English Orrery Simulator - 2D

What an orrery does. It is a clockwork model of the Sun's planets: one hand crank turns a shaft that represents the Earth's year, and each planet arm is driven from that shaft through gears whose tooth counts approximate the planet's orbital period. In this page one crank turn is one month, so twelve turns advance the Earth arm one full circle.

Each planet gets a two-stage train, driver a on the year shaft meshing with b, then c (on the same axle as b) meshing with d on the arm sleeve. Equal centre distance (a + b = c + d) lets d sit on the central axis. The search tries every train with 12 to N_max teeth and keeps the smallest period error; with 100 teeth Mars gets 35/48 × 35/48 = 0.53168 revolutions per year against the true 0.53170 (error 0.00002 %), but Saturn only reaches 0.03% because 29.46 years is hard to reach with whole teeth.

Periods and alignments. The time between two oppositions (Earth passing between the Sun and an outer planet) is the synodic period:

The page finds alignments by running the model (a sign change of the longitude difference, refined by bisection) and compares the measured Mars interval with the formula. Inner planets align with Earth at inferior conjunction, outer planets at opposition. Kepler's third law, T² = a³ with a in AU and T in years, is checked from the semi-major axes of the six planets (log-log slope 1.5).

Gears against the sky. The gear arms rotate uniformly on circles. Real planets move on ellipses: the true heliocentric longitude is found from the Kepler equation M = E − e sin E (Newton iteration, residual below 1e-12) and differs from uniform motion by the equation of centre, about 2 e sin M (up to 10 degrees for Mars, 23 for Mercury). Switch "Arms follow" to see the arm jump to the true sky position, and watch the graph of the difference grow over the decades because of the small period errors of whole-tooth gearing. The Moon turns at its mean sidereal rate; its phase comes from the angle to the Sun (mean elongation), so it ignores the lunar orbit's eccentricity and inclination.

Honest simplifications. Orbit radii on the plate are not to scale (they are evenly spread so all six planets fit; Jupiter's real orbit is 5.2 times Earth's, not 1.7 times), and the planets and Moon are drawn far larger than real. In the true-Kepler mode the arm angle is exact for the stated elements but the arm length stays fixed. Planet data are J2000 mean elements rounded to a few digits; the start date is 7 Nov 2026, and far from J2000 the gear arms drift away from the true positions. The gears turn the right way round only because a real orrery adds idler pairs; the model shows the ratio, not a manufacturable casing.