Specific Intensity

Consider a point source emitting radiation isotropically in all directions. By placing a spherical shells and at radii and , we can write, by conserving the energy:

If we assume one of the shells, say , is fixed, then we can write

which is the inverse square law. Now represents a quantity which is representative of all the rays of radiation passing through the surface. A single ray is a mathematical idealization and by definition has zero cross-section area to carry some of that energy. However, we may consider some infinitesimal area through which a bundle of rays is passing perpendicularly. Now this bundle can carry energy as it has some cross-sectional area, which we can write as

We define as the specific intensity and is the angle between the vector normal to and direction of . The way to understand this expression is quite simple. We want to know how much energy

1. is going to cross the infinitesimally small area ,
2. towards direction given by ,
3. in time ,
4. in the frequency range .

Because it gives a more detailed description of the radiation field, it is much more instructive to derive other quantities (like flux) from the specific intensity. One way to derive other quantities is to calculate the moments of .

Moments of Specific Intensity

Before we derive any moments, let us write equation for a bit more generally in terms of the unit normal .

This essentially means all the rays are aligned with . Next is to ask about the nature of the independent variable for which we will be calculating the moments for. For specific intensity, it is more constructive to think in terms of direction. From the definition, it is clear that is not a spatial density, but rather a function of direction¹. Therefore, it makes sense to derive angular moments.

Zeroth moment (Radiation Energy Density)

By definition, zeroth moment of a quantity is something which is does not depend of our independent variable (direction in our case). So we want to derive the quantity which is describes how much radiation is present locally, regardless of the direction. This essentially defines energy density at that location.

To calculate this energy density, we can once again use the isotropic scenario, and imagine a point which is absorbing incoming radiation from all directions. If in time , they cross an area while traveling at speed , then, we can write energy density for the photons in the frequency bin as

Now, this energy density is only the contribution from the rays aligned with , so we can integrate over all directions to get the total energy density:

A closely associated quantity to is the mean intensity, which is defined as:

Consequently, we can write:

Finally, we can also calculate total radiation density

First moment (Flux)

Footnotes

1. Note that we are trying to define as a density function. So although, it does depend on , it is not a density function in . ↩