source: trunk/MagicSoft/GRB-Proposal/Timing.tex@ 6224

\section{Timing considerations}

The first hint for delayed HE $\gamma$-ray emission from GRBs came with the detection of GRB940217 by the EGRET instrument on board of the Compton Gamma Ray Observatory (CGRO). It was a 18\,GeV photon detected 90\,min. after the burst onset~\cite{EGRET}.

\par

There are different models that predict prompt and delayed HE $\gamma$-ray emission.
Most of the models predict HE photons parallel to the keV-MeV burst but also delayed emission is possible. Our main goal should be observation the GRB location as quickly as possible, this is the reason why the \ma telescope is build to slew fast. However, in order to confirm, or rule out different predictions, we should observe the position for a longer period. Our time estimates are based on the following models:

\begin{itemize}

\item Taking into account the fireball model~\cite{REES1,REES2} there are two efficient mechanisms for the generation of VHE photons~\cite{DERISHEV} in GRBs. (1) 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. (2) UHE photons generated due to inverse Compton (IC) scattering in relativistic shocks are strongly absorbed by infrared background radiation and cannot be observed from source at cosmological distances. With the presence of an dense ambient medium close to the GRB, the VHE 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 timeline including both prozesses 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 is peaking at 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 burst onset. 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 timeline as verified in~\cite{DERISHEV}}
\label{fig:timeline}
\end{figure}

Based on the model in~\cite{DERISHEV}, only the (2) component is promising to be detectable by \ma and other ground based $\gamma$-ray detectors.

\par

For reprocessing of UHE photons below 1\,TeV where radiation can reach the Earth without interaction with the IBL the minimal magnetic field B$_{min}$ is necessary:

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

When the magnetic field is much stronger than B$_{min}$, the delay of reprocessed VHE emission may 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 $\epsilon_{2ph}=1\,TeV$, duration time of GRB main pulse $t_{GRB}=10^{2}\,s$ 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 $B_{\perp}=0.1\,Gauss$, 3.6 hours for $B_{\perp}=1.0\,Gauss$ and 17.3 hours for $B_{\perp}=10\,Gauss$.\\

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 fix a reasonable observation time of a GRB based on the described models. Every bust has its own characteristic and time profile. Hovewer, obseration of the GRB coordinates for/or within 5 hours after the alert may put constraints on model parameters of GRB sources.\\

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