Derivation of the solar geometric relationships using vector analysis
Introduction
Spherical trigonometry was developed very early in the history of mathematics to understand the motion of astronomical objects. The derivation of equations that describe the position and apparent motion of the Sun usually relies upon the use of spherical trigonometry to determine the equations for the Sun's declination, altitude, azimuth and right ascension; and to determine the Equation of Time [1], [2], [3], [4], [5]. Similarly, the derivation of the equations describing the intensity of the direct beam of the solar radiation on a tilted surface of arbitrary tilt angle and orientation for solar energy applications relies upon the same spherical trigonometry approach [2], [6]. Undoubtedly, the resulting equations which are derived in this way describe the solar geometry and direct beam intensity very accurately. However, for many students and practitioners in the field of solar energy, spherical trigonometry is a subject that is generally not well known. In this case, the equations are taken as given and as a result, there is very little understanding or appreciation of the origin of these equations, which may hinder a fuller understanding of the subject.
There have been a few authors who have utilised vector or direction cosine approaches to derive some aspects of the solar geometry and solar intensity on panels. However, typically a mixture of spherical trigonometry and the vectorial approach has been used to derive some of the necessary equations (for example, see [1], [7]). In other cases, only some of the equations have been derived using either a direction cosine approach [8] or a vectorial approach [9].
Therefore, it is the purpose of this paper to present a comprehensive treatment of the vector derivation of all the equations required to describe the Earth–Sun geometry accurately (altitude, azimuth and intensity of the direct beam on a tilted plane). Further, utilising vector analysis, and building on previous work [10], this present paper presents the derivation of the geometric relationships needed to describe the declination and right ascension of the Sun that are consistent with the standard relationships used in accurate algorithms for the Sun's apparent position [11], [12]. The vector approach is also used to derive the equation that accurately describes the Equation of Time. Additionally, this paper seeks to clarify the advantages and disadvantages of the various equations that are presented in the literature for determining the Sun's azimuth. Lastly, the nomenclature used in this work attempts to be consistent with standard texts in the field [6], [11].
Essentially, the vector approach utilises the concepts of unit vectors, vector dot products and the transformation of reference frames (e.g. Cartesian to spherical co-ordinates) (for example, see [13]). Unlike spherical trigonometry, the aforementioned are all mathematical tools that should be familiar to anyone with a background in Engineering or the Physical Sciences.
Section snippets
Solar geometry
The Earth's orbit about the Sun is almost circular at an average distance of 149.6 million km. The Earth's axis of rotation is tilted by an angle ε≈23.44° with respect to the normal to the plane of the Earth's orbit around the Sun (see Fig. 1) [14]. The plane of the Earth's orbit is referred to as the plane of the ecliptic or simply the ecliptic. Alternately this can be expressed by saying that the plane passing through the Earth's equator is inclined obliquely to the plane of the ecliptic, at
Reference frames
In order to derive the solar geometry equations, it is important to define suitable reference frames. Three principal reference frames will be used in this paper, the ecliptic, the equatorial and the horizon reference frames. These reference frames are all geocentric reference frames in that they are centred on, or referenced to, the geometric centre of the Earth. All are commonly found in astronomy textbooks (for example, see [14]) and in the solar literature [1], [5]. In these reference
Declination of the Sun
The Sun's declination was previously defined in Section 3 when discussing Eq. (6) and Fig. 3. The equation required to describe the declination at any time during the Earth's yearly orbit of the Sun can be easily derived using Eq. (5). The declination, δ, is simply the angle that the vector makes with the equatorial (x, y) plane (see Fig. 3) and hence:where Sz is the magnitude of the z-component of and S is the magnitude of (which as it is a unit vector is unity). Therefore,
Altitude of the Sun
In a similar fashion, it is possible to derive an expression for the altitude of the Sun, αS. To do this, consider Eq. (12) for in the horizon reference frame and Fig. 5. From inspection of Fig. 5, it can be seen that sin and hence,where SZ is the magnitude of the Z-component of , and is obtained from Eq. (12). Alternately, Eq. (18) can be derived by considering the two expressions for in the horizon reference frame, Eqs. (12), (13). Again the two
Azimuth of the Sun
To calculate the azimuth of the Sun, γS, consider Eq. (12) and Fig. 5. The azimuth is the angle between the horizontal component of (denoted by ), and the unit vector , which points North (sometimes the azimuth is defined relative to South [6]; however, the usual astronomical definition is with respect to North [14]).
From inspection of Fig. 5, the magnitude of , is simply equal to cos αS, as is a unit vector. Therefore, as γS is the angle between the vector and the
Direct beam intensity on a panel of arbitrary orientation and tilt
In many solar engineering and architectural applications, it is important to be able to calculate the direct beam intensity of the solar radiation incident on a panel or surface which is oriented in any direction and tilted from the horizontal. The direct beam intensity incident on any surface, ID, is given bywhere In is the direct beam intensity incident on a plane normal to the direct beam solar radiation, and θ is the angle of incidence of the direct beam radiation onto the surface
Equation of time and right ascension
Time, as measured by an accurate clock, differs from the time determined from the Sun when using a sundial. Part of this difference has to do with the longitude of the observer's location, daylight saving and the remaining difference (termed the Equation of Time E) is due to factors associated with the Earth's orbit. The exact time difference in minutes is given by the following equation:where Lloc (in degrees) is the longitude of the observer, in a local
Conclusions
Vector analysis has been used to derive all equations required to describe the apparent motion of the Sun: that is, the Sun's declination, altitude, azimuth and right ascension. For the first time, vector analysis has been used to derive accurate equations describing the Sun's declination and right ascension. This has led to a simple derivation of the geometric equations required to describe the Equation of Time. Also in using the vector approach to derive the equations for the Sun's azimuth,
Acknowledgements
The author acknowledges the financial support of the University of New South Wales and the support of the Australian Research Council through its Key Centre's Scheme for the establishment of the School for Photovoltaic and Renewable Energy Engineering.
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