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- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/02%3A_Albedo/2.11%3A_Gaussian_Triple_Integral_AlgorithmI=∫ba∫dc∫feF(x,y,z)dzdydx This algorithm may be generalised further by allowing limits e and f to be functions e(x,y) and f(x,y) and c and d to be functions c...I=∫ba∫dc∫feF(x,y,z)dzdydx This algorithm may be generalised further by allowing limits e and f to be functions e(x,y) and f(x,y) and c and d to be functions c(x) and d(x). For the phase integral, there is no need to invoke the likes of equation (32) since the intensity I(α) is explicitly expressed in terms of α and one stage of the integration is with respect to α.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/02%3A_AlbedoAlbedo is a measurement, expressed as a fraction, of the amount of radiation scattered from a surface or an object. In this chapter we describe those albedos most commonly used and describe methods to...Albedo is a measurement, expressed as a fraction, of the amount of radiation scattered from a surface or an object. In this chapter we describe those albedos most commonly used and describe methods to calculate them in cases where analytical solutions are difficult, if not impossible, to obtain. In order to do this we introduce two more photometric quantities, namely exitance M and intensity I.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/01%3A_Principles_of_Planetary_PhotometryThumbnail: Surface of Mercury (Public Domain; NASA).
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/02%3A_Albedo/2.04%3A_Surfaces_-_Single-scattering_AlbedoThe loss of radiance from a beam of radiance L traversing a thickness ds of a medium is and the single scattering albedo is thus the ratio of the scattering coefficient to the extinction coeff...The loss of radiance from a beam of radiance L traversing a thickness ds of a medium is and the single scattering albedo is thus the ratio of the scattering coefficient to the extinction coefficient. Single scattering albedo is the property of a surface or a layer, and may be regarded as the fundamental albedo, since all albedos that will be derived here from a given definition or reflectance rule will contain at least one instance of ϖ 0 .
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/02%3A_Albedo/2.03%3A_Absorption%2C_Scattering_and_Attenuation_CoefficientsThe mass absorption coefficient, mass scattering coefficient and mass extinction coefficient each with units m 2 kg -1 are defined respectively as α/ρ, σ/ρ and ε/ρ, where ρ is the density (kg m -3 ) o...The mass absorption coefficient, mass scattering coefficient and mass extinction coefficient each with units m 2 kg -1 are defined respectively as α/ρ, σ/ρ and ε/ρ, where ρ is the density (kg m -3 ) of the medium. The atomic (or molecular) absorption, scattering and extinction coefficients are respectively α/N, σ/N and ε/N, where N is the number density (atoms or molecules per unit volume), with units of m 2 /atom (or molecule).
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/01%3A_Principles_of_Planetary_Photometry/1.04%3A_Directions_and_NotationMany authors specify the polar direction of this radiation in terms of an angle of incidence, say i or θ i , as the angle between the surface normal and the incident ray, such that i = π - \( \varthet...Many authors specify the polar direction of this radiation in terms of an angle of incidence, say i or θ i , as the angle between the surface normal and the incident ray, such that i = π - ϑ0 and define μ 0 as μ 0 = cos i. Again, many authors define the polar direction of reflection in terms of an angle of reflection, the angle between the surface normal and the reflected ray, say θ r (which is the same as ϑ), and define μ as cos θ r .
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/01%3A_Principles_of_Planetary_Photometry/1.05%3A_Reflectance_FunctionsIn the most general case of diffuse reflection, the reflectance of a surface will depend on both the direction of the incident radiation and that of the reflected radiation. where λ 0 is sometimes ref...In the most general case of diffuse reflection, the reflectance of a surface will depend on both the direction of the incident radiation and that of the reflected radiation. where λ 0 is sometimes referred to as the Lambertian albedo . Although it is not strictly physically correct, it is convenient (Chandrasekhar, p147) to identify λ 0 with the single scattering albedo ϖ 0 , so for Lambert’s law the BRDF is
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/01%3A_Principles_of_Planetary_Photometry/1.02%3A_Radiance_and_the_Equation_of_Transferprojected on a plane at right angles to the line of sight from the observer); or it is the power arriving per unit area at the observer from unit solid angle (subtended at the observer) of the extende...projected on a plane at right angles to the line of sight from the observer); or it is the power arriving per unit area at the observer from unit solid angle (subtended at the observer) of the extended source. Here dLds is the rate of change of radiance with, and in the direction of, position s in a given medium, r is the density of the medium (kg m -3 ) and κ is the mass attenuation coefficient (m 2 kg -1 ).
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/01%3A_Principles_of_Planetary_Photometry/1.06%3A_Diffuse_Reflection_-_the_Lommel-Seeliger_LawThe Lommel-Seeliger reflectance rule is a time-honoured law which is still very much in use today. It is based on a model which is possibly the simplest from which a solution may be readily obtained f...The Lommel-Seeliger reflectance rule is a time-honoured law which is still very much in use today. It is based on a model which is possibly the simplest from which a solution may be readily obtained for the source function and the equation of transfer. As such it is a single scattering model in which the scattering is isotropic.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/02%3A_Albedo/2.05%3A_Surfaces_-_Normal_AlbedoThe normal albedo pn of a point on a surface is the ratio of the normally observed radiance to that of the Lambertian surface, so that The author has found two definitions of normal albedo in the...The normal albedo pn of a point on a surface is the ratio of the normally observed radiance to that of the Lambertian surface, so that The author has found two definitions of normal albedo in the literature. In one, the surface must be radiated normally and observed normally (μ=μ0=1) and the other in which it can be irradiated from any direction, in which case pn is a function of μ0.
- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Planetary_Photometry_(Tatum_and_Fairbairn)/02%3A_Albedo/2.10%3A_A%2C_p_and_q_for_General_Reflectance_Rulesρ(μ0)=∫2π0∫10fr(μ0,μ,α;…)μdμdϕr, \[I(\alpha)=\int_{\alpha-\pi / 2}^{\pi / 2} \int_{0}^{\pi} f_{r}\left(\...ρ(μ0)=∫2π0∫10fr(μ0,μ,α;…)μdμdϕr, I(α)=∫π/2α−π/2∫π0fr(μ0,μ,α;…)μ0μsinΘdΘdΦ, The Bond albedo and the phase integral are, however, triple integrals, so that a method which combines the advantages of speed and accuracy is required; for this reason Gaussian Quadrature is the chosen method.
