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When you look a sphere from a fixed observation point, you can easily mistake it for a circle, so I was wondering: are galaxies really "disk" shaped or we just don't have the means to detect the "$z$-axis" of them?

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We often see them edge-on, so we have a pretty good idea of what general galaxies' z-axes look like. –  Charles Jun 27 at 13:44

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Not all galaxies are disk-shaped, but some certainly are. (Some others are spiral, etc.) For one thing, we see a lot of galaxies, and several of them look exactly like they would if they were disk-shaped and we were just seeing them from different angles. Some seem circular because we are seeing them head-on, while others seem more linear or elliptical because we are seeing them at an angle.

Furthermore, simulations of galaxy formation yield disk and spiral-shaped galaxies thanks to conservation of angular momentum. A cluster of matter (stars, interstellar gas, etc.) starts with some net angular momentum, and collisions between the different matter particles eventually cause almost all the matter in the galaxy to orbit around their net angular momentum axis.

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