Astronomers know about the future collision of the Andromeda Galaxy and the Milky Way primarily through observations of their movements over time. By studying the velocities and trajectories of both galaxies using various observational techniques, such as spectroscopy and imaging, astronomers can predict their future interactions.
Here are a couple of key points supporting this:
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Relative Motion: The Andromeda Galaxy is moving toward the Milky Way at a speed of about 110 kilometers per second. This was determined through measurements of redshift and proper motion. By calculating their relative velocities, astronomers can project where the galaxies will be in the future.
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Future Predictions: Models that simulate the dynamics of galaxy interactions have been created based on current observations and the laws of gravity. These models predict that the Milky Way and Andromeda will collide in about 4.5 billion years.
So, the correct answer from your options is: Astronomers have observed their movements over time.