Retrograde Motion: Apparent Longitude from Relative Motion
Apparent Longitude $\lambda_{\mathrm{app}}$
Continuous means the plotted longitude is unwrapped, so there are no 0 deg / 360 deg jumps.
Direct
Direct: $d\tilde{\lambda}/dt > 0$ | Stationary: $|d\tilde{\lambda}/dt| \le \epsilon$ | Retrograde: $d\tilde{\lambda}/dt < 0$
Line
Retrograde band
Cursor
Orbit View (top-down)
Sky View
Day $t$
0.00days
$\lambda_{\mathrm{app}}$
—deg
$d\tilde{\lambda}/dt$
—deg/day
State
—
Geometry
—
Retrograde
—days
- 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.
- Apparent longitude: $\lambda_{\mathrm{app}}(t) = \operatorname{atan2}(y_t - y_o,\, x_t - x_o)$, wrapped to $[0^\circ, 360^\circ)$.
- 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.
Quick guide
Drag the plot to scrub time.
Step 1 of 3