The easiest way to experience the Doppler effect is to listen to a moving train, which, as it moves towards a person, it sounds like it has a higher tone, since the frequency of the sound is squeezed together a little bit. As the train speeds away, the sound gets stretched out, and is lower in tone. The same happens when a light-emitting object moves very fast. An object, like a star or a galaxy, that is far away and moving toward us, will look more blue than it normally does. This is called blue shift. A star or galaxy moving away from us will look more red than it would if the source were not moving in our frame of reference. This is where red shift got its name, since the colors are shifted towards the red end of the spectrum.
The reason astronomers can tell how far the light gets shifted is because certain chemical elements, like the calcium in bones or the oxygen people breathe, have a unique "fingerprint" of light that no other chemical elements have. They can see what colors of light are coming from a star, and see what it is made of. Once they know that, they check to see the difference between where the fingerprint, called spectral lines, are actually at, and then look at where they are supposed to be. When they see that, they can tell how far away the star is, whether it is moving toward us or away from us, and also how fast it is going, since the faster it goes, the farther the distance the spectral lines are from where they should be.
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