Question

Diffraction and the Resolution of Optical Instruments The Hubble Space Telescope (HST) has excellent resolving power because there is no atmospheric distortion of the light. Its 2.4 -m-diameter primary mirror can collect light from distant galaxies that formed early in the history of the universe. How far apart can two galaxies be from each other if they are 10 billion light-years away from Earth and are barely resolved by the HST using visible light with a wavelength of \(400 \mathrm{nm} ?\)

Short answer

Answer: To find the distance between the galaxies, follow these steps: 1. Calculate the angular separation of the galaxies using Rayleigh's criterion: $$ θ = 1.22 \frac{(400 \times 10^{-9}\,\text{m})}{(2.4\,\text{m})} $$ Calculate the value of θ. 2. Convert the distance to the galaxies (10 billion light-years) to meters: $$ \text{Distance to galaxies} = (10 \times 10^9\,\text{light-years}) \times (9.461 \times 10^{15} \, \text{m/light-year}) $$ Calculate the value of the distance in meters. 3. Use the small-angle approximation and the formula relating angular separation, distance between galaxies, and distance to the galaxies from Earth to find the distance (d) between the galaxies: $$ θ = \frac{d}{\text{Distance to galaxies}} $$ Solve for d using the value of θ from Step 1 and the distance to the galaxies in meters from Step 2. By following these steps, you'll find the distance between the two galaxies barely resolved by the Hubble Space Telescope.

Step-by-step solution

Calculate the angular separation of the galaxies

To calculate the angular separation θ, use the Rayleigh criterion: $$ θ = 1.22 \frac{λ}{D} $$ We are given the wavelength λ and the diameter of the telescope's aperture D. Plug these values into the equation and solve for θ. Remember to convert the wavelength (400 nm) to meters. $$ λ = (400 \times 10^{-9})\,\text{m} $$ $$ D = 2.4\,\text{m} $$

Convert the distance to the galaxies to meters

The distance of the galaxies from Earth is given in light-years. Convert this to meters using the fact that 1 light-year is approximately equal to \(9.461 \times 10^{15}\,\text{m}\). $$ \text{Distance to galaxies} = (10 \times 10^9\,{\rm light\text{-}years}) \times (9.461 \times 10^{15} \, \text{m/light-year}) $$

Use the angular separation and distance to the galaxies to find the distance between the galaxies

Use the small-angle approximation to find the distance (d) between the galaxies: $$ \tan θ ≈ θ $$ Then, use the formula relating angular separation, distance between the galaxies(resolved by HST), and the distance to the galaxies from Earth: $$ θ = \frac{d}{\text{Distance to galaxies}} $$ Solve this equation for d using the angular separation obtained in Step 1 and the distance to the galaxies in meters from Step 2.