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 is a photograph of the Milky Way in the night sky over Black Rock Desert, Nevada taken on 7/22/2007 (Credit Wikipedia). It is a 54 second eposure taken with a tripod mounted Cannon EOS 5D digital camera with a 16mm lens, wide open at f2.8 and ISO800.
If we were to take a panorama photo of the entire sky and stitch them into a standard geometric projection wwe would get this beautiful image.
This magnificent 360-degree panoramic image (Wikipedia:Milky Way), covering the entire southern and northern celestial sphere, reveals the cosmic landscape that surrounds our tiny blue planet. This gorgeous starscape serves as the first of three extremely high-resolution images featured in the GigaGalaxy Zoom project, launched by ESO within the framework of the International Year of Astronomy 2009 (IYA2009). The plane of our Milky Way Galaxy, which we see edge-on from our perspective on Earth, cuts a luminous swath across the image. The projection used in GigaGalaxy Zoom place the viewer in front of our Galaxy with the Galactic Plane running horizontally through the image — almost as if we were looking at the Milky Way from the outside. From this vantage point, the general components of our spiral galaxy come clearly into view, including its disc, marbled with both dark and glowing nebulae, which harbours bright, young stars, as well as the Galaxy’s central bulge and its satellite galaxies. As filming extended over several months, objects from the Solar System came and went through the star fields, with bright planets such as Venus and Jupiter.
In the 1990s, 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. Here is one that summarizes more of the essential details revealed by studying the interstellar medium.
Below is an artists rendition of what the Milky Way might look like from outside:
Viewed from above, we can now see that our gaze takes across the Perseus Arm (toward the constellation Cygnus), parts of the Sagittarius and Scutum-Centaurus arms (toward the constellations Scutum, Sagittarius and Ophiuchus) and across the central bar. Interstellar dust obscures much of the center of the galaxy. Credit: NASA
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