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
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