4 Derivation of ${<{\vec{V}}/{\vec{V}}_{\bf max}\gt}$

The most important property of the GRBs in the sample is the euclidean value of $<V/V_{\rm max}\gt$. We will use its value to derive a distance scale for GRBs in a subsequent communication.

In most evaluations of $V/V_{\rm max}$ for individual GRBs, it has been assumed that it equals $(C_{\rm max}/C_{\rm min})^{-3/2}$ where $C_{\rm min}$ is the minimum detectable count rate and $C_{\rm max}$ is the count rate at the maximum amplitude of the burst. This may not be correct. If we remove the source to the largest distance at which it is just still detectable, it is likely that the detection of the burst will occur later, and therefore that $C_{\rm min}$ will include some burst signal. Also, in some cases the part of the burst containing $C_{\rm max}$ will not cause a trigger when we remove the burst. Due to these inescapable effects, $V/V_{\rm max}$ will be larger than follows from the values of $C_{\rm max}$ and $C_{\rm min}$ at detection. This situation was encountered by Higdon & Schmidt (1990) in their discussion of GRBs from the Venera 11 and 12 KONUS experiments.

We have derived the euclidean value of $V/V_{\rm max}$ for each GRB as follows. Upon detection, we derive the burst time profile by subtracting the background (interpolated between the two background stretches, see Fig. 1) from the counts. We multiply this original time profile by a factor of X and add it to the background. Next we apply the search algorithm to see whether we detect a burst. If so, we repeat the process with a smaller value of X, until we do not detect a burst anymore. This represents the situation (as best we can) when the source has been removed in distance by a factor X^-1/2 in euclidean space. Therefore $V/V_{\rm max}$ for the source is X^3/2. Application of this process to all bursts in the sample produces $<V/V_{\rm max}\gt~ = 0.334~ \pm ~0.008$.

Acknowledgements

During the many years in which this work came to fruition, I have had much appreciated help from or discussions with D. Chakrabarty, M. Finger, G. Fishman, J. Gunn, J. Higdon, J. Horack, C. Meegan, T. Prince, and B. Vaughan.