# 22.8: Homework Exercises

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# 22.8.1. Broaden Knowledge & Comprehension

B1. Search the web for images of the following atmospheric optical phenomena:

 a. 22° halo b. 46° halo c. sun dogs d. sub sun dogs e. sub sun f. sun pillar g. supersun h. upper tangent arc i. parhelic circle j. lower tangent arc k. white clouds l. circumzenithal arc m. Parry arcs n. primary rainbow o. red sunset p. secondary rainbow q. blue sky r. Alexander’s dark band s. corona t. crepuscular rays u. iridescence v. anti-crepuscular rays w. glory x. fata morgana y. green flash z. mirages aa. halos other than 22° and 46° ab. circumhorizonal arc ac. circumscribed halo ad. Lowitz arcs ae. Scheiner’s halo af. Moilanen arc ag. Hastings arc ah. Wegener arc ai. supralateral arc aj. infralateral arcs ak. subhelic arc am. heliac arc an. diffuse arcs ao. Tricker arc ap. 120° parhelia (120° sun dogs) aq. various subhorizon arcs ar. fog bow as. blue moon at. reflection rainbow au. twinned rainbow av. supernumerary bows

B2. Search the web for images and descriptions of additional atmospheric phenomena that are not listed in the previous question.

B3. Search the web for microphotographs of ice crystals for ice Ih. Find images of hexagonal plates, hexagonal columns, and hexagonal pyramids.

B4. Search the web for diagrams or microphotographs of ice crystals for cubic ice Ic.

B5. Search the web for lists of indices of refraction of light through different materials.

B6. Search the web for highway camera or racetrack imagery, showing inferior mirages on the roadway.

B7. Search the web for computer programs to simulate atmospheric optical phenomena. If you can download this software, try running it and experimenting with different conditions to produce different optical displays.

B8. Search the web for images that have exceptionally large numbers of atmospheric optical phenomena present in the same photograph.

B9. Search the web for information on linear vs. circular polarization of sky light. Polarizing filters for cameras can be either circular or linear polarized. The reason for using circular polarization filters is that many automatic cameras loose the ability to auto focus or auto meter light through a linear polarizing filter. Compare how clear sky would look in a photo using circular vs. linear polarization filters.

B10. Search the web for literature, music, art, or historical references to atmospheric optical phenomena, other than the ones already listed in this chapter.

B11. Search the web for info on the history of scientific understanding of atmospheric optical phenomena. (E.g., Snell’s law, etc.)

# 22.8.2. Apply

A1. Calculate the angle of reflection, given the following angles of incidence:

 a. 10° b. 20° c. 30° d. 40° e. 45 f. 50° g. 60° h. 70° i. 80° j. 90°

A2. Find the refractive index for the following (medium, color) for T = 0°C and P = 80 kPa.

 a. air, red b. air, orange c. air, yellow d. air, green e. air, blue f. air, violet g. water, red h. water, orange i. water, yellow j. water, green k. water, blue m. ice, red n. ice, orange o. ice, yellow p. ice, green q. ice, blue r. ice, violet

A3. Find the speed of light for the medium and color of the previous exercise.

A4. Find the ratio µ of refractive indices for the following pairs of medium from exercise A2:

 a. (a, g) b. (a, m) c. (g, m) d. (b, h) e. (b, n) f. (h, n) g. (c, i) h. (c, o) i. (i, o) j. (d, j) k. (d, p) m. (j, p) n. (e, k) o. (e, q) p. (k, q)

A5. For the conditions in Fig. 22.2, find the angle of refraction in water, given an incident angle of 45° in air for λ (µm) of

 a. 0.4 b. 0.45 c. 0.5 d. 0.55 e. 0.6 f. 0.65 g. 0.7 h. 0.42 i. 0.47 j. 0.52 k. 0.57 m. 0.62 n. 0.67

A6. For green light, calculate the angle of refraction for: (i) water, and (ii) ice, given the following incident angles in air, for conditions in Fig. 22.2:

 a. 5° b. 10° c. 15° d. 20° e. 25° f. 30° g. 35° h. 40° i. 45° j. 50° k. 55° m. 60° n. 65° o. 70° p. 75° q. 80° r. 85°

A7. For Snell’s law in 3-D, given the following component angles in degrees (α1, β1), find the total incident angle (θ1) in degrees.

 a. 5, 20 b. 5, 35 c. 5, 50 d. 5, 65 e. 5, 80 f. 20, 35 g. 20, 50 h. 20, 65 i. 20, 80 j. 35, 50 k. 35, 65 m. 35, 80 n. 50, 65 o. 50, 80 p. 65, 80

A8. Using the incident components in air from the previous problem, find the refraction components of blue-green light in ice. Check your answers using Snell’s law for the total refraction angle. Assume the refractive index in air is 1.0 and in ice is 1.31.

A9. Use the data in Fig. 22.2 to calculate the critical angles for light moving from ice to air, for λ (µm) of

 a. 0.4 b. 0.45 c. 0.5 d. 0.55 e. 0.6 f. 0.65 g. 0.7 h. 0.42 i. 0.47 j. 0.52 k. 0.57 m. 0.62 n. 0.67

A10. Same as the previous exercise, but for light moving from water to air.

A11. Find the reflectivity from spherical raindrops, given impact parameters of:

 a. 0.0 b. 0.1 c. 0.2 d. 0.3 e. 0.4 f. 0.5 g. 0.6 h. 0.7 i. 0.75 j. 0.8 k. 0.85 m. 0.9 n. 0.95 o. 0.98

A12. What value(s) of the impact parameter gives a primary-rainbow viewing angle of θ1 = 42.0° for light of the following wavelength (µm)?

 a. 0.4 b. 0.45 c. 0.5 d. 0.55 e. 0.6 f. 0.65 g. 0.7 h. 0.42 i. 0.47 j. 0.52 k. 0.57 m. 0.62 n. 0.67

A13. Find the output viewing angle for a primary rainbow with the conditions in Fig. 22.2 for green light and the following impact parameters.

 a. 0.5 b. 0.55 c. 0.6 d. 0.65 e. 0.7 f. 0.75 g. 0.8 h. 0.85 i. 0.9 j. 0.95

A14.(§) Use a spreadsheet to calculate and plot primary-rainbow viewing angles for a full range of impact parameters from 0 to 1, for the conditions in Fig. 22.2. Do this for light of wavelength (µm):

 a. 0.4 b. 0.45 c. 0.5 d. 0.55 e. 0.6 f. 0.65 g. 0.7 h. 0.42 i. 0.47 j. 0.52 k. 0.57 m. 0.62 n. 0.67

A15.(§) Same as previous exercise but for a secondary rainbow.

A16. For yellow light in a 22° halo, what is the viewing angle for the following input angles? Use the conditions in Fig. 22.2.

 a. 15° b. 20° c. 25° d. 30° e. 35° f. 40° g. 45° h. 50° i. 55° j. 60° k. 65° m. 70° n. 75° o. 80° p. 85°

A17. Same as the previous exercise, but for blue light.

A18.(§) Use a spreadsheet to generate curves for green light, similar to those drawn in Fig. 22.25 for the 22° halo. Use the conditions in Fig. 22.2.

A19.(§) Same as previous exercise, but for the 46° halo.

A20. For yellow light in a 46° halo, find the viewing angle for the following input angles. Use the conditions in Fig. 22.2.

 a. 58° b. 60° c. 62° d. 64° e. 66° f. 68° g. 70° h. 72° i. 74° j. 76° k. 78° m. 80° n. 82° o. 84° p. 86° q. 88°

A21. Same as the previous exercise, but for blue light.

A22. Find the minimum viewing angle (i.e., the nominal halo angle) for green light using the conditions in Fig. 22.2 for the following ice-crystal wedge angle:

 a. 25° b. 52.4° c. 56° d. 60° e. 62° f. 63.8° g. 70.5° h. 80.2° i. 90° j. 120° k. 150°

A23. For a solar elevation of 10° and conditions as in Fig. 22.2, calculate the viewing angle components for a circumzenithal arc for red light with a crystal rotation of β =

 a. 10° b. 12° c. 14° d. 16° e. 18° f. 20° g. 22° h. 24° i. 26° j. 28° k. 30° m. 32° n. 34°

A24.(§) For conditions as in Fig. 22.2, calculate and plot circumzenithal arcs similar to Fig. 22.33 for red and violet light, but for solar elevation angles of

 a. 10° b. 12° c. 14° d. 16° e. 18° f. 20° g. 22° h. 24° i. 26° j. 28° k. 30° m. 0° n. 2° o. 4° p. 6° q. 8°

A25. For a solar elevation of 10°, calculate the viewing angle components for a sun dog for red light with the following crystal rotations β. Use the conditions in Fig. 22.2.

 a. 15° b. 20° c. 25° d. 30° e. 35° f. 40° g. 45° h. 50° i. 55° j. 60° k. 65° m. 70° n. 75° o. 80° p. 85°

A26.(§) Calculate and plot sun dog arcs similar to Fig. 22.35, but for the following solar elevation angles. Use the conditions of Fig. 22.2 for green light.

 a. 5° b. 15° c. 25° d. 30° e. 35° f. 45° g. 50° h. 55° i. 60°

A27.(§) For γ = 80°, plot upper tangent arc elevation and azimuth angles for a ray of orange-yellow light, for a variety of evenly-space values of rotation angle. Use a solar elevation of:

 a. 10° b. 12° c. 14° d. 16° e. 18° f. 8° g. 22° h. 24° i. 26° j. 28° k. 30° m. 32° n. 34° o. 36° p. 38° q. 40°

A28. Find the fraction of incident light that is scattered for an optical thickness of:

 a. 0.01 b. 0.02 c. 0.05 d. 0.10 e. 0.2 f. 0.5 g. 1. h. 2 i. 3 j. 4 k. 5 m. 7 n. 10 o. 20 p. 50

A29. What fraction of incident red light is scattered from air molecules along a horizontal path near the Earth’s surface, with path length (km) of

 a. 0.5 b. 1 c. 2 d. 5 e. 10 f. 20 g. 50 h. 100 i. 200 j. 500 k. 1000 m. 2000

A30.(§) Calculate and plot the relative fraction of scattering as a function of wavelength, for Rayleigh scattering in air.

A31. For cloud droplets of 5 µm radius, find the corona fringe viewing angles for the 2nd through 8th fringes, for light of the following wavelength (µm):

 a. 0.4 b. 0.45 c. 0.5 d. 0.55 e. 0.6 f. 0.65 g. 0.7 h. 0.42 i. 0.47 j. 0.52 k. 0.57 m. 0.62 n. 0.67

A32.(§) Calculate and plot the viewing angle vs. wavelength for the:

1. second corona fringe.
2. third corona fringe.
3. fourth corona fringe.

A33. (§) Calculate and plot the ratio of refractive index in air to a reference refractive index in air, for a variety of values of ratio of air density to reference air density.

A34. If the temperature decreases 10°C over the following altitude, find the mirage radius of curvature of a light ray. Assume a standard atmosphere at sea level.

 a. 1 mm b. 2 mm c. 3 mm d. 4 mm e. 5 mm f. 6 mm g. 7 mm h. 8 mm i. 9 mm j. 1 cm k. 2 cm m. 3 cm n. 4 cm o. 5 cm p. 10 cm q. 20 cm

# 22.8.3. Evaluate & Analyze

E1. If light is coming from the water towards the air in Fig. 22.1 (i.e., coming along the θ2 path but in the opposite direction), then sketch any incident and reflected rays that might occur.

E2. Salt water is more dense than fresh water. Sketch how light wave fronts behave as they approach the salt-water interface from the fresh water side.

E3. Is it possible for a material to have a refractive index less than 1.0 ? Explain.

E4. In general, how does the refractive index vary with temperature, pressure, and density?

E5. Use geometry and trig to derive eq. (22.6).

E6. Consider Fig. 22.5. If you were in a boat near the words “no escape here” and were looking into the water towards an object in the water at the black dot near the bottom of that figure, what would you see?

E7. Speculate on why rainbows are common with rain from cumulonimbus clouds (thunderstorms) but not with rain from nimbostratus clouds.

E8. If you were standing on a mountain top in a rain shower, and the sun was so low in the sky that sun rays were shining at a small angle upward past your position, could there be a rainbow? If so, where would it be, and how would it look?

E9. Use the relationship for reflection of light from water to describe the variations of brightness of a wavy sea surface during a sunny day.

E10. How does the intensity and color of light entering the raindrop affect the brightness of colors you see in the rainbow?

E11. Draw a sketch of a sun ray shining through a raindrop to make a primary rainbow. If red light is bent less than violet as a sun ray refracts and reflects through a raindrop, why isn’t red on the inside of the primary rainbow instead of the outside?

E12. Use geometry to derive eq. (22.13).

E13. Use geometry to derive eq. (22.14).

E14. Draw a sketch of a sun ray path through a raindrop for a:

 a. Tertiary rainbow b. Quaternary rainbow

Your sketch must be consistent with the viewing angle listed for these rainbows.

E15. During the one reflection or two reflections of light inside a raindrop for primary and secondary rainbows, what happens to the portion of light that is not reflected? Who would be able to see it, and where must they look for it?

E16. Why are higher-order rainbows fainter than lower-order ones?

E17. If neither the primary or secondary rainbows return light within Alexander’s dark band, why is it not totally black?

E18. Compare and contrast a twinned rainbow and supernumerary bows.

E19. Often hexagonal column ice crystals in the real atmosphere have indentations in their basal faces. These are known as hollow columns. How might that affect ice-crystal optical phenomena? E20. For hexagonal columns, what other crystal angles exist besides 60° and 90°? For these other crystal angles, at what viewing angles would you expect to see light from the crystal?

E21. In Fig. 22.18, why are plate-crystal wobble and column-crystal rotation necessary to explain a sun pillar?

E22. Fig. 22.18d shows a dendrite ice crystal. In addition to the sun pillar, for what other halos & optical phenomena might such dendrites be important?

E23. Fig. 22.22a has only reflection while Fig. 22.22b also has two refractions. How might this affect the color of subsuns?

E24. Eq. (22.15) does not consider the finite size of a hexagonal column. What range of input angles would actually allow rays to exit from the face sketched in Fig. 22.24 for 22° halos?

E25. Use geometry to derive eq. (22.15) for the 22° halo.

E26. Derive equation (22.16) for the minimum viewing angle for halos. (You might need calculus for this).

E27.(§) Suppose a halo of 40° was discovered. What wedge angle βc for ice would cause this?

E28. When the sun is on the horizon, what is the angle between the 22° halo and the bottom of the circumzenithal arc?

E29. Is it possible to see both a circumzenithal and a circumhorizonal arc at the same time? Explain.

E30 Why might only one of the sun dogs appear?

E31. Using geometry, derive the minimum thickness to diameter ratio of hexagonal plates that can create sun dogs, as a function of solar elevation angle.

E32. For both sun dogs and rainbows, the red color comes via a path from the sun to your eyes that does not allow other colors to be superimposed. However, for both phenomena, as the wavelength gets shorter, wider and wider ranges of colors are superimposed at any viewing angle. Why, then, do rainbows have bright, distinct colors from red through violet, but sundogs show only the reds through yellows, while the large viewing angles yield white color rather than blue or violet?

E33. Why are the tangent arcs always tangent to the 22° halo?

E34. Sometimes it is possible to see multiple phenomena in the sky. List all of the optical phenomena associated only with

1. large hexagonal plates
2. large hexagonal columns
3. small hexagonal columns
4. small hexagonal plates
5. large hexagonal pyramids

E35. If optical depth is defined as the optical thickness measured vertically from the top of the atmosphere, then sketch a graph of how optical depth might vary with height above ground for a standard atmosphere.

E36. If you take photographs using a polarizing filter on your camera, what is the angle between a line from the subject to your camera, and a line from the subject to his/her shadow, which would be in the proper direction to see the sky at nearly maximum polarization? By determining this angle now, you can use it quickly when you align and frame subjects for your photographs.

E37 In the Solar & Infrared Radiation chapter, Beer’s law was introduced, which related incident to transmitted light. Assume that transmitted light is 1 minus scattered light.

1. Relate the Rayleigh scattering equation to Beer’s law, to find the absorption coefficient associated with air molecules.
2. If visibility is defined as the distance traveled by light where the intensity has decreased to 2% of the incident intensity, then find the visibility for clean air molecules.
3. Explain the molecular scattering limit in the atmospheric transmittance curve of Fig. 8.4a in the Satellite & Radar chapter.

E38. Discuss the problems and limitations of using visibility measurements to estimate the concentration of aerosol pollutants in air.

E39. For atmosphere containing lots of 1.5 µm diameter sulfuric acid droplets, describe how the sky and sun would look during a daily cycle from sunrise past noon to sunset.

E40. To see the brightest fringes furthest from the sun during a corona display, what size cloud droplets would be best?

E41. If you were flying above a cloud during daytime with no other clouds above you, describe why you might not be able to see glory for some clouds.

E42. Contrast and compare the refraction of light in mirages, and refraction of sound in thunder (see the Thunderstorm chapter). Discuss how the version of Snell’s law in the Thunderstorm chapter can be applied to light in mirages.

E43. Microwaves are refracted by changes in atmospheric humidity. Use Snell’s law or Huygens’ principle to describe how ducting and trapping of microwaves might occur, and how it could affect weather radar and air-traffic control radar.

E44. What vertical temperature profile is needed to see the Fata Morgana mirage?

E45. Green, forested mountains in the distance sometimes seem purple or black to an observer. Also, the mountains sometimes seem to loom higher than they actually are. Discuss the different optical processes that explain these two phenomena.

E46. Paint on traffic signs and roadway lines is often sprinkled with tiny glass spheres before the paint dries, in order to make the signs more reflective to automobile headlights at night. Explain how this would work. Also, would it be possible to see other optical phenomena from these glass spheres, such as rainbows, halos, etc? Justify your answer.

# 22.8.4. Synthesize

S1. Suppose atmospheric density were (a) constant with height; or (b) increasing with height. How would optical phenomena be different, if at all?

S2. How would optical phenomena be different if ice crystals were octagonal instead of hexagonal?

S3. Suppose the speed of light through liquid and solid water was faster than through air. How would optical phenomena be different, if at all?

S4. After a nuclear war, if lots of fine Earth debris were thrown into the atmosphere, what optical phenomena would cockroaches (as the only remaining life form on Earth) be able to enjoy?

S5. Using the data in Fig. 22.2, fit separate curves to the refractive index variation with wavelength for air, water, and ice. Can you justify any or all of the resulting equations for these curves based on physical principles?

S6. Knowing the relationship between optical phenomena and the cloud (liquid or water) microphysics, and the relationship between clouds and atmospheric vertical structure, cyclones and fronts, create a table that tells what kind of weather would be expected after seeing various optical phenomena.

S7. If large raindrops were shaped like thick disks (short cylinders) falling with their cylinder axis vertical, how would rainbows be different, if at all?

S8. If large ice crystals were shaped like cylinders falling with their cylinder axis horizontal, how would ice-crystal optics be different, if at all?

S9. Consider Fig. 22.5. Could a thin layer of different fluid be floated on the liquid water to allow all light ray angles to escape from liquid to air? Justify your proposal.

S10. If there were two suns in the sky, each with light rays going to the same raindrop but arriving from different angles, is there a special angle between the arriving sun rays such that the two separate rainbows reinforce each other to make a single brighter rainbow?

S11. Design an ice-crystal shape that would cause: a. different halos and arcs than exist naturally. b. a larger number of natural halo and arcs to occur simultaneously.

S12. Suppose raindrops and ice crystals could cause refraction but not reflection or diffraction. What atmospheric optical phenomena could still exist? Justify your answers.

S13. Suppose that spacecraft landing on other planets can photograph optical phenomena such as halos, arcs, bows, etc. Describe how you could analyze these photographs to determine the chemicals and/ or temperature of these planetary atmospheres.

S14. Design equipment to be used in the Earth’s atmosphere that could determine the vertical temperature profile by measuring the characteristics of mirages.

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