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I've learned recently that the rotation axis of Earth (axial tilt or obliquity) oscillates between 22.1 and 24.5 degrees.

I would like to understand why does that happen.

Having a basic understanding of classical mechanics, (and very little of orbital mechanics) I would have expected that the conservation of angular momentum would keep the rotation axis constant.

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The reason is given in Wikipedia

Lunisolar precession is caused by the gravitational forces of the Moon and Sun on Earth's equatorial bulge, causing Earth's axis to move with respect to inertial space. Planetary precession (an advance) is due to the small angle between the gravitational force of the other planets on Earth and its orbital plane (the ecliptic), causing the plane of the ecliptic to shift slightly relative to inertial space. Lunisolar precession is about 500 times greater than planetary precession.[4] In addition to the Moon and Sun, the other planets also cause a small movement of Earth's axis in inertial space, making the contrast in the terms lunisolar versus planetary misleading, so in 2006 the International Astronomical Union recommended that the dominant component be renamed the precession of the equator, and the minor component be renamed precession of the ecliptic, but their combination is still named general precession.[5] Many references to the old terms exist in publications predating the change.

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