Position of the sun seen by an observer on the ground
An observer on the ground perceives a movement different from that described above in Figure 1.2 (Chapter 1). He does not feel the movement of the earth around the sun. For him, the sun describes an apparent movement in the sky, also called the course of the sun. For most latitudes, the sun rises approximately in the east, ascends into the sky until solar noon, then descends, and sets approximately in the w'est. The exact course depends on the latitude, the time of day, the day in the year and, somewhat, the year. Figure 2.1 is a schematic view of this course for an observer located in the northern hemisphere. The sun is in the south of the observer at solar noon. If I had represented an observer in the southern hemisphere, I could have used the same diagram by reversing the symbols N and S.
The hour angle to is a very convenient quantity for calculating the position of the sun in its apparent path. Knowing that the sun is at its highest at solar noon, this angle measures the trajectory of the sun between this highest point and the point corresponding to an instant t (Figure 2.2). Said differently, со measures the angular arc between the plane formed by the vertical and by the longitude of the location of the observer P, and the position of the sun at time t.
The hour angle со is zero at solar noon. It is counted negatively in the morning and positively in the afternoon, со describes the course of the sun during a day of 24 h, including night. Therefore, со ranges between -л (midnight solar time) and +л (midnight but 24h later). In Figure 2.2, the hour angle at the sunrise is noted coSMrisl, (negative) and that at the sunset is noted cosunse, (positive). These angles are calculated in the later section.
The definition of the hour angle is independent of the hemisphere and is the same for the southern hemisphere. In Figure 2.2, the point P is located in the northern hemisphere. For a point in the southern hemisphere, Figure 2.2 w'ould be the same, except that the north should be interchanged with the south.
Figure 2.1 Schematic view of the course of the sun in the sky for an observer located at P in the northern hemisphere.
Figure 2.2 Hour angle со of the sun for an observer located at P in the northern hemisphere. «sunrise is the hour angle of sunrise, and cosunset is that of sunset
(^sunset = — ^sunrise)'
As a first approximation, the speed of rotation of the Earth on itself is uniform. Since со goes around 2л during 24h, the relationships between со (in rad) and the true solar time tfST (expressed in h) are:
Solar zenithal angle – solar elevation – azimuth
The apparent position of the sun can be described by two angles: the solar zenithal angle 9$ and the azimuthal angle 4/5, or azimuth (Figure 2.3).
The solar zenithal angle 9$ is the angle formed by the direction of the sun and the local vertical (Figure 2.3). It reaches its minimum when the sun is at its highest at
Figure 2.3 Solar angles depicting the course of the sun in the sky for an observer located at P in the northern hemisphere. 9$ is the solar zenithal angle, ys is the solar elevation and vps is the azimuth. N, £, S and W identify the cardinal points north, east, south, and west, respectively.
solar noon. If the sun is at the vertical of the place, i.e., if it is at the zenith of the observer, then #5 is zero. The angle 6$ increases when the sun goes down on the horizon and reaches its maximum at sunrise and sunset.
The definition of this angle corresponds to that of the angle of incidence for a flat horizontal surface at the top of the atmosphere. These two angles are the same in this case.
Assuming that the sun seen by the observer on the ground is a point and not a sphere, and neglecting the refraction of solar rays by the atmosphere, the maximum of в$ is n/2 (90°). In fact, on the one hand, the apparent diameter of the sun is about 32' of arc, or half a degree. The limb of the sun, that is to say, the edge of the solar disk, is therefore 16' of arc above the horizon when the center of the sun passes under the horizon; solar rays are therefore perceived by the observer. On the other hand, the atmosphere refracts the solar rays, and for this reason, the solar rays do not have a rectilinear trajectory during the crossing of the atmosphere, because of the variation of the density of the air with altitude. The greater the length of the atmosphere crossed, or optical path, the more intense the refraction. The error made by neglecting the refraction is zero when = 0°, about Г of arc when = 45°, and about 34' of arc
when the sun is on the horizon. By the combination of the two effects (sphere and refraction), the solar rays emitted by the limb of the sun are visible as long as the center of the sun is within 50' of arc (= 16' + 34') below the horizon. The maximum of в$ is then 1.5853 rad (90.83°). The maximum relative error made on в$ by neglecting these two effects is of the order of 1 %.
A practical consequence of this calculation is that the relative error in the perception of the direction of the sun by an observer is almost zero and is at most 1 % on the horizon, at times of sunrise and sunset. In other words, two observers, one on the ground and the other at the top of the atmosphere, see the sun in the same direction. Consequently, the solar angles are nearly identical on the ground and at the top of the atmosphere and, more generally, at any altitude.
Another angle is often used instead of the solar zenithal angle: the solar elevation angle, also called the solar altitude angle and noted y$. This angle is formed by the direction of the sun and the horizon assuming that there is no obstacle and no relief. It is zero when the sun is on the horizon (sunrise and sunset) and reaches its maximum when the sun is at its highest. The solar elevation is more intuitive than the solar zenithal angle, because it is this elevation that you perceive when you observe the course of the sun. The two angles в$ and ys are complementary to each other:
At a given time trsr in true solar time, the solar zenithal angle is given by: where со is given by equation (2.1), Ф is the latitude, and 6 is the solar declination.
The second solar angle is the azimuth Ту, or solar azimuthal angle. It is defined as the angle formed by the projection of the direction of the sun on the horizontal plane and the north (Figure 2.3). The azimuth ranges from 0 to 2л and increases clockwise. It is 0 northward, nil (90°) eastward, л (180°) southward, and Зл/2 (270°) westward.
The definition of the reference direction for azimuth is arbitrary. Here, it is from the north, as recommended by the ISO 19115 standards for geographic information.
The equations below relate the two solar angles #y and Ту:
If $s is different from 0 and if the latitude Ф is different from ±л/2, the solar azimuth Ту is given by the following equations:
where cos_l() is the arc cosine function. The azimuth is unknown when 0y = 0, and it can be set to л by convention. At the poles, Ту is unknown for any position of the sun; any value can be taken. The following equations may have an interest in subsequent calculations involving the azimuth Ту:
Other standards than the ISO standard are possible for azimuth. For example, several publications in solar engineering have adopted the following standard. Let the azimuth be noted Tengineering in this standard. It is measured from south in the northern hemisphere and is positive toward the west. Thus, it behaves similarly to the hour angle ox It is positive during the afternoon (west) and negative during the morning (east). The solar azimuth Tengineering is 0 southward, nil westward, л northward, and -nil eastward. In the southern hemisphere, it is counted from the north and is positive westward. It is 0 northward, nil westward, л southward, and -nil eastward. The azimuth Tengineering is equal to (Ту-л) in the northern hemisphere and (-Ту) in the southern hemisphere where Ту is defined as above following the ISO 19115 standard.
In summary, given the true solar time and the geographic coordinates of the site of interest, the solar angles are obtained as follows:
- • use equation (2.1) to obtain the hour angle from the true solar time,
- • then, compute the solar zenithal angle with equation (2.6), and
- • finally, compute the azimuth with equation (2.10).
Direction of the sun in the case of an inclined plane
In the previous section, I have computed the solar zenithal angle that gives the direction of the sun with respect to the vertical in the case of a horizontal surface. In practice, surfaces are often inclined: natural slopes, hillsides, photovoltaic panels, roofs, windows... How to compute the angle between the direction of the sun and the normal to an inclined plane? An inclined plane is described with two angles: the inclination jS and the azimuth a (Figure 2.4). The inclination /? varies between 0 in the case of a horizontal plane and n/2 in the case of a vertical plane. The azimuth a. of the plane, also called orientation, is the angle between the projection of the normal to the plane at the surface of the Earth and the north direction, similarly to the azimuth of the sun.
The angle в under which the sun is seen is formed by the normal to the plane and the direction of the sun. It is given by:
This and the following equations are valid regardless of the convention adopted for the azimuth. It is important to use the same convention for the azimuth of the plane a and that of the sun ¥$. Assume someone has adopted the above-mentioned solar engineering convention: a measurement of the solar azimuth ¥engineering from the south in the northern hemisphere and counted positively to the west. If this person describes the orientation a of a photovoltaic panel as being equal to 62°, that is to say, that the panel is oriented toward the southwest, the azimuth of the panel in the ISO convention is 242° (= 180° + 62°). If the orientation of the panel is -118°, toward the northeast, then the azimuth of the panel is 62° (= 180°-118°) in the ISO convention. If the panel is located in the southern hemisphere with an orientation of 62°, toward the northwest,
Figure 2.4 Angle в under which the sun is seen relative to the normal to an inclined plane. The inclination and azimuth of the plane are /3 and a, respectively.
the orientation is 298° (= 360°-62°) in the ISO convention. If the orientation of this panel is -118°, toward the southeast, the azimuth is 118° in the ISO convention.
It is convenient to express в as an explicit function of со for easier calculations for inclined planes. By exploiting (2.6)—(2.12) and given that