Introduction to Radiative Transfer Theory

A basic property of electromagnetic radiation is its ability to transfer energy.  Satellite radiometers measure the magnitue of radiant energy.  As we have illustrated in a previous lecture where we assessed the balance of incoming solar and outgoing terrestrial radiation, we often refer to radiation in units of energy.  We refer to the solar constant as a measure of the energy flux density, which is just the amount of solar energy per unit time (expressed in Joules per second) incident at the top of our atmosphere divided by the surface area of the intercepting disk (m²).  We typically express this in units of watt per m² and refer to it most generally as radiancy.  Depending on which direction this energy is flowing (into or out of a defined area) we give it a different name:

• Irradiance is the radiant flux density incident on an area (like the solar energy striking the surface of the earth).
• Radiant  exitance is the radiant flux density emerging from an area.  
• Radiance is the radiant exitance per unit solid angle, defined by the symbol L.  

In previous lectures I have emphasized the fact that both irradiance (e.g., solar radiation) and radiance (e.g., terrestrial radiation) vary with temperature and wavelength.   Our goal in this lecture is to describe the precise mathematical form of this relationship.

1. Planck's Function and the Birth of Quantum Theory
adapted from Fundamentals of Physics, David Halliday and Robert Resnick,  John Wiley and Sons, New York, 1974.

If we analyze a light source using a spectrometer, (as many physicists were doing in the late 19th century) we can learn how strongly it radiates at various wavelengths.  Spectral radiancy is just the rate at which energy is radiated per unit surface area for a small wavelength interval.  We measure spectral radiancy in energy flux density units of watts/cm² per unit wavelength.

The total energy radiated, without regard to wavelength, is the integral over all wavelengths, or the area under the curve of spectral radiancy.

Since everything radiates according to its temperature, for any material, we can define a family of spectral radiancy curves, one for each temperature.  

From a theoretical perspective, it proved useful  for scientists to consider an idealized heated solid which emits radiation (much like an ideal gas was conceived  to formulate gas laws without analyzing the properties of an infinite number of real gases).  This ideal solid is called a "cavity radiator".  In 1900, Max Planck (1858-1947) laid the basis for the development of modern quantum physics using a theoretical study of cavity radiation.  And he formulated the principle function of radiative transfer, which now bears his name, the Planck function.  

A theoretical explanation describing cavity radiation was one of the outstanding unsolved problems in physics at the end of the 19th century.  Here is an example of the cavity radiation "problem"  or puzzle which physicists were engaged with:

Consider three blocks of three differnt types of metal, and drill a small hole into one wall of each block (the blocks are made of materials like tungsten, and molybdenum).  Raise each block to the same uniform temperature (eg., 2000K).  Observe the light emitted by the blocks in a dark room.  Consider the radiance and the spectral radiance of the blocks, and this is what you observe:

1. There is always more intense radiation from the cavity interior than from the outside wall of the three different blocks.
   
2. At a given temperature, the radiance from the hole is identical for all three radiators, in spite of the fact that the radiance from the outer surface of the three different materials is very different.
   
3. In contrast to the radiance measured at the outer surface of each block, the cavity radiance varies as a relatively simple function of temperture:
   
                                                          
   
   as was shown by Josef Stefan and Ludwig Boltzman, where sigma is called the Stefan-Boltzman constant (5.67x10^-8 watt/meter² K^-4).
   
   The radiancy of each outer surface varies in a more complicated way which is also different for each of the materials.  It can be written as
    
                                              
   
   e  is defined as the emissivity a property which depends on both  the temperature and type of material.
    
4. The spectral radiancy of the cavity radiation varies with temperature only, with curves that look like the figure I have drawn on the board (you will make several of these curves in a lab activity).  It is independent of the shape, size, and material of the cavity.    

We can show that in fact the cavity radiation from two blocks must be equal.  Consider the simple diagram on the board showing two cavities in two blocks made of different materials (A and B), but heated to the same temperature, and brought together.  If the radiation in cavity A was greater than the radiation in cavity B, then radiant energy would flow from cavity A to cavity B, causing block B to heat up, and block A to cool down, which violates the second law of thermodynamics, where we have heat exchange without a temperature gradient resulting in a temperature gradient (this is an "unnatural process", one in which the entropy of the system decreases).

Based on the experimental spectral radiance data from real "cavity radiator" experiments, an empirical fit of the data was generated by Wilhelm Wien (1864-1928), a German Physicist who received the Nobel Prize in Physics for his contributions to radiative transfer.  Wein's formula was:

However, it was Max Planck who modified the formula slightly and got a precise fit to the experimentally determined spectral raidiancy curve data.  Planck's empirical formulation was:

 Essentially, Planck devised a mathematical formula that described the curves exactly; but then he went on to deduce a physical hypothesis that could explain the phenomena.  He conceived that atoms in the wall of  the cavity radiators act like small electromagnetic oscillators each with a characteristic frequency of oscillation.  The oscillators emit electromagnetic energy into the cavity and absorb electromagnetic energy from it.  The cavities radiate like blackbodies, absorbing and emitting all of the radiation entering the cavity, behaving like perfect radiators.  He assumed he could deduce the characteristics of cavity radiation based on this equilibrium with these atomic oscillators.  He made the following "radical" assumptions:

1. Oscillator energy is quantized,  
                                                
   where v is the oscillator frequency and h is the quantum of action, which is now called Plancks constant (6.626x10 ^-34 J sec).
   
2. These atomic oscillators do not radiate continuously, but do so in jumps or quanta.  These quanta of energy result when an atomic oscillator changes from one to another quantized energy states.  
   

Based on these assumptions, Planck was able to develop a theoretical expression for the empirical constants c₁ and c₂, which were

and

where c is just the speed of light, h is Planck's constant, and k is Boltzman's constant  which related changes in the energy for individual molecules in an ideal gas to changes in temperature (1.38x10 ^-23 J K^-1),this fundamental constant arose from Boltzmans work in kinetic theory which sought to derive macroscopic properties of matter (like temperature and pressue) by applying the laws of mechanics (ie., momentum) to molecules.

Planck presented his theory to the Berlin Physical Society on December 14, 1900, and quantum physics dates from that day.

2. Einstein's Contribution

The next important developments in quantum mechanics was the work of Albert Einstein. Using Planck's concept of the quantum, Einstein was able to explain certain properties of  an experimental phenomenon in which electrons are emitted from metal surfaces when radiation falls on these surfaces, this is known as the photoelectric effect.

Based on classical theory, the energy of electrons emitted from a metal surface, as measured by a voltage, should be proportional to the intensity of the radiation. Experimentally, however, it was shown that the energy of the electrons was independent of the intensity of radiation.  Radiation intensity was only able to determine the number of electrons emitted.  The energy of electrons was found to depend solely on the radiation frequency. Higher frequencies of incident radiation result in more electron energy.   There is a certain critical frequency below which no electrons will be emitted. Einstein was able to explain these facts based on the  assumption that a single quantum of radiant energy ejects a single electron from the metal. Because the energy of the quantum is proportional to the frequency, therefore the energy of the electron ejected also depends on the frequency.

Einstein assumed that radiant energy travels through space in these concentrated bundles, quantum units we now think of as photons.  This was the first real introduction of the dual nature of electromagnetic radiation, which has the properies of  both a wave and a particle.  

As we will see in further lectures, both of these concepts (Planck's function of spectral radiance, and the quantized energy of photons) are important to our understanding of where radiant energy comes from (the quantized changes of energy states of specific trace gases in our atmosphere for example), and how the radiant energy is measured and quantified onboard the satellite platform.

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loop of combined GOES8 and GOES9 Water Vapor Channel

GOES8 WV from  5:15 this morning

GOES8 WV image from 15:15 (ten hours later)

GOES8 IR image from 5:15

GOES8 WV image from 15:15 (ten hours later)

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