Humble Pi: When Math Goes Wrong in the Real World

by Matt Parker

As the Earth orbits, the direction it is leaning also changes, going from pointing directly at the sun to pointing away every 13,000 years. A calendar perfectly matching the Earth’s orbit will still swap the seasons every 13,000 years. If we factor the Earth’s axial precession (the change in how it leans) into its orbit, the time between summer solstices is 365 days, 5 hours, 48 minutes and 45.11 seconds. The movement of the Earth’s tilt buys us an extra 20 minutes and 24.43 seconds per orbit. So the true sidereal (literally, “of the stars”) year based on the orbit is longer than the Julian calendar, but the tropical year based on the seasons (which we actually care about) is shorter. It’s because the seasons depend on the tilt of the Earth relative to the sun, not on the actual position of the Earth.