Retrograde, Prograde Motion and Station of Planets (Calculation Formulas)

Due to the Earth’s rotation and revolution from west to east, most celestial bodies seen from Earth rise in the east and set in the west. Therefore, we call the movement of celestial bodies from east to west in the sky “prograde” (direct motion), and from west to east “retrograde” motion. Due to the combined motion of the Earth and the planets, superior planets will exhibit “retrograde” motion for a period of time around “opposition” (the opposite is true for inferior planets). The day (or rather, the moment) between retrograde and prograde motion is called the “station.” That is to say, after a planet “stations,” its motion in the sky changes from prograde to retrograde, or from retrograde to prograde.

The following is a quote from Wikipedia regarding the prograde and retrograde phenomena of planets:

From Earth, the path a planet takes through the sky periodically changes direction. Although all stars and planets appear to move from east to west every night in response to Earth’s rotation, outer planets usually move slowly from west to east relative to the stars. This motion is the normal motion of the planet and is therefore considered prograde. However, because Earth’s orbital period is shorter than that of the outer planets, it periodically overtakes them, much like a faster car on a multi-lane highway. When this happens, the planet that was moving eastward will first stop, then appear to move westward, and then, after Earth has passed the planet in its orbit, it appears to resume its normal west-to-east motion. The inner planets, Mercury and Venus, also exhibit retrograde motion through a similar mechanism; however, their retrograde cycles are also tied to their synodic periods with the Sun. The mechanism explaining apparent retrograde motion is the same as for outer planets. Asteroids and Kuiper Belt objects (including Pluto) also exhibit apparent retrograde motion.

So, how long does each retrograde motion of a planet last? Let us consider the simplest model: planets moving in circular orbits within the same orbital plane. Choosing the time of planetary opposition as the origin, we can list the equations of planetary motion relative to the Earth: \begin{aligned} y &= R_2 \sin(\omega_2 t) - R_1 \sin(\omega_1 t) \\ x &= R_2 \cos(\omega_2 t) - R_1 \cos(\omega_1 t) \end{aligned}

Where R and \omega = \frac{2\pi}{T} are the orbital radius and angular velocity, respectively (subscript 1 refers to Earth, and 2 refers to the superior planet). Obviously, the prograde or retrograde motion of a planet depends on the sign of \dot{\lambda} = \frac{d\lambda}{dt} (positive for prograde, negative for retrograde), and the “station” is the case where \dot{\lambda} equals 0. In a coordinate system with the Earth as the origin: \tan\lambda = \frac{R_2 \sin(\omega_2 t) - R_1 \sin(\omega_1 t)}{R_2 \cos(\omega_2 t) - R_1 \cos(\omega_1 t)}

Differentiating both sides: \begin{aligned} \frac{\dot{\lambda}}{\cos^2 \lambda} &= \Big\{ \Big[R_2 \omega_2 \cos(\omega_2 t) - R_1 \omega_1 \cos(\omega_1 t)\Big] \Big[R_2 \cos(\omega_2 t) - R_1 \cos(\omega_1 t)\Big] \\ &\quad + \Big[R_2 \omega_2 \sin(\omega_2 t) - R_1 \omega_1 \sin(\omega_1 t)\Big] \Big[R_2 \sin(\omega_2 t) - R_1 \sin(\omega_1 t)\Big] \Big\} \\ &\quad \div \Big[R_2 \cos(\omega_2 t) - R_1 \cos(\omega_1 t)\Big]^2 \end{aligned}

Setting the numerator to zero, we have: \omega_1 R_1^2 + \omega_2 R_2^2 - (\omega_1 + \omega_2) R_1 R_2 \cos(\omega_1 t - \omega_2 t) = 0

Thus, the time from opposition to the station is: t_0 = \frac{\arccos\left(\frac{\omega_1 R_1^2 + \omega_2 R_2^2}{(\omega_1 + \omega_2) R_1 R_2}\right)}{\omega_1 - \omega_2}

This only calculates the retrograde time after the “opposition.” Before the opposition, there should be a process that is symmetric and equal in duration. Therefore, the total duration of the planet’s retrograde motion is: t = \frac{2 \arccos\left(\frac{\omega_1 R_1^2 + \omega_2 R_2^2}{(\omega_1 + \omega_2) R_1 R_2}\right)}{\omega_1 - \omega_2}

For inferior planets, although they move faster than Earth at inferior conjunction, from Earth’s perspective, their direction of motion is still “clockwise,” which means it is retrograde. The above formula can also be used!

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