The assumptions of the equilibrium of pressures and equality of velocities along the contact discontinuity lead to 





 and 





, respectively. With the jump condition for the shocks and the equilibrium of pressures, we can obtain
1





The Lorentz factor of the reverse shock 





 could be approximated as
2





as long as 





 and 





. Substituting Equation (2) into (1), we can obtain the following equation:
3





Because 





 , 





, and 





, we ignore the constant 1/2 term in Equation (3) and thus we can obtain the solution of this equation (ignoring the negative solution, also see Panaitescu & Kumar 2004),
4





Here we obtain the relation between the Lorentz factor of the shocked fireball shell 





 and the initial Lorentz factor 





 (





), which depends on the ratio of these two comoving densities. The number density of the ambient medium is assumed to be 





 (Dai & Lu 1998; Mészáros et al. 1998; Chevalier & Li 2000; Wu et al. 2003, 2005; Yi et al. 2013); such a circumburst medium is a homogeneous ISM for k = 0, and a typical stellar wind environment for k = 2. The fireball shell is characterized by an initial kinetic energy Ek, initial Lorentz factor 





, and a width Δ in the lab frame attached to the explosion center, so the number density of the shell in the comoving frame is 





. The ratio of the comoving number density of the relativistic shell 





 to the number density of the ambient medium 





 defined in Sari & Piran (1995) is
5





where 





. The difference between the lab frame speed of the unshocked fireball shell and that of the reverse shock is (Kumar & Panaitescu 2003),
6





Considering the thin shell case 





, we can calculate the radius 





 where the reverse shock finishes crossing the fireball shell,
7





The substitution of Equations (5) and (6) into (7) leads to
8





So the comoving density ratio at 





 is
9





Substituting Equation (9) into (4), we can obtain the Lorentz factor of the reverse shock as it finishes crossing the shell:
10





Therefore, the relation between 





 and the initial Lorentz factor 





 is
11





and
12





For the thin shell case, the reverse shock crossing time 





 almost corresponds to the deceleration time Tdec, i.e., 





. Therefore, we can derive the initial Lorentz factor in the ISM and wind-type cases (also see Panaitescu & Kumar 2004). For k = 0 (ISM),
13





and for k = 2 (wind),
14





With the isotropic-equivalent energy 





 and the peak time of the afterglow onset 





, we can estimate the initial Lorentz factor of GRBs, where 





. Liang et al. (2010) discovered a tight correlation between 





 and 





 using 20 GRBs which show a deceleration feature in the early afterglow light curves. Other work also confirmed this correlation, but with different methods and power-law indices (Ghirlanda et al. 2012; Lü et al. 2012). Using the data on 





 and 





 from Liang et al. (2010, 2013) and Lü et al. (2012), we re-constrain the initial Lorentz factor, and also discover a tight 





 and 





 correlation for the ISM and wind cases. The 





 and 





 correlation in the wind case is even tighter than that in the ISM case, as shown in Figures 4 and 5.
Citation: Yi et al. 2013
Citation function: Uses