Question

Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. The electromagnetic radiation we receive from this hydrogen gas has a Doppler-shifted wavelength of 1915 nm when it reaches earth. From experiments done on earth, we know the source wavelength is 1875 nm. What is the speed of the gas relative to earth

Answer 1

Answer:

The speed of the gas relative to the earth = 6.314 * $10^{6} m/s$

Explanation:

Let the observed wavelength be $\lambda_{0} = 1915 nm$

The wavelength of the source, $\lambda_{s} = 1875 nm$

Let the speed of the gas relative to the earth be = v

The observed wavelength in relation to the source wavelength, according to doppler shift in relativity can be expressed as:

$\lambda_{0} = \lambda_{s}\sqrt{\frac{1-\frac{v}{c} }{1+\frac{v}{c} } }$

Where the speed of light, c = 3 * 10⁸m/s

$\frac{\lambda_{0}}{\lambda_{s}} = \sqrt{\frac{1-\frac{v}{c} }{1+\frac{v}{c} } }\\(\frac{\lambda_{0}}{\lambda_{s}})^{2} = \frac{1-\frac{v}{c} }{1+\frac{v}{c} }\\(\frac{1915}{1875}) ^{2}( {1+\frac{v}{c}) = ({1-\frac{v}{c} })$

$1.043 +\frac{1.043v}{c} = 1 - \frac{v}{c} \\\frac{1.043v}{c} + \frac{v}{c} = 1-1.043\\2.043v = -0.043c\\v = \frac{-0.043c}{2.043} \\v = -0.02105c\\v = -0.02105 * 3 * 10^{8} \\v = -6314243.76 m/s\\The absolute value, v = 6314243.76 m/s\\v = 6.314 * 10^{6} m/s$