# 2.6: Homework Exercises

- Page ID
- 9811

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)# 2.7.1. Broaden Knowledge & Comprehension

(Don’t forget to cite each web address you use.)

B1. Access a full-disk visible satellite photo image of Earth from the web. What visible clues can you use to determine the current solar declination angle? How does your answer compare with that expected for your latitude and time of year.

B2. Access “web cam” camera images from a city, town, ski area, mountain pass, or highway near you. Use visible shadows on sunny days, along with your knowledge of solar azimuth angles, to determine the direction that the camera is looking.

B3. Access from the web the exact time from military (US Navy) or civilian (National Institute of Standards and Technology) atomic clocks. Synchronize your clocks at home or school, utilizing the proper time zone for your location. What is the time difference between local solar noon (the time when the sun is directly overhead) and the official noon according to your time zone. Use this time difference to determine the number of degrees of longitude that you are away from the center of your time zone.

B4. Access orbital information about one planet (other than Earth) that most interests you (or a planet assigned by the instructor). How elliptical is the orbit of the planet? Also, enjoy imagery of the planet if available.

B5. Access runway visual range reports from surface weather observations (METARs) from the web. Compare two different locations (or times) having different visibilities, and calculate the appropriate volume extinction coefficients and optical thickness. Also search the web to learn how runway visual range (RVR) is measured.

B6. Access both visible and infrared satellite photos from the web, and discuss why they look different. If you can access water-vapor satellite photos, include them in your comparison.

B7. Search the web for information about the sun. Examine satellite-based observations of the sun made at different wavelengths. Discuss the structure of the sun. Do any of the web pages give the current value of the solar irradiance (i.e., the solar constant)? If so, how has it varied recently?

B8. Access from the web daytime visible photos of the whole disk of the Earth, taken from geostationary weather satellites. Discuss how variations in the apparent brightness at different locations (different latitudes; land vs. ocean, etc.) might be related to reflectivity and other factors.

B9. Some weather stations and research stations report hourly observations on the web. Some of these stations include radiative fluxes near the surface. Use this information to create surface net radiation graphs.

B10. Access information from the web about how color relates to wavelength. Also, how does the range of colors that can be perceived by eye compare to the range of colors that can be created on a computer screen?

B11. Search the web for information about albedos and IR emissivities for substances or surfaces that are not already listed in the tables in this chapter.

B12. Find on the web satellite images of either forestfire smoke plumes or volcanic ash plumes. Compare the intensity of reflected radiation from the Earth’s surface as it shines through these plumes with earlier satellite photos when the plumes were not there. Use these data to estimate the extinction coefficient.

B13. Access photos and diagrams from the web that describe how different actinometers are constructed and how they work. Also, list any limitations of these instruments that are described in the web.

# 2.7.2. Apply

(Students, don’t forget to put a box around each answer.)

A1. Given distances R between the sun and planets compute the orbital periods (Y) of:

- Mercury (R = 58 Gm)
- Venus (R = 108 Gm)
- Mars (R = 228 Gm
- Jupiter (R = 778 Gm)
- Saturn (R = 1,427 Gm)
- Uranus (R = 2,869 Gm)
- Neptune (R = 4,498 Gm)
- Pluto (R = 5,900 Gm)
- Eris: given \(\ \gamma\) = 557 Earth years, estimate the distance R from the sun assuming a circular orbit. (Note: Eris’ orbit is highly eccentric and steeply tilted at 44° relative to the plane of the rest of the solar system, so our assumption of a circular orbit was made here only to simplify the exercise.)

A2. This year, what is the date and time of the:

a. perihelion | b. vernal equinox |

c. summer solstice | d. aphelion |

e. autumnal equinox | f. winter solstice |

A3. What is the relative Julian day for:

a. 10 Jan | b. 25 Jan | c. 10 Feb | d. 25 Feb |

e. 10 Mar | f. 25 Mar | g. 10 Apr | h. 25 Apr |

i. 10 May | j. 25 May | k. 10 Jun | l. 25 Jun |

m. 10 Jul | n. 25 Jul | o. 10 Aug | p. 25 Aug |

q. 10 Sep | r. 25 Sep | s. 10 Oct | t. 25 Oct |

u. 10 Nov | v. 25 Nov | w. 10 Dec | x. 25 Dec |

y. today’s date | z. date assigned by instructor |

A4. For the date assigned from exercise A3, find:

- mean anomaly
- true anomaly
- distance between the sun and the Earth
- solar declination angle
- average daily insolation

A5(§). Plot the local solar elevation angle vs. local time for 22 December, 23 March, and 22 June for the following city:

- Seattle, WA, USA
- Corvallis, OR, USA
- Boulder, CO, USA
- Norman, OK, USA
- Madison, WI, USA
- Toronto, Canada
- Montreal, Canada
- Boston, MA, USA
- New York City, NY, USA
- University Park, PA, USA
- Princeton, NJ, USA
- Washington, DC, USA
- Raleigh, NC, USA
- Tallahassee, FL, USA
- Reading, England
- Toulouse, France
- München, Germany
- Bergen, Norway
- Uppsala, Sweden
- DeBilt, The Netherlands
- Paris, France
- Tokyo, Japan
- Beijing, China
- Warsaw, Poland
- Madrid, Spain
- Melbourne, Australia
- Your location today.
- A location assigned by your instructor

A6(§). Plot the local solar azimuth angle vs. local time for 22 December, 23 March, and 22 June, for the location from exercise A5.

A7(§). Plot the local solar elevation angle vs. azimuth angle (similar to Figure 2.6) for the location from exercise A5. Be sure to add tic marks along the resulting curve and label them with the local standard times.

A8(§). Plot local solar elevation angle vs. azimuth angle (such as in Figure 2.6) for the following location:

- Arctic Circle
- 75°N
- 85°N
- North Pole
- Antarctic Circle
- 70°S
- 80°S
- South Pole

for each of the following dates:

- 22 Dec
- 23 Mar
- 22 Jun

A9(§). Plot the duration of evening civil twilight (difference between end of twilight and sunset times) vs. latitude between the south and north poles, for the following date:

a. 22 Dec | b. 5 Feb | c. 21 Mar | d. 5 May |

e. 21 Jun | f. 5 Aug | g. 23 Sep | h. 5 Nov |

A10. On 15 March for the city listed from exercise A5, at what local standard time is:

- geometric sunrise
- apparent sunrise
- start of civil twilight
- start of military twilight
- start of astronomical twilight
- geometric sunset
- apparent sunset
- end of civil twilight
- end of military twilight
- end of astronomical twilight

A11. Calculate the Eq. of Time correction for:

a. 1 Jan | b. 15 Jan | c. 1 Feb | d. 15 Feb | e. 1 Mar |

f. 15 Mar | g. 1 Apr | h. 15 Apr | i. 1 May | j. 15 May |

A12. Find the mass flux (kg·m^{–2}·s^{–1}) at sea-level, given a kinematic mass flux (m s^{–1}) of:

a. 2 | b. 5 | c. 7 | d. 10 | e. 14 | f. 18 | g. 21 |

h. 25 | i. 30 | j. 33 | k. 47 | l. 59 | m. 62 | n. 75 |

A13. Find the kinematic heat fluxes at sea level, given these regular fluxes (W·m^{–2}):

a. 1000 | b. 900 | c. 800 | d. 700 | e. 600 |

f. 500 | g. 400 | h. 300 | i. 200 | j. 100 |

k. 43 | l. –50 | m. –250 | n. –325 | o. –533 |

A14. Find the frequency, circular frequency, and wavenumber for light of color:

a. red | b. orange | c. yellow | d. green |

e. cyan | f. blue | f. indigo | g. violet |

A15(§). Plot Planck curves for the following blackbody temperatures (K):

a. 6000 | b. 5000 | c. 4000 | d. 3000 | e. 2500 |

f. 2000 | g. 1500 | h. 1000 | i. 750 | j. 500 |

k. 300 | l. 273 | m. 260 | n. 250 | h. 240 |

A16. For the temperature of exercise A15, find:

- wavelength of peak emissions
- total emittance (i.e., total amount of emissions)

A17. Estimate the value of solar irradiance reaching the orbit of the planet from exercise A1.

A18(§).

- Plot the value of solar irradiance reaching Earth’s orbit as a function of relative Julian day.
- Using the average solar irradiance, plot the radiative flux (reaching the Earth’s surface through a perfectly clear atmosphere) vs. latitude. Assume local noon.

A19(§). For the city of exercise A1, plot the average daily insolation vs. Julian day.

A20. What is the value of IR absorptivity of:

a. aluminum | b. asphalt | c. cirrus cloud |

d. conifer forest | e. grass lawn | f. ice |

g. oak | h. silver | i. old snow |

j. urban average | k. concrete average | l. desert average |

m. shrubs | n. soils average |

A21. Suppose polluted air reflects 30% of the incoming solar radiation. How much (W m^{–2}) is absorbed, emitted, reflected, and transmitted? Assume an incident radiative flux equal to the solar irradiance, given a transmissivity of:

a. 0 | b. 0.05 | c. 0.1 | d. 0.15 | e. 0.2 |

f. 0.25 | g. 0.3 | h. 0.35 | i. 0.4 | j. 0.45 |

k. 0.5 | l. 0.55 | m. 0.6 | n. 0.65 | o. 0.7 |

A22. What is the value of albedo for the following land use?

a. buildings | b. dry clay | c. corn | d. green grass |

e. ice | f. potatoes | g. rice paddy | h. savanna |

i. red soil | j. sorghum | k. sugar cane | l. tobacco |

A23. What product of number density times absorption cross section is needed in order for 50% of the incident radiation to be absorbed by airborne volcanic ash over the following path length (km)?

a. 0.2 | b. 0.4 | c. 0.6 | d. 0.8 | e. 1.0 | f. 1.5 | g. 2 |

h. 2.5 | i. 3 | j. 3.5 | k. 4 | l. 4.5 | m. 5 | n. 7 |

A24. What fraction of incident radiation is transmitted through a volcanic ash cloud of optical depth:

a. 0.2 | b. 0.5 | c. 0.7 | d. 1.0 | e. 1.5 | f. 2 | g. 3 |

h. 4 | j. 5 | k. 6 | l. 7 | m. 10 | n. 15 | o. 20 |

A25. What is the visual range (km) for polluted air that has volume extinction coefficient (m^{-1}) of:

a. 0.00001 | b. 0.00002 | c. 0.00005 | d. 0.0001 |

e. 0.0002 | f. 0.0005 | g. 0.001 | h. 0.002 |

i. 0.005 | j. 0.01 | k. 0.02 | l. 0.05 |

A26.

- What is the value of solar downward direct radiative flux reaching the surface at the city from exercise A5 at noon on 4 July, given 20% coverage of cumulus (low) clouds.
- If the albedo is 0.5 in your town, what is the reflected solar flux at that same time?
- What is the approximate value of net longwave radiation at that time?
- What is the net radiation at that time, given all the info from parts (i) - (iii)?

A27. For a surface temperature of 20°C, find the emitted upwelling IR radiation (W m^{–2}) over the surface-type from exercise A20.

# 2.7.3. Evaluate & Analyze

E1. At what time of year does the true anomaly equal:

a. 45° | b. 90° | c. 135° | d. 180° |

e. 225° | f. 270° | g. 315° | h. 360° |

E2(§)

- Calculate and plot the position (true anomaly and distance) of the Earth around the sun for the first day of each month.
- Verify Kepler’s second law
- Compare the elliptical orbit to a circular orbit.

E3. What is the optimum angle for solar collectors at your town?

E4. Design a device to measure the angular diameter of the sun when viewed from Earth. (Hint, one approach is to allow the sun to shine through a pin hole on to a flat surface. Then measure the width of the projected image of the sun on this surface divided by the distance between the surface and the pin hole. What could cause errors in this device?)

E5. For your city, plot the azimuth angle for apparent sunrise vs. relative Julian day. This is the direction you need to point your camera if your want to photograph the sunrise.

E6.

- Compare the length of daylight in Fairbanks, AK, vs Miami, FL, USA.
- Why do vegetables grow so large in Alaska?
- Why are few fruits grown in Alaska?

E7. How would Figure 2.6 be different if daylight (summer) time were used in place of standard time during the appropriate months?

E8(§). Plot a diagram of geometric sunrise times and of sunset times vs. day of the year, for your location.

E9(§). Using apparent sunrise and sunset, calculate and plot the hours of daylight vs. Julian day for your city.

E10.

- On a clear day at your location, observe and record actual sunrise and sunset times, and the duration of twilight.
- Use that information to determine the day of the year.
- Based on your personal determination of the length of twilight, and based on your latitude and season, is your personal twilight most like civil, military, or astronomical twilight?

E11. Given a flux of the following units, convert it to a kinematic flux, and discuss the meaning and/or advantages of this form of flux.

- Moisture flux: gwater·m
^{–2}·s^{–1}. - Momentum flux: (kgair· m·s
^{–1})·m^{–2}·s^{–1}. - Pollutant flux: gpollutant·m
^{–2}·s^{–1}.

E12.

- What solar temperature is needed for the peak intensity of radiation to occur at 0.2 micrometers?
- Remembering that humans can see light only between 0.38 and 0.74 microns, would the sun look brighter or dimmer at this new temperature?

E13. A perfectly black asphalt road absorbs 100% of the incident solar radiation. Suppose that its resulting temperature is 50°C. How much visible light does it emit?

E14. If the Earth were to cool 5°C from its present radiative equilibrium temperature, by what percentage would the total emitted IR change?

E15(§). Evaluate the quality of the approximation to Planck’s Law [see eq. (a) in the “ A Scientific Perspective • Scientific Laws — the Myth” box] against the exact Planck equation (2.13) by plotting both curves for a variety of typical sun and Earth temperatures.

E16. Find the solar irradiance that can pass through an atmospheric “window” between λ_{1} = 0.3 µm and λ_{2} = 0.8 µm. (See the “ A Scientific Perspective • Seek Solutions” box in this chapter for ways to do this without using calculus.)

E17. How much variation in Earth orbital distance from the sun is needed to alter the solar irradiance by 10%?

E18. Solar radiation is a diffuse source of energy, meaning that it is spread over the whole Earth rather than being concentrated in a small region. It has been proposed to get around the problem of the inverse square law of radiation by deploying very large mirrors closer to the sun to focus the light as collimated rays toward the Earth. Assuming that all the structural and space-launch issues could be solved, would this be a viable method of increasing energy on Earth?

E19. The “sine law” for radiation striking a surface at an angle is sometimes written as a “cosine law”, but using the zenith angle instead of the elevation angle. Use trig to show that the two equations are physically identical.

E20. Explain the meaning of each term in eq. (2.21).

E21.

- Examine the figure showing average daily insolation. In the summer hemisphere during the few months nearest the summer solstice, explain why the incoming solar radiation over the pole is nearly equal to that over the equator.
- Why are not the surface temperatures near the pole nearly equal to the temperatures near the equator during the same months?

E22. Using Table 2-5 for the typical albedos, speculate on the following:

- How would the average albedo will change if a pasture is developed into a residential neighborhood.
- How would the changes in affect the net radiation budget?

E23. Use Beer’s law to determine the relationship between visual range (km) and volume extinction coefficient (m^{–1}). (Note that extinction coefficient can be related to concentration of pollutants and relative humidity.)

E24(§). For your city, calculate and plot the noontime downwelling solar radiation every day of the year, assuming no clouds, and considering the change in solar irradiance due to changing distance between the Earth and sun.

E25. Consider cloud-free skies at your town. If 50% coverage of low clouds moves over your town, how does net radiation change at noon? How does it change at midnight?

E26. To determine the values of terms in the surface net-radiation budget, what actinometers would you use, and how would you deploy them (i.e., which directions does each one need to look to get the data you need)?

# 2.7.4. Synthesize

(Don’t forget to state and justify all assumptions.)

S1. What if the eccentricity of the Earth’s orbit around the sun changed to 0.2 ? How would the seasons and climate be different than now?

S2. What if the tilt of the Earth’s axis relative to the ecliptic changed to 45° ? How would the seasons and climate be different than now?

S3. What if the tilt of the Earth’s axis relative to the ecliptic changed to 90° ? How would the seasons and climate be different than now?

S4. What if the tilt of the Earth’s axis relative to the ecliptic changed to 0° ? How would the seasons and climate be different than now?

S5. What if the rotation of the Earth about its axis matched its orbital period around the sun, so that one side of the Earth always faced the sun and the other side was always away. How would weather and climate be different, if at all?

S6. What if the Earth diameter decreased to half of its present value? How would sunrise and sunset time, and solar elevation angles change?

S7. Derive eq. (2.6) from basic principles of geometry and trigonometry. This is quite complicated. It can be done using plane geometry, but is easier if you use spherical geometry. Show your work.

S8. What if the perihelion of Earth’s orbit happened at the summer solstice, rather than near the winter solstice. How would noontime, clear-sky values of insolation change at the solstices compared to now?

S9. What if radiative heating was caused by the magnitude of the radiative flux, rather than by the radiative flux divergence. How would the weather or the atmospheric state be different, if at all?

S10. Linearize Planck’s Law in the vicinity of one temperature. Namely, derive an equation that gives a straight line that is tangent to any point on the Planck curve. (Hint: If you have calculus skills, try using a Taylor’s series expansion.) Determine over what range of temperatures your equation gives reasonable answers. Such linearization is sometimes used retrieving temperature soundings from satellite observations.

S11. Suppose that Kirchhoff’s law were to change such that a_{λ} = 1 – e_{λ}. What are the implications?

S12. What if Wien’s law were to be repealed, because it was found instead that the wavelength of peak emissions increases as temperature increases.

- Write an equation that would describe this. You may name this equation after yourself.
- What types of radiation from what sources would affect the radiation budget of Earth?

S13. What if the solar “constant” were even less constant than it is now. Suppose the solar constant randomly varies within a range of ±50% of its present value, with each new value lasting for a few years before changing again. How would weather and climate be different, if at all?

S14. What if the distance between sun and Earth was half what it is now. How would weather and climate be different, if at all?

S15. What if the distance between sun and Earth was double what it is now. How would weather and climate be different, if at all?

S16. Suppose the Earth was shaped like a cube, with the axis of rotation perpendicular to the ecliptic, and with the axis passing through the middle of the top and bottom faces of the cube. How would weather and climate be different, if at all?

S17. Suppose the Earth was shaped like a narrow cylinder, with the axis of rotation perpendicular to the ecliptic, and with the axis passing through the middle of the top and bottom faces of the cylinder. How would weather & climate be different, if at all?

S18. Derive eq. (2.21) from the other equations in this chapter. Show your work. Discuss the physical interpretation of the hour angle, and what the effect of truncating it is.

S19. Suppose that the Earth’s surface was perfectly reflective everywhere to short-wave radiation, but that the atmosphere absorbed 50% of the sunlight passing through it without reflecting any. What percentage of the insolation would be reflected to space? Also, how would the weather and climate be different, if at all?

S20. Suppose that the atmosphere totally absorbed all short wave radiation that was incident on it, but also emitted an exactly equal amount of short wave radiation as it absorbed. How would weather and climate be different, if at all?

S21. Consider Beer’s law. If there are n particles per cubic meter of air, and if a vertical path length in air is ∆s, then multiplying the two gives the number of particles over each square meter of ground. If the absorption cross section b is the area of shadow cast by each particle, then multiplying this times the previous product would give the shadowed area divided by the total area of ground. This ratio is just the absorptivity a. Namely, by this reasoning, one would expect that a = n·b·∆s.

However, Beer’s law is an exponential function. Why? What was wrong with the reasoning in the previous paragraph?

S22. What if the atmosphere were completely transparent to IR radiation. How would the surface net radiation budget be different?

S23. Existing radiometers are based on a bolometer, photovoltaic cell, or charge-coupled device. Design a new type of actinometer that is based on a different principle. Hint, think about what is affected in any way by radiation or sunlight, and then use that effect to measure the radiation.