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Wien's displacement law states the following $$\lambda_{max}\propto\frac{1}{T}$$ However, after learning about redshift and blueshift, it doesn't make sense to me why we can use this law. Supposedly, we can predict the temperature of a star given the peak wavelength. However, this wavelength is being observed from earth, so won't this wavelength be subject to redshift or blueshift? It seems to me then, that this displacement law will give only a rough estimate.

How can we use this law, when the wavelength is changing with the expansion of the universe?

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Stars are not perfect blackbodies. They have elements in their atmosphere that absorb and emit light at particular wavelengths. So the general blackbody pattern is slightly modified with lines of different intensity.

Even if it were redshifted by some amount, we can still identify the the exact lines based on the patterns. Given the known lines in the spectrum, we can tell if there is any significant redshift (or blueshift) and correct.

If we could view emission from a perfect blackbody and had no other information about it, then we could not distinguish between a temperature change and a redshift.

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    $\begingroup$ "then we could not distinguish between a temperature change and a redshift." Are you sure? The ratio of two wavelengths from a blackbody tells you the temperature. Then you can use Wien's law to find what the max should be and determine if the spectrum is redshifted. $\endgroup$
    – HiddenBabel
    Commented Sep 7 at 6:50
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    $\begingroup$ A redshift blackbody spectrum is identical to the spectrum of a cooler non-redshifted blackbody. jb.man.ac.uk/distance/frontiers/cmb/… $\endgroup$
    – BowlOfRed
    Commented Sep 7 at 17:00

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