Presets
Playback
Planet orbit
Orbital elements of the observed planet (presets for the five naked-eye planets).
Earth and display
Earth eccentricity and sky-plot scale.
Layers
Model checks
The science
Why planets seem to move backwards
Every planet moves around the Sun in the same direction, so seen from the Sun it never reverses. Seen from the moving Earth it can: the apparent direction is the direction of the vector R = rplanet − rEarth, and the ecliptic longitude is λ = atan2(Ry, Rx). For a planet outside the Earth's orbit the Earth overtakes it near opposition and the planet slides backwards against the stars (retrograde motion) between two stationary points where dλ/dt = 0.
Orbits and the sky
Each body follows a Kepler ellipse: mean anomaly M = M0 + n t with n = 2π/P and P = 365.256 d · a3/2, Kepler's equation E − e sin E = M (solved by Newton iteration), then x = a(cos E − e), y = a√(1−e2) sin E, rotated by the argument of perihelion, the inclination and the node into ecliptic coordinates. The inclination gives the planet an ecliptic latitude, which turns the back-and-forth into the familiar loop or zigzag in the sky track.
1/S = |1/PE − 1/P|, stationary when cosφ = (r2ω + rE2ωE) / [r rE (ω + ωE)] (circular, coplanar)Here φ is the heliocentric angle between planet and Earth measured from opposition, S the synodic period (the time between oppositions: 780 days for Mars) and the retrograde interval lasts 2φ/(ωE − ω).
Honest simplifications
- Unperturbed two-body Kepler orbits with J2000 mean elements, rounded; no perturbations by other planets, no precession of the elements and no light-time or aberration, so dates drift by days to weeks relative to an ephemeris after decades.
- The Earth's orbit is taken in the ecliptic plane (inclination 0) with the Sun at the focus, not the Earth-Moon barycentre.
- The latitude in the sky plot is exaggerated (slider) because the loops are only a few degrees tall.
- The stationary-point formula is valid only for circular coplanar orbits and outer planets; the table readout is therefore labelled as a circular model while the graphs use the true elliptical orbit.
Checks run in the page
(1) Newton's solution of Kepler's equation leaves a residual below 10-10 for 400 test cases and the planet returns to its position after one period. (2) The mean interval between oppositions (or inferior conjunctions) over about 25 events equals 1/|1/PE − 1/P|. (3) For circular coplanar orbits the numerically found stationary point equals the closed-form angle above.
What to try
- Run Mars at 60 d/s: the sky track draws a loop each 780 days, red while the planet is retrograde (about 72 days); graph 2 dips below the zero line.
- Choose Jupiter and Saturn: the loops come more often (399 and 378 days) but are shorter in angle.
- Switch to Venus: an inferior planet; it appears to loop near inferior conjunction.
- Pick Mars on a circular orbit and compare the retrograde length in the readout with the numbers from the real eccentric orbit.
- Raise the planet inclination to 10 degrees to watch the loop open up in latitude.