In order to understand how to collect energy from the sun, one must first be able to predict the location of the sun relative to the collection device. In this chapter we develop the necessary equations by use of a unique vector approach. This approach will be used in subsequent chapters to develop the equations for the sun’s position relative to a fixed or tracking solar collector, (Chapter 4) and the special case of a sun-tracking mirror reflecting sunlight onto a fixed point (Chapter 10). Once developed, the sun position expressions of this chapter are used to demonstrate how to determine the location of shadows and the design of simple sundials. In outline form, our development looks like this:

• Earth-sun angles

• Time

• Standard time zones
• Daylight savings time
• Sidereal time

• Solar altitude, zenith and azimuth angles
• Geometric view of sun’s path
• Daily and seasonal events

• Shadows and Sundials

• Simple shadows
• Sundials

• Notes on Transformation of Vector Coordinates
• Summary

Although many intermediate steps of derivation used to obtain the equations described in this chapter have been omitted, it is hoped that there are adequate comments between steps to encourage the student to perform the derivation, thereby enhancing understanding of the materials presented. Brief notes on the transformation of vector coordinates are included as Section 3.5 and a summary of sign conventions for all of the angles used in this chapter is given in Table 3.3 at the end of this chapter. Figures defining each angle and an equation to calculate it are also included.

One objective in writing this chapter has been to present adequate analytical expressions so that the solar designer is able to develop simple computer algorithms for predicting relative sun and collector positions for exact design conditions and locations. This will eliminate the need to depend on charts and tables and simplified equations.

Figure 3.1 The ecliptic plane showing variations in the earth-sun distance and the equinoxes and solstices. The dates and day numbers shown are for 1981 and may vary by 1 or 2 days.

The earth rotates about its own polar axis, inclined to the ecliptic plane by 23.45 degrees, in approximately 24-hour cycles. The direction in which the polar axis points is fixed in space and is aligned with the North Star (Polaris) to within about 45 minutes of arc (13 mrad). The earth’s rotation about its polar axis produces our days and nights; the tilt of this axis relative to the ecliptic plane produces our seasons as the earth revolves about the sun.

We measure the passage of time by measuring the rotation of the earth about its axis. The base for time (and longitude) measurement is the meridian that passes through Greenwich, England and both poles. It is known as the Prime Meridian.

Today, the primary world time scale, Universal Time (previously called Greenwich Mean Time), is still measured at the Prime Meridian. This is a 24-hour time system, based on mean time, according to which the length of a day is 24 hours and midnight is 0 hours.

Mean time is based on the length of an average day. A mean second is l/86,400 of the average time between one complete transit of the sun, averaged over the entire year. In fact, the length of any one specific day, measured by the complete transit of the sun, can vary by up to 30 seconds during the year. The variable day length is due to four factors listed in order of decreasing importance (Jesperson and Fitz-Randolph, 1977):

• The earth’s orbit around the sun is not a perfect circle but elliptical, so the earth travels faster when it is nearer the sun than when it is farther away.

• The earth’s axis is tilted to the plane containing its orbit around the sun.

• The earth spins at an irregular rate around its axis of rotation.

• The earth `wobbles´ on its axis.

Standard Time Zones - Since it is conventional to have 12:00 noon be approximately in the middle of the day regardless of the longitude, a system of time zones has been developed. See (Blaise 2000) for an interesting story about how this unification developed. These are geographic regions, approximately 15 degrees of longitude wide, centered about a meridian along which local standard time equals mean solar time. Prior to about 1880, different cities (and even train stations) had their own time standards, most based on the sun being due south at 12:00 noon.

Time is now generally measured about standard time zone meridians. These meridians are located every 15 degrees from the Prime Meridian so that local time changes in 1-hour increments from one standard time zone meridian to the next. The standard time zone meridians east of Greenwich have times later than Greenwich time, and the meridians to the west have earlier times.

Figure 3.2 Time zone boundaries within the United States. From Jesperson and Fitz-Randolph 1977.

Daylight Savings Time -To complicate matters further in trying to correlate clock time with the movement of the sun, a concept known as daylight savings time was initiated in the United States in the spring of 1918 to "save fuel and promote other economies in a country at war" (Jesperson and Fitz-Randolph, 1977). According to this concept, the standard time is advanced by 1 hour, usually from 2:00 AM on the first Sunday in April until 2:00 AM on the last Sunday in October. Although various attempts have been made to apply this concept uniformly within the country, it is suggested that the designer check locally to ascertain the commitment to this concept at any specific solar site.

Sidereal Time - So far, and for the remainder of this text, all reference to time is to mean time, a time system based on the assumption that a day (86,400 seconds) is the average interval between two successive times when a given point on the earth faces the sun. In astronomy or orbital mechanics, however, the concept of sidereal time is often used. This time system is based on the sidereal day, which is the length of time for the earth to make one complete rotation about its axis.

The mean day is about 4 minutes longer than the sidereal day because the earth, during the time it is making one revolution about its axis, has moved some distance in its orbit around the sun. To be exact, the sidereal day contains 23 hours, 56 minutes, and 4.09053 seconds of mean time. Since, by definition, there are 86,400 sidereal seconds in a sidereal day, the sidereal second is slightly shorter than the mean solar second is. To be specific: 1 mean second = 1.002737909 sidereal seconds. A detailed discussion of this and other time definitions is contained in another work (Anonymous, 1981, Section B).

Solar Time - Solar time is based on the 24-hour clock, with 12:00 as the time that the sun is exactly due south. The concept of solar time is used in predicting the direction of sunrays relative to a point on the earth. Solar time is location (longitude) dependent and is generally different from local clock time, which is defined by politically defined time zones and other approximations.

Solar time is used extensively in this text to define the rotation of the earth relative to the sun. An expression to calculate the hour angle from solar time is

where t_s is the solar time in hours.

The difference between solar time and local clock time can approach 2 hours at various locations and times in the United States, For most solar design purposes, clock time is of little concern, and it is appropriate to present data in terms of solar time. Some situations, however, such as energy demand correlations, system performance correlations, determination of true south, and tracking algorithms require an accurate knowledge of the difference between solar time and the local clock time.

Knowledge of solar time and Universal Time has traditionally been important to ship navigators. They would set their chronometers to an accurately adjusted tower clock visible as they left port. This was crucial for accurate navigation. At sea a ship's latitude could be easily ascertained by determining the maximum altitude angle of the sun or the altitude angle of Polaris at night. However, determining the ship’s longitude was more difficult and required that an accurate clock be carried onboard. If the correct time at Greenwich, England (or any other known location) was known, then the longitude of the ship could be found by measuring the solar time onboard the ship (through sun sightings) and subtracting from it the time at Greenwich. Since the earth rotates through 360 degrees of longitude every 24 hours, the ship then has traveled 1 degree of longitude away from the Prime Meridian (which passes through Greenwich) for every 4 minutes of time difference. An interesting story about developing accurate longitude measurements may be found in Sobel, 1999.

The level of accuracy required in determining the equation of time will depend on whether the designer is doing system performance or developing tracking equations. An approximation for calculating the equation of time in minutes is given by Woolf (1968) and is accurate to within about 30 seconds during daylight hours.

EXAMPLE: Let us find the clock time for solar noon at a location in Los Angeles, having a longitude of 118.3 degrees on February 11. Since Los Angeles is on Pacific Standard Time and not on daylight savings time on this date, the local clock time will be:

Accurate knowledge of the declination angle is important in navigation and astronomy. Very accurate values are published annually in tabulated form in an ephemeris; an example being (Anonymous, 198l). For most solar design purposes, however, an approximation accurate to within about 1 degree is adequate. One such approximation for the declination angle is

Other latitude angles of interest are the Tropic of Cancer (+23.45 degrees latitude) and the Tropic of Capricorn (- 23.45 degrees latitude). These represent the maximum tilts of the north and south poles toward the sun. The other two latitudes of interest are the Arctic circle (66.55 degrees latitude) and Antarctic circle (-66.5 degrees latitude) representing the intersection of a perpendicular to the earth-sun line when the south and north poles are at their maximum tilts toward the sun. As will be seen below, the tropics represent the highest latitudes where the sun is directly overhead at solar noon, and the Arctic and Antarctic circles, the lowest latitudes where there are 24 hours of daylight or darkness. All of these events occur at either the summer or winter solstices.

When we observe the sun from an arbitrary position on the earth, we are interested in defining the sun position relative to a coordinate system based at the point of observation, not at the center of the earth. The conventional earth-surface based coordinates are a vertical line (straight up) and a horizontal plane containing a north-south line and an east-west line. The position of the sun relative to these coordinates can be described by two angles; the solar altitude angle and the solar zenith angle defined below. Since the sun appears not as a point in the sky, but as a disc of finite size, all angles discussed in the following sections are measured to the center of that disc, that is, relative to the "central ray" from the sun.

The reader should be warned that there are other conventions for the solar azimuth angle in use in the solar literature. One of the more common conventions is to measure the azimuth angle from the south-pointing coordinate rather than from the north-pointing coordinate. Another is to consider the counterclockwise direction positive rather than clockwise. The information in Table 3.3 at the end of this chapter will be an aid in recognizing these differences when necessary.