Related Quantities

In addition to the Stokes vector, there are two other important radiant quantities: irradiance and mean radiance.

Irradiance, or vector irradiance, is defined as the integral over the entire spherical solid angle of the normal component of the radiance, relative to some surface. Irradiance follows the cosine law in that a collimated beam of photons intercepting a plane surface produces an irradiance that is proportional to the cosine of the angle between the photon direction and the surface normal. It has units of $\mu$W$\cdot$cm $^{-2} \cdot$nm^-1 and may be written as,

$$\v{E}(\hat{n};\v{s};\lambda) = \int_{\Omega} I(\v{s};\Omega;\lambda) \hat{n}\cdot \hat{\Omega} \, d\Omega$$ (3.14)

where $\v{s}$ is the position vector, $\hat{n}$ is the surface normal unit vector, $d\Omega=\sin{\theta} d\theta d\phi$ is the solid angle differential expressed in polar coordinates with $\theta$ being the polar angle and $\phi$ being the azimuthal angle.

In a plane-parallel atmosphere (see section 2.6), only the irradiance along the $\hat{z}$-direction is relevant so the vector representation is not necessary. The irradiance in polar coordinates is,

$$E(z;\lambda) = \int_0^{2 \pi} \int_0^{\pi} I(z;\theta,\phi;\l... ...eta d\phi = E^{\downarrow}(z;\lambda)-E^{\uparrow}(z;\lambda)$$ (3.15)

which is also the difference between the downwelling, $E^{\downarrow}$, and the upwelling, $E^{\uparrow}$, irradiances. Irradiance is also the first moment of the radiation field. In terms of the fields, the irradiance is equivalent to the monochromatic Poynting vector. Irradiance is used in the calculation of heating rates and is usually used in determining the radiance reflected by a surface.

The mean radiance, F, also referred to as actinic flux, mean intensity, flux density, or flux (Madronich, 1987), describes the number of photons converging upon a volume element per unit time, area and wavelength. The mathematical expression for mean radiance is equivalent to the zeroth moment of the radiation field,

$$F(z;\lambda) = \frac{1}{4\pi} \int_0^{2 \pi} \int_0^{\pi} I(z;\theta,\phi;\lambda) \sin{\theta} d\theta d\phi$$ (3.16)

and hence is analogous to a flux density. The units usually used for mean radiance are photons$\cdot$s $^{-1} \cdot$cm $^{-2} \cdot$nm $^{-1} \cdot$sr^-1. The mean radiance is an important quantity in the calculation of photodissociation coefficients.