# Orientation and Exposure Efficiency of a Solar Tracking Surface in Clear Sky
## 2 Materials and Method
Presentation of a flux of energy in the form of hourly data is convenient, as its value in watts per hour is numerically equal to the average energy irradiation of the surface in W/m\({}^{2}\) and to the density (intensity) of a flux of solar energy, which is measured in the same units. Hence, such practically important calculations and simple equations are satisfactory performed. Equation (3) is as follows:  
\[I\approx E\Delta\tau\approx G_{b}\ cos\theta\Delta\tau \tag{3}\]Thus, daily energy exposure is equal to the hourly total: \(H=\Sigma I\). A current value \(cos\theta\) features a momentary (orientation) efficiency of the tracking surface, while energy exposure depicts an integral (energy) efficiency. Hence, accordance of a modeled flux to the natural is convenient to compare with actinometrical data, i.e., hourly solar energy, coming to the surface, which is normal to beams in a clear sky. Under cloudless weather, intensity of surface direct flux of solar energy ceases due to light interaction with the components of air, having mass \(M\) on the beam's direction in the atmosphere. The vertical direction of the air of mass \(M_{0}\), which can be calculated by the known barometric equation used in physics, is taken as a unit of comparison. Hence, the mass \(M\) of an inclined air column is convenient to be manifested by a relative optical mass \(m=M/M_{0}\). In actinometry, that integral value is called an optical mass, which serves for calculation of air transparency. However, unequal density and unclear definition of the criterion of an upper boundary of the atmosphere cause the value \(m\) to be assessed by approximate correlations, depending on physical interpretation of the interaction process. The simplest correlation between the air mass and length of the inclined air column is obtained for its hydraulic model, i.e., a flat layer of uncompressed liquid above the height \(H_{0}\). According to the known equations of hydrostatics (4):  
\[H_{0}=\frac{p_{0}}{\rho_{0}g}=\frac{M_{0}}{\rho_{0}gS} \tag{4}\]  
Under the normal air pressure \(p_{0}=101.325\) kPa and temperature \(0\ ^{\circ}\)C, density of air on the Earth's surface \(\rho_{0}=1293\) kg/m\({}^{3}\) height \(H_{0}=7988\) m. In that model, the relative atmospheric mass does not depend on pressure and density of air, and under zenith deflection of a beam by the angle \(\theta_{z}\), correlation of the masses is equal to the correlation of lengths and is manifested by the equation for atmospheric mass (\(AM\)) (5):  
\[m=\frac{M}{M_{0}}=\frac{H}{H_{0}}=\frac{1}{cos\theta_{z}} \tag{5}\]  
According to the equation, the value \(m\) asymptotically tends to infinity under the Sun approaching the horizon. To overcome this discrepancy, the atmosphere is modeled as a spherical envelope of a homogeneous density outside the globe. Under its inclined illumination, the value \(m\) is calculated by the equation, proven by [PERSON] [27; 28] (6):  
\[m=\frac{1}{H_{0}}\bigg{(}\sqrt{R^{2}sin^{2}h+2 RH_{0}+H_{0}^{2}}-R\ sin\ h\bigg{)} \tag{6}\]  
Under normal conditions at sea level, \(H_{0}=7994\approx 8000\) m. Under such a geometric model, at the moments of sunrise or sunset \(h=0\), the value \(m\) is limited and does not exceed the maximum 39.9. This serves as a reference while modifying the Equation (4) for assessment of the integral transparency of real atmosphere with non-homogeneous distribution of pressure, temperature, humidity, extension of the trajectory of beams due to refraction, etc. The present work applies a modified equation of [PERSON] [29; 30; 31], recommended by the European Solar Radiation Atlas (ESRA) [32; 33] (7):  
\[m=\frac{exp\left(-\frac{z}{8334.5}\right)}{cos\theta_{z}+0.50572(96.07995- \theta_{z})^{-1.6354}} \tag{7}\]  
where 8334.5 is the coefficient corresponding to the altitude, at which the air pressure reduced \(e=2.7193\) times. Table 1 presents values of the optical mass of the air in the intervals of zenith angle from \(0^{\circ}\) to \(90^{\circ}\), calculated by Equations (3) and (5) for the surface at sea level.