We look at the density of the stars and the distance for those we can measure. We also can see other spiral galaxies and from that infer how it would look like if we are inside such a galaxy. The fact that it is at least disc shaped can be seen with your own eyes on a proper dark night. ( And no clouds obviously).
From measuring the different wavelength of light coming from a star (or lack of) we can determine what type of star it is. Different stars emit a different type of light and can be vastly different in size. We can also determine roughly what direction the star is moving in depending on how red-shifted or blue shifted the light is.
Using this information along with the “size” it appears in our sky/telescopes we can map out how far each star is. We do that for all the stars and over time we find that there are some areas that are denser with stars and some areas less dense, we plot all that out and it shows the shape of our galaxy. As we keep measuring more and more stars it then fine tunes our models.
We can use telescope to see distant stars and analyse their light to know their distance. By plotting this on a map, we can see that stars are clustered into galaxies, and the stars within about 100,000 ly of us are arranged into a spiral galaxy.
One caveat is that the galactic core is dense with opaque dust, making it hard to know if ours is a bar galaxy, or what’s on the other side of the galaxy.
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