7 Shock with permanent energy injection

The particular case of the generalized shock solution for astrophysical application is a permanent energy injection shock produced by a continuous pumping of energy into the shock from the central source of constant luminosity, $L={\rm const}$. It corresponds to the case of k=1 and $E=L t\propto t$ in contrast to the usually applied Sedov-Taylor shock with $E={\rm const}$, which is produced by a instant implosion of energy into the shock. The possible astrophysical implications of injection shock solutions with $k\neq0$, that is with a central energy source varying in time, are the early phase of SN explosion, rarefied bubbles in the interstellar medium after SN explosions, strong wind from stars and young pulsars, non-steady spherical outflow from accreting black holes and dense stellar clusters near collapse with frequent neutron star collisions. Different analytical and numerical approaches were applied by Falle (1975), Castor et al. (1975), Weaver et al. (1977) to the modeling of permanent energy injection shocks in the case of stellar winds, and on the interstellar medium and interstellar bubbles.

The external radius of the expanding k=1 injection shock evolves with time according to Eq. (5) as $R(t)=\beta(A/\rho_1)^{1/5}t^{(2+k)/5}\propto t^{3/5}$, where L=A is the luminosity of the central source and $\rho_1={\rm const}$ is the density of the ambient gas medium. So the injection shock expands faster than the instant Sedov-Taylor shock (k=0, $R\propto t^{2/5}$). This is because of a constant pumping of energy into the shock, $L={\rm const}$. The corresponding velocity of injection shock expansion is $u_1(t)= {\rm d}R/{\rm d}t=(R/t)(k+2)/5\propto t^{-2/5}$. From the last two equations it is very clear the physical meaning and the difference between the instant shock (k=0) and injection shock (k=1) cases respectively is very clear:

 

$$\left[(5/2)^{2}\beta(\gamma,0)^{-5}\right] R^3 \left(\rho u^2\right)={\rm const};$$ (37)

$$\left[(5/3)^{3}\beta(\gamma,1)^{-5}\right]R^2 u \left(\rho u^2\right)={\rm const}.$$ (38)

In these expressions $R^3\sim V$ is shock volume and $R^2\sim S$ is the shock surface. At k=0 we have the expansion law for instant shock which corresponds to constant energy carried by the swept out gas. The expansion law for a k=1 injection shock corresponds to constant energy flux (or constant source luminosity) carried by the swept out gas.

The discussed strong shock solution is valid only in the region where the shock expansion velocity $c_{\rm s}\ll u(R) \ll c$, where $c_{\rm s}$ is the sound speed in the ambient gas. The expanding strong shock becomes weak and disappears when its expansion velocity drops below the the sound speed $c_{\rm s}$. The maximum radius of the expanding strong shock $R_{{\rm sh}}$ is obtained from the equality $u(R_{{\rm sh}})=c_{\rm s}$ by using Eqs. (5) and (6):

$$R_{{\rm sh}}=\left[\left(\frac{2+k}{5c_{\rm s}}\right)^{2+k}\beta^{5}~ \frac{A}{\rho_1}\right]^{\frac{1}{3-k}}\cdot$$ (39)

The maximum time of a strong shock expansion is

$$t_{{\rm sh}}=\left[\left(\frac{2+k}{5c_{\rm s}}\beta\right)^{5} \frac{A}{\rho_1}\right]^{\frac{1}{3-k}}\cdot$$ (40)

The minimal radius of Newtonian motion of the shocked gas, which is called the Sedov length $l_{\rm S}$, is defined by equality $u(l_{\rm S})=c$. The corresponding expression for the Sedov length, $l_{\rm S}=R_{{\rm sh}}(c_{\rm s}/c)^{(2+k)/(3-k)}\ll R_{{\rm sh}}$, is reproduced from Eq. (39) by substituting cfor $c_{\rm s}$. So the region of applicability of the strong shock solution is $l_{\rm S}\ll r\ll R_{{\rm sh}}$.