It is actually a rather complicated deductive exercise using lots of data in radio astronomy and other fields. The most compelling reason is based on a very simple observation. Take a look at the picture below to get your first clue about the Milky Way's shape:
This image was taken with the Kuiper Airborne Observatory in the direction of the constellation Sagittarius. I can't post the most common view that you get of the Milky Way because that kind of picture, taken by a simple camera pointed up at the sky, is usually copyrighted.
Recently, astronomers using the 2MASS infrared sky survey imaged the entire sky, and put together this dramatic image of the complete Milky Way:
It is very clear from these images, and a bit of detective work, that the Milky Way is a flat system of stars, with a central bulge located in the constellation Sagittarius. So, we know the Milky Way is a very flat system in 3-dimensions, because from the inside, its projection on the sky is a very narrow band of faint stars. We also know it does not have a gigantic nucleus because when we look along the Milky Way on the sky, we do not see a 'ball' of light peeking up from the Milky Way. Using our knowledge of the shapes of other galaxies, we easily conclude it must be a version of a spiral-type galaxy with an unremarkable central nucleus. Here is a similar type of galaxy seen edge-on:
This is the galaxy NGC 4565 taken by Dr. Crowe and his summer students at the University of Hawaii. Note, the Milky Way image was taken from INSIDE our galaxy. We cannot travel outside our galaxy to see what it really looks like because the distance (over 100,000 light years) is too far to travel.
Detailed studies of the motions of interstellar gas clouds fills in this picture considerably by letting us plot the locations and speeds of these clouds, and in most versions of these studies, a distinct spiral pattern emerges like the sketch below:
This map is one of the first made by astronomers when they began to map the locations of major neutral Hydrogen clouds using radio telescopes. By knowing how a spiral galacy rotates (following the laws of Kepler) you can relate the velocity of a cloud and the direction of detection to a specific location in this map. Many such maps have been created using different spiral arm tracers (HII regions, molecular clouds etc) and the results are pretty consistent. Below is an artists rendition of what the Milky Way might look like from outside:
It is available at an excellent web site called Atlas of the Universe There is some dispute over exactly where the spiral arms are located, and their pitch angles, but the essential details support the idea we live in a 'late-type' dusty spiral with perhaps a weak bar-like feature in its core.
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