\section{Timing considerations}

The first experimental hint for delayed HE $\gamma$-ray emission from GRBs 
came from the detection of a 18\,GeV photon from GRB940217 by the EGRET detector 
-- 90\,min. after the onset of the burst~\cite{EGRET}.

\par

Different models predict prompt and delayed HE $\gamma$-ray emission.
Most of them predict HE photons to be simultaneous with the keV-MeV burst, 
but also a delayed emission is possible. 
Our main goal should be to observe the GRB location as quickly as possible. 
However, in order to confirm or rule out different predictions, 
we should observe the position for a longer period of time. \\

Our time estimates are based on the following models:

\begin{itemize}

\item Regarding the fireball model~\cite{REES1,REES2},
two efficient mechanisms are available for the generation of HE photons (from sub-GeV to 100\,TeV)~\cite{DERISHEV}:

\begin{enumerate}
\item The prompt emission of $\sim$100\,GeV photons is expected before and during the keV-MeV peak.
This emission should have their highest luminosity together with the main GRB peak.
\item VHE photons generated due to inverse Compton (IC) scattering in relativistic shocks.
\end{enumerate}

With the presence of a dense ambient medium close to the GRB, 
the UHE photons will be reprocessed into a softer spectral range. 
This would lead to VHE emission delayed by few minutes to hours with 
respect to the beginning of GRB. 
The time-line including both processes is illustrated in figure~\ref{fig:timeline}.

\item In~\cite{DERMER}, two peaks in the GeV light curve are calculated. 
The first is coincident with the keV-MeV peak, some seconds after the burst onset. 
The second maximum peaks between $\approx$ 1.5 hours up to $\approx$ 25 hours after the burst onset.

\item Models in~\cite{LI, WANG} suggest GeV emission after pion production and some thermalization 
of the UHE component with radiation maxima of up to one day or even one week after the onset of the burst. 
This radiation is accompanied by long-term neutrino emission.

\end{itemize}

\begin{figure}[htp]
\centering
\includegraphics[width=0.6\linewidth]{GRBbrigthness.eps}
\caption{A possible example of GRB time-line as depicted in~\cite{DERISHEV}}
\label{fig:timeline}
\end{figure}

Based on the model in~\cite{DERISHEV}, three different components of VHE emission exists in an GRB. 
The corresponding components are illustrated in figure~\ref{fig:timeline}.
\renewcommand{\theenumi}{\alph{enumi}}
\begin{enumerate}
\item There is the prompt 100\,GeV peak before and during the first keV-MeV peak,
\item the VHE emission due to Inverse Compton scattering lasting for the whole duration of the GRB pulse and 
\item the reprocessed Inverse Compton emission which may last up to hours after the GRB onset.
\end{enumerate}
\renewcommand{\theenumi}{\arabic{enumi}}
(b) and (c) are the components which may be detectable by \ma and other ground based $\gamma$-ray detectors.

\par

To achieve significant emission due to inverse Compton scattering 
of the sub-MeV radiation, a minimal magnetic field $B_{min}$ is necessary:

\begin{equation}
B_{min} \sim \frac{5\times10^{-2}}{\Gamma^{3}}\,
             \frac{\epsilon_{2ph}}{1\,\mathrm{TeV}}\,
             \frac{t_{\mathrm{GRB}}}{10\,\mathrm{s}}\,\mathrm{G}
\label{eq:minimal}
\end{equation}

If the magnetic field is much stronger than $B_{min}$, 
the delay of reprocessed photons may become observable. 
Taking into account only the components of $B$ orthogonal to the electron path,
the delay can be calculated via the following asymptotic expression:

\begin{equation}
t_{d} \simeq \frac{2^{4/3}}{3} \biggl(\frac{B_{\perp}}{B_{min}}\biggl)^{2/3}
\label{eq:duration}
\end{equation}

For typical values of the absorption threshold $\epsilon_{2ph}=1\,TeV$, 
the duration time of GRB main pulse $t_{\mathrm{GRB}}=10^{2}\,\mathrm{s}$ and Lorentz factor of the GRB shell 
$\Gamma=10^{2}$, the duration of delayed VHE emission will be 0.8 hours for the component of magnetic 
field perpendicular to electron's trajectory $B_{\perp}=0.1\,\mathrm{G}$, 
3.6 hours for $B_{\perp}=1.0\,\mathrm{G}$ and 17.3 hours for $B_{\perp}=10\,\mathrm{G}$.\\

The observation of the delayed VHE emission and the time correlation will give informations 
about the density of the surrounding interstellar gas, the magnetic field and 
the Lorentz factor of the GRB shell.\\

It is not easy to determine a reasonable observation time of a GRB based on the described models. 
Every burst has its own characteristic and time profile. 
However, observation of the GRB coordinates for/within 5 hours after the alert may set 
constraints on model parameters of GRB sources.\\

In case of a \textcolor{red}{\bf Red Alarm}, we propose to take data for {\bf 5 hours}.
\par
In case of a \textcolor{yellow}{\bf Yellow Alarm}, we propose to observe the source
from the time when it will become observable until {\bf 5 hours} after the GRB beginning.

%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "GRB_proposal_2005"
%%% TeX-master: "Timing"
%%% End: 
