It is possible to approximate the average global temperature of a planet using some simple physics. To do so, one must understand the concept of a "blackbody radiator".

A blackbody radiator is a hypothetical physical object that absorbs **all** electromagnetic radiation which strikes it. Since light is a form of electromagnetic (EM) radiation, all light striking a blackbody radiator would be absorbed by the object, so the object would appear perfectly black. Hence the "blackbody" part of the name.

The law of conservation of energy tells us that the amount of energy emitted by any isolated system or object must precisely balance the amount of energy absorbed by the object. In simpler language, this can be summarized as "energy in" = "energy out". A blackbody radiator must, therefore, give off exactly as much energy as it absorbs.

Any object with a temperature above absolute zero emits electromagnetic radiation. Most objects emit infrared (IR) "light". Objects that are very hot can have peak emissions in the visible light portion of the EM spectrum, which is why the burners on an electric stove or hotplate glow red when they are turned on high. Many stars, which are far hotter still, glow with higher-energy yellow or blue light, or sometimes even ultraviolet "light".

If light shines on a blackbody radiator, the object absorbs all of the energy from that light. The energy is converted to heat, warming the blackbody. Warm objects emit energy as EM radiation, usually as IR "light". Warmer objects emit more EM radiation than cooler ones. The blackbody warms in response to the incoming light until it is hot enough to radiate all of that energy away as infrared. At a certain temperature the outgoing emissions balance the incoming light; the blackbody radiator reaches a state of thermal equilibrium.

Although a perfect blackbody radiator is just a hypothetical "thought experiment", many actual objects behave in a way that is pretty close to a perfect blackbody radiator. Planets are enough like theoretical blackbody radiators that we can use this concept to approximate the temperature of a planet, including Earth.

## Some Math

Minimally, your students need to understand that the **warmer** an object gets, the more **energy it gives off** as electromagnetic radiation. They also must realize that objects that they think of as cold - a snowball for instance - are actually warm in the sense that they have temperatures well above absolute zero (-273° C or -460° F). Thus, even a snowball emits infrared radiation due to its heat!

More precisely, the amount of energy emitted by a blackbody is proportional to the temperature of that object raised to the 4th power. This relationship, called the Stefan-Boltzmann law, was determined experimentally by Joseph Stefan in 1879 and theoretically derived by Ludwig Boltzmann in 1844. Mathematically, the Stefan-Boltzmann law is expressed as:

j^{*} = energy flux = energy per unit time per unit area (joules per second per square meter or watts per square meter)

σ = Stefan-Boltzmann constant = 5.670373 x 10^{-8} watts / m^{2} K^{4} (m = meters, K = kelvins)

T = temperature (in the Kelvin scale)

## Sample Calculations

Calculations using the Stefan-Boltzmann law must be done using the Kelvin (**not** Celsius or Fahrenheit) temperature scales. The Kelvin scale is an absolute measure of temperature, with the zero point set at absolute zero. The zero point in the Fahrenheit scale is pretty much arbitrary, based on quirks of history; the zero value on the Celsius scale represents the freezing point of water, but is also not an absolute scale. A temperature of 100 kelvins is exactly twice as warm as 50 kelvins, but a temperature of 100° F is not twice as hot as 50° F and 100° C is not twice as hot as 50° C. When making calculations using the Stefan-Boltzmann law, if students are supplied temperature values expressed using the Fahrenheit or Celsius scales, they **must** remember to convert those values to kelvins before plugging them into the Stefan-Boltzmann law.

### Example

Suppose you have two iron spheres that behave pretty much like ideal blackbody radiators. Sphere A is as cold as ice, with a temperature of 0° C. Sphere B is as hot as boiling water, with a temperature of 100° C. Which sphere is giving off more electromagnetic radiation? How much energy is each sphere emitting?

**Answer:** Sphere B is emitting more energy as electromagnetic radiation, because it is warmer than sphere A.

Convert temperatures to the Kelvin scale:

**Sphere A:** T_{Kelvin} = T_{Celsius} + 273 = 0 + 273 = 273 kelvins**Sphere B:** T_{Kelvin} = T_{Celsius} + 273 = 100 + 273 = 373 kelvins

Plug these temperatures into the Stefan-Boltzmann law:

**Energy emitted by sphere A:** j* = σ T^{4} = 5.67 x 10^{-8} x 273^{4} = 315 W / m^{2} K^{4}**Energy emitted by sphere B:** j* = σ T^{4} = 5.67 x 10^{-8} x 373^{4} = 1,098 W / m^{2} K^{4}

Note that sphere B emits about 3.5 times as much energy as sphere A (1,098 / 315).

## Student Activities

Depending on the age and background knowledge of your students, and whether you want them to delve into the math of blackbody radiators or not, you can use some of the questions in the assessment section below for in-class demonstration, homework, or quizzes.

If you don't want your students to perform calculations, you can use our blackbody radiation simulation to explore the relationship between temperature and incoming and outgoing radiation:

- go to the
**Planetary Energy Balance web page** - click the "Blackbody" tab
- use the slider to adjust the amount of energy flowing into the blackbody from the flashlight as visible light
- recall that "energy in" = "energy out"
- note the relationship between temperature and emitted energy (higher temperature = more energy emitted)

If you want, ask your students to record and graph several values for energy emissions (e.g. 100, 200, 300 and 400 watts) versus temperature. Advanced students should notice that the graph is not a straight line (since energy emissions vary as temperature to the 4th power).

## Assessment

Here are a few multiple choice questions about blackbody radiators and the law of energy conservation, as well as a couple of calculations covering blackbody radiators and the Stefan-Boltzmann law. An answer key is also provided.