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Intersections of the elliptic and the equatorial planes on the celestial sphere are the Vernal Point and the Autumnal Point. Are these two points two antipodes on the celestial sphere of the Earth?

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Yes… and no. If the Sun–Earth system was alone in the Universe (or at least in the Solar System), it could indeed be the case. However, the Earth’s orbit around the Sun slowly rotates with respect to the rest of the Universe, so that technically, the points where the Sun seems to cross the celestial equator (a projection of the Earth’s equator on the celestial sphere) are not exactly opposite each other.

However, for all practical purposes, yes, these two points are opposite, being defined by the instant orbit of the Earth around the Sun.

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    $\begingroup$ When you say the Earth’s orbit around the Sun slowly rotates with respect to the rest of the Universe, it might be easier to just say precession and nutation? $\endgroup$ Oct 1, 2022 at 10:58
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    $\begingroup$ @BarryCarter: Yes indeed, but not everyone knows what these are. $\endgroup$ Oct 1, 2022 at 22:48
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    $\begingroup$ True, but it's nice to provide jargon/terms of art so the OP can do further research. $\endgroup$ Oct 2, 2022 at 10:08
  • $\begingroup$ I’ll keep that in mind for next time. 😉 $\endgroup$ Oct 3, 2022 at 4:47
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Yes. The points (in fact directions) are defined by the intersection of the two planes (at a given time), not by the position of the Sun at the real equinoxes (six months apart).

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