1: Definitions of and Relations between Quantities used in Radiation Theory
- Page ID
- 6649
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: Introduction to Radiation Theory
- This page explains the theory of radiation, focusing on key terms like radiant flux, intensity, and irradiance. It emphasizes standardized definitions and symbols, noting deviations in astronomical usage, particularly regarding "flux" and "intensity." Additionally, it discusses the use of SI and CGS units and their relation to electrical measurements, highlighting potential confusion in their application.
- 1.2: Radiant Flux or Radiant Power, φ or P
- This page explains radiant power, measured in watts, as the energy output from a source, noting terminology variations across scientific fields. Astronomers refer to this as "luminosity" (\(L\)) for stars, differing from its use in other disciplines. An example is presented with the Sun's radiant power at \(3.85 \times 10^{26} \text{W}\).
- 1.3: Variation with Frequency or Wavelength
- This page covers the concepts of radiant flux and intensity, emphasizing their relationships in frequency and wavelength intervals. It defines key notations and stresses the importance of integrating across all wavelengths. The difference between radiant and luminous quantities is clarified, noting that luminous flux is measured in lumens, which is significant for light pollution and lighting efficiency.
- 1.4: Radiant Intensity, I
- This page explains that not all bodies emit radiation uniformly, with some, like rapidly-rotating stars, radiating non-isotropically. It defines the intensity of a radiation source in a specific direction, represented as \(I\), which measures radiant flux per unit solid angle in W sr\(^{-1}\). For visible radiation, this is termed luminous intensity, measured in candelas.
- 1.5: Per unit
- This page focuses on the "per unit" concept in physics, illustrating its use in terms like density, flux, and intensity. It defines density as mass per unit volume, emphasizing its intensive nature at infinitesimally small points. Flux is articulated as "per unit wavelength interval," although this can imply unrealistic intervals. Intensity as flux "per unit solid angle" faces similar limitations.
- 1.6: Relation between Flux and Intensity
- This page covers the calculation of radiant flux (\(\Phi\)) for both isotropic and anisotropic radiators. For isotropic types, \(\Phi\) is calculated using the equation \(\Phi=4\pi I\). For anisotropic types, it involves integrating intensity over angles, with simplifications for axially symmetric distributions. An example demonstrates modeling bulb intensity, highlighting the connection between total radiant flux, forward intensity, and the flux within specific angular ranges.
- 1.7: Absolute Magnitude
- This page explains magnitude scales in astronomy, detailing their logarithmic relationship to flux and intensity of stars. It includes equations defining zero points for magnitude based on radiant flux values, and discusses absolute bolometric magnitudes across all wavelengths. The text also addresses challenges in determining these zero points and emphasizes the relationship between key constants involved in the calculations.
- 1.8: Normal Flux Density F
- This page explains normal flux density as the energy rate passing per unit area, measured in watts per square meter. It details the calculation for isotropic point sources using radiant flux divided by the surface area of a sphere. For non-isotropic sources, it discusses how normal flux density can also be determined using intensity and distance, emphasizing that it decreases with the square of the distance.
- 1.9: Apparent Magnitude
- This page outlines the relationship between astronomical magnitude scales, highlighting the connection between absolute and apparent magnitude, which depends on distance. It introduces formulas linking magnitude to flux density and suggests a method to establish a zero point. The page also discusses the concept of apparent bolometric magnitude, focusing on the interrelation of these concepts rather than providing extensive detail on the scales.
- 1.10: Irradiance E
- This page covers the principles of irradiance and illuminance in relation to point source radiation. It explains that irradiance \(E\) is influenced by the intensity \(I\) and the angle \(\theta\) of a surface, expressed through the formula \(E = (I \cos \theta) / r^2\). Additionally, it clarifies the connection between luminous intensity and illuminance, with definitions for lux and foot-candle, supported by examples that calculate illuminance across various angles and surface areas.
- 1.11: Exitance M
- This page covers exitance, defined as energy radiated per unit area by a surface, symbolized as \(M\). It emphasizes that exitance is intrinsic and observer-independent. The page explains the Stefan-Boltzmann law relating total exitance of a black body to temperature, introduces the Planck equation for wavelength-specific exitance, and clarifies that "emittance" is an outdated term now replaced by exitance.
- 1.12: Radiance L
- This page explores the equivalence of radiance \(L\) and surface brightness \(B\) in quantifying brightness, detailing their properties, including dependence on viewing direction and projected area. It clarifies terminology variations across fields and introduces concepts of visible light measurement.
- 1.13: Lambertian Surface
- This page discusses Lambertian radiating surfaces, which adhere to Lambert's Law and exhibit isotropic radiance with intensity varying by angle. It distinguishes between perfectly diffusing Lambertian surfaces and reflective specular surfaces, emphasizing the importance of consistent radiance regardless of viewing or illumination angles.
- 1.14: Relations between Flux, Intensity, Exitance, Irradiance
- This page discusses calculating radiant flux from a point light source, the relationship between radiance and exitance with relevant equations, and the determination of irradiance within a hollow hemisphere by connecting radiance to intensity.
- 1.15: A= πB
- This page examines the relationships in radiation theory involving specific quantities and the factor of \(\pi\). It highlights three scenarios related to Lambertian surfaces, illustrating how exitance (M) and irradiance (E) are linked to radiance (L) through the equations \(M = \pi L\) and \(E = \pi L\). The discussion emphasizes the significance of these relationships and their physical units for a clearer understanding of radiation principles.
- 1.16: Radiation Density (u)
- This page covers radiation energy density, which is defined as energy per unit volume and measured in joules per cubic meter (J m⁻³). The symbol \(u\) is introduced to represent this quantity.
- 1.17: Radiation Density and Irradiance
- This page explains the relationship between irradiance \(E\) and energy density \(u\) in an isotropic photon radiation-filled hemisphere, deriving the formula \(E = uc/4\). It illustrates how to calculate energy per unit area over time by considering photon motion and distribution, as well as integrating over solid angles to accommodate different photon arrival directions, while maintaining consistent energy content despite variations in photon energies.
- 1.18: Radiation Pressure (P)
- This page discusses how photons exert pressure through momentum, with defined relationships between pressure \(P\) and energy density \(u\), varying based on surface types: \(P = 2u\) for reflective and \(P = u\) for absorptive surfaces. It also addresses the role of momentum in isotropic radiation fields, particularly in stellar atmospheres, and explains derivation methods for these relationships through photon interactions and angle integrations.