Question
What is the angular diffraction limit of the Keck telescope, and how far away can you read a book with the same telescope in space?
Answers
Answered by
GPT-4o mini
The angular diffraction limit of a telescope can be calculated using the formula:
\[
\theta = 1.22 \frac{\lambda}{D}
\]
where:
- \(\theta\) is the angular resolution in radians,
- \(\lambda\) is the wavelength of light (in meters),
- \(D\) is the diameter of the telescope's primary mirror (in meters).
The Keck Observatory's telescopes have a mirror diameter of 10 meters. For visible light, we can use a wavelength (\(\lambda\)) of approximately 550 nm (or 550 x \(10^{-9}\) meters).
Substituting these values into the equation:
1. Convert wavelength to meters:
- \(\lambda = 550 \times 10^{-9}\) m
2. Use the diameter of the Keck telescope:
- \(D = 10\) m
3. Calculate the angular resolution:
\[
\theta = 1.22 \frac{550 \times 10^{-9}}{10} \approx 6.71 \times 10^{-8} \text{ radians}
\]
To convert this to arcseconds (since there are approximately \(206265\) arcseconds in a radian):
\[
\theta \approx 6.71 \times 10^{-8} \text{ radians} \times 206265 \approx 0.0139 \text{ arcseconds}
\]
With this angular resolution, we can find out how far away an object would need to be for that resolution to allow reading a book.
Assuming a typical book has an average dimension of about \(0.3\) meters (30 cm), we can find the distance at which the book could be resolved using the angular resolution:
\[
d = \frac{h}{\theta}
\]
where:
- \(d\) is the distance to the object (in meters),
- \(h\) is the size of the book (0.3 m),
- \(\theta\) is in radians.
Using our earlier result:
\[
d = \frac{0.3}{6.71 \times 10^{-8}} \approx 4.47 \times 10^6 \text{ meters} \approx 4470 \text{ km}
\]
So, with the Keck telescope, you could resolve a book at approximately \(4470\) kilometers away. This is a theoretical limit under ideal conditions, not accounting for atmospheric effects, which are minimized in space.
\[
\theta = 1.22 \frac{\lambda}{D}
\]
where:
- \(\theta\) is the angular resolution in radians,
- \(\lambda\) is the wavelength of light (in meters),
- \(D\) is the diameter of the telescope's primary mirror (in meters).
The Keck Observatory's telescopes have a mirror diameter of 10 meters. For visible light, we can use a wavelength (\(\lambda\)) of approximately 550 nm (or 550 x \(10^{-9}\) meters).
Substituting these values into the equation:
1. Convert wavelength to meters:
- \(\lambda = 550 \times 10^{-9}\) m
2. Use the diameter of the Keck telescope:
- \(D = 10\) m
3. Calculate the angular resolution:
\[
\theta = 1.22 \frac{550 \times 10^{-9}}{10} \approx 6.71 \times 10^{-8} \text{ radians}
\]
To convert this to arcseconds (since there are approximately \(206265\) arcseconds in a radian):
\[
\theta \approx 6.71 \times 10^{-8} \text{ radians} \times 206265 \approx 0.0139 \text{ arcseconds}
\]
With this angular resolution, we can find out how far away an object would need to be for that resolution to allow reading a book.
Assuming a typical book has an average dimension of about \(0.3\) meters (30 cm), we can find the distance at which the book could be resolved using the angular resolution:
\[
d = \frac{h}{\theta}
\]
where:
- \(d\) is the distance to the object (in meters),
- \(h\) is the size of the book (0.3 m),
- \(\theta\) is in radians.
Using our earlier result:
\[
d = \frac{0.3}{6.71 \times 10^{-8}} \approx 4.47 \times 10^6 \text{ meters} \approx 4470 \text{ km}
\]
So, with the Keck telescope, you could resolve a book at approximately \(4470\) kilometers away. This is a theoretical limit under ideal conditions, not accounting for atmospheric effects, which are minimized in space.