Isothermal self-similar blast wave theory of supernova remnants driven by relativistic gas pressure

Abstract

We investigate the spherically symmetric, self-similar flow behind a blast wave from a point explosion in a medium whose density varies with distance asr−ω with the assumption that the flow is both isothermal and contains a relativistic component of pressure. A self-similar solution is shown to exist only if both the blast wave speed,u s ,and the local sound speed,w, are constant. If Ω[≡ω(1−w2/c2)] lies in 1>Ω>0, there exists a critical point in the radial distance-flow velocity plane. To be physically acceptable, the solution must pass through the origin and through the critical point and then through to the blast front; solution branches between these points exist, although a proper connection at the critical point has not been demonstrated. If Ω Ω>1, the critical point is beyond the blast curve and the flow is subsonic everywhere. For 2 Ω>3/2 or 0 Ω>1 unless the flow meets the blast front atprecisely the velocity (normalized) of (2Ω−1)1/2/(3−2Ω)1/2. The solutions are also unstable for all Ω in 1>Ω>0 near the critical point. Since there is no characteristic time scale in the system, all the instabilities grow as a power law in time rather than exponentially. The existence of these instabilities implies that initial deviations do not decay and the system does not tend to a self-similar form. We conclude that isothermal self-similar blast waves do not provide a valid model for a supernova remnant driven by a relativistic gas pressure. Since the validity of the adiabatic blast wave models has elsewhere been shown to be questionable, it is doubtful whether the self-similar property can be involved at all in the case of supernova remnants. This raises serious questions of interpretation of quantities deduced for supernova remnants on the basis of the use of self-similar models.