Question

Alpha Centauri, the nearest star to our solar system is $4.3$light years away. assume that alpha centauri has a planet with an advanced civilization. professor Dhg, at the planet's Astronomical Institute, wants to build a telescope with which he can find out whether any planets are orbiting our sun.

(a). What is the minimum diameter for an objectives lens that will just barely resolve Jupiter and the sun? The radius of Jupiter's orbit is $780millionkm.$Assume $\lambda =600nm.$

(b). Building a telescope of the necessary size does not appear to be a major problem. What practical difficulties might prevent professor Dhg's experiment from succeeding?

Short answer

a. The minimum diameter is$1.9297\times {10}^{-5}$

b. The practical difficulties might prevent professor Dhg's experiment from succeeding are described below.

Step-by-step solution

Part(a) Step 1: Given information 

We have given that:

The radius of Jupiter's orbit$S=780millionkm$

distance of nearest star to solar system$r=4.3ly$

and the wavelength is$\lambda =600nm$

We need to find the diameter for an objective lens that will just barely resolve Jupiter and the sun.

Part(a) Step 2: Simplification

Firstly we need to solve the minimum angle,

$S=r.\theta$

$\theta =\frac{S}{r}$

$\theta =\frac{780millionkm}{4.3ly}=9.4x{10}^{12}km/ly$

$\theta =1.9297x{10}^{-5}$.

part (b) Step 1: Given Information

We need to find what practical difficulties might prevent professor's experiment from succeeding.

part (b) step 2: Calculation

The angular resolution is,

$\theta =\left(\frac{1.22x\lambda }{d}\right)$

$1.9297x{10}^{-5}=\left(\frac{1.22x(600x{10}^{-9}}{d}\right)$

$d=\left(\frac{1.22x(600x{10}^{-9}}{1.9297x{10}^{-5}}\right)$

$d=3.8cm.$