Solar Geometry¶
Multi-year monthly averaged solar geometry parameters are available for any latitude/longitude via the "Data Tables for a particular location" web application. The call-out below lists the solar geometry parameters provided to assist users in setting up solar panels. In the sections below the equations are provided for calculating each of the parameters, and the methodology for calculating the multi-year monthly averages is described.
Solar Geometry Parameters
- Solar Noon
- Daylight Hours
- Daylight Average of Hourly Cosine Solar Zenith Angles
- Cosine Solar Zenith Angle at Mid-Time Between Sunrise and Solar Noon
- Declination
- Sunset Hour Angle
- Maximum Solar Angle Relative to The Horizon
- Hourly Solar Angles Relative to The Horizon
- Hourly Solar Azimuth Angles
Monthly Average Declination Table
The solar geometry parameters are calculated for the "monthly average day"; consequently each parameter is the monthly "averaged" value for the respective parameter for the given month. The "monthly average day" is the day in the month whose solar declination (δ) is closest to the average declination for that month (Klein, 1977). The table below lists the date and average declination, δ, for each month.
Month Day | δ (°) | Month Day | δ (°) |
---|---|---|---|
January, 17 | -20.9 | July, 17 | 21.2 |
February, 16 | -13.0 | August, 16 | 13.5 |
March, 16 | -2.4 | September, 15 | 2.2 |
April, 15 | 9.4 | October, 15 | -9.6 |
May, 15 | 18.8 | November, 14 | -18.9 |
June, 11 | 23.1 | December, 10 | -23.0 |
Monthly Averaged Solar Noon (UTC time)¶
Equation: Monthly averaged solar noon
Monthly Averaged Daylight Hours (hours)¶
The Monthly Averaged Daylight Hours is from Solar Engineering of Thermal Process, 3rd Edition. Please see the reference box beneath the Sunset Hour Angle section below.
Equation: Monthly averaged daylight hours
Monthly Averaged Cosine Solar Zenith Angle¶
The Cosine Solar Zenith Angle is the average cosine of the angle between the Sun and directly overhead during daylight hours. The determination of monthly averaged daylight average of hourly cosine solar zenith angles for each month is based on the monthly average day (i.e. calculated for the monthly averaged day).
The following equations may need the angles expressed in radians for the trigonometric functions
Depending on the expected input of calculation system, to convert angles in degrees to radians multiply by rpd. This includes the result of the solar declination function.
Equation: Monthly Average of Daily Average of the Cosine Solar Zenith Angle
Equation: Monthly Average of Daylight Average of the Cosine Solar Zenith Angle
Equation: Monthly Averaged Cosine Solar Zenith Angle at Mid-Time Between Sunrise and Solar Noon
Monthly Averaged Declination¶
Declination is the angular distance of the Sun north (positive) or south (negative) of the equator. Declination varies through the year from 23.45° N to 23.45° S and reaches the minimum/maximum at the southern/northern summer solstices. The determination of monthly averaged declination for each month is based on the monthly average day.
The following equations may need the angles expressed in radians for the trigonometric functions
Depending on the expected input of calculation system, to convert angles in degrees to radians multiply by rpd. This includes the result of the solar declination function.
Equations for computation of declination
Sunset Hour Angle¶
The Sunset Hour Angle equation is from Solar Engineering of Thermal Process, 3rd Edition.
Equation: Sunset Hour Angle
Reference
John A. Duffie and William A. Beckman, 2006. Solar Engineering of Thermal Process, 3rd edition, Wiley-Interscience Publication.
Maximum Solar Angle Relative to The Horizon¶
The maximum solar angle relative to the horizon occurs at local solar noon.
Equation: Maximum Solar Angle Relative to The Horizon
Hourly Based Equations¶
The methodologies outlined in the papers below are used to compute the hourly solar angles relative to the horizon and hourly azimuth angles.
Reference
Seidelmann, P.K. (Ed.), 1992. Explanatory Supplement to the Astronomical Almanac. A revision to the Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. University Science Books, Mill Valley, CA (USA), 1992, 780 p., ISBN 0-935702-68-7.
Zhang, Taiping; Stackhouse, Paul W.; Macpherson, Bradley; Mikovitz, J. Colleen (2021). A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function. Renewable Energy. Elsevier BV. 172: 1333–1340. doi:10.1016/j.renene.2021.03.047