2.8: Intensity

The intensity of a source in a given direction is the power radiated per unit solid angle about the specified direction, i.e.

$$I = dP/d \omega.$$

The SI units are watts per steradian (W sr^-1). The intensity of an element of area is the product of its radiance and its projected area., and the intensity of a surface in a given direction is the integral of the radiance over the projected area of the surface. As an example, the shape of an irregularly shaped asteroid can be approximated as a set of connected planar triangular facets; two such facets are shown in figure 1.

For each facet of area ΔA_k the contribution to the intensity in the direction of the observer is

$$\Delta I_k = L_{obs, k} \Delta A_k \cos \theta_k$$

where θ_k is the angle between the surface normal vector n_k and the (fixed) direction to the observer. The total intensity (in the direction toward the observer) of the asteroid is then

$$I = \sum_{k = 1}^{N} \Delta A_k$$

where N is the total number of facets both irradiated and visible to the observer.

Of particular interest is the intensity of a sphere as a function of solar phase angle α. If we consider a sphere of radius α centred in an Oxyz frame with directional spherical coordinates (Θ, Φ) irradiated from the x-direction with flux density F, an element of surface area is α² sin ΘdΘdΦ and its projected area in the direction μ is μα² sin ΘdΘdΦ.

The irradiance of a point (α, Θ, Φ) of a point on the surface is E = Fμ₀, where it may be shown that

$$\mu_0 = \sin \Theta \cos \Phi,$$

and for an observer at phase angle α in the xy-plane

$$\mu = \sin \Theta \cos ( \alpha - \Phi ),$$

in which case the intensity as a function of phase angle is given by

$$I ( \alpha) = \alpha^2 \textbf{F} \int_{ \alpha - \pi/2}^{ \pi/2} \int_0^{ \pi} f_r \mu_0 \mu \sin \Theta d \Theta d \Phi.$$

We will return to this equation, with more detail, in §9.