The Griffith Observatory in Los Angeles includes an astronomical refracting telescope (Griffith telescope) with an objective lens of diameter 305 mm and focal length 5.03 m. Calculate the wavelength of light for which the Griffith telescope has a minimum angular resolution of 1.8 × 10⁻⁴ rad. 1.2 The Griffith telescope is used to observe two point objects which subtend an angle of 1.8 × 10⁻⁴ rad at the unaided eye. The typical human eye has a minimum angular resolution of approximately 3.2 × 10⁻⁴ rad. Calculate the focal length of the eyepiece lens so that an observer can just resolve the two objects when observing them through the Griffith telescope. 1.3 The asteroid Apophis has a diameter of 325 m. It has been calculated that, in 2029, its distance of closest approach to the Earth’s surface will be 3.0 × 10⁶ km. The Griffith telescope may be used to view Apophis using the eyepiece calculated in question 1.2. Deduce whether this telescope is suitable to obtain a detailed view of Apophis. Support your answer with a calculation.

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The Griffith Observatory in Los Angeles includes an astronomical refracting telescope (Griffith telescope) with an objective lens of diameter 305 mm and focal length 5.03 m - AQA - A-Level Physics - Question 1 - 2018 - Paper 4

Question 1

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The Griffith Observatory in Los Angeles includes an astronomical refracting telescope (Griffith telescope) with an objective lens of diameter 305 mm and focal length... show full transcript

Worked Solution & Example Answer:The Griffith Observatory in Los Angeles includes an astronomical refracting telescope (Griffith telescope) with an objective lens of diameter 305 mm and focal length 5.03 m - AQA - A-Level Physics - Question 1 - 2018 - Paper 4

Step 1

Calculate the wavelength of light for which the Griffith telescope has a minimum angular resolution of 1.8 × 10⁻⁴ rad.

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Answer

To find the wavelength ( ( \lambda )) of light, we can use the formula for angular resolution given by Rayleigh's criterion:

$\theta = \frac{1.22 \lambda}{D}$

Where:

( \theta = 1.8 \times 10^{-4} ) radians
( D = 0.305 ) m (diameter of the lens)

Rearranging the formula to calculate ( \lambda ), we get:

$\lambda = \frac{\theta D}{1.22}$

Substituting the values:

$\lambda = \frac{(1.8 \times 10^{-4})(0.305)}{1.22} = 4.47 \times 10^{-5} \text{ m}$

Thus, the wavelength is approximately 4.47 × 10⁻⁵ m.

Step 2

Calculate the focal length of the eyepiece lens so that an observer can just resolve the two objects when observing them through the Griffith telescope.

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Answer

Given:

Minimum angular resolution of the human eye ( = 3.2 \times 10^{-4} ) rad.
Angular resolution required for the telescope's eyepiece should be the same as the human eye.

Using the formula:

$M = \frac{\theta_t}{\theta_e}$

Where:

( M ) is the magnification
( \theta_t = 1.8 \times 10^{-4} ) rad (angular resolution at the telescope)
( \theta_e ) = 3.2 × 10⁻⁴ rad (angular resolution of human eye)

We find:

$M = \frac{1.8 \times 10^{-4}}{3.2 \times 10^{-4}} = 0.5625$

Next, using the magnification formula related to the focal lengths:

$M = \frac{f_t}{f_e}$

Where:

( f_t ) is the focal length of the telescope ( f_t = 5.03 ) m
( f_e ) is the focal length of the eyepiece lens

Rearranging gives:

$f_e = \frac{f_t}{M} = \frac{5.03}{0.5625} \approx 8.95 \text{ m}$

So, the focal length of the eyepiece is approximately 8.95 m.

Step 3

Deduce whether this telescope is suitable to obtain a detailed view of Apophis.

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Answer

Calculation:

First, calculate the angular resolution required to observe Apophis:

$\text{Angular resolution} = \frac{D}{d}$

Where:

( D = 325 ) m (diameter of Apophis)
( d = 3.0 \times 10^{6} \text{ km} = 3.0 \times 10^{9} \text{ m} )

Calculating gives:

$\text{Angular resolution} = \frac{325}{3.0 \times 10^{9}} \approx 1.08 \times 10^{-7} \text{ radians}$

Comparing this with the angular resolution of the telescope:

Minimum angular resolution of the telescope ( \approx 1.8 \times 10^{-4} \text{ radians} )
Angular resolution required to view Apophis ( \approx 1.08 \times 10^{-7} \text{ radians} )

Since the required resolution (1.08 × 10⁻⁷ rad) is significantly smaller than the angular resolution of the telescope (1.8 × 10⁻⁴ rad), it can be concluded that the Griffith telescope is not suitable for obtaining a detailed view of Apophis.

The Griffith Observatory in Los Angeles includes an astronomical refracting telescope (Griffith telescope) with an objective lens of diameter 305 mm and focal length 5.03 m - AQA - A-Level Physics - Question 1 - 2018 - Paper 4

Worked Solution & Example Answer:The Griffith Observatory in Los Angeles includes an astronomical refracting telescope (Griffith telescope) with an objective lens of diameter 305 mm and focal length 5.03 m - AQA - A-Level Physics - Question 1 - 2018 - Paper 4

Calculate the wavelength of light for which the Griffith telescope has a minimum angular resolution of 1.8 × 10⁻⁴ rad.

Calculate the focal length of the eyepiece lens so that an observer can just resolve the two objects when observing them through the Griffith telescope.

Deduce whether this telescope is suitable to obtain a detailed view of Apophis.

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