Flux, Radiance, and Irradiance
In other sections, we have described light from the standpoint of
individual photons. Each photon carries a definite amount of energy
at a definite speed. Photons may be considered as indivisible
particles. Now we wish to consider the amount of electromagnetic
radiation that is moving, or, similarly, the amount of energy being
transported.
In this chapter, we will look at:
4.5.1 Energy
4.5.2 Radiant Flux
4.5.3 Irradiance
4.5.4 Radiance
4.5.5 Spectral radiant flux, radiance, and irradiance
Recall that the SI unit of energy is the Joule (J), equal to 1 kg m2 s-2. We usually think of mechanical energy, which is force times distance. (E = F ( d)). And Newton's First Law says that force is mass times acceleration ( F = m ( a)). And acceleration is the time rate of change of velocity (a= dv/dt). And velocity is the time rate of change of position (v = dx/dt). Now, working backward through this, if we measure position and distance in meters (m) and time in seconds (s), and mass in kilograms (kg), in accordance with the SI system of units, then velocity has units of meters per second, or [m s-1]. Acceleration has units of [m s-2]. Force has units of [kg m s-2], and so energy has units [kg m2 s-2]. Hence, a Joule is the amount of energy it would require to accelerate a 1 kg mass by 1 m s-2 over a distance of 1 m.
Once we have this definition of the Joule, it may be used as a
unit for any kind of energy- not just mechanical energy that is
actually accelerating actual masses. Photons don't do that as they
propagate through space, but they do have energy.
If energy is being moved from one place to another, it is sensible to
describe the rate at which energy is being transported.
Another way to think about this is this: Say you have an object (like
the Sun or the Earth), and you draw an imaginary boundary box around
that object. Then it is of interest to know the time rate of change
of the energy inside the box. Because energy is conserved (that is,
it is neither created nor destroyed), the time rate of change of
energy inside the box is equivalent to the rate at which energy is
flowing through the boundary. The unit of energy flow (or
flux) is the Watt (W). One Watt is equal to one Joule per
second.
Of course, it is always important to look at the Conservation of Energy
Radiant flux is the rate at which energy is being transported across some boundary by electromagnetic radiation. The Sun loses energy through radiation, but it also loses energy by other processes, such as by blowing off hot particles in the solar wind. The Sun's radiant flux is just the part of the energy loss that is due to radiation. The appropriate SI unit for radiant flux is the Watt (W).
We often want to characterize the amount of energy flux that is passing through or striking a particular part of a boundary surface. For example, we might want to know the amount of energy being transported through the top of the atmosphere in a certain location, or the amount of energy striking the surface of the earth, in a given unit of time. The quantity of interest is called the irradiance, and is defined as the radiant flux flowing through some surface, divided by the area of that surface. Note that this can be any surface at all; it doesn't have to be an actual, physical surface, but can be an imaginary surface anywhere in space.
To illustrate the difference between radiant flux and irradiance, suppose we wished to build a solar collector for electrical power generation. If you make the collecting surface a square, one meter on a side, you will collect a certain amount of energy per second (at some time of the day). If you double the area of the collector (say by making it a 2 m ( 1 m rectangle)), you will collect twice the amount of energy per second than your first collector did. The radiant flux striking the surface has doubled. What is not different between these two collectors is how much energy falls on each square meter per second. That is, the irradiance is the same in both cases. Radiant flux is said to be an extensive quantity (that is, it depends on the size of the thing you are looking at), while irradiance is said to be intensive (that is, it is size-independent).
The SI units for irradiance must be the units of radiant flux divided by the units of area, or W m-2 = J s-1 m-2 .
When we are considering the amount of radiation that falls on the surface of the Earth, or the surface of your skin, you need to know the relationship between the intensity of the light through a surface that is at right angles to the direction of propagation of the light, and the irradiance at the physical, horizontal surface of the Earth. In Figure ____ we show the situation that we have when the sun is at some angle to the zenith (the imaginary line that goes straight up and down through some point on the Earth's surface). We know the solar irradiance at the top of the atmosphere from Earth-based and satellite-based measurements. That is, if we imagine a surface of area 1 m2 that is oriented at a right angle to the line between the Earth and the Sun, we know that ______ Joules of electromagnetic energy passes through this surface every second. Now, if we slide that imaginary surface down to the surface of the Earth without changing its orientation, it will make an angle with the Earth's surface that is just equal to the solar zenith angle, . Now imagine blocking out or just ignoring all the light that doesn't go through our 1 m2 imaginary surface. The light that goes through the imaginary surface falls on the real surface of the Earth, but because of the angle between the two surfaces, the light that falls on the Earth's surface is spread out over an area that is larger than the 1 m2 of the imaginary surface. That means that the irradiance hitting the surface of the Earth is smaller by a factor that is just the ratio of the imaginary surface to the illuminated real surface. That factor turns out to be cos(). (For more, see "Why do we use a cosine?")
Recall that when =0 (that is, when the sun is directly overhead) then cos()=1. That is, our imaginary surface lies flat on the real surface of the Earth, and the irradiances at the two surfaces is the same. When the solar zenith angle is =45o, then cos()=1/_2ë0.707. So the irradiance at the Earth's surface is 0.707 times the irradiance through the imaginary surface. Of course, as gets closer and closer to 90o (that is, near sunrise and sunset), cos() gets closer and closer to zero. Of course, it is the irradiance at the Earth's surface that determines how much energy is available for the Earth to absorb. A point on the Earth's surface that, at noon, passes directly below the Sun, will have an irradiance that is much greater than a point that is in the polar regions, where the solar zenith angle is large throughout the day. That is why the polar regions are much colder than the tropics. It is also why the Northern Hemisphere in January is considerably colder than the Southern Hemisphere, and vice-versa in July.
Radiance is the amount of energy that passes through a surface (real or imagined) in a unit of time coming from a particular direction. On a clear day, or a day with scattered clouds, you receive a certain amount of radiant energy from the direction of the Sun. But, thanks to Rayleigh scattering and Mie scattering in the atmosphere, you also receive a certain amount of radiant energy from any other direction you choose.
Let us suppose that you take a narrow tube, like a drinking straw, and you point the tube at some position in the sky, and look through it. A certain amount of light enters the tube and exits where your eyeball is. Actually, what you see through the tube is a round patch of sky. The "direction" you point it corresponds to the center of that patch. If we now do the same thing with a smaller-diameter tube, we can certainly point the smaller tube in the direction we were looking before, but when we look through it, we will see a smaller patch of sky. And less light enters the tube. We can use a tube that is of smaller diameter still, and we will see less light, coming from a smaller patch of sky. Eventually, as we proceed to infinitesimal diameters, we get infinitesimal amounts of light through the tube.
What we want in defining the quantity radiance is something that does not depend on the size of a tube we might be looking (or measuring) through. What we saw with our smaller and smaller tubes was less light coming from a smaller patch of the sky. Recall our discussion of extensive versus intensive quantities in connection with irradiance. What we are looking for here is an intensive quantity: The amount of radiant energy that comes from a certain-sized patch on the sky. Of course, the patch could be any shape at all.
Geometrically, when we look at the sky, we can point our eyes (or a measuring instrument) in any direction, and the direction has to be specified by two angles. There are different ways of choosing two angles. One would be to start out pointed due north, and rotate through a certain azimuth angle, still pointing at the horizon. This would be like specifying a ship's or aircraft's bearing on the compass. Then, to get the second angle, you rotate straight up from the horizon to a certain elevation (degrees above the horizon). If you now trace out the shape of a small patch of sky, you move your instrument (or eyes or finger) through a continuous sequence of (azimuth, elevation) values. The object you describe by doing this is called a solid angle. The unit of a solid angle is called a steradian (sr). You can now think of making that patch bigger and bigger, until the boundary of the patch is the horizon itself. Assuming you are doing this on flat terrain, or on the open ocean, the solid angle size of the whole hemisphere bounded by the horizon is 2" steradians. Likewise, all the sky that is to the east of your meridian (the arc that goes from due north to due south directly over your head) has a solid angle of " steradians.
The definition of radiance is the amount of radiant energy per unit time that crosses through a unit area of an imaginary (or real) surface, coming from a unit of solid angle centered on some direction. The units are thus [W m-2 sr-1].
4.5.5 Spectral radiant flux, radiance, and irradiance
We have just explained the definitions of radiant flux, radiance, and irradiance. These three quantities refer to the total amount of radiant energy that passes through some surface. Different wavelengths of light, as we have seen, have different effects in the atmosphere. We are therefore interested in knowing not only the total amount of energy, but the amount of energy at some wavelength. Another way of looking at that is this. Remember that photons each carry a definite amount of energy. If we multiply the number of photons having a certain energy E that pass through the surface every second by the energy E, we will have the total amount of energy being carried by just those photons, regardless of what other photons might be passing through the surface as well.
If we choose a single, definite photon energy (or, equivalently, definite wavelength), and watch our surface, waiting for photons of exactly that energy, we will be waiting a long, long time. We will occasionally see photons that have nearly the chosen energy, but almost never will we see a photon with exactly that energy.
If, instead of looking for photons having a definite wavelength, we look for those whose wavelength falls in a very narrow range (say 1 nm wide) then we shall find plenty of photons that meet this criterion. And if we keep separate tallies of the energy carried by all photons having wavelengths in the range 250 to 251 nm, those in the range 251-252 nm, those in the range 252-253 nm, etc., then we can plot these range energies versus wavelength. This kind of plot is commonly called a spectrum.
Recall, the radiant flux is the rate at which energy passes through some boundary surface, and has units of Watts. If we choose a 1 nm wide wavelength range, then the amount of flux that is comprised of photons that fall in that range will have units of Watts per nanometer (W nm-1). This quantity is called the spectral flux, or the monochromatic flux.
Likewise, the irradiance is the rate at which energy passes through a unit area of surface per unit of time, and has units [W m-2]. Then the amount of that irradiance that is due to photons in a certain 1 nm - wide range is called the spectral (or monochromatic) irradiance, and has units [W m-2 nm-1].
Finally, radiance is the rate at which energy that came from a unit of solid angle, centered on a certain direction passes through a unit of surface area, and has units [W m-2 sr-1]. The amount of radiance due to photons in a certain 1 nm - wide range is called the spectral (or monochromatic) radiance, and has units [W m-2 nm-1 sr-1].