Estimation of the detectability of optical orphan afterglows
  By neglecting sideways expansion of gamma-ray burst (GRB) jets and assuming their half-opening angle distribution, we estimate the detectability of orphan optical afterglows. This estimation is carried out by calculating the durations of off-axis optical afterglows whose flux density exceeds a certain observational limit. We show that the former assumption leads to more detectable orphans, while the latter suppresses the detectability strongly compared with the model with half-opening angle $\theta_j=0.1$. We also considered the effects of other parameters, and find that the effects of the ejecta energy $E_j$ and post-jet-break temporal index $-\alpha_2$ are important but that the effects of the electron-energy distribution index $p$, electron energy equipartition factor $\epsilon_e$, and environment density $n$ are insignificant. If $E_j$ and $\alpha_2$ are determined by other methods, one can constrain the half-opening angle distribution of jets by observing orphan afterglows. Adopting a set of "standard" parameters, the detectable rate of orphan afterglows is about $1.3\times 10^{-2} {deg}^{-2} {yr}^{-1}$, if the observed limiting magnitude is 20 in R-band. 
Y. C. Zou, X. F. Wu, and Z. G. Dai