Retrograde Motion: Geocentric vs Heliocentric (2D)

Orbits

Outer / inner period ratio 1.88×

Time speed 0.25 yr/s

Live Readouts

Elapsed time

0.00 yr

Earth–planet distance

— AU

Apparent motion

—

Synodic period

— yr

The synodic period is how often Earth laps the outer planet — that cadence sets how often a retrograde loop appears in the lower panel.

How it works

Both bodies orbit the Sun on circular paths at speeds set by their period. The outer planet's real orbital radius follows Kepler's Third Law from the period ratio you set:

T² ∝ a³   (with Earth: T_E = 1 yr, a_E = 1 AU)
a_outer = (T_outer / T_E)^(2/3) AU

Every frame the engine integrates both orbits forward, then computes the outer planet's true bearing as seen from Earth's current position — real vector geometry, recomputed live:

λ(t) = atan2(y_outer(t) − y_Earth(t), x_outer(t) − x_Earth(t))
dλ/dt < 0  →  retrograde (apparent motion reverses)

The lower panel plots this apparent bearing λ against time. Because Earth (inner, faster) periodically overtakes the outer planet on the inside track, λ normally drifts forward but briefly reverses — a "retrograde loop" — right as Earth passes. Ptolemy's geocentric universe needed a small circle (epicycle) bolted onto a large one (deferent) for every planet to fake this. The heliocentric model needs nothing extra: the loop falls straight out of two circular orbits at different speeds, viewed from a moving Earth.

Period ratio — how much slower the outer planet orbits than Earth. 1.88× reproduces Mars; larger ratios (Jupiter ≈ 11.9×) give shorter, more frequent loops relative to the outer planet's own period.
Synodic period — the true real-world cadence of the loops: 1/T_syn = 1/T_Earth − 1/T_outer.
The red segment of the trace marks exactly the interval where dλ/dt < 0 — verified to land once per synodic period, each time Earth passes the outer planet on the top-down view above.

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