The spectrum of the Sedov–Taylor point explosion linear stability

The linear stability of the spherical self-similar Sedov–Taylor blast wave (BW) with a front expanding in a uniform ideal gas with adiabatic index $γ$⁠, according to $RST(t)∝t2/5$⁠, is studied. The Sedov–Taylor BW (STBW) is crucial to understand the complex structures of late supernova remnants as the STBW has been shown to give rise to the Vishniac instability (VI) [E. T. Vishniac, Astrophys. J. 274, 152 (1983)] and to the Ryu–Vishniac instability (RVI) [D. Ryu and E. T. Vishniac, Astrophys. J. 313, 820 (1987)]. However, these approaches are questionable for several reasons, and especially because they do not provide the same result, in opposition to what could be expected from a physical viewpoint, in the limit $γ→1$⁠. We have revisited the RVI and the VI in great detail by taking great care of the behavior close to the center of symmetry of the configuration where the perturbation of the STBW might diverge. Our method allows one to find new spectra for the growth rate $s$ of the instability in terms of the mode number $ℓ$⁠. Two spectrum types are derived: (i) a continuous spectrum for which no dispersion relation $s(ℓ,γ)$ can be found, and (ii) a discrete spectrum for which a dispersion relation $s(ℓ,γ)$ can be derived. The case (i) is new and could provide the explanation why a set of various numerical simulations (or experiments) of the same STBW problem will not most likely give the same result. The second aspect (ii) is also new for at least two reasons aside the strange structure of the discrete spectrum. First, any dispersion curve $s(ℓ,γ)$ contains two types of portions: some portions correspond to growth rates $s$ with no singularity at all for the perturbed solution at the origin, while for the other portions of the dispersion curve, divergences of the perturbed STBW might exist except for the pressure. Second, it is shown that for any given value of $γ$⁠, no growth rate $s$ can exist above an upper limit for the mode number $ℓ$⁠. Finally, our model reconciles the VI and the RVI, and it is demonstrated that both analyses lead to a common analytical dispersion relation $s(ℓ)$ for $γ→1$⁠.