Orrery Gear Train

What is modelled

A hand crank turns the Earth once a year. Every other planet is driven through a pair of gears whose tooth counts approximate the ratio of the Earth period to its own period. A planet with period 11.86 years needs a ratio close to 0.0843 turns per crank turn, which a pair of small gears can only match approximately, so the brass planet slowly drifts from the real one. The page finds the best ratio for the gear size you allow and shows the drift.

Equations

• Kepler third law P2 = a3 / M in years, AU and solar masses.
• Kepler equation E − e sin E = M, true anomaly from tan(ν/2) = √((1 + e)/(1 − e)) tan(E/2), orbit r = a(1 − e2)/(1 + e cos ν).
• Gear error 360 ° × years × (1/P − p/q) with p, q ≤ largest gear.
• Gravity v2 a = G Msun for a circular orbit.

Checks run in the page

• The log-log slope of period against distance and the law for all six planets.
• Newton solution of the Kepler equation and the equal-area law by numerical integration.
• An exhaustive search of all tooth pairs against the chosen ratios and the continued-fraction convergents.
• The centripetal acceleration of each planet against the Sun GM.

Honest simplifications

Orbits are circles except for the one eccentric Mars you can switch on, the distances on the screen are compressed, the planets do not perturb each other and the gear train is a single ratio per planet. Real orreries use compound trains and the real orbital elements.

• Press Coarse gears, then Very fine: watch the error per century fall by orders of magnitude.
• Press Heavier Sun: every true planet runs faster and the gears can no longer follow.
• Press Eccentric Mars and see the true planet speed up near the Sun while the gear ring turns evenly.