Radiation and Planetary Energy Balance

Take away ideas and understandings:

  1. Understand the conservation of energy law. It is important to this week's work and the course as a whole.
  2. Understand black body radiation and the concept of Effective Temperature
  3. The relationship between the absolute temperature of a black body and the energy flux it radiates.
  4. The relationship between the wave length of the maximum energy output of a black body and its absolute temperature.
  5. You should know why the intensity of a light dims as you move away from it.
  6. You should be able to use these relatively simple relationsips to calculate the effective temperature of the Earth or any other planet if you know the temperature of the sun and the distance of the planet from it.
  7. You should know why the Earth is warmer at the equator than it is at the poles.
  8. You should know, in a qualitative way, why the energy of a beam of electromagnetic energy is proportional to its frequency and why this is important.

1.0 What is Energy?

Energy is an abstract quantity that matter or a waves possess. We can measure its effects but we can not measure it directly. It is always conserved. In other words, it is never destroyed but it can be transformed from one kind of energy to another.

For example, if chemical energy is released through the rapid oxidation of carbon in the flame of a natural gas burner and some of that energy is transferred to a pot of water placed over the flame, the temperature of the water will rise. This rise of temperature is due to the increased energy of the water molecules.

Energy can be transferred from the water to a thermometer if a thermometer is placed in the water. The transfer of energy from the water molecules to the mercury molecules causes the mercury molecules to move faster and in turn causes the mercury to expand (rise) within the thermometer.

Notes on energy, work, and power are here.

1.1 Kinds of energy.

  • Kinetic Energy
  • Chemical energy
  • Gravitational energy
  • Electrical energy
  • Mass energy
  • Thermal energy
  • Elastic energy
  • Nuclear energy
  • Radiant energy

1.2 Electromagnetic Energy Generation.

Solar energy arrives at the Earth after travelling through empty space as electromagnetic radiation. We can sense this radiant energy with both our eyes (light) and our skin (heat). The only difference between the light we sense with our eyes and the heat we sense with our skin is the wavelength of the radiation, heat having a longer wavelength than light.

Electromagnetic waves are generated by moving electrons. An electron generates an electric field which we can visualize as lines radiating from the electron (below, left). If the electron moves, say it vibrates back and forth, then this motion will be transfered to the field lines and they will become wavy (below, right). In turn, the moving electron generates a magnetic field that will also become wavy from the motion of the electron.

These combined electrical and magnetic waves reinforce one another. This kind if wave is called an electromagnetic wave and light is such a wave. Since all matter contains electrons and all these electrons are in motion, as are the atomic nucleii they spin around, all matter generates electromagnetic waves.

Since all electromagnetic waves travel at the same speed (c= speed of light) the frequency of the waves is determined by the frequency of the vibrating electrons that generate them. Hot substances have more energy and the atoms vibrate more rapidly than cold bodies. Thus the peak energy radiated by hot bodies has a higher frequency, shorter wavelength, than that of cooler bodies. The relationship of the peak frequency of a black body to its absolute temperature is expressed by Wein's Law.

1.3 Wein's Law:

lmax = a/T

a = 2898 if l is measured in microns (mm, 10-6 meters), T measured in °K.

Electromagnetic (EM) radiation is emitted by all matter and consists of orthogonal electrical and magnetic waves. All EM travels at the same speed through a vacuum (c = 186,000 miles per second or 300,000 kilometers per sec.). This radiation is generated by a moving charge or charges. All matter consists of atoms in motion and these in turn consist of positively charged protons surrounded by a cloud of negatively charged electrons.

The vibrating motion of the atoms causes the cloud of electrons to oscillate and this oscillation generates electromagnetic radiation. Since all electromagnetic radiation travels at the same velocity the frequency and wavelength of the generated radiation depends on the frequency of the oscillating electron cloud. Thus, on average, cool objects, say those at room temperature, generate longer wavelength (low frequency) radiation, than hot objects, such as the sun, which generate shorter wavelength (higher frequency) radiation.

Homework problem 1: If you measure the the wavelength (in nm) of radiation emitted by the Sun to be 0.5 mm, what is its temperature?

1.4 Wave Properties.

EM waves travel at the speed of light in a vacuum, or roughly 300,000 km/s. The wave-nature of EM radiation is expressed in terms of wavelength (commonly in microns, or 10-6 , or the inverse of wavelength - frequency. The "power" of a wave can be quantified in terms of its amplitude.


1.5 EM Frequency Spectrum.

There is a spectrum of electromagnetic waves which ranges in wave length from kilometers (radio waves) to billionths of a meter (X-rays) (See below). Aside from their varying wavelengths, they are identical. However, the difference in wavelength makes a great difference in their behavior.


2.0 Relationship of radiation intensity to distance: The "r-squared law".

Electromagnetic energy emitted by the sun radiates outward in all directions from the sun's surface. So, the farther you are away from the sun the less energy you receive. This relationship is illustrated below, and its effect can be calculated using the "r-squared law". The r-squared law simply states that as you move away a distance r from a radiating object, the radiation emitted from it decreases as 1/r2. So, if you move away from a planet by, say, 2 units, the radiation decreases by 1/(22), or 1/4 of the orginal value. If you move away by 4 units, the radiation decreases to 1/16 of the orginal value.

Because energy is conserved, electromagnetic radiation is not diminished as it travels through empty space. Think of a strobe light suspended in empty space. It emits a flash that consists of a specific amount of energy. That energy travels outward away from the strobe light in all directions at the speed of light (c), like a rapidly expanding balloon (see Figure 15 below). At any moment the total energy in the expanding sphere is exactly the same as the energy initially emitted by the strobe. However, what does change is the amount of energy in a square meter of the surface of the sphere. As the sphere expands the initial energy is spread over a larger and larger area of space. So the intensity of radiation in a square meter is related to the total surface area of the sphere. The surface of the sphere is related to its distance from the source (the strobe) because the area of a sphere is related to its radius by the following equation.

Surface area of a sphere = 4pr2

As the sphere gets larger, the energy per square meter becomes less and, since 4 and p are constants, the intensity per square meter is inversly proportional to r2, 1/r2, or to the distance to the source, 1/D2.

3.0 Calculating Earth's Surface Temperature (assuming no atmosphere).

We can calculate what is called the Effective Temperature of the earth just knowing the temperature of the sun, the radii of the sun and earth, and the earth-sun distance. The effective temperature of the earth (or any other planet) is the temperature the planet should have if it acts like a black body - that is, if it absorbs incoming and reradiates outgoing energy at 100% efficiency. For a body to be in thermal equilibrium with a source of radiation it must radiate as much energy as it receives at a particular wavelength. This is a way to formalize the conservation of energy law for planets (energy in = energy out).

Physical scientists commonly measure vibrational energy of a solid by measuring its temperature. Measured temperature is proportional to the square of the vibrational energy of atoms. The lower the temperature, the lower the vibrational energy of the solid's molecules. The point of least molecular motion is defined as absolute zero, or zero degrees Kelvin. A temperature change corresponding to a degree Kelvin is the same as a temperature change of one Celsius degree. Zero degrees Celsius corresponds to 273 degrees Kelvin and therefore zero degrees Kelvin (the temperature at which all atomic vibration ceases) is equivalent to -273 degrees Celsius.

Step 1: Determine the energy emitted by the Sun

To calculate the effective temperature of the Earth, you first have to calculate how much energy the sun is emitting, and then how much of that energy is hitting the earth. If we also assume the Sun is a black body emitter, then we can calculate its total energy flux using the Stefan-Boltzmann Law.

The total energy (flux) released by a black body is proportional to the fourth power of its absolute temperature. This is because the internal energy of the body is related to its temperature i.e. the more energy the atoms and molecules of the substance have (vibrational energy for example), the higher the temperature of the body. This is known as the Stefan-Boltzmann law named after the men who formulated it. If we know the temperature of a black body we can calculate the energy it will radiate by the following equation.

I = (Stefan-Boltzmann law)

where Ts is the Sun's temperature (6000K), and s is the Stefan-Boltzmann constant (5.67x10-8 Watts/m2)


Step 2: Determine how much of that energy reaches the Earth

Here we take advantage of the "r-squared law" and compute how much of the Sun's emitted energy reaches the surface of the earth. The Sun's radiation per unit area diminishes with distance away from the Sun, and this effect can be calculated using the ratio of the Sun's radius (Rs) to the Earth-Sun distance (rs):

where Rs is the radius of the Sun, and rs is the Earth-Sun distance. Now, the sunlight hitting the Earth only hits the area of the Earth "disk", not the full sphere (light doesn't bend around the dark side!). So, the earth area this sunlight is illuminating is calculated using the area of a circle, or A = pr2, where re = earth radius. Therefore, the sunlight received by the Earth's surface is calculated by the following equation:

Step 3:Determine how much energy the Earth is re-radiating (losing) back out to space (remembering it's a black body).

Here, we also use the Stefan-Boltzmann law and the fact that the Earth is radiating from its entire surface (a sphere, not a disk: A = 4pr2). Te is the temperature of the Earth.

Step 4: Calculate the Effective Temperature of Earth: Incoming energy equals outgoing energy

The last step uses the conservation of energy principle: Energy in has to equal enrgy going out. So, we set the Sun's energy hitting the Earth to equal the Earth's re-radiation of energy back out to space:

Energy input = energy output

After simplification:

Solving for Te (the Earth's effective temperature) using these data yields 283K, or roughly 10°C

The actual average temperature of the Earth is closer to 300K (~27°C), , so why is this calculated Effective Temperature so much lower? Greenhouse gases such as CO2 and water vapor trap a great deal of heat in the atmosphere, and this is the main reason for the difference on Earth.

Homework Problem 2: Calculate the effective temps for Mars and Venus using the equations above and these data. How do these compare with the actual measured surface temperaures. Why are the black body (effective) temperatures so different from the measured temperatures? Using the internet, research and summarize (one paragraph) how surface temperatures are measured on other planets.

4.0 Black Body Radiation Spectrum for the Sun and Earth.

A body at a single temperature generates electromagnetic radiation over a range of wavelengths. The hotter the body the higher the frequency (shorter the wavelength) of the peak frequency, (frequency with the maximum energy output). Compare the frequency ranges and peak frequencies of solar and Earth radiation (below). Note that the Sun's maximum emission is approximately 3 million times greater than the Earth's.

5.0 Interaction of EM radiation with matter.

Matter can reflect, transmit or absorb electromagnetic (radiant) energy. If matter is transparent to radiant energy the energy will pass through it unchanged. If matter is a perfect reflector, the energy will not be changed except to change the direction it is moving. If electromagnetic radiation is absorbed by matter then there is a transfer of energy from the radiant energy to the medium that is doing the absorbing. This may result in an increase in the vibrational energy of the molecules of the absorbing medium if it is a solid, or increased molecular velocities if the medium is a gas, or chemical change if the radiant energy is of sufficiently high energy (high frequency) to break chemical bonds or change the energy level of electrons. In general all of these types of absorption results in a rise in temperature of the absorbing medium.

Radiant energy can interact with matter in three extreme modes. Most often its behavior is a combination of two or more of these modes, but for the sake of explanation we will look at them one at a time. If matter does not interact with the incident radiation i.e. there is no change in the matter because of the radiant energy that strikes it and it does not let the energy pass through it i.e. it is opaque to the radiant energy, then it reflects the energy. Reflection only changes the direction of the beam of radiant energy not its wavelength or amplitude. If matter allows radiant energy to pass through it unchanged, the matter is described as transparent to the incident radiation. Again, as with reflection, there is no change in any of the properties of the radiant energy. On the other hand, if there is some interaction between the incident radiation and the matter e.g. some energy is transferred from the radiant beam to the matter resulting in an increase in molecular energy of the matter, then we describe this transfer of energy from the radiant beam to the matter as absorption.

Black bodies by definition absorb and re-radiate radiant energy equally and completely at all wavelengths they intercept. Thus as a perfect reflector reflects all radiation it intercepts, and a perfectly transparent substance transmits unchanged all radiant energy striking it, so a black body absorbs all radiation that it intercepts.

Gases on the other hand are not black bodies; they absorb and re-radiate only at very specific wavelengths. Our atmospheric gases absorb different narrow bands of incoming solar radiation. Each absorption band is a response to a different mechanism of energy transfer. The smaller molecules of oxygen and nitrogen absorb very short wavelengths of solar radiation while the larger molecules of water vapor and carbon dioxide absorb primarily longer infrared radiant energy.

Updated February 11, 2002

©2002 P. deMenocal (LDEO, Columbia Univ.)