In the hydrogen spectrum the wavelength of H λ line is 656 nm whereas, in the spectrum of a distant galaxy the H λ line has a wavelength of 706 nm . The estimated speed of the galaxy with respect to t

Question

In the hydrogen spectrum the wavelength of H λ line is 656 nm whereas, in the spectrum of a distant galaxy the H λ line has a wavelength of 706 nm . The estimated speed of the galaxy with respect to the earth is

TestHub · Accepted Answer

Since, the wavelength observed is more than the true wavelength, we can say that the galaxy is receding. Doppler effect in this case gives us
$λ^{'} = λ \sqrt{\frac{c + v}{c - v}}$
$706 = 656 \sqrt{\frac{c + v}{c - v}}$
or, $\frac{c + v}{c - v} = {(\frac{706}{656})}^{2} = 1.16$
or, $c + v = 1.16 c - 1.16 v$
$v = \frac{0 .16 c}{2 .16} = \frac{0 .16 \times 3 .0 \times 10^{8}}{2 .16} = 0 .22 \times 10^{8} m s^{- 1}$
$v = 2 .2 \times 10^{7} m s^{- 1}$

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