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\subsection{Radiation Basics}
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\subsubsection{Introduction}
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For years, the warming mechanism in real greenhouses,
paraphrased as \lq\lq the greenhouse effect\rq\rq,
has been commonly misused to explain the conjectured 
atmospheric greenhouse effect. 
In school books, in popular scientific
articles, and even in high-level scientific debates,
it has been stated that the mechanism observed
within a glass house bears some similarity to
the anthropogenic global warming.
Meanwhile, even mainstream climatologists admit
that the warming mechanism in real glass houses
has to be distinguished strictly from the 
claimed ${\rm CO}_2$ greenhouse effect.

Nevertheless, one should have a look 
at the classical glass house problem
to recapitulate some fundamental principles
of thermodynamics and radiation theory.
Later on, the relevant radiation dynamics 
of the atmospheric system will be elaborated on
and distinguished from the glass house set-up. 

\textit{Heat} is the kinetic energy of molecules and atoms
and will be transferred by contact or radiation.
Microscopically both interactions are mediated by photons.
In the former case, which is governed by the
Coulomb resp.\ van\,der\,Waals interaction 
these are the virtual or off-shell photons,
in the latter case these are the real or on-shell
photons. The interaction between photons and
electrons (and other particles that are electrically
charged or have a non-vanishing magnetic momentum)
is microscopically described by the laws of quantum 
theory. Hence, in principle, thermal conductivity
and radiative transfer may be described in a unified framework.
However, the non-equilibrium many body problem is 
a highly non-trivial one and subject to the discipline
of physical kinetics unifying quantum theory
and non-equilibrium statistical mechanics.

Fortunately, an analysis of the problem by applying the
methods and results of classical radiation theory already leads 
to interesting insights.  
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\subsubsection{The infinitesimal specific intensity}
\label{Sec:ISI}
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In classical radiation theory\TschSpace%
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\cite{Chandrasekhar1960}
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%
the main quantity is the 
\textit{specific intensity} 
$I_\nu$.
It is defined in terms of the 
\textit{amount of radiant energy}
$dE_\nu$ 
in a specified frequency interval 
$[\nu,\nu+d\nu]$
that is transported 
across an area element 
$d\textbf{F}_1$
in direction of another area element 
$d\textbf{F}_2 $ 
during a time
$dt$:
 
%
\begin{equation}
dE_\nu
=
I_\nu\,
d\nu\,
dt\,
\frac { (\textbf{r}\,d\textbf{F}_1)\,(\textbf{r}\,d\textbf{F}_2) } 
      { |\textbf{r}|^4 }
\label{Eq:dEv1}      
\end{equation}
% 
where $\textbf{r}$ is the distance vector 
pointing from $d\textbf{F}_1$ to $d\textbf{F}_2$
(Figure \ref{Fig:dF1dF2}).
%
\begin{figure}[htbp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_specific_intensity_.eps,width=3.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=2.00]{PictureSmall_specific_intensity_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=0.66]{PictureLarge_specific_intensity_.png}}
}{}
\vspace*{8pt}
\caption{The geometry of classical radiation:
         A radiating infinitesimal area 
         $d\textbf{F}_1$
         and 
         an illuminated infinitesimal area 
         $d\textbf{F}_2$
         at distance 
         $\textbf{r}$.}
\label{Fig:dF1dF2}
\end{figure}
%

For a general radiation field one may write
%
\begin{equation}
I_\nu = I_\nu(x,y,z;l,m,n;t)
\end{equation}
%
where 
$(x,y,z)$ denote the coordinates,
$(l,m,n)$ the direction cosines,
$t$ the time, respectively, 
to which $I_\nu$ refers.

With the aid of the definition
of the scalar product
Equation (\ref{Eq:dEv1}) 
may be cast into the form 
%
\begin{equation}
dE_\nu
=
I_\nu\,
d\nu\,
dt\,
\cdot
\frac { (\cos\vartheta_1\,d{\rm F}_1) \cdot 
        (\cos\vartheta_2\,d{\rm F}_2) } 
      { {\rm r}^2 }
\label{Eq:dEv2}      
\end{equation}
%
A special case is given by
%
\begin{equation}
\vartheta_2 := 1
\end{equation}
%
With 
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\begin{eqnarray}
\vartheta &:=& \vartheta_1 \nonumber\\
d\sigma   &:=& dF_1        \nonumber\\
d\omega   &:=& dF_2/r^2    \nonumber\\
\end{eqnarray}
%
Equation~(\ref{Eq:dEv2}) becomes
%
\begin{equation}
dE_\nu 
= 
I_\nu\,
d\nu\,
dt\,
\cos\vartheta\,
d\sigma\,
d\omega
\label{Eq:dEv3} 
\end{equation}
%
defining the \textit{pencil of radiation}%
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\cite{Chandrasekhar1960}.
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Equation (\ref{Eq:dEv2}), which will be used below,  
is slightly more general than Equation (\ref{Eq:dEv3}), 
which is more common in the literature.
Both ones can be simplified 
by introducing an \textit{integrated intensity}
%
\begin{equation}
I_0 = \int_0^\infty I_\nu\,d\nu
\end{equation}
%
and a \textit{radiant power} $dP$.
For example, Equation (\ref{Eq:dEv2})
may be cast into the form 
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\begin{equation}
dP
=
I_0
\cdot
\frac { (\cos\vartheta_1\,d{\rm F}_1) \cdot 
        (\cos\vartheta_2\,d{\rm F}_2) } 
      { {\rm r}^2 }
\label{Eq:I}      
\end{equation}
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\subsubsection{Integration}
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When performing integration 
one has to bookkeep the dimensions 
of the physical quantities involved. 
Usually, the area $dF_1$ is integrated
and the equation is rearranged in such 
a way, that there is an intensity $I$
(resp.\ an intensity times 
an area element $IdF$) 
on both sides of the equation.
Three cases are particularly interesting:
%
\begin{itemize}
\item[(a)] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\textit{Two parallel areas with distance $a$.}
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\begin{figure}[htbp]
\begin{center}
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_specific_intensity_parallel_areas_.eps,width=2.50in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_specific_intensity_parallel_areas_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=0.33]{PictureLarge_specific_intensity_parallel_areas_.png}}
}{}
\vspace*{8pt}
\caption{Two parallel areas with distance $a$.}
\label{Fig:sipa}
\end{center}
\end{figure}
%
According to Figure~\ref{Fig:sipa} one may write
%
\begin{equation}
\vartheta_1 = \vartheta_2 =: \vartheta
\end{equation}
%
By setting
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\begin{eqnarray}
r^2             &\,=& r_0^2+a^2 \\
2rdr            &\,=& 2r_0dr_0 \\
\cos\,\vartheta &\,=& \frac{a}{r}
\end{eqnarray}  
%
one obtains
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\begin{eqnarray}
I_{\mbox{\scriptsize\rm parallel areas}}
&=&
\int_0^{2\pi} 
\int_0^{R_0}
I_0 \, \frac{(\cos\,\vartheta)^2}{r^2} \, r_0 dr_0 d\varphi 
\nonumber\\
&=&
\int_0^{2\pi} 
\int_0^{R_0}
I_0 \, \frac{a^2}{r^4} \, r_0 dr_0 d\varphi 
\nonumber\\
&=&
\int_0^{2\pi} 
\int_a^{\sqrt{R_0^2+a^2}}
I_0 \, \frac{a^2}{r^4} \, r dr d\varphi 
\nonumber\\
&=&
2\pi \cdot I_0 \cdot a^2 \cdot 
\int_a^{\sqrt{R_0^2+a^2}}
\frac{1}{r^3} \, dr 
\nonumber\\
&=&
2\pi \cdot I_0 \cdot a^2 \cdot 
\left. - \frac{1}{2r^2} \right|_a^{\sqrt{R_0^2+a^2}}
\nonumber\\
&=&
\pi \cdot I_0 \cdot a^2 \cdot 
\left( \frac{1}{a^2} - \frac{1}{R_0^2+a^2} \right)
\nonumber\\
&=&
\pi \cdot I_0 \cdot 
\frac{R_0^2}{R_0^2+a^2}
\end{eqnarray}
%
\item[(b)] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\textit{Two parallel areas with distance $a \rightarrow 0$}
\\ 
If the distance $a$ is becoming very small whereas $R_0$ is kept finite
one will have
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\begin{equation}
I_{{\mbox{\scriptsize\rm parallel areas\ }}{(a\rightarrow 0)}} 
=
\lim_{a\rightarrow 0}
\left(
\pi\cdot I_0 \cdot
\frac{R_0^2}{R_0^2+a^2}
\right)
=
\pi I_0
\end{equation}
%
This relation corresponds to 
the total half-space intensity 
for a radiation from a unit surface. 
% 
\item[(c)] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  
\textit{The Earth illuminated by the Sun}
\\
With $I_0^{\mbox{\scriptsize\rm Sun}}$ 
being the factor $I_0$ for the Sun
the solar total half-space intensity 
is given by
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\begin{equation}
I_{\mbox{\scriptsize\rm Sun's surface}} = \pi \cdot I_0^{\mbox{\scriptsize\rm Sun}} 
\end{equation} 
%
Setting 
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\begin{eqnarray}
a 
&=&
{{\rm R}_{\mbox{\scriptsize\rm Earth's orbit}}}
\\
R_0
&=&
{{\rm R}_{\mbox{\scriptsize\rm Sun}}}
\end{eqnarray}
%
one gets for the solar intensity at the Earth's orbit  
%
\begin{eqnarray}
I_{\mbox{\scriptsize\rm Earth's orbit}}
&=&
\pi \cdot
I_0^{\mbox{\scriptsize\rm Sun}}
\cdot
\frac{{\rm R}^2_{\mbox{\scriptsize\rm Sun}}}
     {{\rm R}^2_{\mbox{\scriptsize\rm Sun}} +
      {\rm R}^2_{\mbox{\scriptsize\rm Earth's orbit}}}
\nonumber \\
&=&
I_{\mbox{\scriptsize\rm Sun's surface}}
\cdot
\frac{{\rm R}^2_{\mbox{\scriptsize\rm Sun}}}
     {{\rm R}^2_{\mbox{\scriptsize\rm Sun}} +
      {\rm R}^2_{\mbox{\scriptsize\rm Earth's orbit}}}
\nonumber \\
&=&
I_{\mbox{\scriptsize\rm Sun's surface}}
\cdot
\frac{{\rm R}^2_{\mbox{\scriptsize\rm Sun}}}
     {{\rm R}^2_{\mbox{\scriptsize\rm Earth's orbit}}}
\nonumber \\
&=&
I_{\mbox{\scriptsize\rm Sun's surface}}
\cdot
\frac{1}{(215)^2}
\end{eqnarray}
% 
\end{itemize}
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\subsubsection{The Stefan-Boltzmann law}
%
\label{SBlaw}
%
For a perfect black body
and a unit area positioned
in its proximity we can compute
the intensity $I$ with the aid of
the the Kirchhoff-Planck-function,
which comes in two versions
%
\begin{eqnarray}
{\rm B}_\nu (T) 
&=&
\frac {2{\rm h}\nu^3} {c^2}
\left[ \exp 
\left( \frac { {\rm h}\nu } { {\rm k} T } \right) - 1 
\right]^{-1}
\\
{\rm B}_\lambda (T) 
&=&
\frac {2{\rm h}{\rm c}^2} {\lambda^5}
\left[ \exp 
\left( \frac { {\rm h}{\rm c} } { \lambda {\rm k} T } \right) - 1 
\right]^{-1}
\end{eqnarray}
%
that are related to each other by 
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\begin{eqnarray}
{\rm B}_\nu (T) \, d\nu 
=
{\rm B}_\nu (T) \, \frac{d\nu}{d\lambda} \, d\lambda
=
- {\rm B}_\nu (T) \, \frac{{\rm c}}{\lambda^2} \, d\lambda
=:
- {\rm B}_\lambda (T) \,d\lambda   
\end{eqnarray}
%
with
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\begin{equation}
\nu = c / \lambda
\end{equation}
%
where $c$ is the speed of light, $h$ the Planck constant,
$k$ the Boltzmann constant, $\lambda$ the wavelength,
$\nu$ the frequency, and $T$ the absolute temperature, 
respectively. 
Integrating over all frequencies or wavelengths
we obtain the Stefan-Boltzmann $T^4$ law
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\begin{equation}
I
=
\pi \cdot
\int_0^\infty {\rm B}_\nu(T) \, d\nu 
=
\pi \cdot
\int_0^\infty {\rm B}_\lambda (T) \, d\lambda
=
\sigma\,T^4
\end{equation}
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with 
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\begin{equation}
\sigma
=
\pi 
\cdot
\frac{2\pi^4k^4}{15 c^2 h^3}
=
5.670400 
\cdot
10^{-8}
\,
\frac{{\rm W}}{{\rm m}^2{\rm K}^4}
\end{equation}
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One conveniently writes
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\begin{equation}
\sigma
=
5.67
\cdot
\left(
\frac{T}{100}
\right)^4
\,
\frac{{\rm W}}{{\rm m}^2{\rm K}^4}
\end{equation}
%
This is the net radiation energy 
per unit time per unit area 
placed in the neighborhood 
of a radiating plane surface 
of a black body.
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\subsubsection{Conclusion}
\label{Sec:Conclusion}
%
Three facts should be emphasized here:
%
\begin{itemize}
\item In \textit{classical radiation theory}
      radiation is \textit{not} described by a vector field
      assigning to every space point a corresponding vector.
      Rather, with each point of space many rays
      are associated (Figure~\ref{Fig:radiation}). 
      This is in sharp contrast to the modern 
      description of the radiation field as an
      electromagnetic field with the Poynting 
      vector field as the relevant quantity\TschSpace%
      \cite{Jackson1962}.
%
\begin{figure}[htbp]
\ifthenelse{\equal{IJMPB}{\TschStyle}}{%
\centerline{\psfig{file=PictureLarge_specific_intensity_integrated_.eps,width=3.00in}}
}{}%
\ifthenelse{\equal{arXiv}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureSmall_specific_intensity_integrated_.png}}
}{}%
\ifthenelse{\equal{TeX4ht}{\TschStyle}}{%
\centerline{\includegraphics[scale=1.00]{PictureLarge_specific_intensity_integrated_.png}}
}{}
\vspace*{8pt}
\caption{The geometry of classical radiation: 
         Two surfaces radiating against each other.}
\label{Fig:radiation}
\end{figure}
% 
\item The constant $\sigma$ appearing in the $T^4$ law
      is \textit{not} a universal constant of physics.
      It strongly depends on the particular geometry 
      of the problem considered.%
\footnote{For instance, to compute the radiative transfer 
          in a multi-layer setup, the correct point of departure
          is the infinitesimal expression for the radiation intensity,
          not an integrated Stefan-Boltzmann expression already computed
          for an entirely different situation.}
%                    
\item The $T^4$-law will no longer hold
      if one integrates only over a filtered spectrum, 
      appropriate to real world situations. 
      This is illustrated in Figure~\ref{Fig:nb001} .
\end{itemize}
%  
\input{GerlichTscheuschner2007a_210_real_greenhouses_radiation_basics_fig_nb001_}
%    
Many pseudo-explanations 
in the context of global climatology
are already falsified 
by these three fundamental observations 
of mathematical physics.  
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