Question

Sometime around 2022, astronomers at the European Southern Observatory hope to begin using the E-ELT(European Extremely Large Telescope), which is planned to have a primary mirror 42 m in diameter. Let us assume that the light it focuses has a wavelength of 600 nm. (1 light-year = 9.461×1015 m) Note: Jupiter's Diameter dj=1.43×108 m 1) What is the most distant Jupiter-sized planet the telescope could resolve, assuming it operates at the diffraction limit? (Express your answer to two significant figures.)

Answer 1

Answer:

$8.2\times 10^{15}\ m$

Explanation:

$\lambda$ = Wavelength = 600 nm

d = Diameter of mirror = 42 m

D = Distance of object

x = Diameter of Jupiter = $1.43\times 10^8\ m$

Angular resoulution is given by

$\Delta\theta=1.22\frac{\lambda}{d}\\\Rightarrow \Delta\theta=1.22\frac{600\times 10^{-9}}{42}\\\Rightarrow \Delta\theta=1.74286\times 10^{-8}\ rad$

We also have the relation

$\Delta\theta\approx=\frac{x}{D}\\\Rightarrow D\approx\frac{x}{\Delta\theta}\\\Rightarrow D\approx\frac{1.43\times 10^8}{1.74286\times 10^{-8}}\\\Rightarrow D\approx 8.2049\times 10^{15}\ m$

The most distant Jupiter-sized planet the telescope could resolve is $8.2\times 10^{15}\ m$