# Solar Geometric Angles

These angles are all critical to the calculation of solar energy on any given surface at some orientation. To calculate the energy balance across any time scale with respect to the incident solar radiation, you must know the geometric relationships that define the spherical realities of planetary motion relative to the sun for specific tilted surfaces on the planet. Each piece is defined relative to another piece. For example, a solar panel may have a tilt, beta, but that tilt is relative to the horizontal surface on the planet, not relative to the sun. Greek symbols, often with a subscript, are used to ensure clarity through unique identifiers for each angle. Additionally, the sign convention for each angle is critical, as an incorrect sign or origin can lead to drastically incorrect results. Be careful when performing calculations to ensure that each angle and sign is correct.

Table 1.1 – General Angles
Angular Measure
Symbol
Range and Sign Convention
Altitude Angle
α (alpha)
0o to + 90o; horizontal is zero
Azimuth Angle
γ (gamma)
0o to + 360o; clockwise from North origin
Azimuth (alternate)
γ (gamma)
0o to ±180o; zero (origin) faces the equator, East is + ive, West is – ive
Table 1.2 – Earth-Sun Angles
Angular Measure Symbol Range and Sign Convention
Lattitude ϕ (phi) 0o to ± 90o; Northern hemisphere is +ive
Longitude λ (lambda) 0o to ± 180o; Prime Meridian is zero, West is -ive
Declination δ (delta) 0o to ± 23.45o; Northern hemisphere is +ive
Hour Angle ω (omega) 0o to ± 180o; solar noon is zero, afternoon is +ive, morning is -ive
Table 1.3 – Sun-Observer Angles
Angular Measure
Symbol
Range and Sign Convention
Solar Altitude Angle (complement)
αs = 1 – θz (alphas is the complement of thetaz)
0o to + 90o
Solar Azimuth Angle
γs (gammas)
0o to + 360o; clockwise from North origin
Zenith Angle
θz (thetaz)
0o to + 90o; vertical is zero
Table 1.4 – Collector-Sun Angles
Angular Measure Symbol Range and Sign Convention
Surface Altitude Angle α (alpha) 0o to + 90o; horizontal is zero
Slope or Tilt (of collector surface) β (beta) 0o to + 360o; clockwise from North origin
Surface Azimuth Angle γ (gamma) 0o to ±180o; zero (origin) faces the equator, East is + ive, West is – ive
Angle of Incidence θ (phi) 0o to + 90o
Glancing Angle (complement) α=1−θ(alpha) 0o to + 90o

What are the key solar geometric angles required for calculating solar energy on a surface?
The key solar geometric angles required for calculating solar energy on a surface include the tilt angle (β), solar altitude angle (α), solar azimuth angle (γ), incident angle (θ), and surface azimuth angle (ψ). These angles are critical for determining the energy balance across any time scale with respect to the incident solar radiation.
Why is it important to define each solar geometric angle relative to another angle?

Defining each solar geometric angle relative to another angle ensures that the calculations are accurate and consistent. For example, the tilt angle (β) of a solar panel is defined relative to the horizontal surface on the planet, not relative to the sun. This approach helps to avoid errors and ensures that the angles are correctly referenced to each other.

What is the significance of Greek symbols with subscripts in solar geometric angle calculations?

Greek symbols with subscripts are used to ensure clarity and uniqueness in identifying each solar geometric angle. This notation helps to avoid confusion between similar-looking angles and ensures that each angle is correctly referenced in calculations. For example, β (beta) is used to represent the tilt angle, while α (alpha) represents the solar altitude angle.

Why is the sign convention critical in solar geometric angle calculations?

The sign convention is critical in solar geometric angle calculations because an incorrect sign or origin can lead to drastically incorrect results. For example, a negative sign for the tilt angle (β) may indicate a south-facing surface, while a positive sign may indicate a north-facing surface. Care must be taken to ensure that each angle and sign is correct to avoid errors in calculations.

How do solar geometric angles vary with respect to the time of day and year?

Solar geometric angles vary with respect to the time of day and year due to the Earth’s rotation and orbit around the sun. The solar altitude angle (α) and solar azimuth angle (γ) change throughout the day, while the tilt angle (β) remains constant for a fixed surface. The incident angle (θ) and surface azimuth angle (ψ) also vary with the time of day and year. Accurate calculations must account for these changes to determine the energy balance across any time scale.

Can solar geometric angles be calculated using astronomical formulas?

Yes, solar geometric angles can be calculated using astronomical formulas that take into account the Earth’s rotation, orbit, and axial tilt. These formulas can be used to calculate the solar altitude angle (α), solar azimuth angle (γ), and other solar geometric angles for a given location and time. However, care must be taken to ensure that the formulas are correctly implemented and that the input parameters are accurate.

How do solar geometric angles impact the performance of solar panels?

Solar geometric angles have a significant impact on the performance of solar panels. The tilt angle (β) and surface azimuth angle (ψ) affect the amount of incident solar radiation that reaches the panel, while the incident angle (θ) affects the angle at which the radiation strikes the panel. Optimizing these angles can improve the energy output of solar panels, while incorrect angles can lead to reduced performance and energy losses.