\subsection{Absorption/Emission is not Reflection}
\label{Sec:Radiation}
%
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%
\subsubsection{An inconvenient popularization of physics}
\label{AIPP}
%
 
Figure~\ref{fig:AnInconvenientTruth}
is a screenshot from a controversial award-winning
\lq\lq documentary film\rq\rq\ 
about 
\lq\lq climate change\rq\rq, 
specifically \lq\lq global warming\rq\rq, 
starring Al Gore, the former United States 
Vice President, and directed by Davis Guggenheim\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{AIT2006Book,AIT2006}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%
This movie has been supported by managers and
policymakers around the world and has been shown 
in schools and in outside events, respectively.
Lewis wrote an interesting 
\lq\lq A Skeptic's Guide to An Inconvenient Truth\rq\rq\
evaluating Gore's work in detail\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Lewis2006}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%
\begin{figure}[hbtp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_An_Inconvenient_Truth_.eps,width=4.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=4.00]{PictureSmall_An_Inconvenient_Truth_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_An_Inconvenient_Truth_.png}}
}{}
\vspace*{8pt}
\caption{A very popular physical error illustrated
         in the movie \lq\lq An Inconvenient truth\rq\rq\
         by Davis Guggenheim featuring Al Gore (2006).}
\label{fig:AnInconvenientTruth}
\end{figure}
%

From the view of a trained physicist, 
Gore's movie is rather grotesque, 
since it is shockingly wrong. 
Every licensed radio amateur%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{\textit{Callsign of R.D.T.:} DK8HH}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
knows that
what is depicted in 
Figure~\ref{fig:AnInconvenientTruth}
would be true only,
%
\begin{itemize}
\item if the radiation graphically
      represented here was long wave 
      or short wave radiation;
\item if the reflecting sphere was 
      a certain layer of the ionosphere\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Budden1966}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%      
\end{itemize}
%
Short waves 
%
(e.g.\ in the 20\,m/14\,MHz band)
%
are reflected 
by the F layer of the ionosphere
%
(located 120\,-\,400\,km above the Earth's surface)
%
enabling transatlantic connections (QSOs).
Things depend pretty much on the solar
activity, i.e.\ on the sun spot cycle,
as every old man (OM) knows well.
The reflective characteristics of the 
ionosphere diminish above about
% 
${\rm 30}\,{\rm MHz}$.
%
In the very high frequency (VHF) bands 
%
(e.g.\ 2\,m/144\,MHz band)
% 
one encounters the so called Sporadic-E clouds
%
(90\,-\,120\,km above the Earth's surface),
%
which still allow QSOs from Germany to Italy,
for example.
On the other hand at the extremely low 
frequencies (ELF, i.e.\ frequency range 3\,-\,30\,Hz)
%
the atmosphere of the Earth behaves as 
a cavity and one encounters the so called
Schumann resonances\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Schumann1952}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
These may be used 
to estimate a lower bound for the mass of the photon%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\footnote{As a teaching assistant at Hamburg University/DESY, R.D.T.\ 
           learned this from Professor Herwig Schopper.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
and, surprisingly, appear in the climate change discussion\TschSpace% 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Fullekrug2006}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 

However, the radio signal of Al Gore's cellular phone 
(within the centimeter range) does not travel around 
the world and so does not Bluetooth, Radar, microwave 
and infrared radiation (i.e.\ electromagnetic waves 
in the sub millimeter range). 

Ionosphere Radars typically work
in the $6\,{\rm m}$ Band,
i.e.\ at $50\,{\rm MHz}$.
Meteorological Radars work 
in the 0.1\,-\,20\,cm range 
(from 90\,GHz\ down to 1.5\,GHz),
those 
in the 3\,-\,10\,cm range
(from 10\,GHz\ down to 3\,GHz)
are used for wind finding and weather watch\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Atlas1990}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%
It is obvious, that Al Gore confuses 
the ionosphere with the tropopause,
the region in the atmosphere, that
is the boundary between the troposphere 
and the stratosphere.
The latter one is located 
between 
$6\ {\rm km}$ (at the poles)
and 
$17\ {\rm km}$ (at the equator) 
above the surface of the Earth.%
%
\footnote{Some climatologists claim that there is a
          ${\rm CO}_2$ layer in the troposphere 
          that traps or reflects
          the infrared radiation coming from the ground.}
%

Furthermore, Al Gore confuses
\textit{absorption/emission} 
with \textit{reflection}.
Unfortunately, this is also done implicitly and
explicitly in many climatologic papers, often
by using the vaguely defined terms
\lq\lq re-emission\rq\rq,
\lq\lq re-radiation\rq\rq\
and
\lq\lq backradiation\rq\rq.
%
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%
\subsubsection{Reflection}
%
When electromagnetic waves move 
from a medium of a given refractive index $n_1$ into 
a second medium with refractive index $n_2$, both 
reflection and refraction of the waves may occur\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Born1997}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
In particular, 
when the jump of the refractive index 
occurs within a length of the order
of a wavelength, there will be a reflection.
The fraction of the intensity of incident 
electromagnetic wave that is reflected 
from the interface is given by the 
reflection coefficient $R$, 
the fraction refracted 
at the interface is given by the 
transmission coefficient $T$. 
The Fresnel equations, which are 
based on the assumption that the 
two materials are both dielectric, 
may be used to calculate 
the reflection coefficient $R$ and 
the transmission coefficient $T$ 
in a given situation.

In the case of a normal incidence %
the formula for the reflection 
coefficient is
%
\begin{equation}
R
=
\left(
\frac{n_2 - n_1}{n_2 + n_1}
\right)
^2
\end{equation}
%
In the case of strong absorption 
(large electrical conductivity $\sigma$)
simple formulas can be given for larger 
angles $\gamma$ of incidence, as well 
(Beer's formula): 
%
\begin{equation}
R_s
=
\frac{(n_2-n_1\cos\gamma)^2 + n_2^2\sigma^2}
     {(n_2+n_1\cos\gamma)^2 + n_2^2\sigma^2}
\end{equation}
%
and
%
\begin{equation}
R_p
=
\frac{(n_1-n_2\cos\gamma)^2 + n_2^2\sigma^2\cos^2\gamma}
     {(n_1+n_2\cos\gamma)^2 + n_2^2\sigma^2\cos^2\gamma}
\end{equation}
%
When the jump of the refractive index 
occurs within a length of the order
of a wavelength, there will be a reflection,
which is large at high absorption. 
In the case of gases 
this is only possible 
for radio waves of a comparatively 
long wave length in the ionosphere,
which has an electrical conductivity, 
at a diagonal angle of incidence. 
There is no reflection in the 
homogeneous absorbing range.
As already elucidated 
in Section~\ref{AIPP}
this has been well-known to radio 
amateurs ever since and affects
their activity e.g.\ in the 
15 m band, but never in the 
microwave bands. 
On the other hand,
most glasses absorb the infrared light 
almost completely at approximately
$1\,\mu{\rm m}$
and longer wavelength: 
therefore, the reflection 
of the infrared waves for normal 
glasses is very high.

For dielectric media, whose electrical conductivity is zero,
one cannot use Beer's formulas. This was a severe problem
in Maxwell's theory of light.
%
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% 
\subsubsection{Absorption and Emission}
%
If an area is in thermodynamical equilibrium 
with a field of radiation, 
the intensity 
${\rm E}_\nu$ 
(resp.\ ${\rm E}_\lambda$)
emitted by the unit solid angle
into a frequency unit 
(resp.\ a wavelength unit) 
is equal to the absorptance
${\rm A}_\nu$ 
(resp.\ ${\rm A}_\lambda$)
multiplied with 
a universal frequency function ${\rm B}_\nu(T)$ 
(resp.\ a wavelength function ${\rm B}_\lambda(T)$) 
of the absolute temperature $T$. 
One writes, respectively, 
%
\begin{eqnarray}
{\rm E}_\nu 
&=&
{\rm A}_\nu \cdot {\rm B}_\nu (T)
\\
{\rm E}_\lambda 
&=&
{\rm A}_\lambda \cdot {\rm B}_\lambda (T)
\end{eqnarray}
%
This is a theorem by \textit{Kirchhoff}. 
The function 
${\rm B}_\nu(T)$ 
(resp.\ ${\rm B}_\lambda(T)$)
is called the 
\textit{Kirchhoff-Planck-function.} 
It was already considered in Section~\ref{SBlaw}.

The \textit{reflectance} is, respectively,
%
\begin{eqnarray}
{\rm R}_\nu &=& 1- {\rm A}_\nu \\
{\rm R}_\lambda &=& 1- {\rm A}_\lambda
\end{eqnarray}
%
and lies between zero and one,
like the \textit{absorptance} 
${\rm A}_\nu$. 
If ${\rm R}$ is equal to zero 
and ${\rm A}$ is equal 
to one, the body is called a 
perfect black body. 
The emissivity is largest 
for a perfect black body. 
The proposal to realize a 
perfect black body by using a cavity 
with a small radiating opening 
had already been made by Kirchhoff 
and is visualized 
in Figure~\ref{fig:cavity}.
%
\begin{figure}[hbtp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_cavity_.eps,width=1.50in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_cavity_.png}}
\vspace*{-16pt}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_cavity_.png}}
\vspace*{-16pt}
}{}
\vspace*{8pt}
\caption{A cavity realizing a perfect black body.}
\label{fig:cavity}
\end{figure}
%
For this reason, the emission of 
a black body for 
${\rm A}_\nu=1$ 
(resp.\ ${\rm A}_\lambda=1$)
is called \textit{cavity radiation}. 
The emitted energy comes from the walls, 
which are being held at a fixed temperature. 
If this is realized with a 
part of a body's surface, 
it will become clear, that 
these points of view will only be 
compatible, if the electromagnetic 
radiation is emitted and absorbed 
by an extremely thin surface layer.
For this reason, it is impossible 
to describe the volumes of gases 
with the model of black cavity 
radiation.
Since thermal radiation is electromagnetic 
radiation, this radiation would have to be 
caused by thermal motion in case of gases, 
which normally does not work effectively 
at room temperatures. At the temperatures 
of stars the situation is different:
The energy levels of the atoms are 
thermally excited by impacts.
%
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%
\subsubsection{Re-emission}
%
In case of radiation transport calculations, 
Kirchhoff's law is \lq\lq generalized\rq\rq\ 
to the situation, in which the corresponding 
formula for the emission, or respectively, 
for the absorption 
(per unit length along the direction $ds$) 
is supposed to be applicable
%
\begin{equation}
\varepsilon_\nu ds 
=
\alpha_\nu ds
\cdot
{\rm B}_\nu (T)
\label{Eq:generalizedKirchhoff}
\end{equation}
%
The physical meaning of this 
\lq\lq generalization\rq\rq\ 
can be seen most easily, 
if the above mentioned Kirchhoff law 
is mathematically extracted out 
of this formula. For this,
one may introduce
%
\begin{eqnarray}
\varepsilon_\nu &=& {\rm E}_\nu \delta(s-s_0) 
\\
\alpha_\nu      &=& {\rm A}_\nu \delta(s-s_0)
\end{eqnarray}
%
with a $\delta$-density 
localized at the interface.
Physically, this means that all 
of the absorption and emission 
comes out of a thin superficial plane. 
Just like with the correct Kirchhoff law, 
use is made of the fact, 
that all absorbed radiation 
is emitted again, as otherwise 
the volume area would raise 
its temperature in thermal balance.

This assumption is called 
the assumption of  
\textit{Local Thermodynamical Equilibrium (LTE)}.
Re-emission does never mean reflection, 
but, rather, that the absorption 
\textit{does not cause any rise 
of temperature in the gas.}

An important physical difference 
to the correct Kirchhoff law lies
in the fact, that there is 
no formula for the absorption 
per linear unit analogous to 
%
\begin{equation}
{\rm R}_\nu  = 1 - {\rm A}_{\nu}
\end{equation}  

With $\rho$ being the density of the medium
one can define a 
\textit{absorption coefficient} $\kappa_\nu$
and an 
\textit{emission coefficient} $j_\nu$,
respectively, by setting 
%
\begin{eqnarray}
\alpha_\nu      &=& \kappa_\nu \rho 
\label{Eq:alphanu}
\\
\varepsilon_\nu &=& j_\nu \rho 
\label{Eq:varepsilonnu}
\end{eqnarray}
% 
The ratio of the emission to the absorption coefficient
%
\begin{equation}
{\rm S}_\nu = \frac{j_\nu}{\kappa_\nu}
\label{Eq:sourcefunction}
\end{equation}
%
describes the re-emission of the radiation 
and is called the \textit{source function}.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
\subsubsection{Two approaches of Radiative Transfer}
%
In a gas the radiation intensity 
of an area changes in the direction 
of the path element $ds$ according to
%
\begin{equation}
-
\frac {d{\rm I}_\nu} {ds}
=
\alpha_\nu{\rm I}_\nu
-
\varepsilon_\nu
\end{equation}
%
With the aid of the functions introduced in 
Equations
(\ref{Eq:alphanu}) - (\ref{Eq:sourcefunction}) 
this can be expressed as
%
\begin{equation}
\frac {{\rm 1}} {\kappa_\nu \varrho}
\frac {d{\rm I}_\nu} {ds}
=
{\rm I}_\nu 
-
{\rm S}_\nu
\label{Eq:RTE} 
\end{equation}
%
This equation is called the \textit{radiative transfer equation}.

Two completely different approaches
show that this emission function is 
not just determined by physical laws\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Chandrasekhar1960}:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{enumerate}
%
\item
%
The usual one, i.e.\ the one in case of LTE, 
is given by the ansatz
%
\begin{equation}
{\rm S}_\nu ( x,y,z ; l,m,n )
=
{\rm B}_\nu ( {\rm T} ( x,y,z ; l,m,n ) )
\end{equation}
%
where the coordinates $(x,y,z)$ and the direction cosines $(l,m,n)$ define the point and
the direction to which $S_\nu$ and $B_\nu$ (resp.\ $T$) refer. 
%
This approach is justified 
with the aid of  
the Kirchhoff-Planck-function 
${\rm B}_\nu$
and the \lq\lq generalized\rq\rq\
Kirchhoff law introduced in 
Equation 
(\ref{Eq:generalizedKirchhoff}).
This assumption of 
\textit{Local Thermodynamical Equilibrium (LTE)} 
is ruled out by many scientists
even for the extremely hot atmospheres 
of stars. The reader is referred to
Chandrasekhar's classical book on
radiative transfer\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{Chandrasekhar1960}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
LTE does only bear a certain significance 
for the radiation transport calculations, 
if the absorption coefficients were 
not dependent on the temperature, 
which is not the case at low temperatures. 
Nevertheless, in modern climate 
model computations, this approach is 
used unscrupulously\TschSpace%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cite{DOE1985}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\item
%
Another approach is the 
\textit{scattering atmosphere}
given by
%
\begin{equation}
{\rm S}_\nu
=
\frac{1}{4\pi}
\int_0^{\pi}
\int_0^{2\pi}
{\rm p} ( \vartheta, \varphi ; \vartheta', \varphi' )
\,
{\rm I}_\nu ( \vartheta', \varphi' )
\,
\sin \vartheta'
d \vartheta'
d \varphi'
\end{equation}
%
\end{enumerate}

These extremely different approaches show, 
that even the physically well-founded 
radiative transfer calculations are somewhat arbitrary. 
Formally, the radiative transfer Equation
(\ref{Eq:RTE}) 
can be integrated leading to 
%
\begin{equation}
{\rm I}_\nu(s)
=
{\rm I}_\nu(0)
\exp (-\tau(s,0))
+
\int_0^s
{\rm S}_\nu(s')
\exp (-\tau(s,s'))
\kappa_\nu
\varrho
\,
ds'
\end{equation}
%
with the optical thickness
%
\begin{equation}
\tau(s,s') 
=
\int_{s'}^{s}
\kappa_\nu
\, 
\varrho
\,
ds''
\end{equation}
%
The integrations for the separate directions 
are independent of one another.
In particular, 
the ones up 
have nothing to do 
with the ones down.
It cannot be overemphasized,
that differential equations only allow 
the calculation of changes on the basis 
of known parameters. The initial values 
(or boundary conditions) cannot be derived 
from the differential equations to be solved.
In particular, this even holds for this simple
integral.

If one assumes that the temperature of a volume element
should be constant, one cannot calculate a rising 
temperature.
