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When a telescope observes a distant star, all it sees is the light being emitted by the star. Observing the wavelength of this light, we determine the velocity of the star. If the wavelength is red-shifted (longer) then the star is moving away from us, and if the wavelength is blue-shifted (shorter) then the star is moving towards us.

But how do we know that the wavelength that we see is not the red shifted wavelength but the original wavelength emitted by the star? Do we already know what wavelength of light the star is supposed to emit (if yes, how?)

Also, using this method we should be able to identify only the radial component of velocity. Because the tangential component of velocity of the star would have no effect on the wavelength that we observe? So how do we determine the actual velocity (including the proper direction) of the star?

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The first part of this question is already answered at How Do we know about redshift?

Regarding the second part, measuring tangential velocity is much harder, and indeed impossible for stars that are far away. Basically the method is to observe the position of the star very accurately compared to very distant stars which are assumed not to be moving and then observe it again a few years later. After adjusting for known effects like aberration and parallax, what remains is "proper motion".

The most sensitive current telescope for this kind of measurement is the Gaia mission, which is able to measure the movemnt of stars up to a few thousand light years away with reasonable accuracy.

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  • $\begingroup$ Great answer, which makes this question not a duplicate. For galaxies, tangential velocities are even more impossible to measure, but a recent paper (Hagala+ 2019) discusses how dipole signals in the CMB may be used to gain constraints on this anyhow. Pretty cool :) $\endgroup$
    – pela
    Jul 10 '19 at 7:38

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