Final answer:
Rotation curves plotting orbital velocity against distance from a galaxy's center reveal discrepancies indicating dark matter halos. Orbital speeds measured via the Doppler effect, along with Kepler's third law, allow astronomers to calculate the unseen mass. Dark matter halos can be vastly larger than the visible parts of galaxies like the Milky Way and Andromeda.
Step-by-step explanation:
To objectively determine the patterns of velocity behavior in different galaxies, astronomers use rotation curves, which plot orbital velocity against distance from the galaxy's center. The rotation curve of a galaxy such as Andromeda reveals a discrepancy between the actual measured velocities and what would be predicted based on the luminous mass alone. This discrepancy indicates the existence of a halo of dark matter extending well beyond the visible parts of the galaxy. The flatness of the rotation curves at large distances from the galactic center, rather than a decrease expected if only visible matter were present, implies that dark matter forms a significant part of the galaxy's total mass distribution.
The technique for deriving these curves involves measuring the orbital speeds of stars or gas clouds within the galaxy using the Doppler effect. The observed wavelengths are shifted due to the relative motion of the stars or gas clouds as they orbit the galaxy's center. Using these measurements and applying Kepler's third law, astronomers can calculate the mass inside a given orbit, revealing the unseen, or 'dark,' matter.
Dark matter in clusters of galaxies also supports these findings. When examining the clusters' dynamics, the gravitational effects of the cluster can only be accounted for by the presence of additional unseen mass. It is estimated that the dark matter in a galaxy like the Milky Way or Andromeda extends into halos as large as 300,000 light-years, far beyond the reach of visible stars and gas.