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What does the angle between the plane of the earth's orbit and the ecliptic (now about 23.5 degrees) vary between, and over what time scale?

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  • $\begingroup$ I think you mean the plane of Earth's equator and the ecliptic. $\endgroup$ – James K Mar 29 '18 at 8:06
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By definition, it is zero; the ecliptic is the plane of earth's orbit (or rather, the inverse: the path the Sun seems to follow along the sky).

But you're probably looking for variations in the axial tilt of the Earth. There's a lot of formulas in the article, and on the (astronomically speaking) short timescale of a few million years, it oscillates between 22 and 24.5 degrees with a period of about 41,000 years. See the graph below, the red dot is the year 2000:

enter image description here

(picture by Tfr000 on Wikipedia, data from Laskar (1986))

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