1.2. Solar Geometric Angles (Space)

Recall from EME 810 the following four tables showing the Angular Symbols for Standard Solar Relations. These angles are detailed in the D&B textbook in section 1.6. 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 MeasureSymbolRange and Sign Convention
Altitude Angleα (alpha)0o to + 90o; horizontal is zero
Azimuth Angleγ (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 MeasureSymbolRange and Sign Convention
Lattitudeϕ (phi)0o to ± 90o; Northern hemisphere is +ive
Longitudeλ (lambda)0o to ± 180o; Prime Meridian is zero, West is -ive
Declinationδ (delta)0o to ± 23.45o; Northern hemisphere is +ive
Hour Angleω (omega)0o to ± 180o; solar noon is zero, afternoon is +ive, morning is -ive
Table 1.3 - Sun-Observer Angles
Angular MeasureSymbolRange and Sign Convention
Solar Altitude Angle (complement)αs = 1 - θz (alphas is the complement of thetaz)0o to + 90o
Solar Azimuth Angleγs (gammas)0o to + 360o; clockwise from North origin
Zenith Angleθz (thetaz)0o to + 90o; vertical is zero
Table 1.4 - Collector-Sun Angles
Angular MeasureSymbolRange and Sign Convention
Surface Altitude Angleα (alpha)0o to + 90o; horizontal is zero
Slope or Tilt (of collector surface)β (beta)0o to + 360o; clockwise from North origin
Surface Azimuth Angleγ (gamma)0o to ±180o; zero (origin) faces the equator, East is + ive, West is - ive
Angle of Incidenceθ (phi)0o to + 90o
Glancing Angle (complement)α=1−θ (alpha)0o to + 90o