When the apparent top of the sun is slightly be- low the horizon, the surface of the Earth is not re- ceiving direct sunlight. However, the surface can still receive indirect light scattered from air mole- cules higher in the atmosphere that are illuminated by the sun. The interval during which scattered light is present at the surface is called twilight . Because twilight gradually fades as the sun moves lower below the horizon, there is no precise defini- tion of the start of sunrise twilight or the end of sun- set twilight. Arbitrary definitions have been adopted by dif- ferent organizations to define twilight. Civil twi- light occurs while the sun center is no lower than –6°, and is based on the ability of humans to see objects on the ground. Military twilight occurs while the sun is no lower than –12°. Astronomi- cal twilight ends when the skylight becomes suf- ficiently dark to view certain stars, at solar elevation angle –18°. Table 2-2 summarizes the solar elevation angle Ψ definitions used for sunrise, sunset and twilight. All of these angles are at or below the horizon. Sample Application (§) Use a spreadsheet to find the Pacific standard time (PST) for all the events of Table 2-2, for Vancouver, Canada, during 22 Dec, 23 Mar, and 22 Jun. Find the Answer Given: Julian dates 355, 82, & 173. Find: t = ? h (local standard time) Assume: Pacific time zone: t UTC = t + 8 h. Use eq. (2.8a) and Table 2-2: 22Dec 23Mar 22Jun Morning: PST (h) geometric sunrise 8.22 6.20 4.19 apparent sunrise 8.11 6.11 4.09 civil twilight starts 7.49 5.58 3.36 military twilight starts 6.80 4.96 2.32 astron. twilight starts 6.16 4.31 n/a Evening: geometric sunset 16.19 18.21 20.22 apparent sunset 16.30 18.30 20.33 civil twilight ends 16.93 18.83 21.05 military twilight ends 17.61 19.45 22.09 astron. twilight ends 18.26 20.10 n/a Check : Units OK. Physics OK. Exposition : During the summer solstice (22 June), the sun never gets below –18°. Hence, it is astronomi- cal twilight all night in Vancouver in mid summer. During June, Vancouver is on daylight time, so the actual local time would be one hour later. Figure 2.6 Position (solid lines) of the sun for Vancouver, Canada for vari- ous seasons. September 21 and March 23 nearly coincide. Iso- chrones are dashed. All times are Pacific standard time. 23!>!MPDBM!TUBOEBSE!UJNF 25 27 29 31 5 7 9 21 56 :1 246 291 336 381 426 "[JNVUI!"OHMF!)¡* 81 71 61 51 41 31 21 1 &MFWBUJPO!"OHMF!)¡* 33 +VO 34 .BS 33 %FD
34 CHAPTER 2 • SOLAR & INFRARED RADIATION Approximate (sundial) time-of-day correspond- ing to these events can be found by rearranging eq. (2.6): (2.8a) t t C UTC d e s s = ± − · arccos sin ·sin sin cos ·cos λ φ δ φ δ Ψ where the appropriate elevation angle is used from Table 2-2. Where the ± sign appears, use + for sun- rise and – for sunset. If any of the answers are nega- tive, add 24 h to the result. To correct the time for the tilted, elliptical orbit of the earth, use the approximate Equation of Time : ∆ ·sin( ) ·sin( ) t a M b M c a ≈ − + + 2 (2.8b) where a = 7.659 minutes, b = 9.863 minutes, c = 3.588 radians = 205.58°, and where the mean anomaly M from eq. (2.2) varies with day of the year. This time correction is plotted in the Sample Application.
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