2.8: Intensity
- Page ID
- 8725
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 ΔAk 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 θk is the angle between the surface normal vector nk 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.