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In our Milkyway all the stars are orbitting around the black hole in the center. But what is the cause it this rotation and are there galaxies where the stars aren't orbitting a center?

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Stars aren't really orbiting the center of a galaxy, as much as they're orbiting the gravitational potential of the galaxy. The galaxy doesn't need to have a black hole in the center for the stars to orbit, and even if it does (which in fact it does most of the time), the black hole does not dominate the gravitational potential, except in the most central part (see sphere of influence of a black hole).

All matter in the galaxy contributes to the gravitational potential, but since dark matter comprises 5/6 of the mass, this is the most important contributor. Only in the hypothetical case of a perfectly symmetrical distribution of masses, would the black hole and the center of rotation coincide. And since the paths of the stars are ellipses and not perfect circles, the center of the potential will not lie in the center of their orbits, but in one of the focal points. Furthermore, galaxies are dynamical and the potential continually changes, in turn changing the exact orbit of a star.

Galaxies form due to the collapse of a dark matter halo, and the gas that follows along. Such a halo will in general have a non-zero angular momentum, which is partially conserved during the collapse (they do lose a significant amount, for instance through minor and major merging; see e.g. D'Onghia et al. 2006).

As in the case of a forming star, a forming planet, an ice skater, or you on an office chair, reducing the radius means increasing orbital speed. This is the origin of the rotation.

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    $\begingroup$ At 100,000 LY across, stars are not going to be orbiting the exact center of mass of the milky way. They'll be orbiting around some weird time average of where the center of mass was. Gravity is still thought to travel at speed of light, and galaxy is big relative to that. $\endgroup$ – Wayfaring Stranger Feb 11 '17 at 16:08
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    $\begingroup$ @WayfaringStranger: Yes, just as we orbit the place where the Sun was 8 minutes ago. To a high degree of precision, however, that effect can usually be safely ignored, as the dynamical timescale of a galaxy ($\sim\sqrt{R^3/GM}$) is much larger than the light crossing time ($\sim R/c$). $\endgroup$ – pela Feb 11 '17 at 20:38
  • $\begingroup$ The stars in the galaxy NGC 1132 are rotating in all directions how is this possible? Is this also due to the dark matter? $\endgroup$ – Marijn Feb 11 '17 at 21:15
  • $\begingroup$ @Marijn: In general, galaxies can be dynamically either rotation-dominated (like spiral galaxies), or dispersion-dominated (like elliptical galaxies). The latter are probably formed through a major merger of two galaxies, which disturb the system so much as to "destroy" the orbits and let the stars have random orbits. NGC 1131 is an elliptical galaxy. (major mergers don't necessarily result in ellipticals; it depends on a number of factors, including the relative direction of their individual angular momenta). $\endgroup$ – pela Feb 11 '17 at 21:30
  • $\begingroup$ @Marijn, if you're not satisfied with the answer, let me know what could make it better. $\endgroup$ – pela Feb 16 '17 at 22:08
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The cause of the rotation is that the hydrogen cloud that the galaxy condensed out of had some angular momentum which was conserved as the cloud collapsed into a galaxy.

The stars in a galaxy have to be in orbit around something. (Even if it's just the center of mass of the galaxy.) If they were at rest with respect to the center of mass, they'd fall into it!

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