Thursday, 30 January 2014

Lambert's Cosine Law

The formula known as Lambert's Cosine law is widely used in climate science and solar engineering.  It relates the angle of incidence of ray of light to the irradiance of the surface.  When the noon sun is high in a clear sky, the ground it is shining on is warm.  At sunrise and sunset when the sun is low in the sky the heating effect of the sun's rays are spread over a larger area thus its heating effect is lower.  Seasonal variations in sun-earth geometry is one of the factors influencing soil temperatures.  Similarly, the output of solar panels is higher when they are pointing directly at the sun rather than angled away from it.

The formula is:

Where I is the power of the beam, often expressed in watts/m2 and z is the angle between the beam and a normal to the surface being irradiated.  This is shown in the diagram below:

In solar engineering z is often called the zenith angle.  The diagram below shows the irradiance of surface by a beam whose power is constant at 1,000 watts/m2 for three different values of z:

This relationship was first quantified by the prolific 18th century Swiss scientist Johann Heinrich Lambert (see Wikipedia for a biography).

Cosine Response

The term "cosine response" is used to describe the behaviour of sensor which is irradiated by ray at an angle to its surface.  A device with good cosine response is one which allows the Ibeam to estimated knowing the value of Isurface and the zenith angle.  This is important for radiometers that measure solar irradiance which are mounted in a fixed position and do not track the sun as it moves across the sky (Earth-centred explanation used for brevity!).


Whilst determining if a low cost photocell could be used to estimate solar irradiance, an experiment was set up which involved blue-tacking a photocell to a vertical piece of plywood and pointing a torch at it from different angles whilst maintaining a constant distance between the torch and the photocell.  The wires from the photocell were connected across an ammeter, effectively short-circuiting it, thus the output of the meter was directly proportional to the irradiance of the photocell.  A diagram of the layout is shown below:

The photocell selected was a flat, crystalline device measuring approximately 65mm by 20mm which produces current in the range 0 - 25mA depending on the irradiance:

The results of the experiment are shown in the plot below:

There is a close agreement between the theory and the experimental results.  Three sources of error would account for the deviations.  The source was not a pure point, also, the cell was receiving some diffuse light from the surrounding room and it is probable that there were some measurement errors.


The effect of Lambert's Cosine Law is a major determinant of climate.  In the equatorial regions, the zenith angle is always close to zero at solar noon and there is little seasonal variation.  Moving polewards, the zenith angle at solar noon decreases and seasonal variation increases.  The graph below shows the zenith angle at noon and its cosine for a location in South East England.  The Sun is high in the sky in summer (small z, big cos(z)) and very low in winter (large z, small cos(z)).  This causes the solar irradiance at noon on a clear summer day to be greater than 900 watts/m2, but less than 300 watts/m2 on a clear winter's day.

This variation is irradiance is one of the factors which affects soil temperature.  The graph below shows the soil temperature at a depth of 1 metre in a back yard on the south coast of England.  There is a good relationship between the height of the Sun in the sky and the soil temperature.

Latitude is not the only factor which determine climate, hence the spread of values.  Location is also important, coastal areas tend to have milder, wetter climates than the dry continental interiors and advection can influence surface temperatures.