Solar Angles

This section covers the angles at which solar energy arrives to the surface.
From this we can determine how much energy is available over an area to be used for heat or electricity. From this knowledge we can place photovoltaic panels, building windows, or heat collectors correctly.

The journey of a photon

  • Created in the core of the sun
  • Millions of years to bounce out to surface of sun
  • Zips to earth in about 8 minutes
  • Passes through the atmosphere
  • Strikes a solar panel and dislodges an electron in the PV panel
  • Electron is collected and delivered to the grid

Electromagnetic Spectrum

Blackbody Spectrum

Solar Spectrum

The Solar System

Earth sun distance

d=1.5 \times 10^8 \left(1 + 0.017 sin\left(\frac{360(n-93)}{365}\right)\right)km

  • n is the day of the year
  • Is this close to circular?

Earth orientation and orbit

REEPS

Sun Ray Angles

  • Declination: angle between equatorial plane and sun
  • Elevation: angle between local surface of earth and sun angle
  • Azimuth: angle between perpendicular north-south plane and sun angle

Sun Ray Angles

Schroeder

Sun Ray Angles

Schroeder

Solar declination

  • Measures the elevation of the sun relative to the earth’s equitorial plane
  • From the earth’s perspective, the sun exhibits a sinusoidal up and down motion with a period of one year.

Solar declination

Declination equation

We can estimate the declination of the sun with this equation

23.5 \sin\left(\frac{2\pi(n-81)}{365}\right)

  • n is the day of the year

Declination plot

Sun path diagrams

  • Do we have to calculate these cumbersome formulas anytime we want to find the sun?
  • Oregon Sun Path
  • http://solardat.uoregon.edu/SunChartProgram.html

Sun path for February 4

Sun path for SSU

Python Sun Path

Hour angle

H = \frac{15 degrees}{hour} \cdot \textrm{hours before solar noon}

Common panel tilt

REEPS 4.X

Panels tilted at the latitude so equinox sun strikes directly.

REEPS 4.X

Solar angle calculation

Starting with the day, location, and time, calculate the sun position

(See written notes)

Solar angle measurement

If weather permits, we’ll verify this calculation with a direct measurement.

Solar Angles

  • Elevation
  • Azimuth
  • Relation to architectural features

Solar Elevation

The vertical angle of the sun. Zero degrees at sunset or sunrise. 90 degrees if directly overhead.

Solar Azimuth

The “compass angle” of the sun. Due south at noon (in the northern hemisphere). To the west at sunset. To the east at sunrise.

Solar Declination

The angle of the earths equatorial plane and the plane of the earths orbit. Varies between -23.5 and 23.5 degrees and causes the seasons.

Seasons

  • What causes the seasons?
  • The Earth’s axis of rotation is tilted with respect to the orbital or ecliptic plane
  • This changes the intensity of sunlight in different parts of the globe
  • It also changes the distance the rays travel through the atmosphere

Solar position

  • How many angles do we need to specify the sun’s position in the sky?
  • Elevation
  • Azimuth

Hour angle

  • Angle of earth rotation with respect to solar noon

Elevation

We measure the vertical angle from the plane of the ground to the sun

\sin\beta = \cos L \cos \delta \cos H + \sin L \sin \delta

  • L is the latitude angle
  • \delta is the declination angle
  • H is the hour angle

Azimuth

We measure the horizontal angle of the sun from north

\sin \phi = \cos \delta \sin H / \cos \beta

  • L is the latitude angle
  • \delta is the declination angle
  • H is the hour angle

Shading Measurements

Path Finder

Software Implementation

You can see an example of these equations implemented in a software library at these locations

  • https://github.com/dsoto/PyPVSim
  • https://github.com/pvlib/pvlib-python