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3. Radiation Field Basics II Moments of the Radiation Field
Radiation Field Basics II. • Rutten: 2.1 2.2. • Intensity moments. • Emission
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3. Radiation Field Basics II
•Rutten: 2.1, 2.2 •Intensity moments •Emission, extinction, source function •Equation of radiation transfer •Optical depthMoments of the Radiation Field
We have defined the moment operator Moperating on f: -=1 1)(d21][Mmmnnff
The first three moments of the specific intensity are historically named J(zerothmoment), H(first), and K(second):W=W=W=
dcos41dcos41d
412qpq ppnnnnnnIKIHIJ 1 121
11 1 d),(2
1)(d),(21)(d),(21)(mmmmmmmmnnnnnnzIzKzIzHzIzJ
For a plane parallel atmosphere (no fdependence), we get These moments are used in solving the equation of radiation transfer. As shown previously, physically Jand Hare related to the mean intensity of radiation and the flux respectively. Kis related to the radiation pressure by nnpKcp4=Photon Interactions
•Scattering: change direction (energy slightly) •Absorption: energy added to K.E. of particles: photon thermalized •Emission: energy taken from thermal energy of particlesEmission Coefficient
WºdddddnnntVjE
Energy dEnadded to radiation in the volume dVin the frequency range dnin time dtin direction about solid angle dW. Energy added can be due to stimulated emission, spontaneous emission, thermal emission, or energy scattered into the beam. The intensity contribution to a beam from local emission along a path length dsis:j ndefined by: ds)()(dsjsInn=Extinction Coefficient
The energy removed from a beam along a path length dsdefines the extinction coefficient. It may be defined per particle, permass, or per volume ds)(dnnnaIsI-= where anis in units of m-1. The definition per mass is ds)(dnIsInnns-= snis the monochromatic extinction coefficient or cross section per particle (m2) and nthe particle density (m-3). The definition per path length is ds)(drknnnIsI-= where knhas units m2kg-1and ris the density (kg m-3).Source Function
A very important quantity in radiation transfer theory, whose name will become clear soon, is the Source Function. It is has the same units as intensity and is defined: nn na jSº When multiple processes contribute to the local emission and extinction, the total source function is: nn na jStotFor example, in a spectral line:
nnnn nnnn nhh aa++=++=1totlc lclcSSjjS with hn= anl/ ancthe line-to-continuum extinction ratio; Snc, Snl the continuum and line source functions.Equation of Radiation Transfer
The equation of radiation transfer (ERT) along a rayis: ssIssjsIssIsId)(d)()()d()(dnnnnnna-=-+=3. Radiation Field Basics II
•Rutten: 2.1, 2.2 •Intensity moments •Emission, extinction, source function •Equation of radiation transfer •Optical depthMoments of the Radiation Field
We have defined the moment operator Moperating on f: -=1 1)(d21][Mmmnnff
The first three moments of the specific intensity are historically named J(zerothmoment), H(first), and K(second):W=W=W=
dcos41dcos41d
412qpq ppnnnnnnIKIHIJ 1 121
11 1 d),(2
1)(d),(21)(d),(21)(mmmmmmmmnnnnnnzIzKzIzHzIzJ
For a plane parallel atmosphere (no fdependence), we get These moments are used in solving the equation of radiation transfer. As shown previously, physically Jand Hare related to the mean intensity of radiation and the flux respectively. Kis related to the radiation pressure by nnpKcp4=Photon Interactions
•Scattering: change direction (energy slightly) •Absorption: energy added to K.E. of particles: photon thermalized •Emission: energy taken from thermal energy of particlesEmission Coefficient
WºdddddnnntVjE
Energy dEnadded to radiation in the volume dVin the frequency range dnin time dtin direction about solid angle dW. Energy added can be due to stimulated emission, spontaneous emission, thermal emission, or energy scattered into the beam. The intensity contribution to a beam from local emission along a path length dsis:j ndefined by: ds)()(dsjsInn=Extinction Coefficient
The energy removed from a beam along a path length dsdefines the extinction coefficient. It may be defined per particle, permass, or per volume ds)(dnnnaIsI-= where anis in units of m-1. The definition per mass is ds)(dnIsInnns-= snis the monochromatic extinction coefficient or cross section per particle (m2) and nthe particle density (m-3). The definition per path length is ds)(drknnnIsI-= where knhas units m2kg-1and ris the density (kg m-3).Source Function
A very important quantity in radiation transfer theory, whose name will become clear soon, is the Source Function. It is has the same units as intensity and is defined: nn na jSº When multiple processes contribute to the local emission and extinction, the total source function is: nn na jStotFor example, in a spectral line:
nnnn nnnn nhh aa++=++=1totlc lclcSSjjS with hn= anl/ ancthe line-to-continuum extinction ratio; Snc, Snl the continuum and line source functions.Equation of Radiation Transfer
The equation of radiation transfer (ERT) along a rayis: ssIssjsIssIsId)(d)()()d()(dnnnnnna-=-+=- moments-là
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