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It's been known that the Earth's axial tilt varies from 22.5 to 24.5. Does this result in a wobbling while the value of the tilt varies about these two limits? If so, how does it affect the mechanics of the Earth's movement (its rotation and revolution about the Sun)?

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The Wikipedia Precession article explains this.

Precession of the Earth's axis is caused mainly by the Sun's gravity acting on the oblate spheriod of the Earth. The Earth bulges at the equator and the force of the Sun's gravity on the bulge makes the axis wobble like a child's spinning top - the axis traces out a cone shape in space. It has no effect on the Earth's orbit.

Perihelion precession is caused by the gravity of Jupiter and the other outer planets.

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Based on what I can access right now, the precession of Earth's axes are in a cone shape, so there isn't much wobble. But with all of those Near-Earth Objects, and nearby planets, there is probably some wobble (I'm no professional).

There's also this thing called the Milankovitch Cycle, which shows how the combined effects of precession, external forces, and etc. affect climate. It also affects the eccentricity, or the circularity, of Earth's orbit in a cycle of 430,000 years. This affects seasonal length and season intensity. It also changes orbit orientation, so the perihelion and aphelion move to different locations. If you want to learn more, read https://en.wikipedia.org/wiki/Milankovitch_cycles.

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