Vol.36 (Feb) 2026 | Article no.38 2025
This study focuses on developing a mathematical model to compute geographic coordinates (GCS) for any point on Earth based on the horizontal coordinates of observable celestial bodies with known motion at a specific time, such as the Sun, Moon, planets, and stars. Also, the model is applicable during daylight hours, as it can be implemented using sunlight shadows. Additionally, the position of a celestial body at a given time enables precise determination of geographic directions, such as true north. This facilitates accurate alignment of buildings intended for specific orientations, including scientific facilities, temples, and residential buildings designed to harmonize with wind patterns and sunlight exposure. This method is cost-effective, as it does not rely on Global Navigation Satellite System (GNSS) like GPS, BDS, GALILEO, and GLONASS. In addition, a prototype device was engineered to instantaneously determine the Sun’s horizontal coordinates using photovoltaic cells.
Undoubtedly, navigation is the most crucial aspect of our lives. Without it, we cannot determine directions on our planet Earth and consequently we cannot progress. Precise determination of directions, especially the four cardinal directions, is an essential requirement. It has numerous applications, including in engineering, where many buildings require accurate knowledge of longitude and latitude, as well as precise determination of the four cardinal directions. For example, astronomical observatories cannot fulfill their purpose without determining the north–south direction. Similarly, some temple buildings require specific orientations. Masjids must be constructed facing the direction of the Holy Kaaba in Makka, Saudi Arabia, known as the Qibla direction [1, 2]. Some religious practices require facing east, such as in churches, while others require facing Jerusalem. For buildings to be healthy, they must account for sunlight direction as well as wind patterns, to ensure they are healthy and sustainable. Construction engineers still face significant challenges in determining the correct geographic directions when erecting buildings. Moreover, determining directions is essential for scouting enthusiasts and wilderness survival.
The North Star (Polaris) has been renowned as one of the stars that can be used to determine true north and even calculate a location’s latitude. It remains fixed in the sky—neither rising nor setting—as it is a circumpolar star. Unlike other stars that appear to move, Polaris maintains its position in the sky despite the rotation of other circumpolar stars around it and the rising/setting of all other celestial stars, as it is located almost directly above Earth’s geographic North Pole.
However, using Polaris has limitations. While it hovers near the zenith of the geographic North Pole, observers in the Southern Hemisphere cannot see it, as they lack an equivalent southern pole star. Additionally, Polaris is slightly offset from the exact celestial north pole and gradually shifts position over long time due to phenomena affecting all stars—nutation, precession, and parallax.
However, using Polaris has its limitations as it is not perfectly accurate. Although it appears near the celestial pole (i.e., zenith of Earth’s geographic North Pole), inhabitants of the Southern Hemisphere cannot observe it, as they lack a star at the South Celestial Pole. Moreover, Polaris is slightly offset from the exact geographic North Celestial Pole. Its position also gradually changes over time, since all stars are affected by precession and nutation, and parallax [3]. Over hundreds and thousands of years, the positions of stars change and their constellations shape transform. During the time of the Pharaohs, for instance, the North Star was not one of the current circumpolar stars that neither rise nor set. Instead, the ancient Egyptians had Thuban (α Draconis) as their pole star. Remarkably, what we now consider as north was regarded as south in their celestial orientation [4].
The most common alternative is using a compass. However, compasses have inherent inaccuracies in determining true north due to several factors. The primary issue is magnetic declination, as the magnetic north pole is not fixed but shifts over time. Compass readings are also affected by electromagnetic interference (EMI). Nearby metals and electronic equipment can disrupt the magnetic field measurement. Additionally, vibration and movement of the compass itself introduce directional errors. Space weather and local magnetic anomalies factors, such as solar wind [5], further influence compass accuracy.
Scientists have developed mathematical models to correct these deviations and improve compass accuracy, such as the Kalman Filter [6] and World Magnetic Model (WMM) [7]. While approximate corrections are possible, these models require further development to enhance performance in dynamic environments and need continuous updates.
Recently, satellite systems have been used for real-time, precise directional measurements; systems such as the Global Positioning System (GPS) operate in Medium Earth Orbit (MEO). GPS is a space-based radio-navigation system and one specific example of a Global Navigation Satellite System (GNSS). GNSS is the general term that encompasses all global constellations, including GPS (USA), BDS (China), BeiDou (China), Galileo (Europe), and GLONASS (Russia). However, developing and maintaining these systems remains highly costly.
Stars are absolutely the most important elements, as they can be used to tell time. Astronomically, they are primarily relied upon for precise timekeeping, what astronomers call sidereal time [8]. In ancient times, sundials were used to measure solar time, meaning time relative to the Sun. However, by referring to stars, it became clear that solar time is variable, enabling astronomers to track and study this variation in the length of solar time.
The concept is that the more distant a celestial body is, the more stable it appears relative to us and to nearer celestial bodies—even though it may actually be moving much faster than us. Its immense distance makes its motion imperceptible, rendering it seemingly fixed. Distant stars thus serve as references for tracking the Sun’s and planets’ movements. Through them, planetary orbits (including the Sun and Moon) are determined, solar time variations are understood, and so forth [3].
This principle led to the discovery of parallax. Astronomers observed that as Earth orbits the Sun annually, the apparent positions of stars shift monthly, altering constellations’ shapes. When Earth returns to the same orbital position, constellations resume their original configurations. This stellar displacement became known as the parallax angle.
Thus, time determination using stars has been known to astronomers and remains fundamental to all astronomical calculations to this day, referred to as “sidereal time.” Solar time is calculated using sidereal time. Even modern clocks are synchronized based on it. Through pre-prepared astronomical mathematical models, conversion between sidereal time and solar time has become possible.
The stars themselves can be used for space navigation, as star patterns have been utilized in space navigation. Tracking constellations’ shape allows determining directions in the space environment outside Earth’s atmosphere [9].
The use of stars for precise directional determination on Earth has become unfamiliar and uncommon, despite its supreme importance and exceptional accuracy. Moreover, it costs nothing compared to expensive alternatives like GPS systems, which may provide deviated signals due to natural environmental conditions or intentionally by manufacturers—who deliberately introduce relative errors to restrict usage to specific applications. A previous study demonstrated horizon line determination, twilight width measurement, and east direction identification in darkness using Venus’ position [10]. Similarly, some studies have employed vertical deflection measurements to obtain critical geodetic data such as gravity field determination, which also contributes to GNSS enhancement systems [11, 12].
This article examines all possible navigation methods using the Sun, Moon, planets, and stars, for precise alignment of structures and temples, as well as for use by scouts and amateur astronomers.
In the vast theater of the night sky, the precise location of celestial objects is fundamental to astronomical observation and navigation. However, what we observe from Earth is not always an exact representation of reality. The positions of stars, planets, and other celestial bodies can appear shifted due to various physical and geometric effects, leading to the distinction between their true and apparent positions.
The equatorial coordinate system, which uses right ascension (RA) and declination (Dec), serves as a celestial counterpart to Earth’s longitude and latitude. Right ascension measures the eastward position along the celestial equator from the vernal equinox, expressed in time units (hours). Declination, on the other hand, indicates how far north or south an object lies relative to the celestial equator, measured in degrees. These coordinates provide a fixed reference frame, typically anchored to a specific epoch like J2000.0, which represents the mean equator and equinox of January 1, 2000. However, due to the gradual wobble of Earth’s axis known as precession, these coordinates slowly change over time. Consequently, it must be distinguished between the true position—the object’s geometric location in space—and the apparent position, which accounts for precession, nutation (small periodic oscillations in Earth’s axis), and other effects like aberration of light caused by Earth’s motion.
When observing celestial objects from a specific location on Earth, the horizontal coordinate system becomes more practical. This system uses altitude, the angle above the horizon, and azimuth, the compass direction from true north, to describe an object’s position in the local sky. However, these coordinates are highly dependent on the observer’s location and time, and they are subject to atmospheric effects. The most significant of these is atmospheric refraction, which bends light as it passes through Earth’s atmosphere, making objects appear slightly higher in the sky than they actually are. This effect is most pronounced near the horizon, where refraction can lift the apparent position of the Sun or Moon by more than half a degree. To obtain the true altitude, observers must apply corrections for refraction, which can be calculated using empirical formulas that account for temperature, pressure, and the object’s observed elevation.
Directions and navigation are based on horizontal coordinates, while the positions of celestial bodies are based on equatorial coordinates calculated using numerical models. Therefore, conversion between the two is required [8].
The following equation can be used to know the direction of any city in terms of geographical north with clockwise:
where \({\varphi }_{\square}\) and \({\lambda }_{\square}\) are the latitude and longitudes of destination city, while \(\varphi\) and \(\lambda\) are the latitude and longitudes of the current observer location.
Because azimuth represents the clockwise direction of a celestial body from geographic north, it can be used to determine the exact direction of geographic north at any given moment, and consequently, any direction on Earth, including the four cardinal directions. So, the direction of geographic north can be formulated as:
where A is the azimuth, and N is the direction of geographic north.
Determining any direction on Earth (D) is made possible using the azimuth of any celestial body. If we assume that we want to orient ourselves at a direction (d) from geographic north, the deviation can be formulated in terms of the celestial body’s azimuth. That is:
The value D is the required direction on the horizon. If a celestial body has an angular altitude of say 30°, we take its projection on the horizon, and its angular drift from the geographic north is azimuth A. If the value of D is positive, it means that the required direction is to the right of this direction by that value. If it is negative, it means that the direction of the celestial body is to the left. This means that we have made the direction of the celestial body on the horizon a reference for heading towards any direction on Earth, to the right and left of it. The value of D is clockwise from the celestial body.
It is worth noting that refraction correction is not necessary when knowing directions because the correction is only for the altitude and not for the azimuth angle.
The equatorial coordinates of any celestial body can be converted to any type of terrestrial coordinate system. We will use the Geographic Coordinate System (GCS).
To know the position of a celestial body on the map (over it), i.e., to calculate the Earth’s longitude (λz) and latitude (φz) of destination city at which this celestial body is at the zenith point (i.e., a = 90°), the relationship can be used:
where Ts is the sidereal time, λ and φ are the observer's longitude and latitude, λz and φz are longitude and latitude of destination city, and the value of λz is between − 180° and 180°. z refers that the celestial body is at the zenith point. Also, we have
When the longitude of the current observer is
A spherical triangle can be drawn that simulates both the celestial sphere and the Earth. Point P represents the north pole. Point S represents the celestial body seen by the observer, represented by point C. Angle P equals (Δλ), which can be obtained from Eq. 7 in terms of the right ascension of the celestial body. Angle C equals the azimuth A of the same celestial body (celestial body direction). The great circle great circle arc \(\stackrel{\frown}{PC}\) is equal to (90° − φ), while \(\stackrel{\frown}{PS}\) is equal to (90° − φz) or (90° − δ) according to Eq. 4. We can put \(\stackrel{\frown}{CS}\) as (90° − a), as shown in Fig. 1.
This schematic represents the celestial sphere and the Earth. Point S represents the celestial body and its position above the destination city on Earth when celestial body it is at the zenith point of the terrestrial point. Point C is the observer’s location. Point P is the North Pole
From spherical triangle PCS, we can use the sine formula.
Then, the observer longitude can be estimated from Eq. 8, where \({\lambda }_{z}\) is determined by Eq. 4.
We can apply Napier’s Analogies formula,
Now, we have a relationship that allows us to estimate the longitude, as follows:
The cosine formula can be used directly as an alternative solution, but it requires a numerical solution. The formula is
Then
The latitude value can be calculated using the numerical iteration methods or using the Python function “fsolve” at SciPy library [13], as Table 1.
| import numpy as np from scipy.optimize import minimize # impot values of solar cells/panels alpha = np.radians([30, 45, 60]) beta = np.radians([0, 90, 180]) I = np.array([0.6, 0.9, 0.3]) # solar panels currents I_min = np.min(I) I_prime = I—I_min |
| def error(params): a, A = params model = + np.sin(alpha) * np.sin(a) + np.cos(alpha) * np.cos(a) * np.cos(beta—A) return np.sum((I_prime—model)**2) |
| initial_guess = [np.radians(45), np.radians(90)] # initial guess for values result = minimize(error, initial_guess, bounds = [(0, np.pi/2), (0, 2*np.pi)]) # Improve a_opt, A_opt = np.degrees(result.x) print(f"Estimated Altitude (a): {a_opt:.2f}°") print(f"Estimated Azimuth (A): {A_opt:.2f}°") |
To calculate longitude and latitude, the true position of the celestial body must be determined with precision, taking into account atmospheric pressure, temperature, and elevation above sea level. The higher the celestial body is above the horizon, the more accurate the calculations become, as the effect of atmospheric refraction diminishes, which is more pronounced near the horizon. This refraction requires corrections based on atmospheric and environmental conditions.
The equatorial coordinates of stars utilized in formulas 4 through eq. 10 are derived from real-time observations that incorporate corrections for proper motion, precession, nutation, polar motion, aberration, and atmospheric refraction. Consequently, the geographic coordinates of the observer (referenced to the GSC) calculated at this epoch are not aligned with the International Terrestrial Reference Frame (ITRF). As such, they theoretically require correction for polar motion. The omission of this polar motion correction will introduce deviations at the sub-arc-second level.
Alternatively, equatorial coordinates can be obtained directly from ephemeris models instead of converting them from the observed altitude and azimuth of the celestial body in the above questions. However, when relying on the observed altitude and azimuth angles directly in the aforementioned equations, without conversion, the most precise calculation results can be achieved. Astronomical Models: Provide higher precision but require internet/software access (e.g., Astropy, JPL Horizons).
One essential navigation method involves obtaining a sky map showing what an observer should see when facing a specific direction. This requires a stellar coordinate database, converting equatorial coordinates to horizontal coordinates, then projecting them onto a 2D circle simulating the celestial view as Cartesian (x, y) coordinates represented in an image. This coordinate transformation between equatorial and Cartesian systems heavily depends on the optical projection type. Several methods exist:
Equidistant/linear projection: The vertical azimuth angle is directly proportional to the distance from the image center.
Equisolid angle projection: Preserves area ratios between the sky and image.
Stereographic projection.
Orthographic projection.
Our calculations will use the equidistant/linear projection method.
We now need to plot a full-sky map on a circle that simulates all celestial objects above the horizon. This map helps identify the pattern of the star or celestial body at the target city’s zenith point, enabling easy orientation. To simulate the sky’s appearance on a 2D map, we must convert horizontal coordinates (altitude, azimuth) to Cartesian coordinates (x, y) using equidistant projection. First, we convert altitude to zenith distance as follows:
where Z is Zenith distance in degrees unit, and a is the altitude. Calculate the radial distance r in pixels; this step uses the equidistant projection function.
where r is the radial distance, and R is the radius of the circle which represents the sky. Both variables are in pixel units. We consider
The modulus must be set so that the θ is always between 0 and 360°. The angle is always positive. Then we calculate the relative coordinates of the center (dx, dy):
A mod operation must be applied to ensure the angle is always constrained between 0 and 360°, thus keeping it positive. Then, we calculate the relative coordinates of the center (dx, dy):
Now, we get the (x, y) coordinate of the celestial body on a circle representing the sky map calculated in pixels or image units.
where \({x}_{c}\) and \({y}_{c}\) are the coordinates of the center of the circle (the center of the image) drawn to simulate the visible hemisphere of the sky. This requires excluding celestial objects invisible to the observer. The condition can be set
where \({x}_{c}\) and \({y}_{c}\) are the coordinates of the circle’s center (image center) drawn to simulate the visible hemisphere. This requires excluding celestial objects not visible to the observer, for which a condition can be set.
Now, we need to draw a sky map that simulates what the observer sees in any direction parallel to the horizon circle, resulting in a drawn circle containing celestial bodies where its upper half represents what the observer sees above the horizon, while the lower half represents what is below the horizon. We must exclude anything behind the observer whether above or below the horizon. This map is important for determining directions without needing a compass, as non-specialists may get confused and cannot distinguish between stars. Drawing the sky map helps identify which star to use for calculating directions, longitude, and latitude when its position is known.
The fundamental condition is to select stars located at the predetermined target direction D, and those positioned to its right and left within a range not exceeding 90°. Thus, the condition becomes:
Since the celestial objects’ points are the same ones to be plotted without conversion, they will appear on the map as square-shaped Cartesian coordinates. To transform these equatorial coordinates that appear square-shaped into a circular representation simulating the sky’s appearance, we use either the “Relative Distance Preservation” method or the “Shirley-Chiu transformation” [14]. Accordingly, the altitude and azimuth values are adjusted based on the equation:
where a and A are the altitude and azimuth of the celestial body. \({x}_{crl}\) and \({y}_{crl}\) are corrected position of altitude and azimuth on the plot.
After developing a mathematical model that helps determine the observer’s city longitude and latitude based on either knowing a celestial body’s horizontal coordinates at a given time or its pre-calculated coordinates from mathematical models at a specific time—enabling longitude and latitude calculation—we can also identify principal geographic directions like true north and any required directional orientation. We name this the “Rami Model,” and now, we test it from Cairo city with approximate coordinates (30°, 30°), requiring orientation toward Makka.
The solar azimuth angle and its corresponding shadow direction (360°–azimuth) can be utilized during daytime. The same method can be applied to other celestial bodies such as the Moon, planets, and stars. Figure 2 shows Mecca’s direction relative to the Sun’s position at the specified time. The direction was calculated both from the Sun’s shadow position and from true north. The latitude \({\varphi }_{\square}\) and longitude \({\lambda }_{\square}\) of Makka city are assumed (21.4225°, 39.8262°).
An illustration simulating the direction of Makka city in terms of the geographical north direction, the direction of the Sun, and the direction of the Sun’s shadow during the day (left plot). While the right plot is the direction using shown planets’ directions
Figure 3 represents a sky map in a specific direction. In this example, we have chosen the city of Makka, orienting ourselves towards it from the city of Cairo. The horizontal middle line represents the horizon line. Its right and left sides indicate the right and left directions towards the target, which is the city of Makka. The upper point represents the zenith point, while the lower point represents the nadir (i.e., south zenith). The center of the circle is the observer. Therefore, any celestial object located along the great circle connecting the zenith point and the target direction (i.e., Makka), extending downward, represented by the line from the center of the circle upward, indicates the required direction. Any celestial body located around this line can serve as a guide for the desired direction. For amateurs, it is possible to orient directly toward it. As for specialists, they can apply the method in the previous Sect. (3.1) to accurately determine the angle of deviation from this celestial body.
Sky map in the direction of the Qibla at 11:17:19 on May 3, 2025, from the city of Cairo. Only the brightest stars have been drawn, and the same method can be applied to everything that can be seen in the sky
One of the important sky maps that help determine the direction of the target is the drawing of the sky map as it appears over the target city. Through this, one can identify the star located at the zenith. Heading towards it is considered a correct and accurate direction to the target city.
Figure 4 represents the sky map above the city of Makka at a certain time. The yellow line indicates the direction towards the observer’s city that you want to head towards the target.
Sky map above Makka city at 11:57:00 on May 3, 2025
Similarly, the sky map directly above the observer is crucial, with the target direction marked by a distinctive line. All celestial bodies aligned along this line can help precisely determine the desired direction or at least guide orientation toward it. Figure 5 shows the overhead sky map for Cairo.
Sky map above Cairo at 11:46:00 on May 3, 2025. The right maps are the same as the left ones, but they have added the names of the stars. Only the brightest stars are drawn
There are many devices that allow for tracking the Sun, known as solar trackers. These devices determine the direction of sunlight and are primarily used to orient solar panels to generate the maximum amount of energy. There are two types of these devices: one that relies on a light-dependent resistor (LDR) sensor and another that relies on astronomical calculations. However, these devices do not accurately contribute to determining the primary geographic directions. They may also not precisely help in identifying the user’s geographic coordinates, particularly in obtaining Sun’s altitude and azimuth. Therefore, we propose a new, simple device based solely on photovoltaic cells (PV) or solar panels.
The idea lies in the fact that a PV cell generates maximum current when the Sun’s rays are perpendicular to it, while the current drops to zero when the direction of the sunlight becomes parallel to the panels. This can be formulated as follows:
where \({I}_{\text{max}}\) is the maximum electric current intensity the cell can generate, while the angle \(\theta\) is the inclination of the Sun’s rays on the panel. \({I}_{t}\) is the generated electric current from the cell for the total irradiance, including both direct and indirect sunlight. Since sunlight intensity varies from day to day depending on weather conditions, even if the rays are perpendicular, the generated current may not reach the maximum value \({I}_{\text{max}}\). Additionally, there is irradiance from sources other than direct sunlight, whose direction must be calculated. Some of this is diffuse sky radiation [15], and some is ground-reflected irradiance, called Albedo Radiation. These can be combined into a single variable called sky current \({I}_{\text{sky}}\). Thus, the current intensity equation takes the form:
To determine the unknown parameters in this equation and obtain the angle \(\theta\), we can rely on multiple cells oriented in different directions, all measuring the current intensity from the same sunlight at the same moment. For instance, we can arrange a set of PVs of solar panels—no fewer than three panels forming a pyramidal shape or five panels arranged in a cubic configuration. Each panel is tilted relative to the others at a fixed, predetermined angle (e.g., 90° in the case of a cube). It is essential to align the first panel precisely toward the true geographic north to enable the determination of the solar azimuth angle. Additionally, each panel or cell must have the same flat surface area, meaning they must have the exact same \({I}_{\text{max}}\) value. If a large number of cells are available in all directions, the cell with the highest current output will correspond to \({I}_{\text{max}}\). Otherwise, we can eliminate this maximum value by dividing the equations. Since \({I}_{\text{sky}}\) varies instantaneously due to weather conditions, we can assume its value to be the smallest generated current among the cells, as it represents indirect (diffuse) irradiance. All PV cells will generate approximately equal currents during cloudy times due to isotropic diffuse radiation, with deviations < 5% [15]. This reflects the diffuse light effect, where uniform sky radiation minimizes current disparities between differently oriented cells. Thus, the equation becomes:
Here, \({I}_{\text{min}}\) is the smallest current generated by the cells, where \(i\) is the cell number and the total number of cells is \(n\). In the simplified cubic model, \(n = 5\), while \({I}_{i}\) represents the direct current. However, it is preferable for the tilt angle between each panel to be less than 20° to achieve higher efficiency (i.e., reducing blind spots). This model can be represented as a geodesic dome tiling (i.e., A spherical arrangement of triangular cells for uniform solar exposure), where each cell is triangular in shape, ensuring that all cells have the same flat surface area and the same maximum current value \({I}_{\text{max}}\).
Since the panels are tilted relative to each other at a fixed angle ∅ (such as 90° in the cubic model or 18° in the geodesic dome model), the equation can be formulated as follows:
To calculate the altitude and azimuth, the following relationship can be used:
where \(\alpha_i\) and \({\beta }_{i}\) are the altitude and azimuth (compass direction) of the orientation directions of the solar panels. \(a\) and \(A\) are the Sun’s altitude and azimuth. This is based on the principle that solar irradiance on a panel is proportional to the cosine of the angle of incidence between the Sun’s rays and the panel’s normal vector.
Then,
From this formula, the altitude and azimuth values can be obtained using numerical methods, using the least squares fitting or nonlinear solvers like Levenberg–Marquardt (e.g., using Python’s fsolve), or the Newton–Raphson method [16]. After obtaining the values of α and β, they can be converted to altitude a and azimuth A using the previous equations, which are derived from the spherical triangle that shown in Fig 6.
The horizontal great circle is the observer's horizon, with point Z representing the zenith. The oblique great circle is the plane of the solar panel on the celestial sphere, with its upper head point S. While P is the celestial north pole
After developing a mathematical model that helps determine the geographic longitude and latitude of the observer’s city based on either knowing the horizontal coordinates of a celestial body at a specific time or its pre-calculated coordinates from mathematical models at a given time—enabling longitude and latitude calculation—we can also identify principal geographic directions like true north and any required directional orientation. This contributes to aligning buildings that require specific orientations, such as those needing precise true north alignment or main religious structures like the Qibla direction. It also helps optimize building orientations for wind flow and solar radiation efficiency, resulting in healthier structures.
Additionally, we can plot a sky map over a city to identify celestial bodies at its zenith, enabling more precise geographic orientation toward them. The sky map can also help determine stars located at the desired azimuth angle for accurate geographic navigation toward a target. Directions can alternatively be determined relative to any celestial body by deviating right or left by a specified angle. This method is time-dependent, varying from one time to another.
The python source code that is used in the current study is available at GitHub: “ground based navigation,” https://github.com/prof-ramy/navigation.
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This study is a collaboration between the corresponding author, who is the originator of the study concept and holds an invention patent for the underlying idea (Registration No. EG/T/2025/6 and EG/1/2025/6 at Egyptian Patent Office), and a team from Cairo University, where the research forms part of the first author’s doctoral dissertation. The work is conducted in cooperation with Nusa Putra University, which supports the part of solar tracking and incidence direction. This segment is carried out by the student second author as a graduation project from the Faculty of Engineering, under the supervision of the fourth author. All commercial rights reserved to the corresponding author. The authors are very grateful to Asia Pacific Center for Theoretical Physics (APCTP) for their support in article processing charges (APC) of publishing this research paper.
The publication fees of Article Processing Charges (APC) are supported from Asia Pacific Center for Theoretical Physics (APCTP), Korea.
Conceptualization, R.M.; data curation, I.E., M.R.M.; formal analysis, I.E., M.R.M. and R.M.; funding acquisition, R.M.; investigation, I.E., M.R.M. and R.M.; methodology, I.E., M.R.M. and R.M.; software, M.R.M. and A.R.A.R.; equations, R.M. and I.E.; writing—original draft, M.R.M., I.E., and R.M.; writing—review and editing, R.M., A.R.A.R., A.A., and A.H.O. All authors have read and agreed to the published version of the manuscript.
The authors declare that there are no competing interests.
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