Galaxies fall into several categories, which I think I've discussed before. The simplest two are elliptical and spiral. Those classifications are fairly self-explanatory; an elliptical galaxy is a fuzzy blob of stars, typically in the form of a three-dimensional ellipsoid, while a spiral galaxy has those distinctive spiral arms we all know and love from Hubble pictures. If you want to get even more specific, there are two forms of spiral galaxy: the simple spiral galaxy and the barred spiral. The barred spiral galaxies have a distinctive-looking bar in their centers, and while they look pretty strange, we're actually living in one right now! Yep, the Milky Way is a barred spiral galaxy.
In both kinds of spirals, stars orbit roughly circularly around the center, just like our planets orbit the Sun. But because the galaxy is roughly a disk with mass, the galaxy's stars also exhibit up-down oscillation around the center of the disk.
In an elliptical galaxy, on the other hand, there isn't a nice, uniform central gravitational potential, so stars, especially those far from the center, trace out bizarre three-dimensional shapes. It's a really fascinating problem to analyze the orbits of both individual stars and general classes of stellar orbits in elliptical galaxies.
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