Estimating the Lorentz factor and rest frame spectral peak energy of gamma-ray bursts with the opening angle of jets

We present in this paper an approach to estimate the initial Lorentz factor of gamma-ray bursts (GRBs) without referring to the delayed emission of the early afterglow. Under the assumption that the afterglow of the bursts concerned occurs well before the prompt emission dies away, the Lorentz factor measured at the time when the duration of the prompt emission is ended could be estimated by applying the well-known relations of GRB jets. With the concept of the efficiency for converting the explosion energy to radiation, this Lorentz factor can be related to the initial Lorentz factor of the source. The corresponding rest frame peak energy can accordingly be calculated. Applying this method, we estimate the initial Lorentz factor of the bulk motion and the corresponding rest frame spectral peak energy of GRBs for a new sample where the redshift and the break time in the afterglow are known. Our analysis shows that, in the circumstances, the initial Lorentz factor of the sample would peak at 200 and would be distributed mainly within $(100,400)$, and the peak of the distribution of the corresponding rest frame peak energy would be $0.8keV$ and its main region would be $(0.3keV,3keV)$.