Sun Angles and Daylength

11.1 Sun Angles and Daylength

The location of the sun in the sky is described in terms of its altitude

or zenith angle the from the vertical) and its azimuth

the

from true north or south measured in the horizontal plane). Several coordinate systems are possible with azimuth angles:

zero degrees is south and angles increase in the counter clockwise direction from to

2. compass: zero degrees is north and angles increase from to 360" in

a clockwise direction, and

3. astronomical: zero degrees is south and positive angles increase from to 180" in a clockwise direction; the counter clockwise direction is labeled with negative angles from to - 180".

We use the mathematical coordinate representation. Elevation and zenith angles are related by

90 - (angles in degrees). The zenith angle of the sun depends on the time of day, the latitude of the site, and the time of year. It is calculated from

where is the latitude, 6 is solar declination, t is time, and

is the time of solar noon. The earth turns at a rate of 360" per 24 hours, so the

15 factor converts hours to degrees. Time, t is in hours (standard local time), ranging from to 24. Latitude of a site is found in an atlas. Solar declination ranges from

at summer solstice to -23.45' at winter solstice. It can be calculated from

sin6

(1 1.2) where J is the calendar day with J = 1 at January 1. Some values of are given in Table 1 1.1.

The time of solar noon is calculated from

where LC is the longitude correction and E T is the equation of time. LC is

minutes, or + 1 15 hour for each degree you are east of the standard meridian and - 1 15 hour for each degree west of the standard meridian. Standard meridians are at 0,

Generally time zones run approximately

to -7.5" either side of a standard meridian,

Sun Angles and Daylength 169

T A BLE 1 1.1. Solar declination and Equation of Time for various dates.

Declin. E. T. Date Day Degree hours

Declin. E. T.

Date

Day Degree hours

Jan 1

Jun 29 180 23.26 -0.055 Jan

1 -23.09 -0.057

9 190 22.46 -0.085 Jan 20 20 -20.34 -0.182

10 -22.12 -0.123

19 200 20.97 -0.103 Jan

210 18.96 -0.107 Feb 9

30 -17.88 -0.222

Aug 8 220 16.39 -0.094 Feb

40 -14.95 -0.238

18 230 13.35 -0.065 Mar 1 60

50 -1 1.57 -0.232

240 9.97 -0.022 Mar

-7.91 -0.208

Sep 7 250 6.36 0.031 Mar

70 -4.07 -0.170

Sep 17 260 2.58 0.089 Mar

80 -0.11 -0.122

270 -1.32 0.147 Apr

300 -12.55 0.268 May 10 130

310 -15.76 0.273 May 20 140

16 320 -18.56 0.255 May 30 150

6 340 -22.40 0.151 Jun

16 350 -23.26 0.075 26 360 -23.38 -0.007

but this sometimes varies depending on political boundaries. An atlas can be checked to get both the standard meridian and the longitude. The equation of time is a 15 to 20 minute correction which depends on calendar day. It can be calculated from

sin4 f

where f= 279.575 + 0.98565, in degrees. Some values for E T are also given in Table 1 1.1.

The azimuth angle of the sun can be calculated from - co s sin

where is the zenith angle, calculated from Eq. (1 1.1); and AZ is in degrees, measured with respect to due south, increasing in the counter clockwise direction so 90" is

azimuth angles can be cal- culated by taking 360" minus the A Z calculated from Eq.

or by multiplying the result of Eq. (11.5) by -1; these two cases being only two of many possibilities for labeling the azimuth. Using Eqs. (11.1) and (11.5) the sun paths can be plotted for different latitudes, times of the year, and times of the day. Figure 11.1 shows some examples for latitudes of

Note that the angle scale in Fig. 11.1 is different from that calculated from Eq. (11.5). Several azimuth-scale

and

Radiation Fluxes in Natural Environments

degrees latitude 25 degrees latitude

50 degrees latitude 75 degrees latitude

F IGURE Sun tracks at declination angles of -23.5, -10, 0, 10, and 23.5"

for four different latitudes. Zenith angle grids are the concentric circles. Azimuth angles are shown around the outer circle. North is 0 ° , east is

Large dots are at one hour time increments.

labeling conventions exist, such as mathematical, astronomical or geo- graphical, so that the reader should be prepared to convert between various labeling conventions.

Equation 1.1) can be rearranged to solve for daylength. It works best to write the equation in terms of the half daylength, h,, which is the time (in degrees)

sunrise to solar noon. The half daylength is

sin6 = cos cos 6

cos -

where cos = for a geometric sunset (no atmospheric refraction) and

I Estimating Direct and Diffise Short-wave Irradiance 171

is the time of solar noon minus the half daylength divided by 15 (to convert degrees to hours):

and daylength in hours is twice the half daylength in degrees divided by Therefore, a 12-hour daylength corresponds to

= To find the time of actual sunrise,

in Eq. (1 1.5) is set to 90". Bi- ologically

times, especially for flowering and insect activity, begin in twilight hours just before sunrise and extend to just after sunset. Beginning and ending times for "civil twilight" are sometimes used to define these times of activity. Civil twilight is defined as beginning and

ending when the sun is 6" below the horizon, so =

Example 11.1. Find the sun zenith angle for Pullman, WA at 1

PDT

on June 30. Also find the time of first twilight and the daylength. Solution. Convert the time of observation to standard time by subtract-

ing one hour and convert minutes to decimal hours, so t = 9.75 hrs. The calendar day for June 30 is J

181. Pullman latitude is and longitude is 117.2". The standard meridian is

The local merid- ian is therefore 2.8" east of the standard meridian, so LC = 2.8" =

0.19 hrs. From Eq. (11.4) or Table 11.1, ET = Equation (1 1.3) then gives

11.87 hrs. Declination, from Table 11.1 or Eq. (1 1.2) is

= 12 - 0.19 - (-0.06) =

Substituting these values into Eq. (1 1.1) gives

The result is 34.9". The half daylength (including twilight), from Eq

is

Converting to hours gives

8.56 hrs. The time of first

twilight is 11.87 - 8.56 =

3.31 hrs (solar time). The daylength is 2 x

8.56 17.1 timeofsunriseinPDTis3.31

LC + ET +

1 hr =

(PDT).