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Precession is the phenomenon by which the axis of a spinning object "wobbles" over time. The phenomenon is commonly seen in a spinning toy top, but all rotating objects can undergo precession. As a spinning object precesses, the tilt of its axis goes around in a circle in the opposite direction that the object is spinning.

The precession of the Earth's axis through one complete cycle takes approximately 26,000 years. Over this period, the axis' north pole circles from where it is now, near Polaris in a circle. Polaris isn't well suited for the job, it's a pretty weak star. Never fear, in the year 14000, the brilliant Vega will be the pole star. In 3000BC, the faint star Thuban in the constellation Draco was the pole star. Polaris isn't exactly at the pole, any long exposure unguided shot will show Polaris itself having a short trail. It's close enough, though. The south pole precesses too, of course, always remaining exactly opposite the north pole.

Precession of the Earth's axis is a very slow effect, but at the level of accuracy that astronomers work at, it does need to be taken into account. Note that precession has no effect on Earth's axial tilt. It is 23 degrees and precession doesn't change that. Other factors, however, do, including those factors that cause precession.

Imagine a gyroscope that's slowing down. Its axis wobbles. Exactly the same with the Earth. The tides raised by the Moon and the Sun exert friction against the Earth. The gravity of those two bodies also causes a slowing of Earth's rotation and, more importantly, a torque on the axis, making it precess. A billion years ago, there were 400 days in a year and a year hasn't got shorter, days have got longer. This is what helps cause precession.

What is even more amazing is that Hipparchus first estimated precession in around 130 B.C.
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The orbit of a planet around the Sun is also a form of rotation, and so the axis of a planet's orbital plane will also precess over time. Discrepancies in the precession rate of the planet Mercury compared to those predicted by Classical mechanics were one of the major pieces of evidence leading to the acceptance of Albert Einstein's Theory of Relativity, which predicted the anomalies accurately.