Question

One of the strongest emission lines observed from distant galaxies comes from hydrogen and has a wavelength of 122 nm (in the ultraviolet region). (a) How fast must a galaxy be moving away from us in order for that line to be observed in the visible region at 366 nm? (b) What would be the wavelength of the line if that galaxy were moving toward us at the same speed?

Answer 1

Answers:

a) $24(10)^{7} m/s$

b) $4(10)^{-8} m/s$

Explanation:

This problem can be solved by the Relativistic Longitudinal Doppler Effect equation:

$\frac{\lambda}{\lambda_{o}}=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$ (1)

Where:

$\lambda=366(10)^{-9}m$ is the observed wavelength for the hidrogen emission line in the spectrum of the galaxy

$\lambda_{o}=122(10)^{-9}m$ is the wavelength for the same hidrogen line observed at rest

$v$ is the velocity of the galaxy

$c=3(10)^{8}m/s$ is the speed of light

Knowing this, let's begin with the answers:

a) In this part we have to find the speed of the galaxy. So, we have to isolate $v$ from (1):

$v=\frac{{\lambda_{o}}^{2} - {\lambda}^{2}}{-{\lambda}^{2} - {\lambda_{o}}^{2}} c$ (2)

$v=\frac{{(122(10)^{-9}m)}^{2} - {(366(10)^{-9}m)}^{2}}{-{(366(10)^{-9}m)}^{2} - {(122(10)^{-9}m)}^{2}} 3(10)^{8}m/s$ (3)

$v=24(10)^{7}m/s$ (4) This is the velocity of the galaxy

b) Now we have to find $\lambda$ when the galaxy is moving toward us at the same speed, this means $v=-24(10)^{7}m/s$.

Therefore, we have to isolate $\lambda$ from (1):

$\lambda=\lambda_{o}\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$ (5)

$\lambda=122(10)^{-9}m\sqrt{\frac{1+\frac{-24(10)^{7}m/s}{3(10)^{8}m/s}}{1-\frac{-24(10)^{7}m/s}{3(10)^{8}m/s}}}$ (6)

Finally:

$\lambda=4(10)^{-8}m$ (7) This is the observed wavelength when the galaxy is moving toward us.