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I know that we don't see green stars because in blackbody radiation star doesn't produce one spectrum.

The stars that have peaks in the green spectrum produce other spectrum in nearly same amout. This make them appear white.

If stars have peaks in the infrarad spectrum that has wavelength little longer than red.

We see those stars red because they produce red spectrum more than other visible spectrum.

But if stars have peaks in the ultraviolet spectrum that has wavelength little shorter than violet. Why don't we see those stars purple ?

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Purple and green star exists. For example, a blue supergiant star can have its surface temperature to 50000 K, which peaks at purple in the blackbody spectrum. However, our eyes are more sensitive to blue than purple, and the contrast between blue and purple in the spectrum is insignificant, therefore we see blue.

Also, check this article: https://parade.com/295384/marilynvossavant/can-stars-be-green-or-purple/

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Short answer: most stars emit a very wide range of wavelengths, from MHz (radio) to gamma ray, $10^{19} Hz$ -- albeit rather little of the latter. But what we see is the combined wavelengths that our retina responds to, and due to the blackbody curve, by the time there's enough energy/temperature to produce a lot of blue or violet photons, the number of yellow/red photons swamps them.

The coldest stars appear red, as you mention, because the black body spectral peak is low enough that there isn't much at all in the green-to-violet range.

Now, if you take the light from the hottest stars and run it through a prism or grating spectrometer, you can see that there is blue/purple light included.

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  • $\begingroup$ Individuals experience light at the blue end of the spectrum differently. Some individuals have "red" cones that are only sensitive to red, so all light at the blue end of the spectrum looks blue. Others have "red" cones that are also sensitive to light at the far-blue end of the spectrum. Since this light stimulates both their "red" and blue cones, they perceive it as purple. $\endgroup$ – Gerard Ashton Nov 22 '18 at 0:48

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