It’s often said that the shape of a spiral galaxy follows the curve of a golden spiral. You can see this, for example, in the image above. While it’s often implied that this curve matches exactly, that isn’t the case. The spirals of a spiral galaxy do tend to approximately follow a logarithmic spiral (of which the golden spiral is a special case), it is typically only a rough approximation.
This can be seen in a recent paper in the Monthly Notice of the Royal Astronomical Society, which looks at the spiral shapes of dozens of galaxies. The authors looked at non-barred spiral galaxies taken from the Sloan Digital Sky Survey, and measured the shapes of their spiral arms. While this might seem pretty straight forward, it isn’t. Very few galaxies have a top down view such as the image below (which is actually an artistic representation of the Milky Way galaxy as if viewed from above). Usually galaxies are tilted at some angle relative to us, so the authors had to take the galaxy images and calculate what their spiral shapes would be if viewed from above.
The key characteristic of a logarithmic spiral is known as the pitch angle. The larger the pitch angle, the tighter the spiral. For a perfect logarithmic spiral, the pitch angle is constant at every point in the spiral. For galaxies, the pitch angle can vary along different sections of the spiral, since they are not exactly logarithmic spirals.
As the authors looked at various spiral galaxies, they found that about two-thirds of the galaxies had variations in pitch angle of more than 20%. Typically the pitch angle decreased as you moved outward. Again, this reinforces the idea that galaxies are only approximately log spirals. But the authors also found an interesting relationship between the average pitch angle of a galaxy and the brightness of its center (central bulge). Galaxies with a dim central bulge had lower pitch angles (looser spirals), while those with a bright central bulge had higher pitch angles (tighter spirals). This relation between pitch angle and bulge brightness hints at an underlying mechanism.
Currently, it’s thought that spirals in a galaxy arise from density waves, similar to the way traffic jams can build on a highway. The logarithmic spiral shape arises from the density wave model, but there isn’t a clear connection between the central bulge and the pitch angle.
So it seems we still have a bit of a learning curve to understanding spiral galaxies.