Light from the Sun is used to produce a visible spectrum - Scottish Highers Physics - Question 4 - 2015

The position of the hydrogen line is different in each spectrum because of the redshift effect. As the galaxies are moving away from the Sun, the wavelength of the emitted light increases, causing the hydrogen line to shift towards the red end of the spectrum. The distant galaxy exhibits a more significant redshift compared to the nearby galaxy, indicating a greater velocity away from the observer.

The redshift (z) can be calculated using the formula:

$z = \frac{\lambda_{obs} - \lambda_{emit}}{\lambda_{emit}}$

where

• ( \lambda_{obs} = 450 \times 10^{-9} \text{ m} )
• ( \lambda_{emit} = 410 \times 10^{-9} \text{ m} )

Substituting these values gives:

$z = \frac{450 \times 10^{-9} - 410 \times 10^{-9}}{410 \times 10^{-9}} = 0.098$.

The distance (d) to the distant galaxy can be computed using the formula:

$d = \frac{v}{H_0}$

where ( v = z \cdot c ) and ( H_0 ) is the Hubble constant.

Given that ( z = 0.098 ) and using the speed of light ( c = 3.00 \times 10^8 \text{ m/s} ), we first find:

$v = 0.098 \cdot 3.00 \times 10^8 = 2.94 \times 10^7 \text{ m/s}$.

Assuming ( H_0 = 70 \text{ km/s/Mpc} ) or ( 2.27 \times 10^{-18} \text{ s}^{-1} ):

$d = \frac{2.94 \times 10^7}{2.27 \times 10^{-18}} \approx 1.3 \times 10^{25} \text{ m} \approx 4.1 \times 10^{24} \text{ light years}$.