BU GE 645 - The Radiation Field and the Radiative Transfer Equation

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1 Chapter 2 The Radiation Field and the Radiative Transfer Equation 1. The Radiation Field..................................................................................................................... 1 2. Interaction of Radiation with Matter........................................................................................... 5 3. The Equation of Transfer ............................................................................................................ 6 4. Initial and Boundary Conditions................................................................................................. 9 5. Stationary Radiative Transfer Problem..................................................................................... 10 6. Green’s Function and the Reciprocity Principle....................................................................... 11 7. Operator Notations.................................................................................................................... 12 8. The Equation of Transfer in Integral Form............................................................................... 14 9. Eigenvalues and Eigenvectors of the Radiative Transfer Equation.......................................... 16 10. The Law of Energy Conservation ........................................................................................... 18 11. Uniqueness Theorems............................................................................................................. 20 12. General Case of Asymmetry................................................................................................... 22 Problem Sets ................................................................................................................................. 24 References..................................................................................................................................... 26 Further Readings........................................................................................................................... 26 1. The Radiation Field Photons: The energy in the radiation field is assumed carried by point mass-less neutral particles called photons. The energy of a photon E (in Joules) is where = 6.626176⋅10-34 J s (Joules seconds) is Planck’s constant and is photon frequency (in s−1). Frequency is related to wavelength (in meters) as where m s−1 is speed of light. Photons travel in straight lines between collisions and are regarded as a point particles, with position described in Cartesian coordinates by the vector and direction of travel by the unit vector (Fig. 1). Here and throughout the book the symbol is used to denote the length of the vector , i.e., We will also use a polar coordinate system to specify the unit vector . Cartesian coordinates of can be expressed via the polar angle and the azimuthal angle as , , (Fig. 1). The description of photon distribution requires the consideration of photons traveling in directions confined to a solid angle. A solid angle is a part of space bounded by the line segment from a point (the vertex) to all points of a closed curve. A cone is an example of the solid angle which is bounded by lines from a fixed point to all points on a given circle. The solid angle represents the visual angle under which all points of the given curve can be seen from the vertex.2 A measure, or “size”, of a solid angle is the area of that part of the unit sphere with center at vertex that is cut off by the solid angle. Units of the solid angle are expressed in steradian (sr). For a unit sphere whose area is , its solid angle is sr. In the polar coordinate system, the differential solid angle cuts an area consisting of points with polar and azimuthal angles from intervals and . Figure 1. Representation of the unit vector , , in Cartesian and polar coordinate systems. Here , and are Cartesian coordinates of ; and are the corresponding polar and azimuthal angles in a polar coordinate system. Particle Distribution Function: Let denote the density distribution function such that the number of photons dn at time t in the volume element (in m3) about the point , with frequency in a frequency interval to (in s), and traveling along a direction within solid angle (in sr, see Problem 3) is (1) In the frequency domain, the particle distribution function has units of photon number per m3 per frequency interval per steradian (m−3 s sr−1). In the above definition, one can use the wavelength interval to (in m) instead of its frequency counterpart to define the particle distribution function. In the wavelength domain, therefore, the particle distribution function has units of photon number per m3 per m per steradian (m−4 sr−1). Specific Intensity: Many radiometric devices used in remote sensing respond to radiant energy. It is convenient, therefore, to express the particle distribution in terms of energy that photons transport. Consider a volume element with the base (in m2) perpendicular to a direction and the height dz (in m). The number of photons in this volume traveling along the direction is determined by the number of photons which cross in the time interval t to t+dz/c where c is speed of light since the distance traversed by a photon within the interval dt = dz/c does not exceed dz. Equation (1) can be rewritten as . Since the energy of one photon is the amount, dE, of radiant energy (in J) in a time interval dt and in the frequency interval to , which crosses a surface element perpendicular to within a solid angle is given by . (2)3 The distribution of energy that photons transport is given by specific intensity (or radiance) defined as (3a) Its units are J sr−1 in the frequency domain and J s-1 m−3 sr−1=W m−3 sr−1 in the wavelength domain. Here W (1 watt = Js-1) is the unit of radiant power. Some radiometric devices count photons impinging on a detection area for a certain time interval. It is also convenient to express the photon distribution in terms of number of photons crossing a surface of unit area, per unit time per unit frequency per unit steradian. This


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BU GE 645 - The Radiation Field and the Radiative Transfer Equation

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