Observable: Watch the longitude plot. During the pink-shaded retrograde intervals, the target planet appears to reverse its motion against the background stars.
Model: Both planets orbit the Sun. When Earth overtakes Mars (or Venus overtakes Earth), the line of sight sweeps backward temporarily.
Inference: No planet actually reverses its orbit. Retrograde motion is a viewing-geometry effect. Its duration depends on the orbital-period ratio between observer and target.
Planets follow coplanar Keplerian ellipses with elements $(a, e, \varpi, L_0)$ from JPL Table 1 (Standish, 1800-2050).
Time uses model day $t$ (no calendar claims). One model month $= 30$ days.
The unwrapped series $\tilde{\lambda}(t)$ uses a $180^\circ$ jump rule. Retrograde is defined by $d\tilde{\lambda}/dt < 0$.
Stationary points (where $d\tilde{\lambda}/dt = 0$) are refined by bisection to $10^{-3}$ day tolerance.
Kepler's Laws — see the orbital mechanics behind these ellipses.
Eclipse Geometry — another viewing-geometry effect that depends on orbital alignment.
Historical context: Copernicus used retrograde motion to argue for heliocentrism. Ptolemy's epicycles were a mathematical description of this same apparent reversal, but placed Earth at the center.
The planet appears to reverse! This is retrograde motion — an illusion caused by the difference in orbital speeds between observer and target.