# 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 (GMT time)¶

Equation

\begin{align}\ SN = T_ls + (longitude – meridian) / 15 + longitude / 15 \end{align}
\begin{align} Where: \\ SN: & \text{ Solar Noon in GMT in decimal hour. } \\ longitude: & \text{ Local longitude (user input) } \\ & \text{(positive east of Prime Meridian; negative west of Prime Meridian). } \\ meridian: & \text{ Meridian through the center of the local time zone } \\ & \text{(positive east of Prime Meridian; negative west of Prime Meridian). } \\ T_ls: & \text{ Solar noon in local standard time in decimal hour on the monthly average day. } \\ \end{align}

## Monthly Averaged Daylight Hours (hours)¶

The Monthly Averaged Daylight Hours is from Solar Engineering of Thermal Process 3rd Edition.

All-Sky Monthly Averaged Direct Normal Radiation

$$D = \frac{2\omega_{s}}{2\pi} 24$$
\begin{align} Where: \\ D: & \text{ Monthly averaged daylight hours. } \\ \omega_{s}: & \text{ The sunset hour angle in radian on the monthly average day. } \\ \end{align}

## Monthly Averaged Hourly 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).

Warning

All angles in degrees need to be converted to a radian by multiplying it by rpd. This includes the result of the solar declination function.

\begin{align}\ rpd = \pi/180 = acos(-1.0)/180 \end{align}

Equation

\begin{align}\ csza = (f*acos(-\text{f/g})+g*sqrt(1.0-(f/g)^2))/acos(-f/g) \end{align}
\begin{align} Where: \\ csza: & \text{ Monthly averaged hourly cosine solar zenith angle. } \\ & f = \sin⁡(Φ) * \sin(δ) \\ & g = \cos(Φ) * \cos(δ) \\ Φ: & \text{ Latitude. } \\ δ: & \text{ Sun declination. } \\ \end{align}

### Monthly Averaged Cosine Solar Zenith Angle at Mid-Time Between Sunrise and Solar Noon¶

Equation

\begin{align}\ cos(Θ)_{ZMT} = (f + g * sqrt([(g-f)/(2g)]) \end{align}
\begin{align} Where: \\ Θ_{ZMT}: & \text{ Zenith angle at mid-time between sunrise and solar noon. } \\ & f = \sin⁡(Φ) * \sin(δ) \\ & g = \cos(Φ) * \cos(δ) \\ Φ: & \text{ Latitude. } \\ δ: & \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° north to 23.45° south 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.

Warning

All angles in degrees need to be converted to radian by multiplying it by rpd.

\begin{align}\ rpd = \pi/180 = acos(-1.0)/180 \end{align}

Equation

\begin{align}\ sd = 23.45 * sin((360*rpd/365)*(284 + jd)) \end{align}
\begin{align} Where: \\ sd: & \text{ Declination angle. } \\ jd: & \text{ Julian day. } \\ \end{align}

Reference

Cooper, P.I., 1969. The Absorption of solar radiation in solar stills. Solar Energy 12(3), 333-346.

## Sunset Hour Angle¶

All-Sky Monthly Averaged Direct Normal Radiation

\begin{align} & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ & \cos(\omega_{s}) = \tan(Φ) * \tan(δ) \\ \end{align}
\begin{align} Where: \\ \omega_{s}: & \text{ Sunset Hour angle. } \\ δ: & \text{ Declination angle. } \\ Φ: & \text{ Latitude. } \\ & \text{ } & \text{ } & \text{ } & \text{ } & \text{ } & \text{ }\\ & \text{ If } \cos(\omega_{s}) < -1.0 \text{ set } cos(\omega_{s}) = -1 \\ & \text{ If } \cos(\omega_{s}) > +1.0 \text{ set } cos(\omega_{s}) = +1 \\ & \omega_{s} = acos(\tan(Φ) * \tan(δ)) \\ \end{align}
\begin{align} \scriptsize{ \text{ All angles for each month are based upon the monthly average day. See Monthly Average Declination table above. }} \\ \end{align}

Reference

John A. Duffie and William A. Beckman, 1991. Solar Engineering of Thermal Process, 2nd edition, Wiley-Interscience Publication.

## Maximum Solar Angle Relative to The Horizon¶

All-Sky Monthly Averaged Direct Normal Radiation

\begin{align} & \theta_{z} = acos(\sin(δ) * \sin(Φ) + \cos(δ) * \sin(Φ) \\ \end{align}
\begin{align} Where: \\ \theta_{z}: & \text{ Zenith angle. } \\ δ: & \text{ Declination angle. } \\ Φ: & \text{ Latitude. } \\ & \\ & Max(\theta_{z}) =9.0 -\theta_{z} \\ \end{align}
\begin{align} \scriptsize{ \text{ All angles for each month are based upon the monthly average day. See Monthly Average Declination table above. }} \\ \end{align}

Reference

J. E. Braun & J. C. Mitchell, 1983. Solar Geometry for Fixed and Tracking Surfaces. Solar Energy, 31(5), 339-444.

## Hourly Based Equations¶

The methodology outlined in the paper below is used to compute both 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.