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Chapter 25: Problem 55

The radio telescope at Arecibo, Puerto Rico, has a reflecting spherical bowl of \(305 \mathrm{m}(1000 \mathrm{ft})\) diameter. Radio signals can be received and emitted at various frequencies with appropriate antennae at the focal point of the reflecting bowl. At a frequency of \(300 \mathrm{MHz}\), what is the angle between two stars that can barely be resolved? (Tutorial:radio telescope).

Short Answer

Expert verified
Answer: The angular resolution between two stars that can barely be resolved using the Arecibo radio telescope at a frequency of 300 MHz is approximately 0.229 degrees.

Step by step solution

01

Write down the formula for angular resolution

The formula for angular resolution (θ) can be written as: θ = \frac{1.22 \times \lambda }{D} where, θ – Angular resolution λ – Wavelength of the radio signal D – Diameter of the telescope
02

Compute the wavelength from the given frequency

We are given that the frequency (f) is 300 MHz. We can compute the wavelength (λ) using the relation: λ = \frac{c}{f} where c is the speed of light (\(3\times10^8 m/s\)). So, λ = \frac{3*10^8}{300 * 10^6} = 1 \thinspace m
03

Compute the angular resolution using the formula

Now, we have the values of λ and D (diameter of the telescope). We can substitute the values in the formula for angular resolution. θ = \frac{1.22 \times 1}{305} = 0.004 \thinspace radians
04

Convert the angular resolution from radians to degrees

To convert the angular resolution from radians to degrees, we use the following relation: Degrees = \frac{Radians \times 180}{\pi} In this case, Degrees = \frac{0.004 \times 180}{\pi} ≈ 0.229 \thinspace degrees The angle between two stars that can barely be resolved by the Arecibo radio telescope at a frequency of 300 MHz is approximately 0.229 degrees.

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