longitude and latitude

 Venus: 2009, Jan 1 - Jun elliptic orbit (e=0.0068) retrograde Mar 8 - Apr 18 Venus according to Ptolemy, assuming a circular orbit.

Diagrams: geocentric longitude and latitude

Mercury:

Venus:

Mars:

Jupiter:

Saturn:

 Web Links Venus and the pentagram The Origins of the Pentagram and Hexagram Venus: Umlaufbahn und Resonanzen, Venus-Pentagramm (wikipedia) Venus, the Rose & the Heart Epicycle and Deferent Applet Retrograde Motion Ptolemy's Model Ptolemy's Epicycle Ptolemaic System Simulator Ptolemaic Model Applet Epicycloids and Hypocycloids (Eli Maor, PDF) page 9: illustration "The motion of Saturn Jupiter and Mars in respect of the Earth

Updated: 2023, Oct 04