Irradiance and the inverse square law

What is irradiance?

Irradiance of radiation is a measure of the radiation falling on a surface. It is defined as the energy falling on a surface per unit time (i.e. the power per unit area). This relationship can be summarised:

Why does irradiance matter?

An understanding of irradiance is relevant to a range of applications. For example, NASA monitors solar irradiance to understand the activity of the Sun and climate scientists study solar irradiance to research the impact of solar activity on the Earth’s climate. Interactions between solar radiation and the atmosphere of the Earth can impact on air quality, and understanding of irradiance can allow investigation of the composition of the Earth’s atmosphere. Excessive exposure to sunlight has been linked to the development of a range of skin cancers. The performance of solar cells, an increasingly common use of solar radiation as an energy resource, requires an understanding of irradiance.

Investigating irradiance

The relationship between irradiance of a point source and the distance from that source can be investigated using a simple experimental set up: a light source and a linear light meter.

The graph of a typical set of results from such an experiment is shown below:

It is clear from this graph that the relationship between irradiance and distance is not a linear one.

Graphing average irradiance against 1/d (d = distance) demonstrates that average irradiance is not proportional to 1/d.

The graph of average irradiance against 1/d2 demonstrates a linear relationship.

From the graph:

This is described as an inverse square law.

A point source is one which irradiates equally in all directions, i.e. the volume that will be irradiated will be a sphere. The surface area of a sphere can be calculated using A = 4πr2, i.e. the area which will be irradiated is proportional to r2 (or d2).

Calculating solar irradiance and the power of the Sun: measuring a sunbeam

Watch the clip from ‘Measuring a sunbeam’ (BBC Wonders of the Solar System episode ‘Empire of the Sun’ at http://www.bbc.co.uk/programmes/p006szxm.

Professor Brian Cox uses a simple technique to measure the solar energy falling on the Earth, recreating an experiment first carried out by Sir John Herschel in 1838. Pause the clip and use your understanding of irradiance to make the calculations on the Sun’s irradiance.

Irradiance and the distance to the moon

When Neil Armstrong and Buzz Aldrin walked on the surface of the moon on 21 July 1969 as part of the Apollo mission, they set up a lunar laser ranging reflector array, a 0.6 m panel with 100 mirrors. This is the only experiment from the successful Apollo 11 mission that is still running.

A laser pulse is transmitted from a telescope on Earth. The reflectors in the array are designed to send the pulse back precisely to the location from which it was transmitted. Telescopes on Earth receive the returned pulse.

For more than three decades it was possible to use this technique to make observations of the moon’s orbit, the universal gravitational constant, the composition of the moon, and whether Einstein’s theory of relativity works in practice.

Measuring the time for the pulse to be returned allows the moon’s distance to be calculated very precisely. Until 2005 the measurements allowed calculation to within a few centimetres, a very small uncertainty for a distance of approximately 385 000 000 m! The Apache Point Observatory Lunar Laser-ranging Operation (APOLLO for short) in New Mexico, which came online in 2005, allows these measurements to be made to within a few millimetres.

Among its aims, the APOLLO project will measure the 1/r2 law (i.e. the inverse square law) at the lunar distance scale, i.e. approximately 1012 m.

What makes a laser suitable for these types of experiment over astronomical distances? A laser has a very small beam divergence over a distance, compared with a light source such as an ordinary filament lamp. Nevertheless, the laser beam is approximately 7 km in diameter by the time it reaches the moon.

Irradiance and the electromagnetic spectrum

The inverse square law applies to all electromagnetic radiation, i.e. to visible light and all wavelengths on the electromagnetic spectrum, from radio waves to gamma rays.

© Learning and Teaching Scotland 2011