\section{Proposed Observation Strategies}

\subsection{Estimation of the Required Observation Time}

A rough estimate of the needed observation time for GRBs derives
from the estimated number of GRB follow-up observations which can be 
expressed in the following formula:

\begin{equation}
N_{obs} = N_{alert} \cdot DC \cdot F_{overlap}
\end{equation}

where $N_{obs}$ is the mean number of observed bursts, $N_{alert}$ the mean 
number of sent alerts, $DC$ the duty cycle (including the reduction of sky coverage 
due to the maximum allowed zenith angle) and $F_{overlap}$ a reduction factor due to 
the non-overlapping sky coverage between the satellites and \ma. \\

The claimed GRB observation frequency $N_{obs}(SWIFT)$ is predicted to about 150-200 GRBs/year 
by the \sw collaboration~\cite{SWIFT}. We estimate $DC$ from studies on the \ma duty-cycle 
made by Nicola Galante~\cite{NICOLA}. 
The duty-cycle studies are based on real weather data from the year 2002 taking the following criteria:

\begin{itemize}
\item maximum wind speeds of 10\,m/s
\item maximum humidity of 80\%
\item darkness at astronomical horizon
\end{itemize}

In these duty-cycle studies also full-moon nights were considered (requiring
a minimum angular distance between the GRB and the moon of 30$^\circ$) yielding in 
total 10\%.

\par

The duty-cycle in~\cite{NICOLA} will be increased by taking into account that \ma should also observe the 
afterglow emission of an burst that occurred up to 5 hours before the start of the shift. 
The afterglow observation is equivalent to an increase of the duty-cycle of about 6 days per month.
However, taking off the full-moon time, we remain with the anticipated 10\%.\\

The overlap factor $F_{overlap}$ is difficult to estimate since the \sw satellite will continuously slew 
to new sources or follow detected bursts. Figure~\ref{fig:orbit} shows that the satellite will pass very 
precisely over La Palma during the night. Taking into account that it will not look towards the Sun, 
we expect that $F_{overlap}(SWIFT)$ will be at least 0.5 or higher. \\

In conclusion, we can calculate a worst case scenario with 150 \sw alerts per year and an overlap factor 
of 0.5 yielding $N_{obs}^{min} \sim 0.6$/month. 
An upper limit can be derived from 200 \sw alerts and a complete 
overlap with $F_{overlap}(SWIFT) = 1$ yielding $N_{obs}^{max} \sim 1.6$/month.

\subsection{Determine the Maximum Zenith Angle}

We determine the maximum zenith angle for GRB observations by requiring that the overwhelming 
majority of possible GRBs will have an in principle observable spectrum. Figure~\ref{fig:grh}
shows the gamma-ray horizon (GRH) as computed in~\cite{KNEISKE,SALOMON}. The GRH is defined as the
gamma-ray energy at which a part of $1/e$ of a hypothetical mono-energetic flux gets absorbed after
travelling a distance, expressed in redshift $z$, from the source. One can see that at typical
GRB distances of $z=1$, all gamma-rays above 100\,GeV get absorbed before they can reach the earth.

\par

Even the closest GRB with known redshift ever observed, GRB030329~\cite{GRB030329}, lies at a redshift
of $z=0.1685$. In this case $\gamma$-rays above 200\,GeV get entirely absorbed.

\begin{figure}[htp]
\centering
\includegraphics[width=0.85\linewidth]{f4.eps}
\caption{Gamma Ray Horizon as derived in~\cite{KNEISKE}}
\label{fig:grh}
\end{figure}

\par

We assume now a current energy threshold of 50\,GeV for \ma at a zenith angle of
$\theta = 0$\footnote{As this proposal is going to be reviewed in a couple of months, improvements of the energy threshold will be taken into account then.}. According to~\cite{ecl}, the energy threshold of a Cherenkov telescope scales with zenith angle like:

\begin{equation}
E_{thr}(\theta) = E_{thr}(0) \cdot \cos(\theta)^{-2.7}
\label{eq:ethrvszenith}
\end{equation}

Eq.~\ref{eq:ethrvszenith} leads to an energy threshold of about 5.6\,TeV at $\theta = 80^\circ$,
900\,GeV at $\theta = 70^\circ$ and 500\,GeV at $\theta = 65^\circ$.
Inserting these results into the GRH (figure~\ref{fig:grh}), one gets a maximal observable GRB 
distance of $z = 0.1$ at $\theta = 70^\circ$ and $z = 0.2$ at $\theta = 65^\circ$.
We think that the probability for GRBs to occur at these distances is sufficiently small in order to 
neglect the very difficult observations beyond these limits.

\subsection{GRB Observations in Case of Moon Shine}

{\it gspot} allows only GRBs with an angular distance of $> 30^\circ$ from the moon.
The telescope's slewing in case of a GRB alert will be done
without closing the camera lids, so that the camera could be
flashed by the moon during such a movement. In principle
a fast moon-flash shouldn't damage the PMTs, but the behaviour
of the camera and the Camera Control {\it La Guagua} must
be tested. On the other hand, if such tests conclude that it is not safe
to get even a short flash from the moon, the possibility
to implement a new feature into the Steering System must be considered
which follow a path around the moon while slewing.

\par

There was a shift observing the Crab-Nebula with half-moon at La Palma in December 2004.
That experience showed that the nominal HV could be maintained and gave no
currents higher than 2\,$\mu$A. This means that moon-periods can be used for GRB-observations
without fundamental modifications except for full-moon periods. We want to stress that
these periods increase the chances to catch GRBs by 80\%.
It is therefore mandatory that the shifters keep the camera in fully operational conditions with 
high-voltages switched on from the beginning of a half-moon night until the end. 
This includes periods where no other half-moon observations are scheduled. 
If no other data can be taken during the those periods, the telescope should be pointed 
to a Northern direction, close to the zenith. This increases the probability to overlap 
with the FOV of \sw.

\par

Because of higher background with moon-light, we suggest to decrease the maximum zenith angle from
$\theta_{max} = 70^\circ$ to $\theta_{max} = 65^\circ$, there.

\subsection{Active Mirror Control Behaviour}

To reduce the time before the start of the observation, the use of the look-up tables (LUTs) is necessary.
Once generated, the {\it AMC} will use the LUTs and automatically focus the panels for a given 
telescope position. The {\it CC} should send the burst coordinates to the {\it Drive} and the {\it AMC} 
software in the same time. In this way the panels could be focused already during the telescope movement.

\subsection{Calibration}

For ordinary source observation, the calibration is currently performed in the following way:

\begin{itemize}
\item At the beginning of the source observation, a dedicated pedestal run followed by a calibration run is taken.
\item During the data runs, interlaced calibration events are taken at a rate of 50\,Hz.
\end{itemize}

We would like to continue taking the interlaced calibration events when a GRB alert is launched, but leave out the pedestal and calibration run in order not to loose valuable time.

\subsection{In case of Follow-up: Next Steps}

We propose to analyze the GRB data at the following day in order to tell whether a follow-up observation during the next night is useful. We think that a limit of 3\,$\sigma$ significance should be enough to start such a follow-up observation of the same place. This follow-up observation can then be used in two ways:

\begin{itemize}
\item In case of a repeated outbursts for a longer time period of direct observation.
\item In the other case for having off-data at exactly the same sky location.
\end{itemize}

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