5.3. Typical Radii
R_{i} | Initial Radius | c t | 10^{7} - 10^{8} cm |
R_{} | Matter dominates | R_{i} | 10^{9} cm |
R_{pair} | Optically thin to pairs | [(3E / 4 R_{i}^{3} a)^{1/4} / T_{p}] R_{i} | 10^{10} cm |
R_{e} | Optically thin | (_{T} E / 4 m_{p} c^{2} )^{1/2} | 10^{13} cm |
R_{} | Internal collisions | ^{2} | 10^{12} - 10^{14} cm |
R_{} | External Newtonian Shocks | l ^{-2/3} | 10^{16} cm |
R_{} | External Relativistic shocks | l^{3/4} ^{1/4} | 10^{16} cm |
l or L | Non relativistic external shock | l ^{(a)} or l^{-1/3} ^{(b)} | 10^{17} - 10^{18} cm |
l | Sedov Length | l = (3E / 4 n_{ism} m_{p} c^{2})^{1/3} | 10^{18} cm |
Figs. 12 and 13 (from [228]) depict a numerical solution of a fireball from its initial configuration at rest to its final Sedov phase.
Figure 12. Fireball evolution from its initial formation at rest to the final Newtonian Sedov solution. The energy extraction is due to the interaction with the ISM via a relativistic forward shock and a Newtonian reverse shock. We have used for this calculations = 43, E_{0} = 10^{52} [erg], _{0} = 50 R_{0} = 3 × 10^{10}[cm]. Shown are the average value of the Lorentz factor (thick solid line), the value at the forward shock (thin solid line), the maximal value (dotted line) and an analytic estimate (dashed dotted line). From [228]. |
Figure 13. Fireball evolufrom its initial formation at rest to the final Newtonian Sedov solution. The energy extraction is due to the interaction with the ISM via relativistic forward and reverse shocks. The parameters for this computation are: = 0.1, E_{0} = 10^{52} [erg], _{0} = 10^{4}, R_{0} = 4.3 × 10^{9} [cm]. Shown are the average value of the Lorentz factor (thick solid line), the value at the forward shock (thin solid line), the maximal value (dotted line) and an analytic estimate (dashed dotted line). From [228]. |