The Milky Way and many galaxies are spirals. To keep the spiral, the outer stars must move faster than the inner stars. From Issac Newton on, we have known that the father a star is from the center, the slower it must travel.
How can spiral galaxies keep their shape?
This is a great question! You're totally right, if we assume most of the mass in a galaxy is concentrated at the center, then the outer regions should move slower than the inner regions, but in order for spiral arms to hold their shape, the outer regions must move faster instead. The sorts of beautifully distinct spirals you are probably thinking of are usually attributed to density wave theory. We previously discussed this briefly in a different question here. I'll go into slightly more detail than the previous answer below. If you want to read more, Wikipedia can always be a good starting point (including a very convincing animation here!).
Instead of galaxies, let's first imagine dropping a pebble into a pond. Circular waves will propagate outwards from where the pebble falls. Of course, water itself isn't propagating away from the center, since otherwise you would have less water in the center and more at the edges after a while. Instead, it is the push and pull of the water with itself that lets water generate these waves. Each water molecule only moves in a confined zone, but together they look like a wave propagating happily away from the center! We just had to give it a little initial push, by dropping the pebble.
Now, by coarse analogy, imagine that each water molecule is instead a star. Then the crests of the wave (where the wave is tall) correspond to zones with more stars, and these zones will appear brighter. And to a faraway observer, who may not be able to see individual stars, they'll just see bright circles moving away from a central point. This isn't really a physical system, but hopefully illustrates that propagating bright zones in galaxies can be caused by stars moving around locally, only causing apparent sustained motion.
So now that we've established waves are one way we can get moving bright zones without requiring moving objects, what sorts of waves can exist in a galaxy? This requires some math that I'm not going to bore you with, but if you consider a self-gravitating disk that has some finite sound speed ("sound" here has the same sense as sound in Earth's atmosphere, regions of higher pressure that propagate at the sound speed), you can show with some work that a two-armed trailing spiral wave pattern forms. This is principally work done by C. C. Lin and Frank Shu in 1964. And now that we know such waves can exist, we just need something to excite them; this is traditionally attributed to galaxy close encounters.
This is a trimphant theory, but as with most astrophysics, it's hard to be 100% certain. There are still scientists verifying certain parts of the theory that have to be numerically correct (e.g. some people believe spiral arms are actually transient, so they break apart and reform, something not quite captured by the Lin-Shu theory), and there are also alternative explanations:
- Excited by tides: two-armed trailing spirals can be explained by tides between two galaxies. This produces the same sorts of arms, but would die out once the galaxies move apart.
- Self-propagative star formation: discussed coarsely in the earlier question, where star formation drives more star formation, so stars tend to be clumped in this way. People generally don't think this produces two-armed patterns but more fuzzy ones.
- Bar-driving: most spiral galaxies have a central bar, which is not necessary for the Lin-Shu spiral density waves. With a central bar and no self-gravity, you can still get a spiral disk.
- Actually solid-body rotation: You can get close to solid-body rotation when there's a lot of dark matter in the galaxy that is spread out far away from the center. Some spiral galaxies seem to exhibit close to solid-body rotation, which means their arms may be material.
So the jury is still out! But the common feature among most of these models is that the arms are not stars moving in synchrony, but are stars passing in and out of the arms.