Radiation transport

Under radiative transfer (also radiative transfer ) refers to the description of the propagation of radiation (i. A. light as an example of electromagnetic radiation ) through a medium. Radiation transport plays an essential role , especially in the fields of astrophysics . The theory of stellar atmospheres , the formation of star spectra or the formation of interstellar line spectra are based on radiation transport. Furthermore, radiation transport is also required to understand spectroscopy in many other physical (e.g. for analyzes in plasma physics ) and technical (e.g. for various non-thermal light sources ) areas. Radiation transport also plays a major role in the greenhouse effect of the earth's atmosphere.

The process

When electromagnetic radiation propagates in a medium (regardless of whether in photon observation or field observation), it is absorbed , scattered or can leave the medium (in particular by its atoms and ions ) . These processes or the description of these processes are called radiation transport . In this process, the radiation of different wavelengths is influenced differently depending on the properties of the medium (especially its atoms and ions). The aim of a radiation transport calculation is to calculate the emitted light (either as a whole spectrum or individual spectral lines ) or the radiation field inside the medium; either to predict a spectrum or to draw conclusions about the composition of the medium.

Radiation transport calculations in the real sense only take into account the effect of radiation on the medium to a limited extent. So z. B. the energy deposition in the medium (that is, its heating) through absorption in the radiation transport is explicitly treated just as little as the cooling in the case of predominant emissions from the medium. If one does not use other physical laws to include these effects, one assumes that other processes (e.g. convection or heat conduction ) still keep the temperature structure in the medium constant. A physically more complete simulation therefore includes, in addition to energy conservation and other physical laws, the radiation transport as part of the more comprehensive model.

The radiation transport equation

The radiation transport equation forms the foundation of the radiation transport. It links the radiation density L with the absorption coefficient , the scatter coefficient and the emission power j of the material to be passed. The absorption and scattering coefficients, as well as the emission power u. a. on the density and temperature of the material. In astrophysics, such as in the following equations, however, the radiation density is L usually as a specific intensity I referred to. ${\ displaystyle \ kappa}$ ${\ displaystyle \ sigma}$

In a simple one-dimensional, time-independent form it reads:

{\ displaystyle {\ frac {{\ rm {d}} I} {{\ rm {d}} z}} = - (\ kappa + \ sigma) I + j}

In a very general form, it is along the direction ${\ displaystyle {\ vec {n}}}$

{\ displaystyle \ left [{\ frac {1} {c}} {\ frac {\ partial} {\ partial t}} + ({\ vec {n}} \ cdot \ nabla) \ right] I = - ( \ kappa + \ sigma) I + j}

properties

Since the emission performance of the material is partly caused by scattering, and since the scattering itself is an integral over the specific intensity to be calculated, the radiation transport equation is an integro-differential equation .

Derivation

It is common to either postulate the radiation transport equation or to derive it from a Boltzmann transport formalism for photons . Ultimately, however, it must then be postulated that photon transport can be described by the Boltzmann equation. Alternatively, the time-independent radiation transport equation can be derived from Maxwell's equations if the microphysical properties of particles of any shape, orientation and independently scattering are used.

Solution of the radiation transport equation

A so-called formal solution of the radiation transport equation can be given analytically. However, this can only be formulated into a real, usable solution for special cases.

The numerical solution of the radiative transport equation is i. A. very expensive. The most modern and stable process is the so-called "Accelerated Lambda Iteration". Mathematically , this corresponds to a Gauss-Seidel method . For three-dimensional systems and simple absorption properties, the radiation transport problem can also be solved with Monte Carlo simulations . The discrete ordinate method originally developed to solve the Boltzmann equation is a tried and tested method in engineering for three-dimensional systems with any properties .

Borderline cases

For monochromatic light and thermal radiation, which is negligible in comparison to its intensity, the solution of the one-dimensional radiation transport equation is converted into the Lambert-Beer law .
Inside stars , the so-called diffusion approximation applies to the transport of radiation. With their help, the radiation density integrated to the luminosity can be expressed as. ${\ displaystyle L (r) = - {\ frac {4 \ pi acr ^ {2}} {3 \ kappa}} {\ frac {\ mathrm {d} T ^ {4}} {\ mathrm {d} r }}}$
For three-dimensional arrangements, certain ray tracing algorithms (see volume graphics or volume scattering ) also represent an approximate solution to the radiation transport problem.

Problems of radiation transport today

3D radiation transport: Three-dimensional, numerical simulations are very memory and computationally intensive; especially if detailed spectra are to be calculated.
Neutrino radiation transport: The passage of neutrinos through a medium can also be treated as a radiation transport problem, analogous to the passage of photons. Neutrino radiation transport plays a central role in supernova explosions.
Radiation hydrodynamics: In cases in which the energy of the radiation is comparable to or greater than the internal energy of the medium, the hydrodynamics are decisively influenced by the radiation. Numerical simulations of such situations must therefore also take appropriate account of the radiation transport.

literature

Dimitri Mihalas, Barbara Weibel Mihalas: Foundations of Radiation Hydrodynamics . Dover Publications, 2000 ISBN 0486409252
Subrahmanyan Chandrasekhar : Radiative Transfer . Dover Publications, 1960 ISBN 0486605906
Dimitri Mihalas: Stellar Atmospheres . WH Freeman & Co. 1978 ISBN 0716703599 (standard work, no longer in print)
Albrecht Unsöld : Physics of the stellar atmospheres with special consideration of the sun . Springer-Verlag, 1955 (standard work, no longer in print)
Rob Rutten: Radiative Transfer in Stellar Atmospheres . Textbook available online ( PDF )
H. Scheffler, H. Elsässer: Physics of the stars and the sun . Wissenschaftsverlag, 1990 ISBN 3-411-14172-7

Web links

Technology lexicon: Radiation transport equation

Detailed literature and sources

↑ Joachim Oxenius: Kinetic Theory of Particles and Photons . Springer-Verlag, 1986 ISBN 038715809X
↑ Mishchenko, Applied Optics, 41, 2002, p. 7114-34
^ BG Carlson, KD Lathrop: "Transport Theory: The Method of Discrete Ordinates", Computing Methods in Reactor Physics, Argonne National Laboratory, 1968

[1] Joachim Oxenius: Kinetic Theory of Particles and Photons . Springer-Verlag, 1986 ISBN 038715809X

[2] Mishchenko, Applied Optics, 41, 2002, p. 7114-34

[3] BG Carlson, KD Lathrop: "Transport Theory: The Method of Discrete Ordinates", Computing Methods in Reactor Physics, Argonne National Laboratory, 1968