$$\require{cancel}$$

1.15: A= $$\pi$$ B

There are several occasions in radiation theory in which one quantity is equal to $$\pi$$ times another, the two quantities being related by an equation of the form $$A = \pi B$$. I can think of three, and they are all related to the three questions asked and answered in section 1.14.

If the source in question i of Section 1.14 is an element of a lambertian surface, then $$I(\theta ,\phi )$$ is given by Equation 1.13.1, and in that case Equation 1.14.1 becomes

$\phi = \pi I (0) \tag{1.15.1} \label{1.15.1}$

If the element $$\delta \ A$$ in question ii is lambertian, $$L$$ is independent of $$\theta$$ and f , and equation 1.14.3 becomes

$M = \pi L \tag{1.15.2} \label{1.15.2}$

This, then is the very important relation between the exitance and the radiance of a lambertian surface. It is easy to remember which way round it is if you think of the units in which $$M$$ and $$L$$ are expressed and think of $$\pi$$ as a solid angle.

If the hemisphere of question iii is uniformly lambertian (for example, if the sky is uniformly dull and cloudy) then $$L$$ is the same everywhere in the sky, and the irradiance is

$E = \pi L \tag{1.15.3} \label{1.15.3}$