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If yes why is it a coincidence or is it so in other systems, ?this would make for an interesting trajectory of planets taking in account their rotation round the galactic center..

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If that were so the visible galaxy would coincide with the ecliptic. On every clear night you can see that that is not the case. This builds on the fact that the sun moves in the galactic plane, like most stars. Note that the galaxy is not entirely flat though.

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  • $\begingroup$ This is an incomplete answer without describing the solar motion with respect to the Galactic plane. $\endgroup$
    – ProfRob
    Oct 29 '19 at 21:50
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The ecliptic plane and the plane of the Galaxy are inclined at angle of 60 degrees (see What angle does our Solar System make with The Milky Way?).

But this isn't quite what you've asked. Instead we need to calculate the angle between the direction of the solar motion and the direction of the ecliptic north pole. Your question amounts to whether these two directions are aligned.

The north ecliptic pole is at celestial coordinates of 18h +66d in the constellation of Draco.

The direction of solar motion is sometimes confused with the solar apex at 18h28m +30d, but this is the direction of motion relative to local stars, which are all whizzing around the Galactic centre around 240 km/s. Now compared with this large speed along a circular path, the peculiar solar motion is small. It moves at about 10 km/s towards the Galactic centre and about 5 km/s up and out of the plane. Therefore, to first order, the solar motion is towards a Galactic latitude of (just over) zero, and a Galactic longitude of (just under) 90 degrees. In celestial coordinates, this corresponds to about 21h +50d.

Thus there is a significant difference (about 30 degrees) between these two directions and so the alignment you suggest is not present.

However, whilst the direction of the north ecliptic pole will remain (approximately) fixed, the direction of motion of the Sun will change as it orbits the Galaxy. For the foreseeable future though, the motion will be mostly towards a position in the Galactic plane and cannot coincide with the north ecliptic pole.

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