Do all galaxies rotate? If so, do they rotate in the same direction?
What observations have been made of galaxy rotation?
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One of the easiest way to show that galaxies rotates are Doppler observations of atomic or molecular lines. For example, if the observed galaxy is seen edge-on and rotates clockwise, the left side of the galaxy is going away from you, and is then Doppler-shifted in the red. The right side, going toward you, is Doppler-shifted in the blue.
This kind of observations was realized already in the '60, in radio.
The spiral patern observed in spiral galaxies is also a clear signature of their rotation. Note that spiral are linked to density waves in the galactic disk (see for example Lin & Shu 1964).
As for their directions, you can convinced yourself that there is no tendency, just looking at this Hubble telescope gallery. You can also think that galaxies probably rotate by angular momentum conservation during their formation history, and that it highly dependent on initial conditions, that are clearly random.
We have not discovered every galaxy in existence, nor have we watched each to see how it is spinning. Most of this answer will heavily be speculation. You have been warned!
We have three classifications of galaxy:
Spiral, Elliptical and Irregular.
Irregular galaxies are thought to be quite young, and predominantly gasses that are yet to combine.
Elliptical galaxies are thought to be like a disc, or elongated disk in shape.
Spiral galaxies are theorised to be the older of the three, and it is heavily theorised that they are spiral in shape due to a super massive black hole at their center.
We can see that it is thought elliptical galaxies will eventually become spiral galaxies
E classifications are elliptical. S classifications are spiral.
Therefore if we extrapolate the pattern and theories it could potentially be said that in the course of a galaxies life it will eventually become a spiral, orbiting a super massive black hole as more and more of its mass is drawn to the centre.
Ok so if it can be theorised that most galaxies will eventually spiral a black hole it leads us to believe that the rotation of a galaxy will then be because of the rotational direction of a black hole, right?
This NASA theorist says there may be such a thing as a backwards black hole, one that spins in the opposite direction to its accretion disks, which in this case I assume would be the galaxy.
Looking at observations of galaxies leads to most of them, spinning with the arms trailing, but there are cases where the arms are facing the direction of the spin!
So Yes, we have observed the spin of galaxies, no, not every galaxy rotates, irregular galaxies have no central point of mass of which to rotate about. As for if they all spin in the same direction, that is extremely hard to observe without more data points and further study.