# 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

\begin{align}\ SN = 12.0 - \frac{\lambda}{15}+\frac{EoT*4}{60} \end{align}
\begin{align} Where: \\ SN: & \text{ Monthly averaged solar noon in decimal UTC hour. } \\ \lambda: & \text{ Local longitude (user input) in degrees. } \\ & \text{(positive east of Prime Meridian; negative west of Prime Meridian). } \\ EoT: & \text{ Equation of Time in degrees and is calculated for the monthly average} \\ & \text{ day (Klein, 1977) of the given month. } \\ \\ \end{align}

## 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

\begin{align}\ D = \frac{2\omega_{s}}{2\pi} 24 \end{align}
\begin{align} Where: \\ D: & \text{ Monthly averaged daylight hours, in decimal form. } \\ \omega_{s}: & \text{ The sunset hour angle in radian on the monthly average day. } \\ & \text{(positive west of Prime Meridian; negative east of Prime Meridian). } \\ & \omega_{s} = cos^{-1}(-tan\phi\ tan \delta), (-1\leq \tan\phi\ tan \delta≤1) \\ & \omega_{s} = 0, (tan\phi\ tan \delta <-1) \\ & \omega_{s} =\pi, (tan\phi\ tan \delta >1) \\ \phi: & \text{ Latitude. } \\ \delta: & \text{ Declination of the Sun on the monthly average day of the given month. } \\ \end{align}

## 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.

\begin{align}\ rpd = \frac{\pi}{180} = \frac{cos^{-1}(-1.0)}{180} \end{align}

Equation: Monthly Average of Daily Average of the Cosine Solar Zenith Angle

\begin{align}\ CSZA_{Mdly} &= \frac{Fcos^{-1}(-\text{F/G})+G\sqrt{(1.0-(F/G)^2)}}{\pi}, (-1 \leq(F/G)\leq1) \\ F &= \sin⁡(\phi) * \sin(\delta) \\ G &= \cos(\phi) * \cos(\delta) \end{align}
\begin{align} Where&: \\ &CSZA_{Mdly}: \text{ Monthly average of daylight average of the cosine of solar zenith angle. } \\ &\phi: \text{ Latitude. } \\ &\delta: \text{ Sun declination. } \\ \end{align}

Equation: Monthly Average of Daylight Average of the Cosine Solar Zenith Angle

\begin{align}\ CSZA_{Mda} &= \frac{Fcos^{-1}(-\text{F/G})+G\sqrt{(1.0-(F/G)^2)}}{cos^{-1}(-F/G)}, (-1 \leq(F/G)\leq1)\\ F &= \sin⁡(\phi) * \sin(\delta) \\ G &= \cos(\phi) * \cos(\delta) \end{align}
\begin{align} Where&: \\ & CSZA_{Mda}:\text{ Monthly average of daylight average of the cosine of solar zenith angle. } \\ &\phi: \text{ Latitude. } \\ &\delta: \text{ Sun declination. } \\ \end{align}

Equation: Monthly Averaged Cosine Solar Zenith Angle at Mid-Time Between Sunrise and Solar Noon

\begin{align}\ CSZA_{ZMT}&= F + G\sqrt{\frac{G-F}{2G}}\\ F &= \sin⁡(\phi) * \sin(\delta) \\ G &= \cos(\phi) * \cos(\delta) \end{align}
\begin{align} Where&: \\ &CSZA_{ZMT}: \text{ Zenith angle at mid-time between sunrise and solar noon } \\ & \text{ on the monthly average of the given month. } \\ & \phi: \text{ Latitude. } \\ & \delta: \text{ Sun declination. } \\ \end{align}

## 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.

\begin{align}\ rpd = \frac{\pi}{180} = \frac{cos^{-1}(-1.0)}{180} \end{align}

Equations for computation of declination

\begin{align}\ \delta&=sin^{-1}(sin(\epsilon*rpd)*sin(\lambda*rpd))/rpd \\ \epsilon&=23.439-0.0000004*n \\ L&= modulo(280.460+0.9856474*n, 360.0) \\ g&= modulo(357.528+0.9856003*n, 360.0) \\ \lambda&=modulo(L+1.915*sin(g*rpd)+0.020*sin(2*g*rpd), 360.0) \end{align}
\begin{align} Where: \\ \delta: & \text{ Declination angle of Sun In degrees. } \\ n: & \text{ Number of days from Julian 2000.0.} \\ L: & \text{ Mean longitude of the Sun, corrected for aberration, in degrees.} \\ g: & \text{ Mean anomaly, in degrees.} \\ \lambda: & \text{ Ecliptic longitude, in degrees.} \\ \epsilon: & \text{ Obliquity of ecliptic, in degrees.} \end{align}

## Sunset Hour Angle¶

The Sunset Hour Angle equation is from Solar Engineering of Thermal Process, 3rd Edition.

Equation: Sunset Hour Angle

\begin{align} \omega_{s} &= cos^{-1}(-tan\phi\ tan \delta), (-1\leq \tan\phi\ tan \delta≤1) \\ \omega_{s} &= 0, (tan\phi\ tan \delta <-1) \\ \omega_{s} &=\pi, (tan\phi\ tan \delta >1) \\ \end{align}
\begin{align} Where: \\ \omega_{s}: & \text{ Sunset Hour angle. } \\ \phi: & \text{ Latitude. } \\ \delta: & \text{ Declination of the Sun on the monthly average day of the given month. } \\ \end{align}

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

\begin{align} & \alpha_{max} = 90. - |\phi - \delta | \\ \end{align}
\begin{align} Where: \\ \alpha_{max}: & \text{ Maximum solar angle relative to the horizon, in degrees. } \\ \phi: & \text{ Latitude, in degrees. } \\ \delta: & \text{ Declination of the Sun on the monthly average day of the given month. } \\ \end{align}

## 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