and Cosmology

A. The Electromagnetic Radiation Field
417
In this appendix, we will briefly review the most important
properties of a radiation field. We thereby assume
that the reader has encountered these quantities already
in a different context.
A.1 Parameters of the Radiation Field
The electromagnetic radiation field is described by the
specific intensity I ν , which is defined as follows. Consider
a surface element of area d A. The radiation energy
which passes through this area per time interval dt from
within a solid angle element dω around a direction described
by the unit vector n, with frequency in the range
between ν and ν + dν,is
dE = I ν d A cos θ dt dω dν ,
(A.1)
where θ describes the angle between the direction n of
the light and the normal vector of the surface element.
Then, d A cos θ is the area projected in the direction of
the infalling light. The specific intensity depends on the
considered position (and, in time-dependent radiation
fields, on time), the direction n, and the frequency ν.
With the definition (A.1), the dimension of I ν is energy
per unit area, time, solid angle, and frequency, and it is
typically measured in units of erg cm −2 s −1 ster −1 Hz −1 .
The specific intensity of a cosmic source describes its
surface brightness.
The specific net flux F ν passing through an area element
is obtained by integrating the specific intensity
over all solid angles,
∫
F ν = dω I ν cos θ.
(A.2)
The flux that we receive from a cosmic source is defined
in exactly the same way, except that cosmic sources
usually subtend a very small solid angle on the sky.
In calculating the flux we receive from them, we may
therefore drop the factor cos θ in (A.2); in this context,
the specific flux is also denoted as S ν .However,
in this Appendix (and only here!), the notation S ν will
be reserved for another quantity. The flux is measured
in units of erg cm −2 s −1 Hz −1 . If the radiation field is
isotropic, F ν vanishes. In this case, the same amount
of radiation passes through the surface element in both
directions.
The mean specific intensity J ν is defined as the
average of I ν over all angles,
J ν = 1 ∫
dω I ν ,
(A.3)
4π
so that, for an isotropic radiation field, I ν = J ν . The
specific energy density u ν is related to J ν according to
u ν = 4π c J ν
(A.4)
where u ν is the energy of the radiation field per volume
element and frequency interval, thus measured in
erg cm −3 Hz −1 . The total energy density of the radiation
is obtained by integrating u ν over frequency. In the
same way, the intensity of the radiation is obtained by
integrating the specific intensity I ν over ν.
A.2 Radiative Transfer
The specific intensity of radiation in the direction of
propagation between source and observer is constant,
as long as no emission or absorption processes are occurring.
If s measures the length along a line-of-sight,
the above statement can be formulated as
dI ν
ds = 0 .
(A.5)
An immediate consequence of this equation is that the
surface brightness of a source is independent of its
distance. The observed flux of a source depends on
its distance, because the solid angle, under which the
source is observed, decreases with the square of the
distance, F ν ∝ D −2 (see Eq. A.2). However, for light
propagating through a medium, emission and absorption
(or scattering of light) occurring along the path
over which the light travels may change the specific intensity.
These effects are described by the equation of
radiative transfer
dI ν
ds =−κ ν I ν + j ν .
(A.6)
The first term describes the absorption of radiation
and states that the radiation absorbed within a length
interval ds is proportional to the incident radiation.
Peter Schneider, The Electromagnetic Radiation Field.
In: Peter Schneider, Extragalactic Astronomy and Cosmology. pp. 417–423 (2006)
DOI: 10.1007/11614371_A © Springer-Verlag Berlin Heidelberg 2006