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\begin{document}

\sloppy

\centerline{\huge\bf Photo-rates in ProDiMo}
{\ }\\[-1.7ex]
\centerline{\large Peter Woitke, December 2008}

\vspace*{1cm}
\noindent The general definition of a photo-rate is
\begin{equation}
  R_{\rm ph} \;=\; \frac{1}{h} \int\!\sigma(\lambda)\,
                   \lambda u_\lambda \,d\lambda
  \label{eq:RphGeneral}
\end{equation}  

\section{Delta peak cross section}

Let's first consider the most simple case $\sigma(\lambda)=\sigma_0
\delta(\lambda-\lambda_0)$ where the absorption is centered around 
a very narrow spectral band
\begin{equation}
  R_{\rm ph} \;=\; \frac{1}{h} \sigma_0\, \lambda_0 u_{\lambda_0}
\end{equation}  

\subsection{UMIST}

Consider this photo-rate in a standard ISM radiation field $\lambda
u_\lambda^{\rm ISM}$ which is shielded by some column of ISM dust
\begin{equation}
  R_{\rm ph}^{\rm ISM} \;=\; \frac{1}{h} \sigma_0\,\l0 u_\l0^{\rm ISM}
                   \exp(-\tau_\l0^{\rm ISM})
  \label{eq:Rgeo}
\end{equation}  
The visual extinction is defined as auxiliary variable  
\begin{equation}
  A_V^{\rm ISM} = 2.5\log\!e\,\tau_V^{\rm ISM}
  \label{eq:AV}
\end{equation}
with visual optical depth $\tau_V^{\rm ISM}=
\frac{\kappa_V^{\rm ISM}}{\kappa_\l0^{\rm ISM}}\tau_\l0^{\rm ISM}$. 
This yields
\begin{eqnarray}
  R_{\rm ph}^{\rm ISM} &=& 
  \underbrace{\frac{1}{h} \sigma_0\,\l0 u_\l0^{\rm ISM}}_{\displaystyle\alpha}
       \exp\Big(-\underbrace{\frac{1}{2.5\log\!e}
      \frac{\kappa_\l0^{\rm ISM}}{\kappa_V^{\rm ISM}}}_{\displaystyle\gamma}
             A_V^{\rm ISM}\Big) \label{eq:gamma}\\
   &=& \alpha \exp(-\gamma A_V^{\rm ISM})
\end{eqnarray}  
as given by UMIST. Remarks
\begin{itemize}
\item Small $\gamma<1$ in UMIST are because of ionisation by soft radiation
      and because ISM dust shows an opacity decrease toward longer wavelength
      $\kappa_\l0^{\rm ISM}<\kappa_V^{\rm ISM}$, \eg $\gamma=0.5$ for H$^-$ bf.
\item The unshielded rate $\alpha$ depends on the assumptions made by
      UMIST about the ISM irradiation at $\l0$. Not only about the
      spectral slope, but first of all about the absolute strength at
      $\l0$.
\item $\l0$ is not known from the UMIST database, but we can calculate the 
      likewise unknown ISM dust opacity at centre wavelength $\l0$
      from $\gamma$ according to Eq.\,(\ref{eq:gamma}) 
      \begin{equation}
        \kappa_\l0^{\rm ISM} \;=\; 2.5\log\!e\;\gamma\;\kappa_V^{\rm ISM}
        \label{eq:kapl0}
      \end{equation}
\end{itemize}

%=======================================================================
\subsection{ProDiMo}
In arbitrary radiation fields, but slab symmetry, we can still use
Eq.~(\ref{eq:Rgeo}) 
\begin{equation}
  R_{\rm ph} \;=\; \frac{1}{h} \sigma_0\,\l0 u_\l0
                   \exp(-\tau_\l0)
  \label{eq:Rph}
\end{equation}  
We define another auxiliary variable here, namely the UV optical depth as
\begin{equation}
  \tau_{\rm UV} = \frac{\kappa_{\rm UV}}{\kappa_\l0}\tau_\l0
\end{equation}
and use Eqs.~(\ref{eq:Rph}) 
\begin{eqnarray}
  R_{\rm ph} 
  &=& \underbrace{\frac{1}{h} \sigma_0\,\l0 
                  u_\l0^{\rm ISM}}_{\displaystyle=\alpha}\; 
      \frac{u_\l0}{u_\l0^{\rm ISM}}
    \;\exp\Big(-\frac{\kappa_\l0}{\kappa_{\rm UV}}\tau_{\rm UV}\Big) 
\end{eqnarray}
Next, we use Eq.(\ref{eq:AV}) to replace $\tau_{\rm UV}$ by the visual
extinction $A_V^{\rm ISM}= 2.5\log\!e\,\frac{\kappa_V^{\rm
ISM}}{\kappa_{\rm UV}^{\rm ISM}}\tau_{\rm UV}$
as would be present if the dust had interstellar transport coefficients
\begin{eqnarray}
  R_{\rm ph} &=& \alpha\;\frac{u_\l0}{u_\l0^{\rm ISM}} \exp\Big(-
    \frac{\kappa_\l0}{\kappa_{\rm UV}} 
    \;\frac{\kappa_{\rm UV}^{\rm ISM}}{\kappa_V^{\rm ISM}}
    \;\frac{1}{2.5\log\!e} A_V^{\rm ISM} \Big)  \\
             &=& \alpha\;\frac{u_\l0}{u_\l0^{\rm ISM}} \exp\Big(-
    \frac{\kappa_\l0}{\kappa_{\rm UV}} 
    \;\frac{\kappa_{\rm UV}^{\rm ISM}}{\kappa_\l0^{\rm ISM}}
    \;\underbrace{\frac{\kappa_\l0^{\rm ISM}}{\kappa_V^{\rm ISM}}
    \;\frac{1}{2.5\log\!e}}_{\displaystyle=\gamma} A_V^{\rm ISM} \Big)  \\    
             &=& \alpha\;
    \underbrace{\frac{u_\l0}{u_\l0^{\rm ISM}}}_{\displaystyle\approx\chi_0} 
    \exp\Big(-\underbrace{\frac{\kappa_\l0/\kappa_\l0^{\rm ISM}}
         {\kappa_{\rm UV}/\kappa_{\rm UV}^{\rm ISM}}}_{\displaystyle\approx1}
    \;\gamma A_V^{\rm ISM}\Big) \label{eq:color}\\
  &\approx& \alpha\;\chi_0 \exp\Big(-\gamma A_V^{\rm ISM}\Big)
    \label{eq:RphProDiMo}
\end{eqnarray}
Equation~(\ref{eq:RphProDiMo}) demonstrates the following
\begin{itemize}
\item As long as the relative opacity ratio
      $\frac{\kappa_\l0/\kappa_\l0^{\rm ISM}} {\kappa_{\rm UV}/
      \kappa_{\rm UV}^{\rm ISM}}\approx 1$, and
      $\frac{u_\l0}{u_\l0^{\rm ISM}} \approx \chi_0$,
      we can use the UMIST parameter $\alpha$ and $\gamma$
      according to Eq.~(\ref{eq:RphProDiMo}), ...
\item ... if we use $A_V^{\rm ISM}$ and not $A_V$ !
\item Otherwise, we can't. I do not see how this could be justified.
      Can you?
\item The ratio $u_\l0\,/\,u_\l0^{\rm ISM}$ can easily differ
      substantially from $\chi_0=\int_{91.2\,{\rm nm}}^{205\,{\rm nm}}
      \lambda u_\lambda^\star\,d\lambda$ / $ \int_{91.2\,{\rm
      nm}}^{205\,{\rm nm}} \lambda u_\lambda^{\rm ISM}\,d\lambda$,
      even if $\l0$ is well inside the interval $91.2\,{\rm
      nm}\,...\,205\,{\rm nm}$, because $u^\star_\lambda$ is a steep
      function of $\lambda$, whereas $u_\lambda^{\rm ISM}$ isn't (may
      go the other way!).  The only solution I see is to integrate
      properly over the stellar spectrum
      \begin{equation}
        R_{\rm ph} \approx \alpha^\star \exp\Big(-
         \frac{\kappa_\l0/\kappa_\l0^{\rm ISM}}
         {\kappa_{\rm UV}/\kappa_{\rm UV}^{\rm ISM}}
         \;\gamma A_V^{\rm ISM}\Big)
        \quad\mbox{with}\quad
        \alpha^\star = \frac{1}{h} \int\!\sigma(\lambda)\,
                   \lambda u_\lambda^\star \,d\lambda
      \end{equation}
\item The relative opacity ratio in the exponent is a problem because it
      sits in the exponent! For larger than ISM grains, the slope in
      the UV is small, but for ISM grains it isn't! This is just
      expressed by the parameter $\gamma$. It simply makes no sense to
      introduce $A_V$ in the first place. It would be better to keep
      it as
      \begin{equation}
        R_{\rm ph} = \alpha^\star   
        \,\exp\Big(-\frac{\kappa_\l0}{\kappa_{\rm UV}}\tau_{\rm UV}\Big) 
        \label{eq:RphDelta}
      \end{equation}
      and $\frac{\kappa_\l0}{\kappa_{\rm UV}}$ stays in the
      memory as additional parameter for every photo-reaction.
\item For both solutions, we must have $\sigma(\lambda)$, or at least 
      an estimate of the center wavelength $\lambda_0$. 
\item Introducing the proper physical $A_V = 2.5\log\!e\,\tau_V$ 
      does not help, but makes things worse: Applying the same formalism 
      results is
      \begin{equation}
        R_{\rm ph} = \alpha\;\chi_0 \exp\Big(-
                   \frac{\kappa_\l0/\kappa_\l0^{\rm ISM}}
                        {\kappa_V/\kappa_V^{\rm ISM}}
                   \;\gamma A_V\Big)
        \label{eq:RphAV}
      \end{equation}
      The ratio $(\kappa_\l0/\kappa_\l0^{\rm
      ISM})/(\kappa_V/\kappa_V^{\rm ISM})$ is even worse determined as
      compared to $(\kappa_\l0/\kappa_\l0^{\rm ISM})/(\kappa_{\rm
      UV}/\kappa_{\rm UV}^{\rm ISM})$ because, in most cases, $\l0$ is
      in the UV and not in the visual. And the more these wavelength
      are apart (for unknown $\l0$) the larger the error is if putting
      this ratio to 1.
\item It is insightful to combine Eq.\,(\ref{eq:RphAV}) with 
      Eq.\,(\ref{eq:kapl0})
      \begin{equation}
        R_{\rm ph} = \alpha\;\chi_0 \exp\Big(-
                   \underbrace{\frac{\kappa_\l0}
                        {\kappa_V}
                   \;\frac{1}{2.5\log\!e}}_{\gamma^\star} A_V\Big)
      \end{equation}
      Equation (\ref{eq:kapl0}) is a implicit equation for $\l0$.
      If one would use it to determine $\l0$ and then calculate
      the disk opacities at that wavelength, one would get
      a pretty good alternative, I think. 
\end{itemize}


%======================================================================
%======================================================================


\section{The general case}

\subsection{UMIST}

The full rate in a standard ISM radiation field shielded by some column of
ISM dust is
\begin{equation}
  R_{\rm ph}^{\rm ISM} \;=\; \frac{1}{h} \int\!\sigma(\lambda)\,
                   \lambda u^{\rm ISM}_\lambda 
                   \exp(-\tau_\lambda^{\rm ISM})\,d\lambda
\end{equation}  
Introducing $A_V^{\rm ISM}$ results in
\begin{eqnarray}
  R_{\rm ph}^{\rm ISM} 
  &=& \frac{1}{h} 
      \int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda 
      \exp\Big(-\frac{1}{2.5\log\!e} 
      \frac{\kappa_\lambda^{\rm ISM}}{\kappa_V^{\rm ISM}}
      A_V^{\rm ISM}\Big)\,d\lambda \\
  &=& \underbrace{\frac{1}{h} \int\!\sigma(\lambda)\,\lambda u^{\rm
                  ISM}_\lambda\,d\lambda}_{\displaystyle\alpha}\;\;
      \underbrace{
      \frac{\int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda 
            \exp\Big(-\frac{1}{2.5\log\!e} 
            \frac{\kappa_\lambda^{\rm ISM}}{\kappa_V^{\rm ISM}}
            A_V^{\rm ISM}\Big)\,d\lambda}
           {\int\!\sigma(\lambda)\,\lambda u^{\rm
            ISM}_\lambda\,d\lambda}}_{\displaystyle
                        \exp(-\gamma A_V^{\rm ISM})} 
  \label{eq:alphaUMIST}
\end{eqnarray}  
I do not see any further possible simplification for abritrary
$A_V^{\rm ISM}$ and
\begin{equation}
   \exp(-\gamma A_V^{\rm ISM}) = \frac{1}{h\alpha}
   \int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda 
            \exp\Big(-\frac{1}{2.5\log\!e} 
            \frac{\kappa_\lambda^{\rm ISM}}{\kappa_V^{\rm ISM}}
             A_V^{\rm ISM}\Big)\,d\lambda
  \label{eq:gammaGeneral}
\end{equation}
states an implicit equation for the function
$\gamma\!=\!\gamma(A_V^{\rm ISM})$ that has be to fitted numerically,
provided that $\sigma(\lambda)$, $\kappa_\lambda^{\rm ISM}$, and $
u^{\rm ISM}_\lambda$ are known.  However, for small $A_V^{\rm
ISM}\!\ll\!1$, with $\exp(-x)\!\approx\!1\!-\!x$, I get the following
result
\begin{equation}
  \gamma = \frac{1}{2.5\log\!e}\;\frac
     {\int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda
      \frac{\kappa_\lambda^{\rm ISM}}{\kappa_V^{\rm ISM}}\,d\lambda}
     {\int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda\,d\lambda}
  \label{eq:gammaSmall}
\end{equation}
which states a generalization of Eq.\,(\ref{eq:gamma}) for the
definition of $\gamma$. Equating these two equations gives us a
hint how to define an effective opacity at 
aborption center wavelength $\l0$:
\begin{equation}
  \frac{\kappa_\l0^{\rm ISM}}{\kappa_V^{\rm ISM}}
  \;\approx\; \frac
     {\int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda
      \frac{\kappa_\lambda^{\rm ISM}}{\kappa_V^{\rm ISM}}\,d\lambda}
     {\int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda\,d\lambda}
\end{equation}


%=======================================================================
\subsection{ProDiMo}
In arbitrary radiation fields and slab symmetry we write
\begin{equation}
  R_{\rm ph} \;=\; \frac{1}{h} \int\!\sigma(\lambda)\,
                   \lambda u_\lambda^\star 
                   \exp(-\tau_\lambda)\,d\lambda \ .
  \label{eq:RphSlabGen}
\end{equation}  
The UV dust opacity in ProDiMo is defined as band-mean according to
\begin{equation}
  \kappa_{\rm UV} = \frac{1}{\scriptstyle\lambda_2\!-\!\lambda_1}
                    \int_{\lambda_1}^{\lambda_2} 
                    \kappa_\lambda\,d\lambda
\end{equation}
with $\lambda_1\!=\!91.2\,$nm and $\lambda_2\!=\!205\,$nm, 
and the unattenuated UV strength is defined as
\begin{equation}
  \chi_0 = \int_{\lambda_1}^{\lambda_2}
         \lambda u_\lambda^\star\,d\lambda \;\;\Bigg{/}\;
         \int_{\lambda_1}^{\lambda_2} 
         \lambda u_\lambda^{\rm ISM}\,d\lambda \ , 
\end{equation}
Putting $\tau_{\rm UV}$ into Eq.\,(\ref{eq:RphSlabGen}) we find
\begin{eqnarray}
  R_{\rm ph} &=& \frac{1}{h} \int\!\sigma(\lambda)\,
                   \lambda u_\lambda^\star
                   \exp\Big(-\frac{\kappa_\lambda}{\kappa_{\rm UV}}
                        \tau_{\rm UV}\Big)\,d\lambda 
   \label{eq:RphStraight}\\
   &=& \underbrace{\frac{1}{h} \int\!\sigma(\lambda)\,\lambda 
                   u^\star_\lambda\,d\lambda}_{\displaystyle\alpha^\star}\;\;
       \underbrace{\frac{\int\!\sigma(\lambda)\,
                   \lambda u_\lambda^\star
               \exp\Big(-\frac{\kappa_\lambda}{\kappa_{\rm UV}}
                   \tau_{\rm UV}\Big)\,d\lambda}
                        {\int\!\sigma(\lambda)\,
       \lambda u^\star_\lambda\,d\lambda}}_{\displaystyle
               \exp(-\beta^{\,\star}\,\tau_{\rm UV})} \\
  &=& \alpha^\star \exp(-\beta^{\,\star}\,\tau_{\rm UV})
  \label{eq:RphGenSlab}
\end{eqnarray}  
Again, this doesn't help much for the definition of $\beta^{\,\star}$, 
unless we consider small $\tau_{\rm UV}\!\ll\!1$
\begin{equation}
  \beta^{\,\star} \,=\, \frac{\displaystyle\int\!\sigma(\lambda)\,
                   \lambda u_\lambda^\star
                   \frac{\kappa_\lambda}{\kappa_{\rm UV}}\,d\lambda}
               {\displaystyle\int\!\sigma(\lambda)\,
                   \lambda u_\lambda^\star\,d\lambda}
  \label{eq:beta}
\end{equation}
Remarks
\begin{itemize}
  \item $\beta^{\,\star}$ is a ``colour correction''. It accounts
        for the deeper penetration of star light into the disk, if
        the typical wavelength of molecular absorption is longer
        than typical UV, and vice versa.
  \item We do not need $\chi$ nor $A_V$ in general.
  \item Eqs.\,(\ref{eq:RphGenSlab}) and (\ref{eq:beta}) are a
        generalisation of Eq.\,(\ref{eq:RphDelta}).
  \item If $\sigma(\lambda)$ is known, $\alpha^\star$ and
        $\beta^{\,\star}$ can be pre-calculated straightforwardly
        for every photo-reaction, unless we allow for
        position-dependent dust opacities ($\to$ dust settling).
\end{itemize}
The {\Large\bf big big} question now is: {\sl Whether and how can we use
$\chi_0$ and the UMIST coefficients $\alpha$ and $\gamma$ in cases
where we have no informations about $\sigma(\lambda)$?}
\clearpage
\noindent We start all over again with Eq.\,(\ref{eq:RphStraight}) and insert
ISM quantities
\begin{eqnarray}
  R_{\rm ph} &=& \frac{1}{h} \int\!\sigma(\lambda)\,
                   \lambda u_\lambda^\star
                   \exp\Big(-\frac{\kappa_\lambda}{\kappa_{\rm UV}}
                        \tau_{\rm UV}\Big)\,d\lambda \\
   &=& \frac{1}{h} \int\!\sigma(\lambda)\,\lambda u_\lambda^\star
      \exp\Big(-\frac{\kappa_\lambda}{\kappa_{\rm UV}}
     \;\frac{\kappa_{\rm UV}^{\rm ISM}}{\kappa_V^{\rm ISM}}
     \;\frac{1}{2.5\log\!e} A_V^{\rm ISM}
     \Big)\,d\lambda   \\
   &=& \alpha^\star \exp(-\delta^{\star}\gamma A_V^{\rm ISM})
  \label{eq:ProDiMo}
\end{eqnarray}
In order to determine $\delta^\star$, let's consider small $A_V$ rightaway
\begin{eqnarray}
  R_{\rm ph} &\approx& 
     \frac{1}{h} \int\!\sigma(\lambda)\,\lambda u_\lambda^\star
      \left(1-\frac{\kappa_\lambda}{\kappa_{\rm UV}}
     \;\frac{\kappa_{\rm UV}^{\rm ISM}}{\kappa_V^{\rm ISM}}
     \;\frac{1}{2.5\log\!e} A_V^{\rm ISM}
     \right)\,d\lambda \\
  &=& \underbrace{\frac{1}{h} \int\!\sigma(\lambda)\,\lambda 
                  u_\lambda^\star\,d\lambda}_{\displaystyle=\alpha^\star} 
     \;-\; \frac{1}{h} \int\!\sigma(\lambda)\,\lambda 
                  u_\lambda^\star\,
        \frac{\kappa_\lambda}{\kappa_{\rm UV}}
     \;\frac{\kappa_{\rm UV}^{\rm ISM}}{\kappa_V^{\rm ISM}}
     \;\frac{1}{2.5\log\!e} A_V^{\rm ISM}\,d\lambda \\
  &=& \alpha^\star (1-\delta^{\star}\gamma A_V^{\rm ISM})
\end{eqnarray}
Inserting the definition of $\gamma$ (Eq.\,\ref{eq:gammaSmall})
yields
\begin{equation}
  \alpha^\star \delta^{\star}
  \frac{\int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda
      \frac{\kappa_\lambda^{\rm ISM}}{\kappa_V^{\rm ISM}}\,d\lambda}
     {\int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda\,d\lambda}
= \frac{1}{h} \int\!\sigma(\lambda)\,\lambda 
        u_\lambda^\star
        \frac{\kappa_\lambda}{\kappa_{\rm UV}}
     \;\frac{\kappa_{\rm UV}^{\rm ISM}}{\kappa_V^{\rm ISM}}\,d\lambda
\end{equation} 
We can cancel $\kappa_V^{\rm ISM}$ and use the definition of the
unattenuated stellar photo-rate $\alpha^\star$
\begin{equation}
  \delta^{\star} = 
    \frac{\kappa_{\rm UV}^{\rm ISM}}{\kappa_{\rm UV}}
  \;\frac{\;\;\;\int\!\sigma(\lambda)\,\lambda 
        u_\lambda^\star
        \kappa_\lambda\,d\lambda \;\Big{/}
       \int\!\sigma(\lambda)\,\lambda 
        u_\lambda^\star \,d\lambda }
    {\int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda
      \kappa_\lambda^{\rm ISM}\,d\lambda \;\Big{/}
       \int\!\sigma(\lambda)\,\lambda u^{\rm ISM}_\lambda
       \,d\lambda}
  \label{eq:delta}
\end{equation} 
This is the desired color correction for the application of the UMIST
parameters, a generalization of Eq.\,(\ref{eq:color}).  

The discussion of Eq.\,(\ref{eq:delta}) is a bit irksome. To
zero-order approximation, everything calcels and
$\delta^{\star}\!\approx\!1$, as long as the integral boundaries are
situated in the UV. Since we only need this description if we do not
have detailed cross sections $\sigma(\lambda)$ at hand, there is no
point in trying to really use Eq.\,(\ref{eq:delta}). However,
Eq.\,(\ref{eq:delta}) can be useful to check the errors made by
assuming $\delta^{\star}\!\approx\!1$.  It gives a better feeling for
the meaning of $\kappa_\l0/\kappa_\l0^{\rm ISM}$  in
Eq.\,(\ref{eq:color}). In addition to the discussion after
Eq.\,(\ref{eq:color}), which demonstrates the main effect, one can
say that things get worse, because the reddish disk opacities are
weighted by the reddish stellar irradiation, and the blueish ISM
opacities with the blueish ISM irradiation. This gives more weight to
the extremes and makes deviations from unity more likely.

In conclusion, the best way to use the UMIST rate coefficients
still seems to be to assume $\alpha/\alpha^{\rm ISM}\approx\chi_0$
\;and\; $\delta^\star\approx 1$ 
\begin{equation}
  R_{\rm ph} \;\approx\; \alpha\,\chi_0\,\exp(-\,\gamma A_V^{\rm ISM})
\end{equation}
although it is hard to justify.


\end{document}
