KTH ROYAL INSTITUTE OF TECHNOLOGY

Solar Fundamentals Sun-Earth Geometry and Solar Position Part 3 Sun-Earth Geometry

The direction of the incident beam radiation depends upon the position of the Sun, which varies throughout the day.

Image Source: J. Spelling, 2009

MJ2411: RENEWABLE ENERGY TECHNOLOGY 2 Sun-Earth Geometry

The path taken by the Sun depends on the observers latitude and the day of the year (through the declination angle δ)

Image Source: J. Spelling, 2009

  n 173 δ: declination angle [rad]   arcsin0.39795cos2    365  n: calendar day (Gregorian)

MJ2411: RENEWABLE ENERGY TECHNOLOGY 3 Sun-Earth Geometry

The path taken by the Sun depends on the observers latitude and the day of the year (through the declination angle δ)

Image based on: W.B.Stine and R.W.Harrigan, 1985 Equinoxes: 21st March / 21st September -> δ = 0 Summer Solstice: 21st June -> δ = +23.45° Winter Solstice: 21st December -> δ = -23.45°

MJ2411: RENEWABLE ENERGY TECHNOLOGY 4 Sun-Earth Geometry

Based on the declination angle and latitude, the length of a given day N can be calculated:

24      N  arccos tan  tan     180  

N: day length [hrs] φ: latitude [°N] ψ: longitude [°W]

Attention: Latitude in °N and longitude in °W.

Image based on: Wikipedia, 2015

MJ2411: RENEWABLE ENERGY TECHNOLOGY 5 Sun-Earth Geometry

The solar position can be defined using a number of angles:

• Solar Azimuth Angle: γs

• Solar Zenith Angle:θz

• Solar Elevation Angle: θs = 90° - θz

Image Source: J. Spelling, 2009

MJ2411: RENEWABLE ENERGY TECHNOLOGY 6 Sun-Earth Geometry

The solar position can be defined using a number of angles: First step, convert local solar time into the:  • Hour Angle:   t 12 12 s

Image based on: W.B.Stine and R.W.Harrigan, 1985

MJ2411: RENEWABLE ENERGY TECHNOLOGY 7 Sun-Earth Geometry

The solar position can be defined using a number of angles:

• Solar Zenith Angle:  z  arccoscos cos cos  sin sin 

 cos z sin  sin  • Solar Azimuth Angle:   sgn() arccos  s  sin cos   z 

Latitude: φ Declination: δ Hour Angle: ω

Image Source: J. Spelling, 2009

MJ2411: RENEWABLE ENERGY TECHNOLOGY 8 Example: Solar Position

It is 15:45 on the clock in Stockholm (59.3°N, 18.0°E) on the 26th July (n = 207). We already know that solar time is: 14h50

What is the solar position?

• Solar Azimuth Angle

• Solar Zenith Angle

MJ2411: RENEWABLE ENERGY TECHNOLOGY 9 KTH ROYAL INSTITUTE OF TECHNOLOGY

Calculate Example: Solar Position

It is 15:45 on the clock in Stockholm (59.3°N, 18.0°E) on the 26th July (n = 207), what is the solar position (azimuth, zenith)?   n 173   arcsin0.39795cos2  18.9   365     t 12  43.0 12 s   59.3

 z  arccoscos cos cos  sin sin   50.2  cos sin  sin   z   s  sgn() arccos   56.1  sin z cos 

MJ2411: RENEWABLE ENERGY TECHNOLOGY 11 Solar Position Maps

The path of the Sun can be plotted on solar maps

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- Image Source: J. Spelling, 2011

MJ2411: RENEWABLE ENERGY TECHNOLOGY 12 Solar Position Maps

The path of the Sun can be plotted on solar maps

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- Image Source: J. Spelling, 2011

MJ2411: RENEWABLE ENERGY TECHNOLOGY 13 Solar Position Maps

When the clock time is used instead of solar time, the effects of longitude and the equation of time can become visible Solar Analemma

Image Source: J. Spelling, 2012

MJ2411: RENEWABLE ENERGY TECHNOLOGY 14 KTH ROYAL INSTITUTE OF TECHNOLOGY

Solar Fundamentals Sun-Earth Geometry and Solar Position

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