Hydrodynamic evolution

The hydrodynamic evolution of the shells prior to, during and after their collision has been presented by Mimica et al. (2004). The only difference between the model of Mimica et al. (2004) and the current ones is the presence of a moving background medium. The motion of the background medium does not change qualitatively the hydrodynamic evolution of the current set of models compared to the previous ones (Mimica et al. 2004). Our simulations show that the evolution of the shells can be divided into three typical stages: the evolution prior to the collision (Figs. 5.3 and 5.4, upper panels), the interaction phase (Figs. 5.3 and 5.4, lower left panels), and the post-collision evolution (Figs. 5.3 and 5.4, lower right panels).

Shells start with sharp discontinuities at their edges so that their pre- collision evolution can be predicted using an exact one-dimensional Riemann solver. In Fig. 5.5 the top two panels show the analytic evolution of the flow conditions. The leading discontinuity of each shell decays into a bow shock (for example, S1b of the leading shell) and a reverse shock (S1a and S2a) separated by a contact discontinuity (e.g., CD1R in the top right panel of Fig. 5.5). The trailing discontinuity of each shell develops into a rarefaction

Figure 5.3: Four stages of the hydrodynamic evolution of shells collision: the pre-collision stage (upper left panel), the beginning of the collision (upper right panel), the formation of internal shocks (lower left panel) and close to the post-collision state (lower right panel). The upper half of each panel shows the logarithm of the rest-mass density, the lower half the logarithm of the pressure. For the purpose of visualization both the density and pressure have been cut-off at 10−23_{g cm}−3 _{and 10}−6 _{erg cm}−3_{, respectively.}

(R1b and R2b) that connects the still unperturbed state inside the shell with a contact discontinuity separating shell matter from the external medium (CD1L in the top left panel of Fig. 5.5), and into a second rarefaction (R1a

andR2a) that connects the state behind the contact discontinuity with the external medium.

The pre-collision evolution is qualitatively similar when instead of sharp discontinuities a more smooth transition between the shells and the external medium is assumed. The Riemann structure emerging from the edges of the shells will be quantitatively the same, i.e, it will consist of the same structure of shocks and rarefactions as with our set up. However, the exact values of the state variables in the intermediate states connecting the conditions in the shells with the external medium will be obviously different.

The pre-collision hydrodynamics has two direct consequences. On the one hand, each shell is heated by a reverse shock (S1a and S2a). On the other hand, both shells are spread inzdirection as external medium shocked in the bow shocks (S1b and S2b) piles up in front of the shells. The latter

Figure 5.4: Four stages of the hydrodynamic evolution of shells collision: the pre-collision stage (upper left panel), the beginning of the collision (upper right panel), the formation of internal shocks (lower left panel) and near the post-collision state (lower right panel). Values at the jet axis of rest-mass density (full line), pressure (dashed line) and Lorentz factor of the fluid moving to the right (triple-dot-dashed line) are plotted.

effect is complicated in case of the faster trailing shell by the fact that its bow shock (S2b) soon starts to interact with the rarefaction (R1a) of the slower leading shell. Thereby the bow shock speeds up, and it eventually catches up with the slower leading shell. Our simulations show that the resulting interaction of the two shells occurs at a distance which is slightly smaller than the distance derived from an analytic estimate (see below). The accelerating bow shock S2b drags along the whole Riemann structure. This explains why the state behind S2b is not uniform (as in case of the slower leading shell), but shows a monotonically decreasing density and pressure distribution (Fig. 5.5). It further explains why the density behind the reverse shock of the faster shell (S2a) is always less than that behind the reverse shock (S1a) of the slower shell.

Before the bow shockS2bof the faster trailing shell can enter the interior of the slower shell, it has to cross the rarefactionR1b, i.e., it has to propagate through a steadily increasing density. Hence, the emission produced by the shock will increase gradually during this epoch until it becomes an internal shock propagating through the slower shell (figs. 5.3 and 5.4, panels at times 160 ks and 220 ks). We point out that in analytic models (e.g., Spada

R2a R2b R1b R1a S1a S2b S2a S1b S1b R1b R1a CD1L CD1R S1a

Figure 5.5: Snapshot illustrating the flow structure along the symmetry axis arising from the shell motion just before the two shells start to interact. The lower panel shows the density (solid line) and pressure (dashed line) distribution measured in units ofρext and

ρextc2, respectively. The dash-dotted line gives the Lorentz factor of the fluid which is

moving towards the right. The upper left (right) panel displays the exact solution of the one dimensional Riemann problem defined by the trailing (leading) edge of the right shell. Labeled are the two bow shocksS1bandS2b, the two reverse shocksS1aandS2a, the four rarefactionsR1a, R1b, R2aand R2b, and (in the top panels only) the contact discontinuitiesCD1LandCD1R.

et al. 2001) the internal shock does appear instantaneously when the two shells touch each other.