All-sky Signaling

If the intention is to eventually broadcast to the entire sky with no preference for particular locations there are two competing constraints:
(i) The use of a narrow beam width will produce both a larger gain and also reduce the maximum possible time delay at the receiver between the detection of a GRB and the transmitted signal.
(ii) The narrower the beam the longer it will take to cover the entire sky.

One consideration may lead to a ``natural'' beam width for transmissions. If the transmitter wishes to broadcast out to a certain distance D, then a beam width can be chosen such that the maximum time delay at D would be equal to the average interval between GRBs for the intensity level that they are using to select GRBs. i.e. $\Delta T = 1 / R$where R is the GRB/transmission rate. This yields $\theta\ = (2c/DR)^{1/2}$

Note that basing a beam width on this consideration means that using additional lower intensity GRBs would not alter the rate at which the sky is illuminated as the time delay and area of sky illuminated both depend on $\theta^{2}$. However, using additional bursts does result in a narrower beam and hence larger gains and smaller maximum time delays. The assumed transmission distance, Dmight vary depending on the transmission direction, for example transmissions perpendicular or parallel to the Galactic plane. In addition, the beam width could potentially also be altered depending on the luminosity of a gamma-ray burst so that the maximum time delay at a certain distance would be the mean recurrence time for bursts of that luminosity or greater.

The time taken to illuminate an area equal to the entire sky, although with non-uniform sky coverage, is given by $T = 4 \pi / (R B)$where B is the area of the beam in steradians. Hence the ``natural'' beam width gives T = 2D/c. Thus, for any significant distance, a considerable time is required to illuminate the entire sky. Alternatively, if this ``natural'' beam width is not used then the number of GRB locations that needs to be monitored is $\sim R \Delta T$ = $R D \theta^2/2c$.