There’s really two questions buried in here – are spirals and ellipticals really fundamentally different objects, and how the spiral arms in spiral galaxies come about, so I’ll tackle these slightly independently, starting with whether spirals and ellipticals are really that different.
And they are! There’s a really easy way to spot this difference; spiral galaxies are incredibly, incredibly thin. The thickness of the spiral disk of our own Milky Way is only about 0.6 light years, whereas it’s about 100,000 light years across. With some straightforward division, we can calculate that the Milky Way is 166,666 times larger from edge to edge than it is vertically. This kind of thinness is hard to wrap your mind around. A standard piece of printer paper, typically clocking in at a width of 0.05 mm, is proportionally 30 times as thick as our galaxy.
To get something as proportionally thin as the Milky Way, you could paint a solid circle on a basketball court, as big a circle as you could fit in the court. The width of an NBA basketball court is 50 feet wide, so you’d have a 50 foot wide circle painted onto the floor. Let your paint dry. The average coat of paint is about 100 microns thick (0.00394 inches) – so relative to the size of the 50 foot circle, the height of one coat of paint on the floor is pretty close to proportional to the height of the Milky Way’s thin disk of stars. If you could suspend this in the air, you’d be looking at a fifty foot circle that’s approximately the same width as a thin human hair. Spiral galaxies are mind-bogglingly thin.
Meanwhile, an elliptical galaxy doesn’t have this stupendously thin disk. Ellipticals tend to be more football shaped than anything else, so they are usually only a few times longer in one direction than in either of the other two, so there’s not much of an “edge” in any direction. Ellipticals and spirals are two very different types of galaxies, and the differences we see just by looking at their shapes are reflected in other properties, like their age (ellipticals are older), color (the old stars in the ellipticals make them much redder), and number of new stars formed (overwhelmingly, new stars are found in spiral galaxies).
All that said, spiral arms are still kind of weird. If the galaxy was a static object, like our circle made of paint, and rotated as a unit, you’d need the spiral arms to have been there from the galaxy’s birth, but once they got there you could keep them fairly easily. Unfortunately, we can see that galaxies don’t rotate as a solid object, like a DVD in a player.
So that idea’s out; what next? The stars and gas which are closest to the center of a spiral galaxy rotate faster around the center than the stars at the outskirts. This difference in rotation speeds means, if you give a galaxy a spiral arm pattern and then let the galaxy just exist for a little while, your nice loose spiral arm pattern will wind up into a really tight spiral, and the lack of space between arms will make it hard to even spot them in the first place. So if this scenario is the case, then the strong spiral arm features shouldn’t last long. Again, there are problems; if spiral arms shouldn’t last very long, then in general you wouldn’t expect to see many strong spiral arms if you look out at the galaxy population. And while there are certainly galaxies without distinct spiral arms, there are a lot of galaxies with strong spiral features. We’ll have to throw out this idea as well.
What we are left with is an idea called spiral density wave theory, which suggests that the spiral arms aren’t a physical “thing”, but are made of stars which are simply passing through, more like a traffic jam than anything else. The apparent spiral arms appear because stars don’t orbit the center of the galaxy in perfect circles. Each star is instead on an elliptical orbit, much like the recurrent comets in our solar system. As stars are moving the slowest at the distant edge of their orbit, if a large number of stars have turnaround points around the same place, you’ll wind up with an extra dense region of stars, creating an apparent spiral arm. Each star will continue along its own orbit, drifting in and out of spiral arms as the galaxy spins.
So your idea of differently rotating matter lining up to form the spiral arms is actually very close to the mark for the stars found within the very thin disks of spiral galaxies. However, because of the 3D nature of galaxies, in that spirals are so very thin, and ellipticals are so extremely round, we can’t account for the differences between spirals and ellipticals this way.
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