Expansion of non-spherical relativistic blast waves is considered in the Kompaneets (the thin shell) approximation. We find that the relativistic motion effectively "freezes out" the lateral dynamics of the shock front: only extremely strongly collimated shocks, with the opening angles $\Delta \theta \leq 1/\Gamma^2$, show appreciable modification of profiles due to sideways expansion. For less collimated profiles the propagation is nearly ballistic; the sideways expansion of relativistic shock becomes important only when they become mildly relativistic.
Deep Dive into On the sideways expansion of relativistic non-spherical shocks and GRB afterglows.
Expansion of non-spherical relativistic blast waves is considered in the Kompaneets (the thin shell) approximation. We find that the relativistic motion effectively “freezes out” the lateral dynamics of the shock front: only extremely strongly collimated shocks, with the opening angles $\Delta \theta \leq 1/\Gamma^2$, show appreciable modification of profiles due to sideways expansion. For less collimated profiles the propagation is nearly ballistic; the sideways expansion of relativistic shock becomes important only when they become mildly relativistic.
Dynamics and corresponding radiative signatures of non-spherical relativistic shocks remains an important unresolved issues in studies on Gamma Ray Bursts (GRBs). Since GRBs produce narrowly collimated outflows that evolve laterally, understanding the overall dynamics -both theoretical and in terms of agreement between different numerical results -is imperative to the interpretation of the broadband observations of GRBs Rhoads (1999); Frail et al. (2001).
Presently, there are two competing views on the lateral evolution of the relativistic outflows. Theoretically, it is typically argued that the lateral evolution of the flow proceeds with relativistic velocities (Piran 1999), (see also Wygoda et al. 2011). This view is contradicted by the results of numerical simulations that show very little lateral evolution in the relativistic regime Cannizzo et al. (2004);Zhang & MacFadyen (2009); Meliani & Keppens (2010); van Eerten & MacFadyen (2011).
In this Letter we argue that this disagreement results from the incorrect theoretical assumptions about the lateral evolution of the flow. What is important for the interpretation of observations is the evolution of a curved shock. Previously the lateral evolution of the non-spherical shocks was incorrectly treated as a free lateral expansion into vacuum (e.g., Wygoda et al. 2011, Eq. 5).. The assumption of the lateral expansion with the sound speed results in a “gramophone-type” profiles and exponential slowing down of the ejecta. This has drastic implications for the underlying light curves (eg Kumar & Panaitescu 2000;Panaitescu & Kumar 2003). In fact the dynamics of the non-spherical shocks is more subtle; the correct treatment, as we argue below, is consistent with slow lateral evolution seen in numerical simulations.
Evolution of strong non-spherical shocks is a well studies problems in fluid dynamics. The two fundamental works that have laid the foundation for non-spherical (two-dimensional) shocks, due to Kompaneets (1960) and to Laumbach & Probstein (1969), were originally designed to treat strong shock waves in the non-isotropic medium. These two complimentary methods have been extensively applied in astrophysics to treat supernova explosions (Bisnovatyi-Kogan & Silich 1995) and non-isotropic winds (e.g., Icke 1988). In the Kompaneets approximation the internal pressure of the gas is assumed to be constant. Then the Rankin-Hugonio conditions determine the normal velocity of the shock in the external inhomogeneous medium. A modification of the Kompaneets approximation -a thin or snowplow shell approximation -has also been used extensively (e.g., Wiita 1978; Mac Low & McCray 1988;Bisnovatyi-Kogan et al. 1989). In a complimentary Laumbach-Probstein approach (Laumbach & Probstein 1969) the streamlines of the shocked material are assumed to be radial, thus neglecting the lateral pressure forces.
The relativistic generalization of the Kompaneets and the Laumbach-Probstein methods have been discussed by Shapiro (1979). Relativistic dynamics provide extra support for the thin shell method, since in the relativistic blast waves the shocked material is concentrated in even narrower region R/Γ 2 than in the non-relativistic Sedov solution. In addition, the limited causal connection (over the angle ∼ 1/Γ) provides a justification for the Laumbach-Probstein method on the angle scale comparable to 1/Γ. As has been pointed out by Shapiro (1979), the two methods -Kompaneets and Laumbach-Probstein -become very similar in the relativistic regime. This is due to the fact the in a relativistic quasi-spherical wave, the typical angle that a shock wave makes with the direction of the velocity is of the order α ∼ 1/Γ 2 . Thus the post shock pressure along the shock differs only by one part in Γ 2 , so that both approximations of constant post-shock pressure and radial post-shock motion become equivalent. In some sense the propagation of strongly relativistic non-spherical shocks becomes trivial: relativistic kinematic effect freeze out the lateral dynamics of the flow so that different parts of the flow behave virtually independently.
In this section we re-derive the relativistic Kompaneets equation Kompaneets (1960); Shapiro (1979) allowing for the arbitrary velocity of the shock and arbitrary (angle-dependent) luminosity and/or external density. Consider a shock propagating with a three-velocity V at an angle α to its normal. There are three generic rest frames in the problem: laboratory frame K, a frame where the shock is normal to the flow K 1 and a shock rest frame K 0 . A frame K 1 is related to the lab frame K by a Lorentz boost along y axis with a Lorentz factor Γ = 1/ 1 -V 2 sin 2 α. In K 1 the velocity of the shock is V 1 = Γ V cos α (along the x direction). Thus, Γ 2 1 = 1/(1 -V 2 1 ) = Γ 2 cos 2 α + sin 2 α (the shock becomes non-relativistic when π/2 -α ∼ 1/Γ). In the frame K 0 V 1 and Γ 1 are the velocity and the Lorentz factor of the unshocked medium. In the lab fr
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