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An amateur astronomer wants to build a telescope with a diffraction limit that will allow him to see if there are people on the moons of Jupiter. (a) What diameter mirror is needed to be able to see 1.00 m detail on a Jovian Moon at a distance of $7.50\times 10^{8}\textrm{ km}$ from Earth? The wavelength of light averages 600 nm. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?
Question by OpenStax is licensed under CC BY 4.0.
  1. $549\textrm{ km}$
  2. This size is impractical
  3. It's unreasonable to expect to see objects so close together at such a great distance.
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OpenStax College Physics for AP® Courses Solution, Chapter 27, Problem 68 (Problems & Exercises) (2:59)

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Video Transcript
This is College Physics Answers with Shaun Dychko. An amateur astronomer wants to see two people separated by only 1.00 meter on a moon of Jupiter, which is 7.50 times 10 to the 8 kilometers away from Earth using a visible wavelength of light of 600 nanometers so I converted both of these lengths into meters. Now this angle here we can get from the Rayleigh Criterion— it's the smallest possible angle that you can still distinguish between these two points— this is the distance from the Earth based observer looking through the telescope and that's 7.50 times 10 to the 11 meters away. Now we can imagine that the distance between these two people is pretty close to the length of this arc on an imaginary circle that has this radius and the angle Θ is defined when you have it in units of radians [it's found] to be this arc length, which we'll call s divided by the radius r but with such a small angle, this arc length will pretty much be the same as the straight line distance between the two points on this circle. So we are going to say that x is approximately equal to the arc length. So if we solved for the arc length here, we would multiply both sides by r and we get s equal to and we are going to say that x is approximately the same as s and so x is approximately equal to r times Θ. So we need to figure out what Θ is using the formula for the Rayleigh Criterion and that is Θ is 1.22 times the wavelength of light divided by the diameter of the aperture or the telescope that's being used to observe this phenomenon. Now this is a reflecting telescope so this is actually going to be the size of the mirror in the telescope. Okay! So x is r times 1.22 times λ over D so we are solving for D by multiplying both sides by D over x and so the size of the mirror is 1.22 times the distance to the moon times the wavelength of light being used divided by the separation between the features... between the people and the moon. 1.22 times 7.50 times 10 to the 11 meters times 600 times 10 to the minus 9 meters divided by 1 meter and that is 549 kilometers that's the size of the mirror that you would need on Earth to distinguish with such great resolution objects so close together at such a great distance. So that size is impractical so that's the answer to part (b) and in part (c), you would say it's unreasonable to expect to see objects so close together at such a great distance.