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How do scientists find out the true color of the star's light as well as the true doppler shift (relative speed)? Seems to me you wouldn't know how to separate out those 2 values.

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Spectroscopy is done on the starlight. Say we look for hydrogen lines. We know where they’ll appear in the spectrum of laboratory hydrogen. If the starlight is redshifted then all the hydrogen lines will be shifted by the same amount. As can be seen in an example spectrum below.

enter image description here

Once the shift is quantified, we can work back what the unshifted spectrum of the star would look like.

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