We can deduce the motion of a source by seeing whether the emission or absorption lines in its spectrum are shifted in wavelength relative to what we would expect their wavelengths to be if the source were at rest. This shift is called the Doppler shift.
The Doppler formula relates the amount of shift to the velocity of the source. When the velocity is small, the formula is simple:
When the source is moving towards us, the wavelengths are shifted to shorter wavelengths, that is, towards the blue. When the source is moving away from us, the wavelengths are shifted to longer wavelengths, that is, towards the red.
The spectrum at the top shows the Hydrogen Balmer lines at their rest wavelengths. Therefore, this source is practically at rest with respect to us.
In the bottom spectrum, all of the lines are shifted towards longer wavelengths. We say the spectrum is redshifted and we infer that the source is moving away from us.
If the source is moving at very high velocities, i.e. close to the speed of light, we need to invoke a slightly more complicated equation to relate the change of wavelength to the source velocity, based on the same principle. It is called the relativistic doppler formula.
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