the geometry $\alpha$. The constant $\epsilon$ is introduced to summarize the behavior of the solutions. Whilst the integral diverges for $\epsilon=0$ and gives negative masses for $\epsilon <0$ valid solutions are given for $\epsilon>0$. Hence, together with $\kappa\in\mathbb{R}_0^+$ we conclude that $$\kappa_\mathrm{max}<\alpha+1\label{eq:strict_limit}$$ is a strict upper limit for the solveability of the system [@Solinger1975; @Lerche1976]. These general findings remain also valid for the EP we discuss in the next section.

![Summarized solutions for the velocity (a,c) and the density (b,d) in different coordinates ($\alpha=1,2$) and $\kappa=1$ at different times $\tau=1,3,6$ (solid, dashed and dashed dotted lines, respectively). The behavior at the origin is caused by numerical uncertainties by computing the RHC. []{data-label="fig:figure7"}](Figure07.pdf)

Isothermal blast waves (BW) {#secBW}
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In analogy to the well known solutions of point explosions by [@Sedov1946] and [@Taylor1950] the isothermal counterpart is derived by [@Korobeinikov1956]. He used the same assumption of a vanishing temperature gradient ahead of the shock as [@Sedov1946], meaning $0\approx\partial_r T_{\mathrm{d}} \ll \partial_r T_{\mathrm{u}}$. But this assumption means that there exists a temperature jump $0=T_\mathrm{d}\neq T_\mathrm{u}$ in contrast to a real isothermal assumption where very efficient cooling causes the same temperature throughout the system $\left(T_\mathrm{d}=T_\mathrm{u}\right)$. Hence, this results in a mixture of adiabaticity 