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Radiative transfer

Radiative transfer is the physical and mathematical framework describing the propagation, emission, absorption, and scattering of electromagnetic radiation through a medium, such as an atmosphere, stellar interior, or other matter, governed by interactions between photons and particles.^[1] This process underpins the exchange of energy across scales, from planetary climates to cosmic phenomena, where virtually all interactions between the Earth-atmosphere system and the broader universe occur via radiative mechanisms.^[2] At its core, radiative transfer quantifies how radiation fields evolve, using key quantities like specific intensity I_\nu—the energy per unit time, area, frequency, and solid angle—and radiative flux F_\nu, which integrates intensity over directions to yield net energy flow.^[3] The foundational radiative transfer equation (RTE) models these dynamics as \frac{dI_\nu}{ds} = -\beta_\nu I_\nu + j_\nu, where s is the path length, \beta_\nu is the extinction coefficient (combining absorption and scattering), and j_\nu is the emission coefficient, often incorporating the Planck function B_\nu(T) for thermal sources.^[1] In optically thin media, radiation travels freely with minimal attenuation, while in optically thick cases, optical depth \tau_\nu = \int \beta_\nu ds determines exponential decay, as in I_\nu(\tau_\nu) = I_\nu(0) e^{-\tau_\nu} for pure absorption.^[3] Scattering introduces complexity via phase functions, such as the single-scattering albedo \omega_\nu, which partitions radiation between absorption and redirection.^[3] Applications span atmospheric physics, where radiative transfer drives climate models by simulating solar shortwave (\lambda < 4 \, \mu\text{m}) absorption and terrestrial longwave (\lambda > 4 \, \mu\text{m}) emission, influencing temperature profiles and trace gas detection.^[2] In astrophysics, it elucidates stellar spectra and planetary albedos, while in oceanography and remote sensing, it enables retrieval of properties like water vapor and aerosols from satellite observations.^[3] Numerical solutions, including Monte Carlo methods or discrete ordinates, address the RTE's complexity for practical computations in these fields.^[3]

Fundamental Concepts

Radiometric Quantities

Radiometric quantities provide the foundational measures for quantifying electromagnetic radiation in the context of its propagation through media, essential for analyzing energy transfer in physical systems. These quantities describe the energy flux, its distribution over surfaces and directions, and its spectral properties, enabling precise modeling of radiation fields in fields such as astrophysics, atmospheric science, and optics.^[4] The primary radiometric quantity is the radiant flux, denoted Φ, which represents the total power emitted, transmitted, or received by radiation, measured in watts (W). It quantifies the overall energy flow without regard to direction or spatial distribution. Irradiance, E, is the radiant flux incident on a surface per unit area, with units of W m⁻², describing how radiation is spread over a receiving plane; for example, solar irradiance at Earth's surface averages about 1000 W m⁻² under clear conditions. Radiant exitance, M, is analogous but applies to flux emitted from a surface per unit area, also in W m⁻². These surface-integrated quantities are crucial for bulk energy balance calculations in radiative transfer scenarios.^[5]^[6] A more detailed measure is radiance, often termed specific intensity and denoted I_ν (or L_ν in some contexts), which specifies the radiant flux per unit area, per unit solid angle, and per unit frequency, with units W m⁻² sr⁻¹ Hz⁻¹. This quantity captures the directional and spectral nature of radiation propagation, representing the brightness of a radiation field along a particular line of sight; physically, it indicates how much energy crosses a small area perpendicular to the beam direction within a narrow cone of solid angle and frequency band. Unlike irradiance, which integrates over all directions, radiance is conserved along rays in free space, making it invariant under translation and a cornerstone for phase-space descriptions in radiative transfer. Bolometric quantities extend these by integrating over all frequencies, such as bolometric flux (total energy flux across the spectrum) or bolometric radiance, useful for total energy assessments without spectral resolution.^[7]^[8]^[9] The terminology and formalization of radiometric quantities emerged in the early 20th century as radiometry developed from photometry, with standardization efforts by organizations like the International Commission on Illumination (CIE); however, foundational concepts trace to 19th-century work on thermal radiation by Gustav Kirchhoff, who in 1859 established the principle that absorptivity equals emissivity for bodies in thermal equilibrium, laying groundwork for quantifying emission and absorption spectra. Kirchhoff's contributions, including the introduction of the blackbody concept, influenced the evolution of these terms from qualitative optics to quantitative measures applicable to electromagnetic propagation.^[10]^[9] A benchmark example is the blackbody radiation spectrum, described by Planck's law for the spectral radiance B_ν(T) of an ideal blackbody at temperature T:

B_\nu(T) = \frac{2 h \nu^3}{c^2} \frac{1}{e^{h\nu / kT} - 1}

where h is Planck's constant, ν is frequency, c is the speed of light, and k is Boltzmann's constant; this formula, derived by assuming quantized energy exchanges, peaks at a frequency scaling with T and integrates to the total blackbody radiance of σ T⁴ / π (with σ the Stefan-Boltzmann constant), illustrating how radiometric quantities encode thermal emission properties. Specific intensity plays a central role in formulating the radiative transfer equation by describing the directional radiation field.

Specific Intensity and Phase Space

The specific intensity, denoted I_\nu(\mathbf{r}, \boldsymbol{\Omega}, t), quantifies the distribution of radiative energy in phase space and is defined as the amount of energy dE per unit time dt, per unit area dA perpendicular to the propagation direction, per unit solid angle d\Omega, and per unit frequency interval d\nu.^[11] This makes its units W m^{-2} Hz^{-1} sr^{-1} in SI or erg s^{-1} cm^{-2} Hz^{-1} sr^{-1} in cgs. In the context of radiative transfer, phase space for photons is parameterized by position \mathbf{r}, propagation direction \boldsymbol{\Omega} (a unit vector), time t, and frequency \nu, reflecting the six-dimensional nature of the photon's position and momentum coordinates, where momentum magnitude is tied to \nu via |\mathbf{p}| = h\nu / c.^[11] The specific intensity thus serves as a fundamental distribution function describing the local radiation field in this extended phase space.^[11] In vacuum or free space, the specific intensity is conserved along individual rays, meaning I_\nu remains constant as radiation propagates without interactions. This conservation arises from Liouville's theorem, which states that the phase space density of photons is invariant along trajectories in the absence of forces or collisions, ensuring no dilution or concentration of the radiation bundle in phase space.^[11] Mathematically, this is expressed as \frac{dI_\nu}{ds} = 0, where s is the path length along the ray in the direction \boldsymbol{\Omega}.^[11] A key property of the specific intensity is its relativistic invariance: the quantity I_\nu / \nu^3 is Lorentz invariant, transforming consistently across inertial frames due to the scaling of energy, frequency, and solid angle under Lorentz boosts.^[11] This invariance stems from the phase space density of photons being proportional to I_\nu / \nu^3, which remains unchanged in special relativity. In relativistic contexts, such as high-speed astrophysical flows, this property facilitates the transformation of radiation fields between frames, preserving the underlying photon distribution.^[11] Macroscopic radiometric quantities, such as irradiance E = \int I_\nu \cos\theta \, d\Omega \, d\nu, are obtained by integrating the specific intensity over directions and frequencies.^[11]

The Radiative Transfer Equation

Derivation

The derivation of the radiative transfer equation (RTE) proceeds from the conservation of radiative energy along a ray path in a medium that absorbs, emits, and scatters radiation. Consider a pencil beam of radiation characterized by the specific intensity I_\nu at frequency \nu and direction \hat{\Omega}, propagating a differential distance ds along the ray. The net change dI_\nu in the specific intensity results from three physical processes: absorption, which removes energy from the beam; thermal emission, which adds energy; and scattering, which redirects energy both into and out of the beam.^[7] Absorption diminishes the intensity by an amount proportional to the local absorption coefficient \kappa_\nu and the incident intensity, yielding a term -\kappa_\nu I_\nu \, ds. Thermal emission contributes an isotropic source term j_\nu \, ds, where j_\nu is the emission coefficient representing the energy added per unit volume, time, frequency, and solid angle. Scattering involves both out-scattering, which removes intensity from the direction \hat{\Omega} at a rate \sigma_\nu I_\nu \, ds (with \sigma_\nu the scattering coefficient), and in-scattering from other directions \hat{\Omega}', given by \frac{\sigma_\nu}{4\pi} \int I_\nu(\hat{\Omega}') P(\hat{\Omega}', \hat{\Omega}) \, d\Omega' \, ds, where P(\hat{\Omega}', \hat{\Omega}) is the phase function normalized such that \frac{1}{4\pi} \int P \, d\Omega = 1. Combining these, the balance equation becomes

dI_\nu = \left( -\kappa_\nu I_\nu + j_\nu - \sigma_\nu I_\nu + \frac{\sigma_\nu}{4\pi} \int I_\nu(\hat{\Omega}') P(\hat{\Omega}', \hat{\Omega}) \, d\Omega' \right) ds.

Dividing by ds yields the steady-state, monochromatic form of the RTE:

\frac{dI_\nu}{ds} = -(\kappa_\nu + \sigma_\nu) I_\nu + j_\nu + \frac{\sigma_\nu}{4\pi} \int I_\nu(\hat{\Omega}') P(\hat{\Omega}', \hat{\Omega}) \, d\Omega',

or equivalently,

\frac{dI_\nu}{ds} = -\kappa_\nu I_\nu + j_\nu + \sigma_\nu \left[ \frac{1}{4\pi} \int I_\nu(\hat{\Omega}') P(\hat{\Omega}', \hat{\Omega}) \, d\Omega' - I_\nu(\hat{\Omega}) \right].

This integro-differential equation describes the evolution of I_\nu along the path s.^[12] The derivation assumes a monochromatic approximation, treating radiation at a single frequency \nu while neglecting frequency redistribution during scattering or emission; in practice, this holds when line widths are narrow compared to the spectrum. It also assumes steady-state conditions, omitting the time derivative \partial I_\nu / \partial t; the time-dependent form includes this term on the left-hand side via the chain rule \partial I_\nu / \partial t + \hat{\Omega} \cdot \nabla I_\nu. Additional assumptions include straight-line propagation of photons (valid for geometric optics, ignoring diffraction and refraction on scales smaller than the wavelength) and incoherent, unpolarized scattering (neglecting wave interference and polarization effects). These simplifications align the equation with classical transport theory while capturing the dominant processes in astrophysical and atmospheric media.^[7] The foundational ideas trace back to Arthur Schuster's 1905 work on radiation through a foggy atmosphere, where he introduced basic two-stream models for absorption and scattering in stellar contexts. Karl Schwarzschild advanced this in 1906 by deriving a differential form incorporating radiative equilibrium in the solar atmosphere, emphasizing absorption and emission without scattering. The modern, comprehensive formulation, including the full scattering integral and invariance principles, was established by Subrahmanyan Chandrasekhar in his 1950 treatise Radiative Transfer.^[12]

Components and Interpretation

The radiative transfer equation (RTE) describes the propagation of radiation through a medium, accounting for interactions that alter the specific intensity I_\nu along a path s. The equation's right-hand side comprises terms representing absorption, emission, and scattering, each capturing distinct physical processes between photons and matter. These terms highlight how radiation is attenuated or enhanced, with the net effect determining the observable intensity at a given point.^[13] The absorption term, -\kappa_\nu I_\nu, where \kappa_\nu is the absorption coefficient (often termed opacity in astrophysical contexts), physically represents the loss of radiant energy due to photon absorption by atoms, ions, or molecules in the medium. This process converts photon energy into internal excitation or ionization of the absorbing particles, reducing the intensity along the ray path proportionally to both the local intensity and the medium's absorptive properties. In frequency-specific treatments, \kappa_\nu quantifies the probability per unit length that a photon at frequency \nu is absorbed, with higher values indicating more opaque conditions.^[13]^[14] The emission term, j_\nu, denotes the addition of radiant energy from the medium, arising from processes such as spontaneous emission, recombination, or thermal re-emission following absorption. The thermal source function S_\nu^\text{thermal} = j_\nu / \kappa_\nu encapsulates the ratio of emitted to absorbed intensity per unit absorption optical depth, providing a measure of the medium's intrinsic luminosity. Emission can be isotropic, as in thermal radiation from local thermodynamic equilibrium where S_\nu^\text{thermal} = B_\nu(T) (the Planck function), or anisotropic in cases like directed fluorescence or non-equilibrium populations, influencing the angular distribution of outgoing radiation. In general, with scattering, the total source function is S_\nu = \frac{\kappa_\nu S_\nu^\text{thermal} + \sigma_\nu \frac{1}{4\pi} \int I_\nu(\Omega') P(\Omega', \Omega) \, d\Omega'}{\kappa_\nu + \sigma_\nu}.^[13]^[14] The scattering term, -\sigma_\nu I_\nu + \sigma_\nu \frac{1}{4\pi} \int I_\nu(\Omega') P(\Omega', \Omega) \, d\Omega', where \sigma_\nu is the scattering coefficient and P(\Omega', \Omega) is the phase function describing the angular redistribution of scattered photons (with \frac{1}{4\pi} \int P \, d\Omega' = 1), accounts for radiation redirected without energy loss. The -\sigma_\nu I_\nu subterm represents removal of intensity in the original direction, while the integral \sigma_\nu \frac{1}{4\pi} \int I_\nu(\Omega') P(\Omega', \Omega) \, d\Omega' adds contributions from incoming radiation I_\nu(\Omega') from direction \Omega'. Isotropic scattering occurs when P = 1, uniformly redistributing photons; forward scattering favors small angles (e.g., in large-particle Mie scattering); Rayleigh scattering by small particles, common in molecular atmospheres, follows P(\theta) = \frac{3}{4} (1 + \cos^2 \theta), peaking at 0° and 180° due to dipole-induced polarization.^[13]^[15]^[16] The absorption optical depth \tau_\nu^\text{abs} = \int \kappa_\nu \, ds provides a dimensionless measure of cumulative absorption; \tau_\nu^\text{abs} \ll 1 indicates optically thin conditions for absorption where radiation passes freely, while \tau_\nu^\text{abs} \gg 1 signifies optically thick media dominated by local emission. For the general RTE with scattering, the extinction optical depth \tau_\nu = \int (\kappa_\nu + \sigma_\nu) \, ds is used, transforming the RTE into \frac{dI_\nu}{d\tau_\nu} = I_\nu - S_\nu, with the formal solution I_\nu(\tau_\nu, \hat{\Omega}) = I_\nu(0) e^{-\tau_\nu} + \int_0^{\tau_\nu} S_\nu(t, \hat{\Omega}) e^{-(\tau_\nu - t)} \, dt, where the exponential decay attenuates the boundary intensity and the integral accumulates source contributions weighted by their optical separation. In scattering-inclusive cases, S_\nu incorporates the scattering integral, and solving requires iterative or numerical methods due to non-locality. For pure absorption (\sigma_\nu = 0), \tau_\nu = \tau_\nu^\text{abs} and S_\nu = j_\nu / \kappa_\nu.^[13]^[14] The extinction coefficient \alpha_\nu = \kappa_\nu + \sigma_\nu combines absorption and scattering into the total interaction rate per unit length, representing all processes that remove photons from the original beam direction. This total opacity governs the overall penetration depth of radiation, with scattering effectively acting like absorption followed by re-emission in a new direction, thus blurring the distinction in highly scattering media like stellar interiors or planetary atmospheres.^[13]^[15]

Solution Methods

Exact Solutions

Exact solutions to the radiative transfer equation (RTE) are analytical expressions that provide precise descriptions of radiation fields under highly idealized conditions, such as plane-parallel geometries, homogeneous media, and simplified scattering or absorption processes. These solutions are invaluable for benchmarking numerical methods and gaining physical insight into radiative processes, though their applicability is limited to cases without complex spatial variations or anisotropic effects.^[17] In plane-parallel atmospheres, the formal integral solution for the specific intensity I(\tau, \mu) of outgoing radiation (\mu > 0) is given by

I(\tau, \mu) = I(0, \mu) e^{-\tau / \mu} + \int_0^\tau S(t) e^{-(\tau - t)/\mu} \frac{dt}{\mu},

where \tau is the optical depth, \mu = \cos \theta is the cosine of the angle from the normal, and S(t) is the source function. This expression arises from integrating the RTE along a ray path, accounting for attenuation of the incident intensity at the boundary and contributions from emission and scattering within the medium. It applies to monochromatic radiation at frequency \nu, denoted I_\nu(\tau_\nu, \mu), in a stratified atmosphere.^[17]^[18] A classic example is Milne's problem, which considers a semi-infinite, homogeneous atmosphere with no incident radiation at the boundary (I(0, \mu) = 0) and isotropic scattering. The exact solution yields the emerging intensity at the surface, expressed in terms of the Hopf function q(\mu), which describes the angular distribution of the radiation field. This solution, first posed by Milne in 1921 and resolved analytically using the Wiener-Hopf method, provides the extrapolation distance for the diffusion approximation and is fundamental for understanding emergent radiation in stellar atmospheres.^[19] The discrete ordinates method offers a semi-exact approach for multi-angle problems by expanding the intensity in a finite set of discrete directions, reducing the RTE to a system of ordinary differential equations. While inherently numerical, it becomes exact in the limit of infinite ordinates and is precisely solvable in simple cases like non-scattering media or slab geometries with constant coefficients.^[20] Exact solvability typically requires restrictive conditions, including isotropic scattering (phase function independent of angle), constant absorption and scattering coefficients, and a gray atmosphere where opacity is frequency-independent. These assumptions simplify the integro-differential RTE to forms amenable to closed-form integration or transformation techniques like Laplace or Fourier methods.^[18]

Approximate Solutions

Approximate solutions to the radiative transfer equation (RTE) are crucial for addressing scenarios where exact analytical or numerical solutions are computationally prohibitive, such as in multi-dimensional geometries or media with varying optical properties. These techniques systematically reduce the complexity of the integro-differential RTE by introducing assumptions that control error while preserving key physical behaviors, including absorption, emission, and scattering. Common approaches include moment expansions, perturbation expansions for extreme optical depths, directional averaging, and adaptive closures, each tailored to specific regimes like optically thin scattering or thick diffusion-dominated transport. Moment methods approximate the specific intensity I_\nu by expanding it in a basis of orthogonal functions, typically spherical harmonics for general geometries or Legendre polynomials in plane-parallel cases, to derive a coupled set of moment equations. In the spherical harmonics P_N method, pioneered by Jeans, the expansion I(\mathbf{r}, \hat{\Omega}) = \sum_{l=0}^N \sum_{m=-l}^l f_{lm}(\mathbf{r}) Y_l^m(\hat{\Omega}) transforms the RTE into a hierarchy of partial differential equations for the moments f_{lm}, with closure achieved by truncating at order N and approximating higher moments (e.g., via maximum entropy or diffusion assumptions). This yields a hyperbolic system suitable for numerical solution in astrophysical simulations, offering higher accuracy than lower-order methods for anisotropic fields while remaining more tractable than full Monte Carlo approaches. For plane-parallel atmospheres, the Legendre expansion I(\mu) = \sum_{n=0}^N \frac{2n+1}{2} \phi_n P_n(\mu) similarly leads to moments \phi_n = \int_{-1}^1 I(\mu) P_n(\mu) d\mu, closed by setting \phi_{N+1} = 0 or other relations, enabling efficient computation of fluxes in stellar interiors. These methods excel in balancing angular resolution and spatial dynamics, with applications in supernova modeling and reactor physics. Perturbation theory exploits asymptotic limits of the optical depth \tau to simplify the RTE, providing series expansions valid for small or large \tau. In the optically thin regime (\tau \ll 1), multiple scattering is negligible, so the intensity approximates the integrated source function along the ray, I(\tau) \approx \int_0^\tau S(t) e^{-(\tau - t)} \, dt \approx \int_0^\tau S(t) \, dt, capturing direct emission without reabsorption. Conversely, in the optically thick limit (\tau \gg 1), radiation behaves diffusively, with the flux given by Fick's law \mathbf{F} = -\frac{1}{3\kappa} \nabla E, where E is the energy density and \kappa the opacity, derived from the second moment of the RTE. For weak scattering perturbations, the Born approximation linearizes the scattering integral by replacing the total field with the incident field, yielding the scattered intensity to first order as I_s(\mathbf{r}, \hat{\Omega}_s) \approx \int V(\mathbf{r}') e^{i \mathbf{k} \cdot (\mathbf{r} - \mathbf{r}')} d\mathbf{r}', useful in granular media or atmospheric aerosols where higher-order scattering is minimal. These limits benchmark more general approximations and are foundational in planetary atmosphere modeling. The two-stream approximation further simplifies the RTE by assuming isotropic intensity within forward and backward hemispheres, averaging over \mu > 0 and \mu < 0 to define hemispheric fluxes F^+ = 2\pi \int_0^1 I(\mu) \mu d\mu and F^- = -2\pi \int_{-1}^0 I(\mu) \mu d\mu. This reduces the azimuthal integral to two coupled ordinary differential equations in optical depth, \frac{dF^+}{d\tau} = -(1 - \omega) (F^+ - S) + \omega \varpi F^- (and analog for F^-), where \omega is albedo and \varpi the asymmetry factor, solvable analytically for homogeneous media. Introduced by Schuster to model radiation penetration in foggy atmospheres, it captures essential backscattering effects with minimal parameters, achieving errors under 10% for solar fluxes in clear skies and serving as a building block for multi-stream extensions in climate simulations. Variable Eddington factors enhance moment methods by replacing the constant Eddington tensor (1/3 in P_1) with a position- and time-dependent tensor f_{ij} = \frac{1}{E} \int I \hat{\Omega}_i \hat{\Omega}_j d\Omega, derived self-consistently from lower moments to interpolate between isotropic diffusion (f = 1/3) and beamed streaming (f \to 1). Levermore and Pomraning formulated this as a flux-limited closure, where f = \frac{1 + \chi}{3} with \chi a limiter function of the flux gradient, ensuring positivity and causality in transitional regimes like supernova shocks. This dynamic approach reduces diffusion errors by up to 50% in heterogeneous media compared to fixed closures, facilitating stable implicit solvers in hydrodynamic-radiation coupled codes.

Local Thermodynamic Equilibrium

Local thermodynamic equilibrium (LTE) is an approximation in radiative transfer where collisional processes between particles dominate over radiative processes, leading to a local Maxwell-Boltzmann distribution of velocities and populations of excited states that follow the Boltzmann distribution at the local kinetic temperature T.^[21]^[22] Under these conditions, the source function S_\nu simplifies to the Planck function B_\nu(T), independent of frequency and direction.^[23] This assumption modifies the radiative transfer equation (RTE) to \frac{dI_\nu}{ds} = -\kappa_\nu (I_\nu - B_\nu(T)), where I_\nu is the specific intensity, \kappa_\nu is the opacity, and s is the path length.^[24] In optically thick media, where the optical depth \tau \gg 1, the solution for I_\nu(\tau) approaches B_\nu(T), meaning the radiation field locally resembles blackbody radiation at temperature T.^[13] LTE holds in regions of high density and low radiation field strength, such as deep stellar interiors, where frequent collisions thermalize the plasma faster than radiative transitions can disrupt equilibrium.^[25] It breaks down in low-density environments with strong radiation fields, like outer stellar atmospheres, where non-LTE effects arise due to radiative excitation and de-excitation dominating.^[26] A key application is the gray atmosphere model under LTE and radiative equilibrium, where frequency-independent opacity leads to a temperature structure T^4(\tau) = \frac{3}{4} T_\mathrm{eff}^4 \left( \tau + \frac{2}{3} \right), with T_\mathrm{eff} the effective temperature defined by the outgoing flux \sigma T_\mathrm{eff}^4.^[27] This integrated form relates T_\mathrm{eff}^4 to the average of T^4(\tau) over optical depth, approximately T_\mathrm{eff}^4 \approx \frac{1}{2} \int_0^\infty T^4(\tau) \, e^{-\tau} d\tau in emergent flux calculations.^[28] The LTE approximation gained prominence in the 1920s through Sverre Rosseland's work on radiative diffusion in stellar interiors, enabling simplified models of energy transport in optically thick regions.

Eddington Approximation

The Eddington approximation, also known as the diffusion approximation, simplifies the radiative transfer equation (RTE) in optically thick media by assuming that the specific intensity is nearly isotropic, allowing the use of a moment closure to reduce the integro-differential RTE to a set of ordinary differential equations.^[29] This approach is particularly useful for modeling radiative transport in stellar interiors and dense atmospheres where photons undergo frequent scattering or absorption, leading to a diffusive behavior akin to heat conduction.^[30] The derivation begins by taking the first two angular moments of the RTE in a plane-parallel geometry, where the optical depth \tau_\nu is defined as \tau_\nu = \int_z^\infty (\alpha_\nu + \kappa_\nu) dz' with \alpha_\nu as the absorption coefficient and \kappa_\nu as the scattering coefficient.^[29] The zeroth moment yields the mean intensity J_\nu = \frac{1}{2} \int_{-1}^1 I_\nu d\mu, and the first moment gives the flux H_\nu = \frac{1}{2} \int_{-1}^1 I_\nu \mu d\mu, where \mu = \cos\theta is the direction cosine and I_\nu is the specific intensity.^[29] Integrating the RTE over these moments produces:

\frac{\partial H_\nu}{\partial \tau_\nu} = J_\nu - S_\nu,

\frac{\partial K_\nu}{\partial \tau_\nu} = H_\nu,

where S_\nu is the source function and K_\nu = \frac{1}{2} \int_{-1}^1 I_\nu \mu^2 d\mu is the second moment representing the radiation pressure tensor.^[29] To close this system, the Eddington approximation assumes an isotropic intensity field, expanding I_\nu \approx J_\nu + 3 H_\nu \mu, which implies K_\nu = \frac{1}{3} J_\nu and introduces the Eddington factor f = \frac{1}{3}.^[29] Substituting this closure yields the diffusion relation \frac{dK_\nu}{d\tau_\nu} = \frac{f}{\kappa_\nu} \frac{dJ_\nu}{d\tau_\nu} = -\frac{1}{3 \rho \kappa_\nu} \frac{dJ_\nu}{dz}, where \rho is density and z is physical depth, and the overall diffusion equation \nabla \cdot \mathbf{F} = -\kappa \rho (J - S), with \mathbf{F} = 4\pi H as the radiative flux.^[29] This approximation is valid in regimes where the optical depth \tau \gg 1, ensuring isotropic radiation and gradual spatial variations over the mean free path, but it breaks down near boundaries or in optically thin media where anisotropy dominates, leading to errors in flux predictions up to 20-50% at \tau \approx 1.^[30] Historically, the method was developed by Arthur Eddington in the 1920s for modeling radiative transport in stellar atmospheres and interiors, as detailed in his seminal work on stellar structure. It is closely tied to the Rosseland mean opacity, which weights frequency-dependent opacities by the derivative of the Planck function to ensure accurate flux diffusion in multi-frequency problems, \frac{1}{\bar{\kappa}} = \int_0^\infty \frac{1}{\kappa_\nu} \frac{dB_\nu}{dT} d\nu / \int_0^\infty \frac{dB_\nu}{dT} d\nu. Improvements to the fixed Eddington factor include variable Eddington factors, which dynamically adjust f(\tau) based on the local radiation field to enhance accuracy in transitional optical depths, as pioneered in non-LTE stellar atmosphere models.^[31]

References

[1]
[PDF] Radiative transfer - NMSU Astronomy
Radiative transfer is a formalism of macroscopic physics that is founded upon the countless microscopic physical interactions between radiation and matter.
[2]
[PDF] 1. Radiative Transfer 2. Spectrum of Radiation 3. Definitions
Virtually all the exchanges of energy between the earth-atmosphere system and the rest of the universe take place by radiative transfer.
[3]
[PDF] Introduction to the Theory of Atmospheric Radiative Transfer
... radiative transfer. (RT) theory have been made to these and other topics in atmospheric physics. The lack of standardization of symbols and terms in current ...
[4]
[PDF] Section 11 Radiative Transfer
Radiometry characterizes the propagation of radiant energy through an optical system. Radiometry deals with the measurement of light of any wavelength; ...
[5]
1.4 Basic Radiometric Quantities - Gigahertz-Optik
Radiometry deals with the measurement of energy per time (= power, given in watts) emitted by light sources or impinging on a particular surface.
[6]
Understanding Radiance (Brightness), Irradiance and Radiant Flux
The three terms discussed above are quantities used to characterize radiation within a certain wavelength band, (UV, VIS and/or IR). It is also common to ...Missing: transfer | Show results with:transfer
[7]
[PDF] Chapter 1 The Radiation Field and the Radiative Transfer Equation
Specific Intensity: Many radiometric devices used in remote sensing respond to radiant energy. ... transport is given by specific intensity (or radiance).<|control11|><|separator|>
[8]
Geometrical Radiometry - Ocean Optics Web Book
Mar 6, 2021 · We ﬁrst deﬁne radiance, the fundamental quantity that describes light in radiometric terms. We then deﬁne various irradiances and other ...
[9]
Radiometry and Photometry: Units and Conversion Factors
1968 Optical Society of America. The following outline is concerned with definitions, symbols, units, and conversion factors as they occur, and are helpful, ...
[10]
Science, Optics and You - Timeline - Gustav Robert Kirchhoff
Nov 13, 2015 · In the course of his studies of radiation, Kirchhoff coined the term black body to describe a hypothetical perfect radiator that absorbs all ...
[11]
[PDF] RADIATIVE PROCESSES IN ASTROPHYSICS
It is easy to relate the phase space density of photons to the specific intensity I, and thus determine the transformation properties of I,. This is done by ...
[12]
Radiative transfer : Chandrasekhar, S. (Subrahmanyan), 1910
Jul 2, 2019 · Radiative transfer, 393 p. : Unabridged and slightly revised version of the work first published in 1950.
[13]
[PDF] Fundamentals of Radiative Transfer
ELEMENTARY PROPERTIES OF RADIATION Electromagnetic radiation can be decomposed into a spectrum of con- stituent components by a prism, grating, or other ...
[14]
[PDF] (AST 461) LECTURE 1: Basics of Radiation Transfer
Specific intensity is constant along a ray in free space. Here's why: Consider all rays passing through both dA1 and dA2 and use conservation of energy: (fig 3).Missing: nu^ | Show results with:nu^
[15]
[PDF] 12.815 Atmospheric Radiation - MIT OpenCourseWare
Radiative Transfer in a Scattering Atmosphere. 1. Coordinate system in a ... We may consider writing the phase function in terms of Legendre polynomials.
[16]
Rayleigh Scattering: Phase Function
The Rayleigh phase function is given in terms of scattering angles or phase angles by P ray ( Theta ) = (3/4) (1 + cos 2 ( Theta )) = (3/4)(1 + cos 2 ( theta ) ...
[17]
Exact solution of the standard transfer problem in a stellar atmosphere
Apr 3, 2005 · It consists in solving the radiative transfer equation in a homogeneous and isothermal plane-parallel atmosphere, with light scattering taken ...
[18]
Exact solution of the standard transfer problem in a stellar atmosphere
It consists in solving the radiative transfer equation in a homogeneous and isothermal plane-parallel atmosphere, with light scattering taken as isotropic and ...
[19]
Exact solution of a problem in a stellar atmosphere using the ...
including the effects of scattering, absorption and thermal emission ̵.
[20]
A New Look at the Discrete Ordinate Method for Radiative Transfer ...
Thus, the present approach offers an efficient and reliable computational scheme that lends itself readily to the solution of a variety of radiative transfer ...
[21]
Local Thermodynamic Equilibrium - an overview - ScienceDirect.com
Such an equilibrium would be reached if the electron–atom collisional excitation rates are much greater than the radiative decay rates. This condition is ...
[22]
[1406.3553] Basics of the NLTE physics - ar5iv - arXiv
If the collisional processes dominate, then the distribution is close to the equilibrium one. On the other hand, if radiation processes dominate, then the ...
[23]
[PDF] Chapter 3: Formal transfer equation
A medium that has τ " 1 is called optically thick while a medium that has τ # 1 is called optically thin. Rather than using mean free paths, in radiative ...Missing: perturbation | Show results with:perturbation
[24]
[PDF] CHAPTER 26 Radiative Transfer Consider an incoming signal of ...
To get some further insight into the source function and radiative transfer in general, consider form II of the radiate transfer equation. If Iν > Sν then.
[25]
[PDF] 7. Non-LTE – basic concepts - Astronomy in Hawaii
Oct 13, 2003 · LTE valid: low temperatures & high densities non-valid: high temperatures & low densities. Page 5. Spring 2016. LTE vs NLTE in hot stars.
[26]
[PDF] INTRODUCTION TO NON-LTE RADIATIVE TRANSFER AND ...
where the additional terms represent absorption and emission of photons by a “continuum” source. Here r is the ratio of continuum to line opacity and Sc is the ...
[27]
[PDF] RADIATIVE TRANSFER IN STELLAR ATMOSPHERES
5. Invariance of the specific intensity. The area element dA emits radiation towards dA'. In the absence of any matter. between emitter and receiver (no ...
[28]
[PDF] PHYS-633: Introduction to Stellar Astrophysics
Radiation Transfer: Absorption and Emission in a Stellar Atmosphere ... a general equation for radiative transfer, let us first consider the case with just ...
[29]
[PDF] ASTR 4003/8003, Class 2: Radiative transfer II
Using the Eddington approximation, we can rewrite our first moment equation,. Equation 28, as. ∂Kν. ∂τν. = Hν = 1. 3. ∂Jν. ∂τ. (33). Taking the derivative of ...
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[PDF] 3. The formal radiative transfer equation
A medium can be optically thick at one wavelength and optically thin at another. • Is the sun optically thick at all wavelengths? • Link between the optical ...
[31]
On the Use of Variable Eddington Factors in Non-LTE Stellar ...
Abstract. It is shown that by use of variable Eddington factors, the accuracy of difference-equation solutions of transfer problems may be greatly improved.