Ultra-Relativistic Blast Wave: Stability and Strong Non-Universality

Linear eigenmodes of a spherically symmetrical ultra-relativistic blast wave (the Blandford-McKee, BMK, solution) are calculated. The BMK solution is shown to be stable and strongly non-universal. It is stable because all the eigenmodes decay. Nonuniversality of the BMK solution (BMK is not an intermediate asymptotic) is a consequence of causality. In terms of eigenmodes – some eigenmodes decay too slowly. For each degree l, there exists an eigenmode which decays at the smallest rate. The amplitude of this eigenmode, defined as relative perturbation of energy behind the shock front, is constant at early time. Later, when the blast wave slows down to Γ ≈ l/10, an oscillatory decay commences, and the amplitude drops to less than 10% of the initial value at Γ ≈ l/100. The non-universality is surprisingly strong. Near the end of the ultra-relativistic stage, perturbations with l ∼< 200, that is more than 10,000 different harmonics, are suppressed by less than a factor of 10. Spherical symmetry is only reached by the time the blast wave slows down to non-relativistic velocities, when the Sedov/Taylor/von Neumann solution sets in. Subject headings: shock waves – gamma rays: bursts 1. Ultra-relativistic blast waves, GRBs, and their aftreglows It is thought that ultra-relativistic blast waves occur naturally, and GRB afterglows come from synchrotron-emitting ultra-relativistic blast waves (Piran 1999). A spherically symmetrical ultra-relativistic blast wave (the BMK solution) was given by Blandford and McKee (1976). We will show that this solution is stable but strongly non-universal (§3). In §2 we derive the basic equations, it includes a simple derivation of the BMK solution. Qualitatively, non-universality follows from causality. For the BMK solution, the shock front moves with Lorentz factor Γ ∝ t−3/2. Consider a light signal propagating along the front, starting at a polar angle θ = 0. Then, with c = 1, (dr/dt) + (rdθ/dt) = 1, which, for Γ ≫ 1, gives θ = 2 3 Γ−1. Two regions of the shock front, separated by angle θ, do not talk to each other until the blast wave slows down to Γ ∼ θ−1. It turns out that non-universality is very strong (§3). The non-universality of ultra-relativistic blast waves should be observed as non-universality of early (ultra-relativistic) GRB aftreglows.