Part of how we determine it is red shifting. You’ve heard an ambulance or other emergency vehicle drive by you before, and heard the frequency change, right? The change in pitch is *precisely* determined by its speed relative to you.
So imagine you lived in a neighborhood where the speed limit on your street was 10 mph, and on the main access road further away was 25, and then on the highway beyond that was 50, and finally the interstate beyond that is 70. You hear a car driving nearby playing a famous song obnoxiously loud, and you notice the song’s frequency is shifted. If you can determine the speed of the car from its music, you might have an educated guess about its location in terms of which road it is driving on and which direction along that road.
Because of a quirk in universal expansion after the big bang, knowing how fast a star is moving actually gives us a lot of information about where that star might be relative to us. It’s very rare for a far-away star to be going slowly, or for a near star to be going fast. And because stars are all filtering light through the same mix of hydrogen and helium, it’s like they’re all playing a “song” we know, and that we can get reliable speed information from.
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