On pulsar-driven isothermal blast-wave models of supernova remanants

Abstract

We investigate the ‘equilibrium’ and stability of spherically-symmetric self-similar isothermal blast waves with a continuous post-shock flow velocity expanding into medium whose density varies asr−ω ahead of the blast wave, and which are powered by a central source (a pulsar) whose power output varies with time astω−3. We show that: (1) for ω<0, no physically acceptable self-similar solution exists; (ii) for ω>3, no solution exists since the mass swept up by the blast wave is infinite; (iii) ϕ must exceed zero in order that the blast wave expand with time, but ϕ<2 in order that the central source injects a finite total energy into the blast wave; (iv) for 3>ωmin(ϕ)>ω>ωmax(ϕ)>0, where $$\begin{gathered} \omega _{\min } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} + {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} + {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \omega _{\max } (\varphi ){\text{ }} = {\text{ }}2[5{\text{ }} - {\text{ }}\varphi {\text{ }} - {\text{ }}(10{\text{ }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 )^{1/2} ]^2 [2{\text{ }} - {\text{ (10 }} + {\text{ 4}}\varphi {\text{ }} - {\text{ 2}}\varphi ^2 {\text{)}}^{{\text{1/2}}} ]^{ - 2} , \hfill \\ \end{gathered} $$ two critical points exist in the flow velocity versus position plane. The physically acceptable solution must pass through the origin with zero flow speed and through the blast wave. It must also pass throughboth critical points if\(\varphi > \tfrac{5}{3}\), while if\(\varphi 5/3) (through whichall solutions pass with thesame slope) has not been established; (v) for 3>ω>ωmin(ϕ) it is shown that the two critical points of (iv) disappear. However a new pair of critical points form. The physically acceptable solution passing with zero flow velocity through the origin and also passing through the blast wave mustby-pass both of the new critical points. It is shown that the solution does indeed do so; (vi) for 3>ωmin(ϕ)>ωmax(ϕ)>ω it is shown that the dependence of the self-similar solution on either ω or ϕ is non-analytic and therefore, inferences drawn from any solutions obtained in ω>ωmax(ϕ) (where the dependence of the solutionis analytic on ω and ϕ) are not valid when carried over into the domain 3>ωmin(ϕ)>ωmax(ϕ)>ω; (vii) all of the physically acceptable self-similar solutions obtained in 3>ω>0 are unstable to short wavelength, small amplitude but nonself-similar radial velocity perturbations near the origin, with a growth which is a power law in time; (viii) the physical self-similar solutions are globally unstable in a fully nonlinear sense to radial time-dependent flow patterns. In the limit of long times, the nonlinear growth is a power law in time for 5 ω+2ϕ, and the square of the logarithm in time for 5=ω+2ϕ.