Chapter summary

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In this chapter we saw how different types of telescopes are used to collect data across the electromagnetic spectrum. We saw the differences and similarities among the techniques used in different parts of the spectrum. Much of the progress in astronomy in the past few decades has come from our ability to make high quality observations in parts of the spectrum other than the visible.

We looked at the important features of any telescope, the collecting area and the angular resolution. The collecting area determines how sensitive the telescope is to faint objects. The angular resolution is limited by diffraction (especially in

Questions

4.1. Describe the factors that limit the angular resolution of an optical telescope. Include estimates of the size of each effect.

4.2. What do we mean when we say that the main reason for building large ground-based optical telescopes is light-gathering power?

*4.3. Explain how improving the seeing at a site might allow you to detect fainter stars. (Hint: Think of what happens to the photons on your detector (film or CCD) when the image is smeared.)

*4.4. Suppose you are observing two stars that are 2 arc sec apart. Draw a diagram illustrating what you observe under conditions of 4, 2 and 1 arc sec seeing. Your diagram should be a series of graphs showing intensity as a function of position on a detector.

4.5. Suppose two stars are 5 arc sec apart on the sky. We clearly cannot resolve them with our eyes, but the angular resolution of even a modest sized telescope is sufficient to resolve them. However, the light from the telescope must still pass through the narrow pupil of the eye. Why doesn't the diffraction of the light entering the eye smear the images too much for us to resolve the two stars?

4.6. 'Faster' photographic emulsions can be made by making the grains larger. Why do you the radio), and in other cases is limited by atmospheric seeing (especially in the visible).

We saw the various techniques for extracting information from the radiation collected by our telescopes. Improving detector efficiency and panoramic ability has been important in all parts of the spectrum.

We saw the importance of site selection for an observatory. As an ultimate site, we saw the advantages of telescopes in space. In space, we can observe at wavelengths where the radiation does not penetrate the Earth's atmosphere. Even in the visible, we can achieve improved sensitivity and angular resolution.

think this works? What are the possible drawbacks to this?

4.7. What are the advantages of CCDs over photographic emulsions and photomultipliers?

4.8. Compare image formation (similarities and differences) in the eye and in a camera.

4.9. Why is chromatic aberration a problem even for black and white photographs?

4.10. If there is no angular magnification in a simple camera, how can using a longer focal length lens give a larger image?

4.11. A higher quality (more expensive) camera lens generally has smaller /-stops than an inferior lens. Why is this?

4.12. Why do some people need to wear eyeglasses while driving at night but not during the day? (What is it about the lower light level that degrades images?)

4.13. Compare the advantages and disadvantages of reflecting and refracting telescopes.

4.14. Compare the advantages and disadvantages of various focal arrangements in reflecting telescopes.

4.15. What does it mean to focus the eye or a camera 'at infinity'?

4.16. If you want to photograph a planet you use your long focal length telescope; if you want to do photometry on a faint star, you use a short focal length telescope. Explain.

4.17. For many observations (both imaging and spectroscopy) it is becoming important to read the data into a computer. Briefly discuss the techniques for doing this that we have mentioned in this chapter.

4.18. What are the important considerations in choosing an observatory site?

4.19. What are currently the best methods of reducing seeing effects at a given site?

4.20. What advantages does HST have over ground-based telescopes for optical observations?

4.21. What are the similarities and differences between ultraviolet and optical observations?

4.22. What are the similarities and differences between infrared and optical observations?

4.23. For infrared observations, we must still live with the fact that parts of a telescope radiate like blackbodies at about 300 K. Why isn't this a problem for optical observations?

4.24. How does a bolometer work? How would you use a bolometer to measure the power received in a small wavelength range, for example between 10 and 11 ^m?

4.25. Why can't balloons get above all of the atmosphere?

4.26. Explain the similarities between trying to display astronomical images and displaying things such as topographical maps.

4.27. Explain why simply using color film in your camera does not give you a true color astronomical photograph.

4.28. In what ways are radio observations similar to and different from (a) optical and

(b) infrared observations?

4.29. Suppose we want to use a radio telescope with a transmitter rather than a receiver of radio waves. Draw a diagram (similar to Fig. 4.26) showing how the transmitted radiation would be spread out on the sky.

4.30. Why is possible to observe with the Aricebo dish even though the surface has holes in it?

4.31. Why is it possible to do radio observations during the day, but not optical observations?

4.32. What are the advantages of observing in the millimeter part of the spectrum? What are the additional difficulties?

4.33. How would an image made by the VLA compare with one made with a single telescope as large as the VLA (assuming you could build one)?

4.34. How does VLBI differ from normal interferometry?

4.35. What are the difficulties in making a mirror to work for X-rays?

Problems

4.1. What is the limiting magnitude for naked eye viewing with a 5 m diameter telescope?

4.2. Estimate the angular resolution of a 5 m diameter telescope in space.

4.3. Compare the collecting areas of 5 and 8 m diameter reflectors. Comment on the significance of this comparison.

4.4. The full Moon subtends an angle of approximately 30 arc min. How large would the image of the Moon be on your film if you used a 500 mm focal length lens for your camera?

4.5. If we have two objects 0(") apart on the sky, how far apart, x, are their images on the film of a camera with a focal length f. (Assume that we wish to express x and f in the same units.)

4.6. The focal length of the objective on your telescope is 0.8 m. You are using a 25 cm focal length eyepiece. In the image you find that the angular separation between two stars is 10 arc sec. What is the actual angular separation on the sky between the two stars?

4.7. The focal length of the objective on your telescope is 0.5 m. (a) What focal length eyepiece would you have to use to have the image of the full Moon (whose actual size is 30 arc min) subtend an angle of 2°? (b) If you then took a photograph with a 500 mm focal length camera lens, how large would the image be on the film?

4.8. Scale the results in Example 4.1 to write an expression for the limiting magnitude of a telescope of diameter, assuming that you will be viewing directly with your eye.

4.9. Suppose some star is at the limit of naked eye visibility (m = 6). How much farther away can we see the same object with a telescope of diameter D? Evaluate your answer for D = 5 m.

4.10. Suppose that we use a reflector in the coudé focus, and each of the three mirrors reflects 95% of the light. What fraction of the light is lost as a result of these three reflections?

*4.11. (a) Using the fact that the limiting magnitude of the eye is 6, derive an expression for the limiting magnitude for direct viewing with a telescope of diameter D. (Ignore the effects of sky brightness.) (b) Use this result to derive an expression for the farthest distance at which a telescope of diameter D can be used to see an object of absolute magnitude M.

4.12. (a) What is the diameter of a single telescope with the same collecting area as the Multiple Mirror Telescope? (b) Astronomers have proposed a new telescope with a total collecting area equal to that of a single 25 m diameter telescope. How many 4 m diameter telescopes would be needed to make up this new telescope? (c) The Very Large Telescope, being built by ESO in Chile, has four telescopes, each with an 8 m diameter area. What would be the diameter of a single telescope with the same collecting area?

4.13. Suppose you have a Cassegrain telescope at home, with a 0.25 m diameter primary mirror and a secondary mirror with a diameter of 5 cm. What fraction of the primary is blocked by the secondary?

4.14. If we want to double the image size in a particular observation, by what amount would we have to change the exposure length to have a properly exposed photo?

4.15. Scale the results in Example 4.2 to write an expression for the angular resolution, in seconds of arc ("), of a telescope of diameter D for viewing the middle of the visible part of the spectrum.

4.16. What is the angular resolution of the HST at 200 nm wavelength?

4.17. Generally CCDs have fewer picture elements (pixels) than do photographic plates, so if you want to image a large field, a single photograph might suffice, but you would need a number of CCD images. Suppose you had to make a 3 X 3 square of CCD images to cover your single photographic field, and that your photograph has a quantum efficiency of 5%

while your CCD has an efficiency of 80%. How long will the needed CCD images take relative to the time for the photograph?

4.18. The sodium D lines in the Sun's spectrum are at wavelengths of 589.594 and 588.997 nm.

(a) If a grating has 104 lines/cm, how wide must the grating be to resolve the two lines in first order? (b) Under these conditions what is the angular separation between the two lines? (c) How would the results in (a) and

(b) change for second order?

4.19. A diffraction grating has N lines, a separation d apart. The spectrum is projected on a screen a distance D (Wd) from the grating. Two lines are A and A + AA apart. How far apart are they on the screen?

4.20. If we want to observe at a wavelength of 10 ^m, what are the largest fluctuations that the mirror surface can have?

4.21. What are the angular resolutions of the KAO, SOFIA, IRAS, ISO and SIRTF at wavelengths of 100 ^m?

4.22. Two infrared sources in the Orion Nebula are 500 pc from us and are separated by 0.1 pc. How large a telescope would you need to distinguish the sources at a wavelength of 100 ^m?

*4.23.Suppose we are observing an infrared source that is 500 pc away. It radiates like a 50 K blackbody and is 1 pc in extent. (a) What is the total energy per second per square meter reaching the Earth from this source? How does that compare with the total amount of solar radiation reaching the Earth per second per square meter. (b) Suppose we observe this source using a satellite with a 1 m diameter mirror, and we observe at a wavelength of 100 ^m. What is the energy/s/Hz striking the telescope? (c) Suppose the telescope radiates like a blackbody at 300 K, but with an efficiency of 1%. (That is, the spectrum looks like that of a blackbody but with an intensity reduced by a factor of 100.) What is the energy/Hz/s given off by the telescope at this wavelength? How does your answer compare with that in (b). (d) Redo part (c), assuming that we can cool the mirror to 30 K (still with a 1% emission efficiency).

4.24. Suppose we are using an interference filter at a wavelength of 10 ^m. (a) How far do you have to move the plates to go from one order maximum to the next? (b) For any given (power/surface area/Hz) in each case? (c) How order, how far do you have to move the plates does this compare with the power you receive for the peak wavelength to shift from 10.00 ^m from a 50 kW radio station 10 km from you?

to 10.01 ^m? 4.30. What are the angular resolutions of (a) the

4.25. What is the angular resolution (in arc VLA in its largest configuration (baselines up minutes) of (a) a 100 m diameter telescope to 13 km) at a wavelength of 21 cm; (b) the operating at 1 cm wavelength, and (b) a 30 m VLA in its most compact configuration (base-telescope operating at 1 mm wavelength? lines up to 1 km) at a wavelength of 1 cm;

4.26. Two radio sources in the Orion Nebula are (c) the VLBA (with baselines up to 3000 km) at 500 pc from us and are separated by 0.1 pc. a wavelength of 21 cm; (d) the Millimeter How large a telescope would you need to dis- Array (with baselines up to 1 km) at 1 mm tinguish the sources at a wavelength of wavelength?

(a) 21 cm? (b) 1 mm? 4.31. How many pairs of telescopes are there in

4.27. We sometimes use as a measure of the quality (a) the VLA with 27 telescopes, (b) the VLBI, of a radio telescope the diameter d, divided by with ten telescopes, (c) the proposed the limiting wavelength Amin. (a) Why is this Millimeter Array, which may have 40 or 75

quantity important? (b) If a telescope has telescopes depending on the final design?

surface errors of size Ax, give an expression 4.32. What is the total collecting area of (a) the VLA

for this quantity in terms of d and Ax. (b) the ALMA with 75 10 m diameter tele-

4.28. How large a collecting area would you need to scopes? (c) For ALMA, how many 12 m diameter collect 1 W from a 1 Jy source over a band- telescopes would you need to have the same width of 109 Hz (1 GHz)? collecting area as 75 10 m diameter telescopes.

4.29. If a radio source emits a solar luminosity 4.33. (a) Show that for an interferometer with N (4 X 1033 erg/s) in radio waves, (a) what would telescopes, the number of independent pairs be the power per surface area reaching us if we of telescopes, at any instant, is N(N — 1)/2. were (i) 1 AU away, (ii) 1 pc away? (b) If that (b) Evaluate this for the VLA, the VLBA and power is uniformly spread out over a frequency ALMA.

range of 1011 Hz, what is the flux density 4.34. Derive equation (4.9).

Computer problems

4.1. Draw a graph of the angular resolution vs. wave length, over the infrared part of the spectrum, for IRAS, KAO, SOFIA, ISO.

4.2. Suppose we have a radio telescope whose (one-dimensional) beam pattern is a gaussian with a full width at half maximum (FWHM) of 1 arc min.

Calculate the observed source intensity as a function of (one-dimensional) position on the sky, assuming that the source has a uniform intensity over a finite size of (a) 1 arc sec (b) 1 arc min (c) 1 degree, and the source has zero intensity outside that region.

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