• Hour angle  
  

• Solar time
• Equation of time
• Time conversion

• Declination angle  
  

• Latitude angle  
  

• Observer-Sun Angles

The earth revolves around the sun every 365.25 days in an elliptical orbit, with a mean earth-sun distance of 1.496 x 10¹¹ m (92.9 x 10⁶ miles) defined as one astronomical unit (1 AU). This plane of this orbit is called the ecliptic plane. The earth's orbit reaches a maximum distance from the sun, or aphelion, of 1.52 × 10¹¹ m (94.4 × 10⁶ miles) on about the third day of July. The minimum earth-sun distance, the perihelion, occurs on about January 2nd, when the earth is 1.47 × 10¹¹ m (91.3 × 10⁶ miles) from the sun.  Figure 3.1 depicts these variations in relation to the Northern Hemisphere seasons.

Ideally, the meridians 7 degrees on either side of the standard time zone meridian should define the time zone. However, boundaries separating time zones are not meridians but politically determined borders following rivers, county, state or national boundaries, or just arbitrary paths. Countries such as Spain choose to be on `European Time (15^o E) when their longitudes are well within the adjacent Standard Time Meridian (0^o). Figure 3.2 shows these time zone boundaries within the United States and gives the standard time zone meridians (called longitudes of solar time).

3.1.2 The Hour Angle  

To describe the earth's rotation about its polar axis, we use the concept of the hour angle . As shown in Figure 3.3, the hour angle is the angular distance between the meridian of the observer and the meridian whose plane contains the sun. The hour angle is zero at solar noon (when the sun reaches its highest point in the sky). At this time the sun is said to be ‘due south’ (or ‘due north’, in the Southern Hemisphere) since the meridian plane of the observer contains the sun. The hour angle increases by 15 degrees every hour.

Figure 3.3 The hour angle . This angle is defined as the angle between the meridian parallel to sun rays and the meridian containing the observer.

                         (3.1)

EXAMPLE: When it is 3 hours after solar noon, solar time is 15:00 and the hour angle has a value of 45 degrees. When it is 2 hours and 20 minutes before solar noon, solar time is 9:40 and the hour angle is 325 degrees (or 35 degrees).

Equation of Time - The difference between mean solar time and true solar time on a given date is shown in Figure 3.4. This difference is called the equation of time (EOT). Since solar time is based on the sun being due south at 12:00 noon on any specific day, the accumulated difference between mean solar time and true solar time can approach 17 minutes either ahead of or behind the mean, with an annual cycle.

                    (3.2)

where the angle x is defined as a function of the day number N

                     (3.3)

with the day number, N being the number of days since January 1. Table 3.1 has been prepared as an aid in rapid determination of values of N from calendar dates.

Figure 3.4 The equation of time (EOT). This is the difference between the local apparent solar time and the local mean solar time.

EXAMPLE: February 11 is the 42^nd day of the year, therefore N = 42 and x is equal to 40.41 degrees, and the equation of time as calculated above is 14.30 minutes. This compares with a very accurately calculated value of -14.29 minutes reported elsewhere (Anonymous, 1981). This means that on this date, there is a difference between the mean time and the solar time of a little over 14 minutes or that the sun is "slow" relative to the clock by that amount.

To satisfy the control needs of concentrating collectors, a more accurate determination of the hour angle is often needed. An approximation of the equation of time claimed to have an average error of 0.63 seconds and a maximum absolute error of 2.0 seconds is presented below as Equation (3.4) taken from Lamm (1981). The resulting value is in minutes and is positive when the apparent solar time is ahead of mean solar time and negative when the apparent solar time is behind the mean solar time:

                     (3.4)

Here n is the number of days into a leap year cycle with n = 1 being January 1 of each leap year, and n =1461 corresponding to December 31 of the 4th year of the leap year cycle. The coefficients A_k and B_k are given in Table 3.2 below. Arguments for the cosine and sine functions are in degrees. 

Table 3.2 Coefficients for Equation (3.4)

k	Ak (hr)	Bk (hr)
0	2.0870 × 10-4	0
1	9.2869 × 10-3	1.2229 ´ 10-1
2	5.2258 ´ 10-2	1.5698 ´ 10-1
3	1.3077 ´ l0-3	5.1 602 ´ 10-3
4	2.1867 ´ l0-3	2.9823 ´ 10-3
5	1.5100 ´ 10-4	2.3463 ´ 10-4

Time Conversion - The conversion between solar time and clock time requires knowledge of the location, the day of the year, and the local standards to which local clocks are set. Conversion between solar time, t_s and local clock time (LCT) (in 24-hour rather than AM/ PM format) takes the form

                     (3.5)

where EOT is the equation of time in minutes and LC is a longitude correction defined as follows:

                     (3.6)

and the parameter D in Equation (3.5) is equal to 1 (hour) if the location is in a region where daylight savings time is currently in effect, or zero otherwise.

3.1.3 The Declination Angle  

The plane that includes the earth 's equator is called the equatorial plane. If a line is drawn between the center of the earth and the sun, the angle between this line and the earth's equatorial plane is called the declination angle , as depicted in Figure 3.5. At the time of year when the northern part of the earth's rotational axis is inclined toward the sun, the earth 's equatorial plane is inclined 23.45 degrees to the earth-sun line. At this time (about June 21), we observe that the noontime sun is at its highest point in the sky and the declination angle = +23.45 degrees. We call this condition the summer solstice, and it marks the beginning of summer in the Northern Hemisphere.

As the earth continues its yearly orbit about the sun, a point is reached about 3 months later where a line from the earth to the sun lies on the equatorial plane. At this point an observer on the equator would observe that the sun was directly overhead at noontime. This condition is called an equinox since anywhere on the earth, the time during which the sun is visible (daytime) is exactly 12 hours and the time when it is not visible (nighttime) is 12 hours. There are two such conditions during a year; the autumnal equinox on about September 23, marking the start of the fall; and the vernal equinox on about March 22, marking the beginning of spring. At the equinoxes, the declination angle is zero.

Figure 3.5 The declination angle . The earth is shown in the summer solstice position when = +23.45 degrees. Note the definition of the tropics as the intersection of the earth-sun line with the surface of the earth at the solstices and the definition of the Arctic and Antarctic circles by extreme parallel sun rays.

The winter solstice occurs on about December 22 and marks the point where the equatorial plane is tilted relative to the earth-sun line such that the northern hemisphere is tilted away from the sun. We say that the noontime sun is at its "lowest point" in the sky, meaning that the declination angle is at its most negative value (i.e., = -23.45 degrees). By convention, winter declination angles are negative.

                     (3.7)

where the argument of the cosine here is in degrees and N is the day number defined for Equation (3.3) The annual variation of the declination angle is shown in Figure 3.5.

3.1.4 Latitude Angle  

The latitude angle is the angle between a line drawn from a point on the earth’s surface to the center of the earth, and the earth’s equatorial plane. The intersection of the equatorial plane with the surface of the earth forms the equator and is designated as 0 degrees latitude. The earth’s axis of rotation intersects the earth’s surface at 90 degrees latitude (North Pole) and -90 degrees latitude (South Pole). Any location on the surface of the earth then can be defined by the intersection of a longitude angle and a latitude angle.

The solar altitude angle is defined as the angle between the central ray from the sun, and a horizontal plane containing the observer, as shown in Figure 3.6. As an alternative, the sun’s altitude may be described in terms of the solar zenith angle which is simply the complement of the solar altitude angle or

                     (3.8)

The other angle defining the position of the sun is the solar azimuth angle (A). It is the angle, measured clockwise on the horizontal plane, from the north-pointing coordinate axis to the projection of the sun’s central ray.

Figure 3.6 Earth surface coordinate system for observer at Q showing the solar azimuth angle Α, the solar altitude angle and the solar zenith angle for a central sun ray along direction vector S. Also shown are unit vectors i, j, k along their respective axes.

It is of the greatest importance in solar energy systems design, to be able to calculate the solar altitude and azimuth angles at any time for any location on the earth. This can be done using the three angles defined in Section 3.1 above; latitude , hour angle , and declination . If the reader is not interested in the details of this derivation, they are invited to skip directly to the results; Equations (3.17), (3.18) and (3.19).

For this derivation, we will define a sun-pointing vector at the surface of the earth and then mathematically translate it to the center of the earth with a different coordinate system. Using Figure 3.6 as a guide, define a unit direction vector S pointing toward the sun from the observer location Q:

                     (3.9)

where i, j, and k are unit vectors along the z, e, and n axes respectively. The direction cosines of S relative to the z, e, and n axes are S_z, S_e and S_n, respectively. These may be written in terms of solar altitude and azimuth as

                     (3.10)

Similarly, a direction vector pointing to the sun can be described at the center of the earth as shown in

Figure 3.7. If the origin of a new set of coordinates is defined at the earth’s center, the m axis can be a line from the origin intersecting the equator at the point where the meridian of the observer at Q crosses. The e axis is perpendicular to the m axis and is also in the equatorial plane. The third orthogonal axis p may then be aligned with the earth’s axis of rotation. A new direction vector S´ pointing to the sun may be described in terms of its direction cosines S´_m , S'_e and S'_p relative to the m, e, and p axes, respectively. Writing these in terms of the declination and hour angles, we have

                     (3.11)

Note that S´_e is negative in the quadrant shown in Figure 3.7.

Figure 3.7 Earth center coordinate system for the sun ray direction vector S defined in terms of hour angle and the declination angle .

These two sets of coordinates are interrelated by a rotation about the e axis through the latitude angle and translation along the earth radius QC. We will neglect the translation along the earth’s radius since this is about 1 / 23,525 of the distance from the earth to the sun, and thus the difference between the direction vectors S and S´ is negligible. The rotation from the m, e, p coordinates to the z, e, n coordinates, about the e axis is depicted in Figure 3.8. Both sets of coordinates are summarized in Figure 3.9. Note that this rotation about the e axis is in the negative sense based on the right-hand rule. In matrix notation, this takes the form

                     (3.12)

Figure 3.9 Composite view of Figures 3.6, 3.7 and 3.8 showing parallel sun ray vectors S and S’ relative to the earth surface and the earth center coordinates.

                     (3.13)

Substituting Equations (3.10) and (3.1)(3.11) for the direction cosine gives us our three important results

                     (3.14)

                     (3.15)

                     (3.16)

Equation (3.14) is an expression for the solar altitude angle in terms of the observer’s latitude (location), the hour angle (time), and the sun’s declination (date). Solving for the solar altitude angle , we have

                     (3.17)

Two equivalent expressions result for the solar azimuth angle (A) from either Equation (3.15) or (3.16). To reduce the number of variables, we could substitute Equation (3.14) into either Equation (3.15) or (3.16); however, this substitution results in additional terms and is often omitted to enhance computational speed in computer codes.

The solar azimuth angle can be in any of the four trigonometric quadrants depending on location, time of day, and the season. Since the arc sine and arc cosine functions have two possible quadrants for their result, both Equations (3.15) and (3.16) require a test to ascertain the proper quadrant. No such test is required for the solar altitude angle, since this angle exists in only one quadrant.

The appropriate procedure for solving Equation (3.15) is to test the result to determine whether the time is before or after solar noon. For Equation (3.15), a test must be made to determine whether the solar azimuth is north or south of the east-west line.

Two methods for calculating the solar azimuth (A), including the appropriate tests, are given by the following equations. Again, these are written for the azimuth angle sign convention used in this text, that is, that the solar azimuth angle is measured from due north in a clockwise direction, as with compass directions. Solving Equation (3.15) the untested result, A^’ then becomes

                     (3.18)

Solving Equation (3.16), the untested result, A^" becomes

                     (3.19)

In summary, we now have equations for both the sun’s altitude angle and azimuth angle written in terms of the latitude, declination and hour angles. This now permits us to calculate the sun’s position in the sky, as a function of date, time and location (N, , ).

EXAMPLE: For a site in Miami, Florida (25 degrees, 48 minutes north latitude/ 80 degrees, 16 minutes west longitude) at 10:00 AM solar time on August 1 (not a leap year), find the sun's altitude, zenith and azimuth angles.....For these conditions, the declination angle is calculated to be 17.90 degrees, the hour angle -30 degrees and the sun's altitude angle is then 61.13 degrees, the zenith angle 28.87 degrees and the azimuth angle 99.77 degrees.

The path of the sun across the sky can be viewed as being on a disc displaced from the observer. This "geometric" view of the sun's path can be helpful in visualizing sun movements and in deriving expressions for testing the sun angles as needed for Equation (3.18) to ascertain whether the sun is in the northern sky.

The sun may be viewed as traveling about a disc having a radius R at a constant rate of 15 degrees per hour. As shown in Figure 3.10, the center of this disc appears at different seasonal locations along the polar axis, which passes through the observer at Q and is inclined to the horizon by the latitude angle pointing toward the North Star (Polaris).

Figure 3.10 A geometric view of the sun’s path as seen by an observer at Q. Each disc has radius R.

The center of the disc is coincident with the observer Q at the equinoxes and is displaced from the observer by a distance of at other times of the year. The extremes of this travel are at the solstices when the disc is displaced by ± 0.424 R along the polar axis. It can be seen that in the winter, much of the disc is "submerged" below the horizon, giving rise to fewer hours of daylight and low sun elevations as viewed from Q.

In the summer the sun path disc of radius R has its center Y displaced above the observer Q. Point X is defined by a perpendicular from Q. In the n - z plane, the projection of the position S onto the line containing X and Y will be where is the hour angle. The appropriate test for the sun being in the northern sky is then

                     (3.20)

The distance XY can be found by geometry arguments. Substituting into Equation (3.20), we have

                     (3.21)

This is the test applied to Equation (3.18) to ensure that computed solar azimuth angles are in the proper quadrant.

The hour angle for sunset (and sunrise) may be obtained from Equation (3.17) by substituting the condition that the solar altitude at sunset equals the angle to the horizon. If the local horizon is flat, the solar altitude is zero at sunset and the hour angle at sunset becomes

                     (3.22)

The above two tests relate only to latitudes beyond ± 66.55 degrees, i.e. above the Arctic Circle or below the Antarctic Circle.  

If the hour angle at sunset is known, this may be substituted into Equation (3.18) or (3.19) to ascertain the solar azimuth at sunrise or sunset.

                     (3.23)

where ω_s is in degrees. It is of interest to note here that although the hours of daylight vary from month to month except at the equator (where ), there are always 4,380 hours of daylight in a year (non leap year) at any location on the earth.

Another limit that may be obtained is the maximum and minimum noontime solar altitude angle. Substitution of a value of into Equation (3.17) gives

                     (3.24)

where denotes the absolute value of this difference. An interpretation of Equation (3.24) shows that at solar noon, the solar zenith angle (the complement of the solar altitude angle) is equal to the latitude angle at the equinoxes and varies by ±23.45 degrees from summer solstice to winter solstice.

In polar coordinates, for a pole height, OP and the shadow azimuth, θ_s defined in the same manner as the previous azimuth angles i.e. relative to true north with clockwise being positive, we have for the shadow length:

                     (3.25)

                     (3.26)

                     (3.27)

                     (3.28)

Figure 3.12  Shadow cast by pole OP showing x and y coordinates of shadow tip, and shadow azimuth (A_s).

Throughout the history of mankind, the sun’s shadow has been used to tell he time of day. Ingenious devices casting shadows on surfaces of all shapes and orientations have been devised. However, all of these can be understood with a simple manipulation of Equations (3.25) through (3.28) developed above.

The most common sundial consists of a flat, horizontal ‘card’ with lines marked for each hour of daylight. These lines spread out from a point with the noon line pointing toward the North Pole. A ‘gnomon’ or shadow-casting device, is mounted perpendicular to the card and along the noon line. The gnomon is tapered at the local latitude angle, with its height increasing toward the north. Figure 3.13 below shows a simple horizontal sundial with the important angles defined.

In order to derive an equation for the directions of the time lines on the card, it must be assumed that the declination angle is zero (i.e. at the equinoxes). Using Equations (3.27) and (3.28) to define the coordinates of the shadow cast by the tip of the gnomon, and Equations (3.14), (3.15) and (3.16) to describe these in terms of the latitude angle , and the hour angle with the declination angle being set to zero. An expression for the angle between the gnomon shadow and its base may be derived. Through simple geometric manipulations, the expression for this angle simplifies to

                     (3.29)

A procedure used often in defining the angular relationship between a surface and the sun rays is the transformation of coordinates. Defining the vector V, we obtain

                     (3.30)

where i, j, and k are unit vectors collinear with the x, y, and z-axes, respectively, and the scalar coefficients (V_x, V_y, V_z) are defined along these same axes. To describe V in terms of a new set of coordinate axes x', y ', and z' with their respective unit vectors i', j' and k' in the form

                     (3.31)

we must define new scalar coefficients . Using matrix algebra will do this. A column matrix made of the original scalar coefficients representing the vector in the original coordinate system is multiplied by a matrix which defines the angle of rotation.

                     (3.32)

where A_ij is defined in Figure 3.14 for rotations about the x, y, or z axis. 

Figure 3.14 Axis rotation matrices A_{I j} for rotation about the three principal coordinate axes by the angle .

It should be noted here that if the magnitude of V is unity, the scalar coefficients in Equations (3.30) and (3.31) are direction cosines. Note also that the matrices given in Figure 3-22 are valid only for right-handed, orthogonal coordinate systems, and when positive values of the angle of rotation, are determined by the right-hand rule.

Title	Symbol	Zero	Positive Direction	Range	Equation No.	Figure No.
Earth-Sun Angles
Latitude		equator	northern hemisphere	+/- 90o	---	3.3
Declination		equinox	summer	+/- 23.45o	(3.7)	3.5
Hour Angle		noon	afternoon	+/- 180o	(3.1)	3.3
Observer-Sun Angles
Sun Altitude		horizontal	upward	0 to 90o	(3.17)	3.6
Sun Zenith		vertical	toward horizon	0 to 90o	(3.8)	3.6
Sun Azimuth	A	due north	clockwise*	0 to 360o	(3.18) or (3.19)	3.6
Shadow and Sundial Angles
Shadow Azimuth	As	due north	clockwise	0 to 360o	(3.26)	3.12
Gnomon Shadow		noon	afternoon	+/- 180o	(3.29)	3.13

Anonymous (1981), "The Astronomical Almanac for the Year 1981," issued by the Nautical Almanac Office of the United States Naval Observatory.