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When looking into the sky at night using my bare eyes, I see that stars appear in different colors. From my understanding this is caused by different chemical compositions of those stars which show up in their spectral lines and therefore defining their color.

Does blue and red shift visibly change the color perception when looked at the stars with your naked eye?

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No, the blue and red shift of stars is not possible to detect with the naked eye. There are a couple of reasons for this. First the effect is slight: even for an object moving at thousands of km/s (way faster than stars in the milky way move), it is only a tiny change in hue. Next, since the light of stars is a mixture of many different colours (a black body spectrum) if the light from a star is red shifted, then ultra-violet is shifted down and takes the place of blue, the blue moves towards green and so on. The end result is a mix of all the colours, and so very little change in the perceived colour.

The colour of stars has relatively little to do with their composition, and a lot to do with their temperature. Cool stars (either small, or large, old and with puffed up cool outer layers) look red or orange. Hot stars (large and powerful stars) are white or blue-white. Stars have a spectrum that approximates a blackbody, and so the actual composition has relatively little effect on the colour.

The composition of the atmosphere of a star will absorb particular wavelengths of light, and this results in a dark line in the spectrum (not enough is absorbed to change the apparent colour). It is by measuring the position of these lines in the spectrum that red- and blueshift is observed.

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    $\begingroup$ +1, but note that emission lines are not always necessary to measure redshift. Although much less precise, the drop-out technique pioneered discoveries of high-redshift galaxies in the 90'ies. SED fitting is another way to find redshifts without emission lines. $\endgroup$ – pela Dec 31 '16 at 8:05
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As James K says, it is not possible with the naked eye. But in fact, it almost is. The quasar CTA 102 lies at a redshift of $z\simeq1$ and usually has an apparent magnitude of +17. Very recently (within the last few months) it had a bright outburst, increasing its magnitude to +11.5, i.e. more than two orders of magnitude in brightness! This is still not visible with the naked eye, but a colleague of mine just photographed it with a regular camera. A 10 sec exposure with no tracking makes it barely visible, but a 30 sec exposure, piggybacking on a telescope for tracking the sky, displays a beautiful reddish dot.

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