Solar energy/Resource availability

The amount of sunlight that will strike the Earth's surface will depend on a few factors, some of which are periodic and others can be considered variable. Some periodic factors:

• Seasonal
• Day/night
• Urban shade

Some variable factors

• Cloud coverage
• Other natural weather events

In order to calculate the daily paths of the sun over the horizon across the year, several formulas and equations are first necessary to solve.

In addition to the position of the planet around the sun, the amount of sunlight available at any point on the Earth's surface will also depend on the longitude and latitude of the geographic position. Latitude is a measure of how far north or south you are, and varies from -90° (90° south) to +90° (90° north). The longitude is a measure of how far east or west one is, and varies from -180° (180° west) to +180° (180° east).

True solar time (TST) is used to convey the relative position of the sun compared to the earth, regardless of your position on earth. The following equation returns the true solar time in hours.

${\displaystyle TST=UTC+{\frac {longitude}{15}}+EoT}$

The equation of time can be approximated by a sum of two sine waves. The first method is precise, and the second method shown here is approximate.

${\displaystyle EoT=\Delta t_{ey}=-7.659\sin(D)+9.863\sin \left(2D+3.5932\right)}$(minutes)

where: ${\displaystyle D=6.240\,040\,77+0.017\,201\,97(365.25(y-2000)+d)}$

where ${\displaystyle d}$ represents the number of days since 1 January of the current year, ${\displaystyle y}$.

For ${\displaystyle EoT}$ in minutes, see the following equation. Ensure that the trigonometry relations are calculated in radians mode.

${\displaystyle EoT(minutes)=\Delta t_{ey}=7.678\sin({B+1.374})-9.87\sin({2B})}$

Where B is:

${\displaystyle B=2\pi {\frac {(n-81)}{365}}}$

Where ${\displaystyle n}$ is the day of the year.

For ${\displaystyle EoT}$ in degrees, see the following equation. Ensure that the trigonometry relations are calculated in degrees mode.

${\displaystyle EoT(degrees)=0.0072cos(J)-0.0528cos(2J)-0.0012cos(3J)-0.1229sin(J)-0.1565sin(2J)-0.0041sin(3J)}$Where J is:

${\displaystyle J=n\times {\frac {360}{365}}}$

Where ${\displaystyle n}$ is the day of the year.

Note that some formulas result in positive answers, while other result in negative times. This is correct and refers to how early or late compared to true solar time the sun will be. For example, the sun can be as much as 14.29 minutes or 3.56 degrees slow (a positive amount) on February 11th (the 42nd day of the year), and as much as 16.40 minutes or -4.10 degrees fast (a negative amount) on November 3rd (307th day of the year).

Throughout the annual orbit of the Earth around the Sun, the distance between the two bodies is around 150 million km. Because the Earth's orbit is elliptical, the actual distance between the Sun and the Earth will vary somewhat. The Earth takes around 365.25 days to completely orbit the Sun.

The solar declination, δ, is the angle at which the sun Earth is inclined relative the plane of orbit around the Sun. For the northern hemisphere, the angle of declination varies from +23.45 degrees in Winter to -23.45 degrees in Summer.

The solar declination angle can be calculated at any point of the year with the following equations:

${\displaystyle \delta _{degrees}=23.45sin(360(n+284)/365)}$

${\displaystyle \delta _{radians}=arcsin(0.369sin(360(n-82)/365+2sin(360(n-2)/365))}$

Where J is the day of the year, with January 1st being n=1.

${\displaystyle J=n\times (360/365)}$, in degrees.

Calculate the hour angle when the altitude angle is equal to 0.

${\displaystyle \omega _{0}({\text{degrees}})=arccos(-(\tan {\delta })*(\tan {Lat}))}$

Using this equation, you can find true solar time (TST) at sunrise and at sunset.

${\displaystyle TST_{sunrise}=12-({\frac {1}{15}})\omega _{0}}$, make sure that ${\displaystyle \omega _{0}}$is in degrees.

${\displaystyle TST_{sunset}=12+({\frac {1}{15}})\omega _{0}}$, make sure that ${\displaystyle \omega _{0}}$is in degrees.

In addition you can find day length, i.e. the time that the sun is above the horizon, using the half hour angle equation:

${\displaystyle S_{0}=({\frac {2}{15}})\omega _{0}}$, make sure that ${\displaystyle \omega _{0}}$is in degrees.

A quick check of the half hour angle and whether or not it is greater or less than ${\displaystyle {90}^{\circ }}$ can tell you if the day is shorter or longer than the night.

if: ${\displaystyle \omega _{0}>90}$, then the day is longer than the night, and the time that the sun is in the sky is at least 12 hours.

if: ${\displaystyle \omega _{0}<90}$, then the day is shorter than the night, and the time that the sun is in the sky is less than 12 hours.

Imagine the sun is up and you are standing outside. Turn your body so that you facing the sun. Then point your arm directly towards the sun. The angle between the ground (the plane surface, if you are standing on a hill ignore the angle of the hill) and your arm is the elevation angle. The function for the angle of elevation in degrees is:

${\displaystyle h=arcsin[cos(\delta -{Lat})]=90^{\circ }-Lat+\delta }$

Where ${\displaystyle \delta }$ is the angle of declination in degrees, ${\displaystyle Lat}$ is the latitude of the location under analysis in degrees.

${\displaystyle h=arcsin(\cos {\delta }\cos {HA}\cos {Lat}+\sin {\delta }\sin {Lat})}$

The hour angle (noted HA) is the conversion of true solar time (TST) to an angle. Where solar midnight is always 0:00am or 24:00 - or +180 degrees and true solar noon is always 12:00 and 0 degrees. The calculation of the half hour angle, HA is:

For HA in degrees:

${\displaystyle HA=(TST-12)\times {\frac {360}{24}}}$

For HA in radians:

${\displaystyle HA=(TST-12)\times {\frac {2\pi }{24}}}$

As can be deduced from the equations, the sun moves 15 degrees (360/24) in the ecliptic plane, no matter what day of the year it is.

This angle is measured by this type of sundial:

The first step before calculating the azimuth angle is determining whether if the ${\displaystyle sin(az)}$ is positive or negative.

The equation for ${\displaystyle sin(az)}$ is the following equation. Ensure that the trigonometry functions are set to degrees mode for calculating ${\displaystyle \delta }$, ${\displaystyle HA}$, and ${\displaystyle h}$, if you converted the variables to degree form.

${\displaystyle sin(az)={\frac {cos(\delta )sin(HA)}{cos(h)}}}$

Where ${\displaystyle \delta }$ is solar declination in degrees, ${\displaystyle HA}$ is half hour angle in degrees and ${\displaystyle h}$ is altitude angle in degrees.

if ${\displaystyle sin(az)<0}$ then use ${\displaystyle az=-arccos({\frac {sin(Lat)sin(h)-sin(\delta )}{cos(Lat)cos(h)}})}$

if ${\displaystyle sin(az)>0}$ then use ${\displaystyle az=arccos({\frac {sin(Lat)sin(h)-sin(\delta )}{cos(Lat)cos(h)}})}$

Where ${\displaystyle Lat}$ is the latitude in degrees, ${\displaystyle \delta }$ is the solar declination angle in degrees, and ${\displaystyle h}$ is the altitude angle in degrees.

Note, that:

• if ${\displaystyle tan(Lat)tan(\delta )>1}$, then the Sun is still up above the horizon (polar summer in the northern hemisphere).

• if ${\displaystyle tan(Lat)tan(\delta )<1}$, then the Sun will not rise (polar winter in the northern hemisphere).

1. Find ${\displaystyle n}$ of year (day # of year, with January 1st being ${\displaystyle n=1}$.)
2. Calculate solar declination angle ${\displaystyle \delta }$ in degrees.
3. Calculate equation of time (${\displaystyle EoT}$) in (minutes). Ensure that ${\displaystyle EoT}$ is converted to hours before the next step using the conversion ratio of 60 minutes per hour.
4. For a select number of true solar times (${\displaystyle TST}$), for example, 8am, 9am, 10am, ... 10pm, etc. calculate the coordinated universal time (${\displaystyle UTC}$) and then the local time (${\displaystyle LT}$).
5. Calculate the hour angle ${\displaystyle HA}$ for each of the true solar times under consideration.
6. Calculate the altitude angle ${\displaystyle h}$ in degrees for each hour angle, ${\displaystyle HA}$.
7. Calculate the ${\displaystyle sin(az)}$ to determine if the value is negative or positive.
8. Based on the relative positivity or negativity of the ${\displaystyle sin(az)}$, choose the correct azimuth (${\displaystyle az}$) formula to use.

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