Clockwork Orrery - 2D Heliocentric Map, Retrograde Loops and Gear Ratios

Presets

Planet tracked from Earth

Time rate Orbit eccentricity scale

sightline trails orbit rings

Live readouts

Date (years after J2000)Heliocentric distance rDistance from EarthGeocentric longitudeApparent motionElongation from Sun

Gear-train designer

Max teeth per gear

Target period ratio T/T_EarthBest gear pair (driver:driven)Gear ratio achievedError

Independent checks

Kepler's third law T²/a³Kepler equation residualSynodic period: simulatedSynodic period: 1/|1/T₁−1/T₂|Retrograde span (simulated)Literature average

Geocentric longitude over ±1.5 years

The science

Each planet follows a Kepler ellipse: M = 2π(t/T) + M₀, E − e sin E = M (solved by Newton iteration), r = a(1 − e cos E), true anomaly from tan(ν/2) = √((1+e)/(1−e)) tan(E/2). Kepler's third law T² = a³ (years, AU) links size and period; the clockwork orrery realises the period ratios with meshing gears, whose speeds obey teeth_A ω_A = teeth_B ω_B.

Seen from Earth, an outer planet appears to loop backwards (retrograde) around opposition, when Earth overtakes it on the inside track; Mercury and Venus loop around inferior conjunction. The synodic period is 1/T_syn = |1/T₁ − 1/T₂|.

Simplifications: all orbits coplanar (so no latitude), two-body Kepler orbits with no mutual perturbations, orbital elements fixed at J2000 values, so dates are illustrative rather than ephemeris-grade. Gear ratios: best rational approximation with both tooth counts within the chosen limit.