Supplement 1.7: Radiation quantities and radiometry      (9/9)

Relationships between radiometric quantities

1. Radiant power $\varphi$ and radiant energy $Q$
   $\varphi =\frac{\text{d}Q}{\text{d}t}Q=\int \varphi \text{d}t$
2. Radiant energy density $U$ and radiant energy $Q$
   $U=\text{}\frac{\text{d}Q}{\text{d}V}Q=\int U\text{d}V$
3. Radiant intensity $I$ and radiant power $\varphi$
   $I=\frac{\text{d}\varphi }{\text{d}\Omega }\varphi =\int I\text{d}\Omega$
4. Irradiance $E_{rad}$ and radiant power $\varphi$
   $E_{rad}=\frac{\text{d}\varphi }{\text{d}A}\varphi =\int E_{rad}\text{d}A$
5. Radiant exitance $M$ and radiant power $\varphi$
   $M=\frac{\text{d}\varphi }{\text{d}a}\varphi =\int M\text{d}a$
6. Radiance $L$ and radiant power $\varphi$
   $L=\frac{{\text{d}}^{\text{2}}\varphi }{\text{d}acos\vartheta \text{d}\Omega }{\text{d}}^{\text{2}}\varphi =L\text{d}acos\vartheta \text{d}\Omega$
7. Radiance $L$ and radiant intensity $I$
   $L=\frac{\text{d}I}{\text{d}acos\vartheta }\text{d}I=L\text{d}acos\vartheta$

Chapter 1 of Jeremy B. Tatum's textbook Stellar Armospheres contains many more examples with detailed calculations.

Re 3 and 4: Radiant intensity, irradiance and the photometric distance law

The radiant intensity of a point light source is given by the quotient of the radiant power and the solid angle: $I=\text{d}\varphi /\text{d}\Omega$.

An irradiated surface element da at a distance r from the light source corresponds to the solid angle element $\text{d}\Omega =\text{d}a/r^2$. The surface element may be inclined at an angle ϑ to the direction of irradiation, in which case:

The irradiance of the surface element is therefore:

It decreases with the square of the distance from the radiation source. This is the photometric law of distance.

Re 7: The radiant intensity and radiance of Lambert emitters

In the case of a Lambertian emitter, the radiant intensity decreases with the cosine of the angle to the normal of the radiating surface:

Inserting this into the relationship for radiance gives:

The radiance $L$ of Lambertian emitters is therefore independent of the viewing angle.