Hubble's law and the Big Bang theory — practice questions

1. Identify the given values:
   • Recessional velocity, $v = 4900 \text{ km s}^{-1}$
   • Hubble's constant, $H_0 = 70 \text{ km s}^{-1}\text{Mpc}^{-1}$
2. State Hubble's Law:
   • $v = H_0 d$
3. Rearrange to solve for distance ($d$):
   • $d = \frac{v}{H_0}$
4. Substitute the values and calculate:
   • $d = \frac{4900 \text{ km s}^{-1}}{70 \text{ km s}^{-1}\text{Mpc}^{-1}}$
   • $d = 70 \text{ Mpc}$

Answer: The distance to the galaxy is $70 \text{ Mpc}$.

1. Calculate the change in wavelength ($\Delta \lambda$):
   • $\Delta \lambda = \lambda_{observed} - \lambda_{original}$
   • $\Delta \lambda = 662.8 \text{ nm} - 656.3 \text{ nm} = 6.5 \text{ nm}$
2. Calculate the redshift (Z):
   • $Z = \frac{\Delta \lambda}{\lambda_{original}}$
   • $Z = \frac{6.5 \text{ nm}}{656.3 \text{ nm}} = 0.009904$
3. Calculate the recessional velocity (v):
   • For small redshifts, $v \approx Zc$
   • $v = 0.009904 \times (3.00 \times 10^8 \text{ m s}^{-1}) = 2.9712 \times 10^6 \text{ m s}^{-1}$
4. Convert velocity to km s⁻¹:
   • $v = (2.9712 \times 10^6 \text{ m s}^{-1}) / 1000 = 2971.2 \text{ km s}^{-1}$
5. Use Hubble's Law to find the distance (d):
   • $d = \frac{v}{H_0}$
   • $d = \frac{2971.2 \text{ km s}^{-1}}{72 \text{ km s}^{-1}\text{Mpc}^{-1}} = 41.26... \text{ Mpc}$

Answer: The distance to the galaxy is approximately $41 \text{ Mpc}$ (to 2 s.f.).