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4 Discussion

We find large odds against the hypothesis that all Type Ib, Ib/c, and Ic supernovae produce observable gamma-ray bursts, the specific hypothesis considered by Wang & Wheeler (1998a, 1998b). The odds against the hypothesis that a fraction of Type Ib-Ic supernovae produce observable bursts are more moderate, because they account for the possibility that f is so small that no SN-produced GRBs are detected, in which case the data can not distinguish between $H_1^\prime$ and H2.

Type Ib, Ib/c and Ic SNe are now being found at a rate of about eight a year, so that the size of the sample of known Type Ib-Ic SNe should triple within about five years. One might hope that future analyses, using the statistical methodology that we have presented here, could either show that the association between Type Ib-Ic SNe and GRBs is rare, or confirm the proposed association. Unfortunately, achieving the former will be difficult: the limit on the fraction $f_{\rm SN}$ of Type Ib-Ic SNe that produce observable GRBs scales like $N_{\rm
SN}^{-1}$ for large $N_{\rm SN}$, and therefore tripling the size of the sample of known Type Ib-Ic SNe without observing an additional possible SN - GRB association would only reduce the 99.7% probability upper limit on $f_{\rm SN}$ to 0.24.

The parameter f represents the fraction of observable SNe which can produce observable GRBs. Some observable SNe might not produce observable GRBs because of intrinsic effects, such as beaming, whereas others might not do so because the sampling distance for SN-produced GRB could be less than the sampling distance for the SNe themselves. It is possible to separate these two effects, by writing $f=f_{\rm
intrinsic}\times f_{\rm sampling}$. Obviously, the more interesting quantity is $f_{\rm intrinsic}$, since it addresses the nature of the GRB sources.

We may attempt to find a constraint on $f_{\rm intrinsic}$ by assuming the correctness of the identification of SN 1998bw with GRB 980425 and using that association to estimate $f_{\rm sampling}$, under the (dubious) assumption that the GRBs produced by Type Ib-Ic SNe are standard candles. The result is $f_{\rm sampling}\approx 
1.9\ 10^{-3}$. This is rather bad news for the prospect of constraining $f_{\rm intrinsic}$, since the product of $f_{\rm intrinsic}$ and $f_{\rm sampling}$ is only constrained by the data to be less than about 0.7. Thus this argument can place no constraint on $f_{\rm intrinsic}$. Elsewhere, we show that placing a meaningful constraint on $f_{\rm intrinsic}$ would require a GRB experiment approximately 80 times more sensitive than BATSE (Graziani et al. 1998).

One can also approach the proposed association between SNe and GRBs from the opposite direction. The interesting question, from this point of view, is what fraction $f_{\rm GRB}$ of the GRBs detected by BATSE could have been produced by Type Ib-Ic SNe? (note that this is different from the question addressed by Kippen et al. 1998, who constrained the fraction of BATSE GRB that could have been produced by known SNe). Such a limit may be derived under the same assumptions as made in the previous paragraph. We find that no more than $\sim 90$ SNe could have produced GRB detectable by BATSE, indicating that $f_{\rm GRB}$ can be no more than about 5%.

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