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In astronomical spectroscopy, Doppler shift is an important parameter. But why is it applicable to spectral lines alone, and not to the continuous spectrum in the background? Shouldn't the motion of the source affect the background continuum too?

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The motion does affect the observed continuous spectrum. The flux at each wavelength is doppler shifted to a new wavelength.

The result is of course a new continuum, so a doppler shifted continuous spectrum is rather hard to distinguish from one that is not. In particular, unless the continuum has some feature or break in its slope, then it can be impossible to determine a redshift based only on continuum spectrum measurements.

An example where this is important is in the afterglow of gamma ray burst sources. The spectrum of the afterglow is formed of a synchrotron radiation continuum with several "breaks" in the slope. The frequency of the breaks is affected by the large cosmological redshift of these sources and must be accounted for in the analysis.

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