Question

Astronomers have discovered a planetary system orbiting a star, which is at a distance of 3.9 × 1014 m from the earth. One planet is believed to be located at a distance of 1.5 × 1012 m from the star. Using visible light with a vacuum wavelength of 556 nm, what is the minimum necessary aperture diameter that a telescope must have so that it can resolve the planet and the star?

Answer 1

Answer:

0.00014456 m

Explanation:

Wavelength = λ = 556 nm

Planet Distance = Δy = 1.5 × 10¹² m

Star Distance = Δy' = 3.9 × 10¹⁴ m

Diameter of lens = D

Distance between star and planet separated by angle θ

$\Delta y=\Delta y'd \theta \\\Rightarrow d \theta=\frac{\Delta y}{\Delta y'}\\\Rightarrow d \theta=\frac{1.5\times 10^{12}}{3.9\times 10^{14}}=0.00384\ radians$

Angular resolution by Rayleigh criteria

$d\theta=1.22\frac{\lambda}{D}\\\Rightarrow D=1.22\frac{\lambda}{d\theta}\\\Rightarrow D=1.22\frac{556\times 10^{-9}}{0.00384}\\\Rightarrow D=0.00014456\ m$

∴Diameter aperture is 0.00014456 m