![]() This is why I ignore the Z-component of the planet's motion. In the pictures below, that corresponds to the points of reversal in the tilted horizontal motion. Procedure: Part 1: Position of Sun and Planets 1. An increase in right ascension represents prograde motion. If it decreases, then the planet is in retrograde motion until the right ascension begins to increase. ![]() Whether the motion is in a loop or an S-curve, what determines when retrograde motion happens is when the component of the vector joining two planets that is parallel to the ecliptic reverses its direction of rotation. An easy way to determine if a planet is in retrograde motion is to look at the right ascension over time. Below are two pictures of time-lapsed retrograde motion of Mars. When the movement of this line reverses direction, the other planet is entering or leaving retrograde motion.įor the purposes of determining when retrograde motion occurs, motion of the planets perpendicular to the ecliptic plane does not matter. Due to Earth's tilt, using an equatorial coordinate system would be more mathematically difficult since the relative motion of the planets would be three-dimensional.Īs for the rest of the answer, the idea is to calculate the angle of a ray from Earth through the other planet, as this represents the position of the planet in the sky with respect to the fixed stars. This simplifies the comparison of orbits since most of the planets orbit the sun very close to the ecliptic plane. ![]() The coordinate system I used was a heliocentric ecliptic coordinate system. Working out the equations for orbits that are not in the same plane would have taken far too much time for too little gain. (Keep in mind that there was not a great deal of real accurate data regarding the positions of the planets at the time. I'm assuming from the upvotes that my answer was correct but incomplete. Eudoxus was the first person to devise a model that could explain the retrograde motion of the planets in the sky. ![]()
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