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Chapter 35: Q. 47 (page 1019)

Alpha Centauri, the nearest star to our solar system is 4.3light 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 λ=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

Expert verified

a. The minimum diameter is1.9297×10-5

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

Step by step solution

01

Part(a) Step 1: Given information 

We have given that:

The radius of Jupiter's orbitS=780millionkm

distance of nearest star to solar systemr=4.3ly

and the wavelength isλ=600nm

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

02

Part(a) Step 2: Simplification

Firstly we need to solve the minimum angle,

S=r.θ

θ=Sr

θ=780millionkm4.3ly=9.4x1012km/ly

θ=1.9297x10-5.

03

part (b) Step 1: Given Information

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

04

part (b) step 2: Calculation

The angular resolution is,

θ=1.22xλd

1.9297x10-5=1.22x(600x10-9d

d=1.22x(600x10-91.9297x10-5

d=3.8cm.

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Most popular questions from this chapter

The resolution of a digital cameras is limited by two factors diffraction by the lens, a limit of any optical system, and the fact that the sensor is divided into discrete pixels. consirer a typical point-and--shoot camera that has a 20-mm-focal-lengthlens and a sensor with 2.5-μm-widepixels.

(a) . First, assume an ideal, diffractionless lens, at a distance of 100m,what is the smallest distance, in cmbetween two point sources of light that the camera can barely resolve? in answering this question, consider what has to happen on the sensor to show two image points rather than one you can use S1=fbecauses>>f.

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(c). what is the f-numberof the lens for the diameter you found in part b? your answer is a quite realistic value of the f-numberat which a camera transitions from being pixel limited to being diffraction limited for f-numbersmaller than this (larger-diameter apertures), the resolution is limited by the pixel size and does not change as you change the apertures. for f-numberlarger than this (smaller-diameter apertures). the resolution is limited by diffraction and it gets worse as you "stop down" to smaller apertures.

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