A distant observer is located at redshift 2. What is the observed wavelength of the 21 cm hyperfine transition line of hydrogen originating from the galaxy?

The observed wavelength of the line of hydrogen originating from the galaxy is 23cm.

Explanation:

The speed of the galaxies using the shifted of the spectral lines can be known by using the Doppler speed:

 (1)

Where is the wavelength shift, is the wavelength at rest, v is the velocity of the source and c is the speed of light.

Therefore, from equation 1 it can be deduce the equation for the redshift

 (2)

Where z is the redshift.

Hence, the redshift represents this shift of the spectral lines to red part in the spectrum of a galaxy or any object which is moving away.

Then, can be isolated from equation 2.

 (3)

For this particular case and

 (3)

 (3)

Hence, the observed wavelength of the line of hydrogen originating from the galaxy is 23cm.

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