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Earth comes closest to the Sun every year around January 3. Winter solstice is at 21-22 December, merely some 12-13 days away. That means projection of Earth rotation axis onto the plane of its orbital movement almost overlaps with its orbit's semi-major axis; the axial tilt direction coincides with the direction Earth orbit is elongated.

I couldn't find corresponding data regarding other bodies in the Solar System.

Is this merely a coincidence, or is there a regularity, with direction of the axial tilt coinciding with eccentricity of the orbit? How does that look like in case of other bodies than Earth?

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It's mere coincidence. In about 10,000 years, perihelion will coincide with the northern hemisphere summer solstice.

There are two precessions of interest here. One is the 26,000 year period axial precession of the Earth's rotation axis, caused mostly by the Moon and the Sun; the other is the 112,000 year period apsidal precession of the Earth's orbit, caused mostly by Jupiter and Venus (but also with a pinch of help from general relativity). This result in three different concepts of a year:

  • The sidereal year is the mean time it takes the Earth to make a complete orbit with respect to inertial space ("the fixed stars").

  • The tropical year (aka solar year) is the mean time it takes for the Sun's ecliptic longitude to increase by 360 degrees, and also the time between one vernal equinox to the next. This is the year used as the basis of our calendar. This year is shorter than the sidereal year by about 20 minutes and 25 seconds.

  • The anomalistic year is the mean time between one perihelion passage of the Earth to the next. This year is longer than the sidereal year by about 4 minutes and 43 seconds, and hence longer than the tropical year by about 25 minutes and 8 seconds.

That 25+ minute difference between the anomalistic and tropical years means that time of perihelion passage advances by about a day every 57 years.

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