The Temperature of the Earth

The following calculation assumes that the energy into the Earth system and the energy out are not affected by the atmospheric gasses. When the system is in equilibrium, that is neither warming nor cooling, Energy In must equal Energy Out.

Energy In

Energy enters the Earth system from the sun mainly in the form of visible light. This solar flux is represented by the letter L. The value is 1361 Watts per square metre (W/m²). About 30% of the light from the sun is reflected out of the Earth system, mainly by clouds and ice. The reflectivity of the planet is called the albedo and is represented by the letter α.

Each square metre then absorbs energy from the sun according to the following formula:
L(1 – α)

We are interested in the total energy absorbed. The solar flux varies over the surface of the Earth, strongest at the equator where the surface is perpendicular to the incoming light and weakest at the poles where the surface is at a large angle to the sun’s rays. Adding up all of these variable amounts is tricky but there is a simple method to calculate it. The light waves from the sun are parallel to each other and the Earth intercepts them and casts a shadow. This shadow is a circle which has a simple formula - πr². So the formula for the total energy from the Sun absorbed by the earth is:
L(1 – α) πr²

Energy out

The formula for the energy that is radiated away from the surface is:
ϵσT⁴ 4 πr²
ϵ is emissivity, how close the source is to a black body (0 not a back body to 1 a black body), σ the Stefan-Boltzman constant, 5.67 X 10^-8 (Watts/square metre), T⁴ is temperature in Kelvin raised to the fourth power, 4πr² is area of a sphere.
If the system is in equilibrium Energy In equals Energy Out
So
L(1 – α) πr² = ϵσT⁴ 4 πr²

Calculate Temperature, ie solve for T

πr² appears on both sides so it can be cancelled

So
L(1 – α) = 4 ϵσT⁴



With L = 1361 W/m² α = 0.3 σ = 5.67 X 10^-8 and ϵ = 1 (given that Earth is relatively close to being a black body)
T = 254.5 K,
(K is degrees Kelvin. The Kelvin Scale starts at absolute zero ie -273⁰ Celsius. Note Kelvin temperatures are indicated by K, and do not by convention use a degree (⁰) sign.)

Consequently 254.5 K equals -18.5⁰ C

Given that the global temperature is about 15⁰ C, according to this calculation the planet should be about 33⁰ C below current temperatures.

This calculation gives a very incorrect answer because it ignores the effect of Greenhouse Gasses.

The Discovery of Infrared by William Hershel

The Discovery of Infrared by William Hershel

Before I started researching this topic I was only aware of Herschel as the discoverer of the planet Uranus in 1781.

It turns out that he made at least one other important discovery, that if infrared light, in 1800.

Herschel's discovery is described at this website in the following way:

He was interested in learning how much heat passed through the different colored filters he used to observe the Sun and noticed that filters of different colors seemed to pass different levels of heat. Herschel thought that the colors themselves might contain different levels of heat, so he devised a clever experiment to investigate his hypothesis.

Herschel directed sunlight through a glass prism to create a spectrum - the "rainbow" created when light is divided into its colors - and measured the temperature of each color. He used three thermometers with blackened bulbs (to better absorb the heat) and, for each color of the spectrum, placed one bulb in a visible color while the other two were placed beyond the spectrum as control samples. As he measured the temperatures of the violet, blue, green, yellow, orange and red light, he noticed that all of the colors had temperatures higher than the controls and that the temperature of the colors increased from the violet to the red part of the spectrum. After noticing this pattern, Herschel decided to measure the temperature just beyond the red portion of the spectrum in a region apparently devoid of sunlight. To his surprise, he found that this region had the highest temperature of all.

Herschel performed further experiments on what he called "calorific rays" (derived from the Latin word for 'heat') beyond the red portion of the spectrum. He found that they were reflected, refracted, absorbed and transmitted just like visible light. What Sir William had discovered was a form of light (or radiation) beyond red light. These "calorific rays" were later renamed infrared rays or infrared radiation (the prefix infra means `below'). Herschel's experiment was important not only because it led to the discovery of infrared light, but also because it was the first time that someone showed that there were forms of light that we cannot see with our eyes.

The video below was produced by FLIR Systems, makers of infrared thermal imaging systems. It begins by describing Herschel's experiment and infrared light in general, before describing the use of thermal imaging systems in the modern world.

Over the next century or so the rest of the electromagnetic spectrum was discovered.

The diagram below shows where light and infra red fit into the the electromagnetic spectrum. Infra red has a wavelength between 0.7 and 300 micrometres. A micrometre (micrometer for Americans. The symbol for a micrometre is µm.