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Chapter 4: Problem 64

What is the minimum angular separation of two stars that are just-resolvable by the \(8.1-\mathrm{m}\) Gemini South telescope, if atmospheric effects do not limit resolution? Use \(550 \mathrm{nm}\) for the wavelength of the light from the stars.

Short Answer

Expert verified
The minimum angular separation of two stars just-resolvable by the 8.1-meter Gemini South telescope without atmospheric effects, using a wavelength of 550 nm, is approximately \(6.14\ \mathrm{arcseconds}\).

Step by step solution

01

Write down the given parameters

We are given: - Diameter of the telescope, D = 8.1 m - Wavelength of light, λ = 550 nm
02

Convert the wavelength to meters

We need to convert the given wavelength from nanometers to meters: \(λ= 550\ \mathrm{nm} \times \dfrac{1\ \mathrm{m}}{10^9\ \mathrm{nm}} = 5.5 \times 10^{-7}\ \mathrm{m}\)
03

Apply the Rayleigh's Criterion to calculate the minimum angular separation

Using the Rayleigh's Criterion formula, \(θ = \dfrac{1.22λ}{D}\) we substitute the values of λ and D to find the minimum angular separation θ: \(θ= \dfrac{1.22 (5.5 \times 10^{-7}\ \mathrm{m})}{8.1\ \mathrm{m}} = 8.27 \times 10^{-8}\ \mathrm{radians}\)
04

Convert the angular separation to arcseconds

We can convert the angular separation from radians to arcseconds for better readability: \(θ_{arcseconds} = θ \times \dfrac{180\degree}{π \ \mathrm{radians}} \times \dfrac{3600\ \mathrm{arcseconds}}{1\degree} = 1.71 \times 10^{-6} \degree \times \dfrac{3600\ \mathrm{arcseconds}}{1\degree} = 6.14\ \mathrm{arcseconds}\) So, the minimum angular separation of two stars that are just-resolvable by the 8.1-meter Gemini South telescope without atmospheric effects is 6.14 arcseconds.

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Most popular questions from this chapter

A diffraction grating produces a second maximum that is \(89.7 \mathrm{cm}\) from the central maximum on a screen \(2.0 \mathrm{m}\) away. If the grating has 600 lines per centimeter, what is the wavelength of the light that produces the diffraction pattern?

(a) Calculate the angle at which a \(2.00-\mu \mathrm{m}\) -wide slit produces its first minimum for 410 -nm violet light. (b) Where is the first minimum for 700 -nm red light?

(a) Show that a 30,000 line per centimeter grating will not produce a maximum for visible light. (b) What is the longest wavelength for which it does produce a firstorder maximum? (c) What is the greatest number of line per centimeter a diffraction grating can have and produce a complete second-order spectrum for visible light?

(a) What is the minimum width of a single slit (in multiples of \(\bar{\lambda}\) ) that will produce a first minimum for a wavelength \(\lambda\) ? (b) What is its minimum width if it produces 50 minima? (c) 1000 minima?

What is the distance between lines on a diffraction grating that produces a second-order maximum for 760 -nm red light at an angle of \(60.0^{\circ}\) ?
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