And then I loved thee,
And showed thee all the qualities of the isle,
The fresh springs, brine pits, barren place and fertile.

-- Caliban, The Tempest, W. Shakespeare.

06/30/08

Format for printing

We wish to learn:

• What does "temperature" really mean?
• How is temperature related to energy and radiation?
• What was the energy balance and temperature of the early Earth?
• What determines the temperature of planets?
• What gases in the atmosphere effect temperature and how?
• What is the Greenhouse Effect?

Jump to: [The Meaning of Temperature and Energy] [The Temperature of the Early Earth] [Greenhouse Effect and Natural Greenhouse Gases] [Summary]

The Meaning of Temperature and Energy

Temperature

All matter (gases, liquids and solids) is made up of atoms in various chemical relationships. Atoms consist of a nucleus orbited by electrons, and are extremely tiny.  There are at least 2.6 E25 atoms (that is, 26,000,000,000,000,000,000,000,000) in a cubic meter of matter; this is much more than the estimated number of stars in our entire universe.  Because the numbers are so large it usually makes sense to talk about average properties.

Atoms move randomly, jiggling furiously (at speeds of tens of kilometers per second) and continually colliding with each other. When a cold object (such as a hand) comes into contact with a hot object (say a hot kettle) energy is transferred by these atomic collisions from the hot to the cold body. The energy flows from hot to cold because the atoms in the cold body are pushed to greater speeds (see Figure).

In a hot body, the atoms move rapidly (individual atoms have different speeds, but the mean speed is high). In a cold body, the atoms have relatively low speeds. If collisions occur between hot and cold atoms, the average speed of the atoms (and therefore the temperature) would be intermediate between the high and low cases.

Temperature is a measure of the average speed of the moving atoms. The faster they jiggle, the higher the temperature. The mathematical definition tells us that temperature increases as the (mean atomic speed)².

Mathematically we say: , where v is the average atomic speed.

Temperature is measured in units called degrees. For science, the most useful scale is the Kelvin scale. The Kelvin scale starts from a value of absolute zero--a temperature corresponds to perfectly stationary atoms (a situation that cannot be attained in practice). Each degree Kelvin is equivalent to one degree Celsius, with 273K equal to 0^oC). Earlier we determined that Earth's mean surface temperature is about 300K.

Energy

There are several types of energy. Some are :

• Kinetic energy: energy associated with motion
• Potential energy: energy associated with position
• Internal energy: energy associated with atomic speed
• Radiant energy: energy associated with radiation
• Chemical energy: energy associated with chemical bonds

Energy has to be conserved; it can change forms but the total amount must remain the same. We cannot create or destroy energy. This is the famous Conservation of Energy Principle.

The Temperature of Early Earth

Let's calculate the temperature of the early Earth. The initial hydrogen and helium have probably been lost through gravitational escape processes and the atmosphere as we know it today has not yet evolved. We are left with a planet that is cold and airless. We can calculate the temperatures expected for airless planets warmed by the Sun's rays in two ways. First, let's use a radiation law known as the "R-squared" Law.

R-squared Law.

"R-Squared" Law

This law states that the farther you are from an emitting object, the less light you receive. In fact, a doubling of the distance away reduces radiation by a factor of four. This is best explained with a picture (see Figure).

We expect planets farther from the Sun to be colder. The R-squared model allows us to calculate how much colder each successive planet would be, based on an estimate of the amount of light received. We can use this law to calculate the temperatures of each planet using the current-day temperature of Mercury as a reference.

Simple Calculation of Early Earth Temperature

If we assume that Mercury's temperature has not changed, we can calculate the reduction in energy received by each planet using the R-Squared Law, knowing only its mean distance from the Sun.

The Figure shows the results of this calculation compared with the actual current day temperatures of the planets. We can see that the simple model does not work very well. Venus is much, much hotter than we would expect. Earth, Mars, Jupiter and Saturn are also hotter than expected - what is going on??

Comparison of the "R-squared" Model temperatures and actual current temperatures. for the planets Mercury is used as the reference. The other planets are all hotter than expected, due to the Greenhouse Effects of their atmospheres. Note the extreme example of Venus!

More Complicated (but more accurate) Calculation of Early Earth Temperature.

Let's see if we can get a more accurate model of the temperature of Early Earth. This model is slightly more complicated. The principle of Conservation of Energy tells us that the energy from the Sun absorbed by a planet must equal the energy lost by the planet. Since the planet floats in space, the only way to add or subtract energy is through radiation. Therefore we can say for the Earth:
Rate of absorption of energy from the Sun
= Rate of emission of radiation to space.

The amount of energy radiated to space from the Early Earth depends on its temperature. Temperature can be calculated using the Stefan-Boltzmann Law. Let's go through the simple arithmetic to calculate the temperature of the Early Earth.

Solar and Terrestrial Radiation.

First, recall the types of radiation that the Sun and Earth emit (Figure).

The Sun emits a lot of light in the visible range of wavelengths. We assume that the portion of this light that hits the Earth is absorbed.

The Early Earth emits in the infra-red. We assume that all of this radiation is lost to space.

The Law of Conservation of Energy tells us that the amount lost (in the infra red) has to equal the amount received (in the visible). Another way of viewing this is to say that the temperature of the Early Earth has to rise until exactly as much energy is lost through radiation as is gained from solar absorption. This equilibrium sets the temperature of any atmosphere-less planet.

To calculate the temperature, we need to know how much light is received by the Earth and how much is lost through emission. Let's do this in easy steps.

Step 1.

According to the Stefan-Boltzmann Law, the amount of light emitted by the Sun equals =,
where T_s is the temperature of the solar surface (assume 6000K).

Step 2.

The amount of this light that is absorbed by the Earth can be determined if we recognize that the sunlight is "diluted" by the time it reaches the Earth's orbit by a factor of

The amount of sunlight falling on the Earth per square centimeter is given by multiplying the intensity at the Earth's orbit by the area cut out of the Sun's radiation beam by the Earth.

Therefore, the light received by the Earth is given by:

(Energy input = amount emitted by the Sun multiplied by the R-squared dilution factor and by the area cut out of the Sun's radiation beam by the earth) (see Figure 5)

Step 3.

We can also write down the amount of radiative energy lost from the Earth, again using the Stefan-Boltzmann law:

,

(Energy output = amount emitted per unit area of the Earth multiplied by the surface area of the Earth)

Step 4.

Using the Law of Conservation of Energy, equate energy input to energy output and solve the equation for T_e:

(Energy input = Energy output)

After cancellation:

Step 5.

Calculate T_e.

For the Early Earth T_e = 283K.

Step 6.

This calculation assumes that all the sunlight falling on the Early Earth was absorbed. If we assume that some of it (say 83%, like modern day Mars) is reflected, the temperature is 260^oK. This is about 40^o colder than the temperature today - for reasons to be explained below.

We have just calculated the Radiative Equilibrium Temperature of the Earth. It is the temperature that the Earth would have with no atmosphere, when infra-red emission exactly balances the radiation received by the Sun.

But, we know that our actual temperature today is ~300K. What is wrong with our calculation? The atmosphere is responsible for increasing the actual temperature above the radiative equilibrium temperature. This increase is the so-called Greenhouse Effect.

3. The Greenhouse Effect and Natural Greenhouse Gases

Results from a more sophisticated calculation than the one described above demonstrate the magnitude of the natural Greenhouse Effect for the planets (see Figure 6).

The natural greenhouse effect is responsible for life as we know it, and should be distinguished from the infamous anthropogenic greenhouse effect that is currently causing so much concern. The natural Greenhouse Effect is beneficial and warms our planet to more livable temperatures.

The Natural Greenhouse Effect. Calculated radiative equilibrium temperatures taking into account variations in planetary albedo are compared with actual surface temperatures. Most planets show a Greenhouse effect due to atmospheric gases.

Why are the planets warmer than expected on the basis of theory? The answer is that certain gases in the atmospheres of these planets act to warm them up. The explanation is best understood by reference again to the type of radiation emitted by the Sun and by the planets (see Figure 7, below). Note that we need an extra 70% leaving the top to achieve equilibrium. Where does it come from?

Simple picture of Radiative Transfer (in percentages).

Over a long term average, the Earth and its atmosphere must radiate as much energy out to space as it receives from the sun. In fact, the same type of balance exists between the Earth and its atmosphere!

Albedo (L) = percentage of incoming radiation that is reflected back into space = 30% for Earth
(higher for Venus)

Let's look at the long wave (IR) component of the planetary radiation budget:

Infrared component of the planetary radiation budget

 At the Earth's surface, we note that the gain and loss in energy is greater than that received from the Sun - how can this be?

The answer makes sense when we consider that the surface of a planet receives a great deal of energy from its own atmosphere. Thus the effect of the atmosphere is to warm the surface over the temperature above that resulting from the Sun's energy.  We then have to ask how does the atmosphere increase the Earth's temperature?

The atmosphere warms the Earth by "trapping" radiation, allowing the surface to warm to 300K. At that temperature, the black body surface radiation is large enough to ensure that an equilibrium condition pertains. The atmosphere traps radiation through the action of certain gases, called Greenhouse Gases. These gases (e.g., CO₂, H₂O, NO, CFCs, CO) are very good at absorbing and re-emitting infrared radiation. They intercept the IR radiation from the ground and reflect some of the energy back to the ground, warming it up more than would occur otherwise.  So, the Greenhouse Effect provides additional heat!

Summary

Temperature is a measure of the square of the mean speed of atoms and molecules.  Energy comes in different types, but must be conserved as a whole.

The temperature of the early Earth (and other atmosphere-less planets and satellites) was determined by the balance between the absorption of solar visible light and the emission of infra-red light. The temperature that allows these energy sources and sinks to balance is called the radiative equilibrium temperature.

The Greenhouse Effect allows certain planets to warm up above their respective radiative equilibrium temperatures. The effect is due to the presence in the atmosphere of Greenhouse gases, such as CO₂, H₂O, NO, etc.

Suggested Readings:

• Ahrens, C. Donald, "Meteorology Today", 4th ed., West Publishing Co., 1991.
• Walker, J. C. G., "Evolution of the Atmosphere", Macmillan Publishing Co., Inc., 1977.
• Lewis, J. S., and R. G. Prinn, "Planets and their Atmospheres: Origin and Evolution", Academic Press, 1984.

All materials © the Regents of the University of Michigan unless noted otherwise.