Example Red Shift Problem

Suppose you abserve a galaxy for which the bluegreen line of hydrogen is red shifted to the red end of the visible spectrum at 700 nm. What is the recession speed of the galaxy?

The bluegreen line of the hydrogen spectrum is the n=4 to n=2 transition for which the photon energy and wavelength are:

The main difficulty I have with this kind of problem is getting the sign wrong on one of the quantities or inverting a term. So, starting from the beginning, the observed wavelength is lengthened by a receding source according to

but the observer sees the period of the light as time-dilated, so the observed wavelength is

So the galaxy is seen to be receding at 0.349c. Note that in this approach, the velocity of recession is treated as a positive number. Problems occur from the fact that some approaches use a negative number for a receding velocity. The usual red shift treatment in astronomy books treat the recession velocity as positive and express the red shift in terms of a z-parameter. In those terms, this calculation becomes

But in standard Doppler shift calculations, a receding velocity is treated as a negative quantity and with that presumption, the Doppler wavelength shift becomes

So care must be taken to keep the signs consistent.

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