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Chapter 6: Problem 34

Several groups of astronomers are making plans for large ground-based telescopes. (a) What would be the diffractionlimited angular resolution of a telescope with a 40 -meter objective mirror? Assume that yellow light with wavelength \(550 \mathrm{~nm}\) is used. (b) Suppose this telescope is placed atop Mauna Kea. How will the actual angular resolution of the telescope compare to that of the 10 -meter Keck I telescope? Assume that adaptive optics is not used.

Short Answer

Expert verified
The diffraction-limited angular resolution of a 40-meter telescope using yellow light of wavelength 550 nm is approximately 0.00346 arcseconds. This is smaller than the resolution of the 10-meter Keck I telescope which is 0.01382 arcseconds - indicating that the 40-meter telescope can resolve more detail compared to the 10-meter Keck I telescope.

Step by step solution

01

Calculate Diffraction-Limited Resolution

To calculate the diffraction-limited resolution of the 40 meter telescope, plug in the given values into the formula for diffraction limit. Here, \(\lambda = 550 \, nm = 550 \times 10^{-9} \, m\) and D = 40 m. So, the diffraction-limited resolution becomes \( \theta = 1.22 ( 550 \times 10^{-9} / 40 ) = 1.675 \times 10^{-8} \, radians\). To convert this to arcseconds (a commonly used measure in astronomy), we know that 1 radian = 206265 arcseconds. Hence, the diffraction limit in arcseconds is \( \theta = 1.675 \times 10^{-8} \times 206265 = 0.00346 \, arcseconds\).
02

Compare With Keck I Telescope

This step is a comparative analysis with the 10 meter Keck I telescope. Using the same concept, let's calculate the resolution for Keck I: \(\lambda = 550 \, nm = 550 \times 10^{-9} \, m\), D = 10 m. Using the formula for diffraction limit again, Keck I's resolution is \( \theta = 1.22 ( 550 \times 10^{-9} / 10 ) = 6.7 \times 10^{-8} \, radians\), which converts to \( \theta = 6.7 \times 10^{-8} \times 206265 = 0.01382 \, arcseconds\). The 40-meter telescope has a smaller resolution value, indicating that it can discern more detail.

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Most popular questions from this chapter

To search for ionized oxygen gas surrounding our Milky Way Galaxy, astronomers aimed the ultraviolet telescope of the FUSE spacecraft at a distant galaxy far beyond the Milky Way. They then looked for an ultraviolet spectral line of ionized oxygen in that galaxy's spectrum. Were they looking for an emission line or an absorption line? Explain.

What is chromatic aberration? For what kinds of telescopes does it occur? How can it be corrected?

The Institute of Space and Astronautical Science in Japan proposes to place a radio telescope into an even higher orbit than the HALCA telescope. Using this telescope in concert with ground-based radio-telescopes, baselines as long as \(25,000 \mathrm{~km}\) may be obtainable. Astronomers want to use this combination to study radio emission at a frequency of \(43 \mathrm{GHz}\) from the molecule silicon monoxide, which is found in the interstellar clouds from which stars form. \((1 \mathrm{GHz}=\) 1 gigahertz \(=10^{9} \mathrm{~Hz}\).) (a) What is the wavelength of this emission? (b) Taking the baseline to be the effective diameter of this radio-telescope array, what angular resolution can be achieved?

Suppose your Newtonian reflector has an objective mirror \(20 \mathrm{~cm}\) ( 8 in.) in diameter with a focal length of \(2 \mathrm{~m}\). What magnification do you get with eyepieces whose focal lengths are (a) \(9 \mathrm{~mm}\), (b) \(20 \mathrm{~mm}\), and (c) \(55 \mathrm{~mm}\) ? (d) What is the telescope's diffraction-limited angular resolution when used with orange light of wavelength \(600 \mathrm{~nm}\) ? (e) Would it be possible to achieve this angular resolution if you took the telescope to the summit of Mauna Kea? Why or why not?

(a) Compare the light-gathering power of the Keck I 10.0-m telescope with that of the Hubble Space Telescope (HST), which has a \(2.4-\mathrm{m}\) objective mirror. (b) What advantages does Keck I have over HST? What advantages does HST have over Keck I?
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