Question

In hydrogen spectrum the wavelength of $H_{\alpha }$ line is 656 nm; whereas in the spectrum of a distant galaxy $H_{\alpha }$ line wavelength is 706 nm. Estimated speed of galaxy with respect to earth is

• A. $2\times 10^{8}m/s$
• B. $2\times 10^{7}m/s$
• C. $2\times 10^{6}m/s$
• D. $2\times 10^{5}m/s$

Answer 1

Answer: (B)

Since, the wavelength $(\lambda )$ is increasing, we can say that the galaxy is receding. Doppler effect can be given by
$\lambda =\lambda \sqrt{\frac{c+v}{c-v}}$...(i)
or $706=656\sqrt{\frac{c+v}{c-v}}$ or $\frac{c+v}{c-v}=(\frac{706}{656}{)}^2=1.16$ $\therefore c+v-1.16c-1.16v$ $v=\frac{0.16c}{2.16}=\frac{0.16\times 3.0\times 10^8}{2.16}=0.22\times 10^{8}m/s$ $v\approx 2.2\times 10^{7}m/s$ If we take the approximation then Eq. (i) can be written as $\Delta \lambda =\lambda \left(\frac{v}{c}\right)$...(ii) From here $v=\left(\frac{\Delta \lambda }{\lambda }\right),c=\left(\frac{706-656}{656}\right)(3\times 10^8)$ $v=0.23\times 10^{8}m/s$ Which is almost equal to the previous answer. So, we may use Eq. (ii) also.