How is the Final Velocity of Star Movements Calculated?

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The final velocity of star movements is calculated by combining radial and proper (tangential) motions, typically using vector addition. While some discussions suggest that the velocities can be simply added using Pythagorean theorem, there are concerns about the relativistic effects that may need to be considered, particularly in cases involving significant velocities. The transverse Doppler shift is mentioned as a potential factor that could complicate the calculations. For example, Bernard's star has a tangential velocity of 90 and a radial velocity of 111, resulting in a calculated space velocity of 142. The debate centers on whether relativistic components should be integrated into these calculations for greater accuracy.
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When calculating the space velocities of stars by combining what we see of both their radial and proper (tangential) motions, how is the final velocity determined?I don't believe they just add these two vectors together do they? Do they make use of the transverse Doppler shift calculation here?
 
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For instance, apparently Bernard star has a tangential velocity of 90 and a radial of 111. Using plain pythagorans, this works out to 142 in space velocity. (see http://en.m.wikipedia.org/wiki/Proper_motion?wasRedirected=true)

why are these two vectors just added together so simply like this when there ought to be a relativistic component to it as used in the transverse Doppler shift calc?
 

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