Question

Certain wavelengths in the light from a galaxy in the constellation Virgo are observed to be 0.4% longer than the corresponding light from Earth sources. (a) What is the radial speed of this galaxy with respect to Earth? (b) Is the galaxy approaching or receding from Earth?

Short answer

(a) The radial speed of galaxy relative to Earth is $1.2\times {10}^6\mathrm{m}/\mathrm{s}$.

(b) The galaxy is receding from Earth.

Step-by-step solution

Identification of given data

The percentage of wavelength greater than corresponding light from Earth sources is $\mathrm{x}=0.4\%=0.004$

The wavelength of the light shift toward increased wavelength if the source is moving away from the observer. If the wavelength shifts towards decreased wavelength then source moves toward observer.

Determination of radial speed of galaxy relative to Earth

(a)

The radial speed of galaxy relative to Earth is given as:

$\mathrm{v}=\left(\frac{{\left(1+\mathrm{x}\right)}^2-1}{{\left(1+\mathrm{x}\right)}^2+1}\right)\mathrm{c}$

Here, c is the speed of light and its value is $3\times {10}^8\mathrm{m}/\mathrm{s}$.

Substitute all the values in the above equation.

$$\begin{gathered} \mathrm{v}=\left(\frac{{\left(1+0.004\right)}^2-1}{{\left(1+0.004\right)}^2+1}\right)\mathrm{c}\left(\frac{3\times {10}^8\mathrm{m}/\mathrm{s}}{\mathrm{c}}\right) \\ \mathrm{v}=1.2\times {10}^6\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, the radial speed of galaxy relative to Earth is $1.2\times {10}^6\mathrm{m}/\mathrm{s}$.

Identification of movement of Galaxy

(b)

As the wavelengths of the light are observed longer than corresponding wavelength on Earth so the light is shifting toward larger wavelength. The galaxy is moving away from the Earth so galaxy is receding from Earth.

Therefore, the galaxy is receding from Earth.