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Yes and no. The stars in the Milky Way galaxy are all moving in
relation to the center of the galaxy; they rotate around the galactic
center in very much the same way the Earth rotates around the
Sun. The earth and the Sun, for that matter, both rotate around
the galactic center, too. This is why the stars do not appear
to move from our point of view; both the Earth and the stars are
rotating around the galactic center in the same direction; hence,
from our perspective, the stars do not move.
We can see the stars move across the sky each night,
however. The visible motion of the stars is due to the motion
of the Earth; the stars remain in the same position relative to
each other, but seem to move across our sky because the Earth
itself is moving around the Sun.
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